Abstract group 147456.rd: $C_2^9.(D_6\times S_4)$

• Label: 147456.rd
• Order: \( 2^{14} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{21} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $27$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_2^9.(D_6\times S_4)$
Order:	\(147456\)\(\medspace = 2^{14} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	Group of order \(18874368\)\(\medspace = 2^{21} \cdot 3^{2} \)
Composition factors:	$C_2$ x 14, $C_3$ x 2
Derived length:	$4$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	5759	1538	53632	38206	6144	39104	3072	147456
Conjugacy classes	1	105	3	158	74	4	85	2	432
Divisions	1	105	3	158	74	4	85	2	432
Autjugacy classes	1	45	3	56	37	2	31	1	176

Minimal presentations

Permutation degree:	$27$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid a^{2}=b^{6}=c^{6}=d^{4}=e^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,2)(3,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(26,27), (1,3)(2,5)(4,9) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^9$ . $(D_6\times S_4)$	$S_3$ . $(C_2^9.S_4.C_2)$	$C_2^7$ . $(C_2^5:S_3^2)$ (2)	$(C_2^9\times S_3)$ . $(C_2\times S_4)$	all 107
Aut. group:	$\Aut(D_{12}:C_2^3)$	$\Aut(C_{12}:C_2^4)$	$\Aut(C_{12}.C_2^4)$	$\Aut((C_4\times D_4).D_6)$

Elements of the group are displayed as permutations of degree 27.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 332 normal subgroups (104 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^7.(S_3\times S_4).C_2$
Commutator:	$G' \simeq$ $C_2^9.A_4.C_3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^5$	$G/\Phi \simeq$ $C_2^6.S_3^2.C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3:(C_2^9.C_2^3)$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_2^9.(D_6\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^7.S_4.C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^4:D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 18 (order at least 8192) are shown, as well as all normal subgroups of any index.

Order 147456: $C_2^9.(D_6\times S_4)$

Order 73728: $C_2^9.(S_3\times S_4)$, $C_2^7.C_2^4.S_3^2$, $C_2^7.C_2^4.S_3^2$, $C_2^7.C_2^4.(C_6\times S_3)$, $C_2^9.A_4.D_6$, $(C_2^8\times C_6).A_4.C_2^2$, $C_2^5.C_2^6.C_3.D_6$

Order 49152: $C_3:(C_2^7.C_2.C_2^6)$, $C_2^9.A_4.C_2^3$

Order 36864: $C_2^7.(D_6\times S_4)$ x 4, $(C_2^8\times C_6).A_4.C_2$, $C_2^6.C_2^2.D_6^2$, $C_2^7.C_2^4.C_3.C_6$, $C_2^9.(C_6\times A_4)$, $(C_2^8\times C_6).A_4.C_2$, $C_2^9.A_4.S_3$, $C_2^9.A_4.C_6$, $C_2^6.D_6^2.C_2^2$, $C_2^9.A_4.S_3$, $C_2^7.(D_6\times S_4)$

Order 24576: $C_3:(C_2^7.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^6)$ x 4, $C_3:(C_2^7.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^3.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^5)$ x 4, $C_3:(C_2^6.D_4^2.C_2)$ x 4, $C_3:(C_2^7.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^5)$ x 4, $C_2^9.A_4.C_2^2$ x 2, $C_2^5.C_2^6.D_6$ x 2, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^7.C_2^4.D_6$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^9.A_4.C_2^2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^9.A_4.C_2^2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^9.A_4.C_2^2$, $C_3:(C_2^9.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $(C_2^4\times D_{12}).D_4^2$

Order 18432: $C_2^6.(D_6\times S_4)$ x 32, $C_2^6.D_6^2.C_2$ x 16, $C_2^6.(D_6\times S_4)$ x 11, $C_2^6.A_4.D_6.C_2$ x 4, $C_2^6.A_4.D_6.C_2$ x 4, $(C_2^5\times C_6).A_4.C_2^3$ x 4, $C_2^6.A_4.D_6.C_2$ x 4, $C_2^6.A_4.D_6.C_2$ x 4, $C_2^7.(S_3\times S_4)$ x 4, $C_2^7.(C_6\times S_4)$ x 4, $(C_2^6\times C_6).A_4.C_2^2$ x 2, $(C_2^6\times C_6).A_4.C_2^2$ x 2, $(C_2^6\times C_6).A_4.C_2^2$ x 2, $C_2^6.(C_6\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.(S_3\times S_4).C_2$, $C_2^6.(S_3\times S_4).C_2$, $C_2^6:C_3.D_6.C_2^3$, $(C_2^5\times C_6).A_4.C_2^3$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.C_4:S_3^2.C_2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_6.C_2^6:C_3.C_2^4$, $(C_2^5\times C_6).A_4.C_2^3$, $C_2^6.(S_3\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.D_6:D_6.C_2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^4.C_2^4.C_3^2.C_2^3$, $C_2^9.A_4.C_3$, $C_2^6:C_3.D_6.C_2^3$, $C_2^6.C_6^2.C_2^3$, $C_2^6.D_6:D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6:C_3.D_6.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^2\times C_2^6:C_3.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^4.C_2^4.C_3^2.C_2^3$, $C_2^6.(C_2\times S_3\times S_4)$, $C_2^6.(C_6\times S_4).C_2$, $C_2^4.C_2^4.S_3^2.C_2$, $C_2^3\times C_2^6.S_3^2$, $C_2^6:C_3.D_6.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^6.(C_2\times S_3\times S_4)$, $C_2^7.D_6^2$

Order 16384: $C_2^4.C_2^4:D_4^2$

Order 12288: $C_3:(C_2^7.C_2^5)$ x 10, $C_3:(C_2^7.C_2^5)$ x 10, $C_3:(C_2^7.C_2^5)$ x 10, $C_3:(C_2^7.C_2^5)$ x 8, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.D_4^2)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^8.C_2^4)$ x 4, $C_3:(C_2^6.C_2^5.C_2)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^8.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^4.C_2^2)$ x 4, $C_3:(C_2^7.C_2^4.C_2)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.D_4^2)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^8.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^5.C_2)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^5.C_2^3.C_2^4)$ x 4, $C_3:(C_2^6.D_4^2)$ x 4, $C_3:(C_2^6.D_4^2)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^7.C_2^4.C_2)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2^5.D_4^2.C_2)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_4^2.C_2^2)$ x 4, $C_3:(C_2^5.D_4^2.C_2)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^5.C_2^4.C_2^3)$ x 4, $C_3:(C_2^8.C_2^4)$ x 4, $C_3:(C_2^6.C_2^5.C_2)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^6)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.D_4^2)$ x 4, $C_3:(C_2^6.C_2^5.C_2)$ x 4, $C_3:(C_2^7.C_2^4.C_2)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3.C_2^3)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2^7.C_2^4.C_2)$ x 4, $C_3:(C_2^6.C_2.C_2^5)$ x 4, $C_3:(C_2^6.C_2^2.C_2^4)$ x 4, $C_3:(C_2.C_2^6.C_2^5)$ x 4, $C_3:(C_2^6.C_2^5.C_2)$ x 4, $C_3:(C_2^7.C_2.C_2^4)$ x 4, $C_2^8.(C_2\times S_4)$ x 4, $C_3:(C_2^6.C_2^6)$ x 3, $C_2^9.S_4$ x 3, $C_2^9.A_4.C_2$ x 3, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2.C_2^4)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^7.C_2.C_2^4)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_2^9.S_4$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^4.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^4.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_2^9.A_4.C_2$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_2^6.A_4.C_2^4$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_2^7.C_2^4.S_3$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_2^6.C_6.C_2^5$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $(D_4\times C_2^6).S_4$

Order 9216: $S_3\times C_2^6:S_4$ x 43, $C_2^2\times C_2^6.S_3^2$ x 21, $(C_2^6\times C_6).A_4.C_2$ x 18, $C_2^4.C_2^4.C_3.D_6$ x 16, $C_2^6.(C_6\times S_4)$ x 16, $(C_2^5\times C_6).A_4.C_2^2$ x 16, $C_2\times C_2^6:C_3.D_6.C_2$ x 16, $C_2^4.C_2^4.S_3^2$ x 16, $C_2^4.C_2^4.S_3^2$ x 16, $C_2^6.A_4.D_6$ x 16, $(C_2^5\times C_6).A_4.C_2^2$ x 11, $C_2^6.(C_6\times S_4)$ x 9, $C_2^6.C_6^2.C_2^2$ x 8, $C_2^6.(S_3\times D_{12})$ x 8, $C_2^6.C_6^2.C_2^2$ x 8, $C_2^6.C_6^2.C_2^2$ x 8, $C_2^6:C_3.D_6.C_2^2$ x 8, $C_2^4.C_2^4.C_3.D_6$ x 8, $C_2^6.C_4:S_3^2$ x 8, $C_2^6.C_6^2.C_2^2$ x 8, $C_2^6.(S_3\times D_{12})$ x 8, $C_6.C_2^6:C_3.C_2^3$ x 8, $C_2^6.D_6:D_6$ x 8, $C_2^6.D_6:D_6$ x 8, $C_2^6.(C_4\times S_3^2)$ x 8, $C_2^8.S_3^2$ x 8, $(C_2^6\times C_6).A_4.C_2$ x 5, $C_2^7.A_4.C_6$ x 4, $C_2^6.(C_6\times S_4)$ x 4, $(C_2^5\times C_6).A_4.C_2^2$ x 4, $C_2^6.C_2^2.C_6^2$ x 4, $C_2^6.A_4.D_6$ x 4, $C_2^7.A_4.S_3$ x 4, $C_2^5.(D_6\times S_4)$ x 4, $C_2^6.A_4.D_6$ x 3, $(C_2^6\times C_6).A_4.C_2$ x 2, $C_2^7.(C_6\times A_4)$ x 2, $(C_2^6\times C_6).A_4.C_2$ x 2, $Q_8:C_2^2.D_6^2.C_2$ x 2, $C_2^6.A_4.D_6$ x 2, $C_2^7.(C_6\times A_4)$ x 2, $(C_2^5\times C_6).A_4.C_2^2$ x 2, $C_2^2\times C_2^6.S_3^2$ x 2, $C_2^6.C_3^2.D_4.C_2$ x 2, $(C_2^6\times C_6).A_4.C_2$ x 2, $Q_8:C_2^2.D_6^2.C_2$ x 2, $C_2^7.(C_6\times A_4)$ x 2, $C_2^4.C_2^4.(C_6\times S_3)$ x 2, $C_2^6.D_6^2$ x 2, $C_2^6.(C_6\times S_4)$, $(C_2^5\times C_6).A_4.C_2^2$, $C_6.C_2^6:C_3.C_2^3$, $C_2^6.(C_6\times D_{12})$, $C_2^6.D_6:S_3.C_2$, $C_2^6.A_4.D_6$, $C_2^6.A_4.D_6$, $(C_2^5\times C_6).A_4.C_2^2$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4)$, $C_2^6.A_4.D_6$, $C_2^3\times C_2^6:C_3.S_3$, $C_2^6.A_4.D_6$, $C_2^6.C_3^2.D_4.C_2$, $C_2^6.(C_6\times S_4)$, $C_2^4.C_2^4.(C_6\times S_3)$, $C_2^4.D_6^2.C_2^2$, $C_2^6.C_2^2.C_6^2$, $C_2^6.A_4.D_6$, $C_2^6.(S_3\times S_4)$, $C_2^6.(S_3\times S_4)$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^4.D_6:S_4.C_2$, $C_2^6.(S_3\times S_4)$, $C_2^7.(C_6\times A_4)$, $C_2^6.(S_3\times S_4)$, $C_2\times C_6\times C_2^6.D_6$, $C_2^6.D_6:S_3.C_2$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.S_3^2$, $(C_2^5\times C_6).C_6.C_2^3$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.C_3.D_6$, $C_2^2\times C_6\times C_2^6.S_3$, $C_2^6:C_3.C_6.C_2^3$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4)$, $C_2^6.(C_2.S_3^2).C_2$, $C_2^6.C_6.C_6.C_2^2$, $C_2^6.(S_3\times S_4)$, $(C_2^6\times C_6).A_4.C_2$, $C_2^2\times C_2^6.C_6\times S_3$, $C_2^6.(C_6\times S_4)$, $C_2^6.A_4.D_6$, $C_2^7.C_6^2.C_2$, $C_2^6.(C_6\times S_4)$, $C_2^6.A_4.D_6$, $C_2^6.C_6^2.C_2^2$, $C_2^6:C_3.D_6.C_2^2$, $(C_2^5\times C_6).A_4.C_2^2$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4)$, $C_2^4.D_6:S_4.C_2$, $C_2^2\times C_2^6:C_3.D_6$, $C_2^6.A_4.D_6$, $C_6.C_2^6:C_3.C_2^3$, $C_2^4.C_2^4.C_3.D_6$, $(C_2^6\times C_6).C_6.C_2^2$, $C_2^6.A_4.D_6$, $C_2^6:C_3.D_6.C_2^2$, $C_2^6.A_4.D_6$, $C_2^6.C_6^2.C_2^2$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.S_3^2$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.(C_6\times S_3)$, $C_2^6.A_4.D_6$, $C_2^6.(S_3\times S_4)$, $C_2^6.(S_3\times S_4)$, $C_2^6.A_4.D_6$, $C_2^6:C_3.D_{12}.C_2$, $C_2^4.C_2^4.C_3.D_6$, $C_2^6.C_2^2.C_6^2$, $C_2^6:C_3.C_6.C_2^3$, $C_2^6.(A_4\times D_6)$

Order 8192: $C_2^7.C_2^5.C_2$ x 31, $C_2^7.C_2^5.C_2$ x 4, $C_2^7.C_2^5.C_2$ x 4, $C_2^6.C_2^5.C_2^2$ x 4, $C_2^6.C_2^6.C_2$ x 4, $C_2^6.C_2^6.C_2$, $C_2^8.C_2^5$, $C_2^6.C_2^5.C_2^2$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^3.C_2^4$, $C_2.C_2^6.C_2^6$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^4.C_2^3$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2.C_2^5$, $C_2^6.C_2^4.C_2^3$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^6.C_2$, $C_2^{10}.D_4$, $C_2^7.D_4^2$, $C_2^7.C_2^6$

Order 6144: $C_3:(C_2^7.C_2^4)$ x 4, $C_2^9.A_4$, $C_3:(C_2^7.C_2^4)$, $C_3:(C_2^7.C_2^4)$, $C_3:(C_2^9.C_2^2)$, $C_3:(C_2^8.C_2^3)$

Order 4096: $C_2^5.C_2^6.C_2$

Order 3072: $C_3:(C_2^7.C_2^3)$ x 6, $C_3:(C_2^7.C_2^3)$ x 4, $C_3:(C_2^7.C_2^3)$ x 4, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^8.C_2^2)$, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^9.C_2)$, $C_2^9\times S_3$

Order 2048: $C_2^9.C_2^2$

Order 1536: $C_6\times C_2^6.C_2^2$ x 6, $C_2\times D_4.C_2^4\times S_3$ x 2, $C_2\times C_2^4.C_2^3\times S_3$ x 2, $C_2^7:D_6$ x 2, $C_2^5\times S_3\times D_4$, $C_2^8\times C_6$

Order 1024: $C_2^6:C_2^4$ x 4, $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^4.C_2^6$

Order 768: $C_2^6:D_6$ x 8, $D_{12}:C_2^5$ x 8, $D_6\times C_2^6$ x 4, $C_6\times C_2.C_2^6$ x 2, $C_6\times C_2^4.C_2^3$ x 2, $C_2\times C_3:(C_2^6.C_2)$ x 2, $C_2^7:C_6$ x 2, $C_2\times C_3:(C_2.C_2^6)$ x 2, $C_2^5\times C_3:D_4$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2^4\times C_6\times D_4$, $C_2^5\times D_{12}$, $C_2^5\times C_4\times S_3$

Order 512: $C_4^2:C_2^5$ x 6, $C_2^9$

Order 384: $C_{12}.C_2^5$ x 8, $D_6\times C_2^5$ x 7, $C_2^6\times C_6$ x 4, $C_2^6:C_6$ x 4, $C_2^6:S_3$ x 4, $C_2^5\times C_{12}$, $C_6.C_2^6$, $C_{12}:C_2^5$

Order 256: $C_2^4:C_2^4$ x 2, $C_2^5:D_4$ x 2, $C_2^5:D_4$ x 2, $D_4\times C_2^5$

Order 192: $D_6\times C_2^4$ x 8, $C_{12}:C_2^4$ x 8, $C_2^5\times C_6$ x 3, $D_6:C_2^4$ x 2, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$

Order 128: $D_4:C_2^4$ x 8, $C_2^7$ x 4, $C_2^3\wr C_2$ x 4, $C_2^5\times C_4$

Order 96: $C_2^3\times D_6$ x 15, $C_2^4\times C_6$ x 8, $C_2^3:D_6$ x 8, $C_{12}:C_2^3$ x 4, $C_2^2\times D_{12}$ x 4, $C_{12}:C_2^3$ x 4, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 64: $C_2^6$ x 3, $D_4\times C_2^3$

Order 48: $C_2^2\times D_6$ x 11, $C_2^3\times C_6$ x 7, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2

Order 32: $C_2^5$ x 8, $C_2^2\times D_4$ x 4, $C_2^3\times C_4$

Order 24: $C_2\times D_6$ x 6, $C_2^2\times C_6$ x 4

Order 16: $C_2^4$ x 7, $C_2^2\times C_4$ x 2

Order 12: $D_6$ x 8, $C_2\times C_6$ x 3

Order 9: $C_3^2$

Order 8: $C_2^3$ x 4

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 18 (order at least 8192) are shown, as well as all normal subgroups of any index.

Order 147456: $C_2^9.(D_6\times S_4)$

Order 73728: $C_2^9.(S_3\times S_4)$, $C_2^7.C_2^4.S_3^2$, $C_2^7.C_2^4.S_3^2$, $C_2^7.C_2^4.(C_6\times S_3)$, $C_2^9.A_4.D_6$, $(C_2^8\times C_6).A_4.C_2^2$, $C_2^5.C_2^6.C_3.D_6$

Order 49152: $C_3:(C_2^7.C_2.C_2^6)$, $C_2^9.A_4.C_2^3$

Order 36864: $(C_2^8\times C_6).A_4.C_2$, $C_2^6.C_2^2.D_6^2$, $C_2^7.C_2^4.C_3.C_6$, $C_2^9.(C_6\times A_4)$, $(C_2^8\times C_6).A_4.C_2$, $C_2^9.A_4.S_3$, $C_2^9.A_4.C_6$, $C_2^6.D_6^2.C_2^2$, $C_2^9.A_4.S_3$, $C_2^7.(D_6\times S_4)$, $C_2^7.(D_6\times S_4)$

Order 24576: $C_2^9.A_4.C_2^2$ x 2, $C_2^5.C_2^6.D_6$ x 2, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^7.C_2^4.D_6$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^9.A_4.C_2^2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^9.A_4.C_2^2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^9.A_4.C_2^2$, $C_3:(C_2^9.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.D_4^2.C_2)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $(C_2^4\times D_{12}).D_4^2$

Order 18432: $C_2^6.(D_6\times S_4)$ x 5, $C_2^6.(C_6\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.(S_3\times S_4).C_2$, $C_2^6.(S_3\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6:C_3.D_6.C_2^3$, $C_2^6.A_4.D_6.C_2$, $(C_2^5\times C_6).A_4.C_2^3$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.C_4:S_3^2.C_2$, $C_2^6.C_6^2.C_2^3$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^6.A_4.D_6.C_2$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $(C_2^5\times C_6).A_4.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_6.C_2^6:C_3.C_2^4$, $(C_2^5\times C_6).A_4.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^6.(S_3\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.D_6:D_6.C_2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^4.C_2^4.C_3^2.C_2^3$, $C_2^9.A_4.C_3$, $C_2^6:C_3.D_6.C_2^3$, $C_2^6.C_6^2.C_2^3$, $C_2^6.D_6:D_6.C_2$, $C_2^7.(S_3\times S_4)$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6:C_3.D_6.C_2^3$, $C_2^6.A_4.D_6.C_2$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^2\times C_2^6:C_3.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^4.C_2^4.C_3^2.C_2^3$, $C_2^6.(C_2\times S_3\times S_4)$, $C_2^6.(C_6\times S_4).C_2$, $C_2^4.C_2^4.S_3^2.C_2$, $C_2^6.D_6^2.C_2$, $C_2^3\times C_2^6.S_3^2$, $C_2^6:C_3.D_6.C_2^3$, $C_2^6.A_4.D_6.C_2$, $C_2^6.(C_2\times S_3\times S_4)$, $C_2^6.(D_6\times S_4)$, $C_2^7.D_6^2$, $C_2^7.(C_6\times S_4)$

Order 16384: $C_2^4.C_2^4:D_4^2$

Order 12288: $C_3:(C_2^7.C_2^5)$ x 4, $C_3:(C_2^7.C_2^5)$ x 4, $C_3:(C_2^7.C_2^5)$ x 4, $C_3:(C_2^6.C_2^6)$ x 3, $C_2^9.S_4$ x 3, $C_2^9.A_4.C_2$ x 3, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^7.C_2.C_2^4)$ x 2, $C_3:(C_2.C_2^6.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^8.C_2^4)$ x 2, $C_3:(C_2^7.C_2.C_2^4)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_2^9.S_4$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^7.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6)$ x 2, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.D_4^2)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^4.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^4.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_2^9.A_4.C_2$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^6)$, $C_2^6.A_4.C_2^4$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.D_4^2)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_2^7.C_2^4.S_3$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^5.C_2^3.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.D_4^2)$, $C_3:(C_2^6.D_4^2)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^5.C_2^6.C_2)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2^5.D_4^2.C_2)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_4^2.C_2^2)$, $C_3:(C_2^5.D_4^2.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^9.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.D_4^2)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^7.C_2^5)$, $C_2^6.C_6.C_2^5$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^5.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^6.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^7.C_2^4.C_2)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^6.C_2.C_2^5)$, $C_3:(C_2^6.C_2^2.C_2^4)$, $C_3:(C_2.C_2^6.C_2^5)$, $C_3:(C_2^7.C_2^5)$, $C_3:(C_2^7.C_2.C_2^4)$, $C_3:(C_2^8.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2)$, $C_3:(C_2^7.C_2.C_2^4)$, $(D_4\times C_2^6).S_4$, $C_2^8.(C_2\times S_4)$

Order 9216: $S_3\times C_2^6:S_4$ x 13, $C_2^2\times C_2^6.S_3^2$ x 7, $(C_2^6\times C_6).A_4.C_2$ x 5, $(C_2^5\times C_6).A_4.C_2^2$ x 5, $(C_2^6\times C_6).A_4.C_2$ x 3, $C_2^6.A_4.D_6$ x 3, $C_2^6.(C_6\times S_4)$ x 3, $C_2\times C_2^6:C_3.D_6.C_2$ x 2, $C_2^6.A_4.D_6$ x 2, $(C_2^5\times C_6).A_4.C_2^2$ x 2, $C_2^2\times C_2^6.S_3^2$ x 2, $C_2^6.C_3^2.D_4.C_2$ x 2, $C_2^4.C_2^4.C_3.D_6$ x 2, $C_2^4.C_2^4.(C_6\times S_3)$ x 2, $C_2^6.(C_6\times S_4)$, $(C_2^6\times C_6).A_4.C_2$, $C_2^7.(C_6\times A_4)$, $(C_2^5\times C_6).A_4.C_2^2$, $C_6.C_2^6:C_3.C_2^3$, $C_2^6.(C_6\times D_{12})$, $C_2^7.A_4.C_6$, $C_2^6.D_6:S_3.C_2$, $C_2^4.C_2^4.C_3.D_6$, $(C_2^6\times C_6).A_4.C_2$, $C_2^6.A_4.D_6$, $C_2^6.(C_6\times S_4)$, $C_2^6.A_4.D_6$, $(C_2^5\times C_6).A_4.C_2^2$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4)$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.A_4.D_6$, $Q_8:C_2^2.D_6^2.C_2$, $C_2^3\times C_2^6:C_3.S_3$, $C_2^6.A_4.D_6$, $C_2^6.C_3^2.D_4.C_2$, $C_2^6.(C_6\times S_4)$, $C_2^4.C_2^4.(C_6\times S_3)$, $C_2^4.D_6^2.C_2^2$, $C_2^6.C_2^2.C_6^2$, $C_2^6.A_4.D_6$, $C_2^6.(S_3\times S_4)$, $C_2^6.(S_3\times S_4)$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4)$, $C_2^6.C_6^2.C_2^2$, $C_2^6.(S_3\times D_{12})$, $C_2^4.D_6:S_4.C_2$, $C_2^6.(S_3\times S_4)$, $C_2^7.(C_6\times A_4)$, $C_2^6.(S_3\times S_4)$, $C_2\times C_6\times C_2^6.D_6$, $C_2^6.D_6:S_3.C_2$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.S_3^2$, $(C_2^5\times C_6).C_6.C_2^3$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.C_3.D_6$, $C_2^2\times C_6\times C_2^6.S_3$, $C_2^7.(C_6\times A_4)$, $C_2^6:C_3.C_6.C_2^3$, $C_2^6.C_6^2.C_2^2$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4)$, $C_2^6.(C_2.S_3^2).C_2$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.C_6.C_6.C_2^2$, $C_2^6.(S_3\times S_4)$, $(C_2^6\times C_6).A_4.C_2$, $C_2^2\times C_2^6.C_6\times S_3$, $C_2^6.(C_6\times S_4)$, $C_2^6.A_4.D_6$, $C_2^7.C_6^2.C_2$, $C_2^6.(C_6\times S_4)$, $C_2^6.A_4.D_6$, $C_2^6.C_6^2.C_2^2$, $C_2^6.C_6^2.C_2^2$, $C_2^6.C_2^2.C_6^2$, $C_2^6:C_3.D_6.C_2^2$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6:C_3.D_6.C_2^2$, $C_2^6.A_4.D_6$, $(C_2^5\times C_6).A_4.C_2^2$, $C_2^6.(C_6\times S_4)$, $C_2^4.D_6:S_4.C_2$, $C_2^2\times C_2^6:C_3.D_6$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.S_3^2$, $C_2^6.C_4:S_3^2$, $C_6.C_2^6:C_3.C_2^3$, $C_2^7.A_4.S_3$, $C_2^4.C_2^4.C_3.D_6$, $(C_2^6\times C_6).C_6.C_2^2$, $C_2^4.C_2^4.S_3^2$, $C_2^6.A_4.D_6$, $C_2^6:C_3.D_6.C_2^2$, $C_2^6.A_4.D_6$, $C_2^6.C_6^2.C_2^2$, $C_2^6.A_4.D_6$, $C_2^4.C_2^4.S_3^2$, $C_2^6.A_4.D_6$, $C_2^6.C_6^2.C_2^2$, $C_2^4.C_2^4.(C_6\times S_3)$, $C_2^6.(S_3\times D_{12})$, $C_2^6.A_4.D_6$, $C_2^6.A_4.D_6$, $C_6.C_2^6:C_3.C_2^3$, $C_2^6.D_6:D_6$, $(C_2^6\times C_6).A_4.C_2$, $C_2^6.(S_3\times S_4)$, $C_2^6.D_6:D_6$, $Q_8:C_2^2.D_6^2.C_2$, $C_2^6.(S_3\times S_4)$, $C_2^6.(C_4\times S_3^2)$, $C_2^6.A_4.D_6$, $C_2^6:C_3.D_{12}.C_2$, $C_2^4.C_2^4.C_3.D_6$, $C_2^6.C_2^2.C_6^2$, $C_2^7.(C_6\times A_4)$, $C_2^6:C_3.C_6.C_2^3$, $C_2^6.(A_4\times D_6)$, $C_2^6.D_6^2$, $C_2^8.S_3^2$, $C_2^5.(D_6\times S_4)$

Order 8192: $C_2^7.C_2^5.C_2$ x 19, $C_2^6.C_2^6.C_2$, $C_2^8.C_2^5$, $C_2^6.C_2^5.C_2^2$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^3.C_2^4$, $C_2.C_2^6.C_2^6$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^4.C_2^3$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^5.C_2^2$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2.C_2^5$, $C_2^6.C_2^4.C_2^3$, $C_2^6.C_2^6.C_2$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^6.C_2$, $C_2^{10}.D_4$, $C_2^7.D_4^2$, $C_2^7.C_2^6$

Order 6144: $C_2^9.A_4$, $C_3:(C_2^7.C_2^4)$, $C_3:(C_2^7.C_2^4)$, $C_3:(C_2^7.C_2^4)$, $C_3:(C_2^9.C_2^2)$, $C_3:(C_2^8.C_2^3)$

Order 4096: $C_2^5.C_2^6.C_2$

Order 3072: $C_3:(C_2^7.C_2^3)$ x 3, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^8.C_2^2)$, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^7.C_2^3)$, $C_3:(C_2^9.C_2)$, $C_3:(C_2^7.C_2^3)$, $C_2^9\times S_3$

Order 2048: $C_2^9.C_2^2$

Order 1536: $C_6\times C_2^6.C_2^2$ x 3, $C_2\times D_4.C_2^4\times S_3$, $C_2^5\times S_3\times D_4$, $C_2\times C_2^4.C_2^3\times S_3$, $C_2^7:D_6$, $C_2^8\times C_6$

Order 1024: $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^4.C_2^6$, $C_2^6:C_2^4$

Order 768: $D_6\times C_2^6$ x 3, $D_{12}:C_2^5$ x 3, $C_2^5\times C_3:D_4$ x 2, $C_6\times C_2.C_2^6$, $C_2^4\times C_6\times D_4$, $C_6\times C_2^4.C_2^3$, $C_2\times C_3:(C_2^6.C_2)$, $C_2^6:D_6$, $C_2^7:C_6$, $C_2\times C_3:(C_2.C_2^6)$, $C_2^5\times D_{12}$, $C_2^5\times C_4\times S_3$, $C_2\times C_3:(C_2^4.C_2^3)$

Order 512: $C_4^2:C_2^5$ x 3, $C_2^9$

Order 384: $D_6\times C_2^5$ x 4, $C_2^6\times C_6$ x 3, $C_{12}.C_2^5$ x 3, $C_2^5\times C_{12}$, $C_6.C_2^6$, $C_{12}:C_2^5$, $C_2^6:C_6$, $C_2^6:S_3$

Order 256: $C_2^4:C_2^4$, $D_4\times C_2^5$, $C_2^5:D_4$, $C_2^5:D_4$

Order 192: $D_6\times C_2^4$ x 4, $C_2^5\times C_6$ x 2, $D_6:C_2^4$ x 2, $C_{12}:C_2^4$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$

Order 128: $C_2^7$ x 3, $D_4:C_2^4$ x 3, $C_2^5\times C_4$, $C_2^3\wr C_2$

Order 96: $C_2^3\times D_6$ x 7, $C_2^4\times C_6$ x 4, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 64: $C_2^6$ x 2, $D_4\times C_2^3$

Order 48: $C_2^2\times D_6$ x 5, $C_2^3\times C_6$ x 4, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 32: $C_2^5$ x 4, $C_2^2\times D_4$, $C_2^3\times C_4$

Order 24: $C_2^2\times C_6$ x 3, $C_2\times D_6$ x 3

Order 16: $C_2^4$ x 4, $C_2^2\times C_4$

Order 12: $D_6$ x 3, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 3

Order 6: $C_6$ x 2, $S_3$

Order 4: $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 147456: $C_2^9.(D_6\times S_4)$ ($C_1$)

Order 73728: $C_2^9.(S_3\times S_4)$ ($C_2$), $C_2^7.C_2^4.S_3^2$ ($C_2$), $C_2^7.C_2^4.S_3^2$ ($C_2$), $C_2^7.C_2^4.(C_6\times S_3)$ ($C_2$), $C_2^9.A_4.D_6$ ($C_2$), $(C_2^8\times C_6).A_4.C_2^2$ ($C_2$), $C_2^5.C_2^6.C_3.D_6$ ($C_2$)

Order 36864: $(C_2^8\times C_6).A_4.C_2$ ($C_2^2$), $C_2^7.C_2^4.C_3.C_6$ ($C_2^2$), $C_2^9.(C_6\times A_4)$ ($C_2^2$), $(C_2^8\times C_6).A_4.C_2$ ($C_2^2$), $C_2^9.A_4.S_3$ ($C_2^2$), $C_2^9.A_4.C_6$ ($C_2^2$), $C_2^9.A_4.S_3$ ($C_2^2$)

Order 24576: $C_2^9.A_4.C_2^2$ ($S_3$), $C_3:(C_2^9.C_2^4)$ ($S_3$)

Order 18432: $C_2^9.A_4.C_3$ ($C_2^3$)

Order 12288: $C_3:(C_2^9.C_2^3)$ ($D_6$), $C_3:(C_2^9.C_2^3)$ ($D_6$), $C_2^9.S_4$ ($D_6$), $C_2^9.S_4$ ($D_6$), $C_3:(C_2^9.C_2^3)$ ($D_6$), $C_2^9.A_4.C_2$ ($D_6$)

Order 6144: $C_3:(C_2^7.C_2^4)$ ($S_4$) x 4, $C_2^9.A_4$ ($C_2\times D_6$), $C_3:(C_2^7.C_2^4)$ ($S_4$), $C_3:(C_2^7.C_2^4)$ ($S_4$), $C_3:(C_2^9.C_2^2)$ ($C_2\times D_6$), $C_3:(C_2^8.C_2^3)$ ($S_4$)

Order 4096: $C_2^5.C_2^6.C_2$ ($S_3^2$)

Order 3072: $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$) x 6, $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$) x 4, $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$) x 4, $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^8.C_2^2)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^9.C_2)$ ($C_2\times S_4$), $C_2^9\times S_3$ ($C_2\times S_4$)

Order 2048: $C_2^9.C_2^2$ ($S_3\times D_6$)

Order 1536: $C_6\times C_2^6.C_2^2$ ($C_2^2\times S_4$) x 6, $C_2\times D_4.C_2^4\times S_3$ ($C_2^2:S_4$) x 2, $C_2\times C_2^4.C_2^3\times S_3$ ($C_2^2:S_4$) x 2, $C_2^7:D_6$ ($C_2^2:S_4$) x 2, $C_2^5\times S_3\times D_4$ ($C_2^2:S_4$), $C_2^8\times C_6$ ($C_2^2\times S_4$)

Order 1024: $C_2^6:C_2^4$ ($S_3\times S_4$) x 4, $C_2^6.C_2^4$ ($S_3\times S_4$), $C_2^6.C_2^4$ ($S_3\times S_4$), $C_2^4.C_2^6$ ($S_3\times S_4$)

Order 768: $C_2^6:D_6$ ($C_2^3:S_4$) x 8, $D_{12}:C_2^5$ ($C_2^3:S_4$) x 4, $D_{12}:C_2^5$ ($C_2^3:S_4$) x 4, $D_6\times C_2^6$ ($C_2^3:S_4$) x 3, $C_6\times C_2.C_2^6$ ($C_2^3:S_4$) x 2, $C_6\times C_2^4.C_2^3$ ($C_2^3:S_4$) x 2, $C_2\times C_3:(C_2^6.C_2)$ ($C_2^3:S_4$) x 2, $C_2^7:C_6$ ($C_2^3:S_4$) x 2, $C_2\times C_3:(C_2.C_2^6)$ ($C_2^3:S_4$) x 2, $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2^3:S_4$) x 2, $C_2^4\times C_6\times D_4$ ($C_2^3:S_4$), $D_6\times C_2^6$ ($C_2^3:S_4$), $C_2^5\times D_{12}$ ($C_2^3:S_4$), $C_2^5\times C_3:D_4$ ($C_2^3:S_4$), $C_2^5\times C_3:D_4$ ($C_2^3:S_4$), $C_2^5\times C_4\times S_3$ ($C_2^3:S_4$)

Order 512: $C_4^2:C_2^5$ ($D_6\times S_4$) x 6, $C_2^9$ ($D_6\times S_4$)

Order 384: $D_6\times C_2^5$ ($C_2^4:S_4$) x 7, $C_{12}.C_2^5$ ($C_2^2\wr S_3$) x 4, $C_{12}.C_2^5$ ($C_2^4:S_4$) x 4, $C_2^6:C_6$ ($C_2^4:S_4$) x 4, $C_2^6:S_3$ ($C_2^4:S_4$) x 4, $C_2^6\times C_6$ ($C_2^2\wr S_3$) x 3, $C_2^6\times C_6$ ($C_2^4:S_4$), $C_2^5\times C_{12}$ ($C_2^4:S_4$), $C_6.C_2^6$ ($C_2^4:S_4$), $C_{12}:C_2^5$ ($C_2^4:S_4$)

Order 256: $C_2^4:C_2^4$ ($C_2^4:S_3^2$) x 2, $C_2^5:D_4$ ($C_2^4:S_3^2$) x 2, $C_2^5:D_4$ ($C_2^4:S_3^2$) x 2, $D_4\times C_2^5$ ($C_2^4:S_3^2$)

Order 192: $C_{12}:C_2^4$ ($C_2^5:S_4$) x 8, $D_6\times C_2^4$ ($C_2^5:S_4$) x 6, $C_2^5\times C_6$ ($C_2^5:S_4$) x 3, $D_6\times C_2^4$ ($C_2^5:S_4$), $D_6\times C_2^4$ ($C_2^5:S_4$), $C_{12}:C_2^4$ ($C_2^5:S_4$), $D_6:C_2^4$ ($C_2^5:S_4$), $D_6:C_2^4$ ($C_2^5:S_4$), $C_2^3\times D_{12}$ ($C_2^5:S_4$), $C_{12}:C_2^4$ ($C_2^5:S_4$)

Order 128: $D_4:C_2^4$ ($C_2^5:S_3^2$) x 4, $D_4:C_2^4$ ($S_3\times C_2^3:S_4$) x 4, $C_2^3\wr C_2$ ($S_3\times C_2^3:S_4$) x 4, $C_2^7$ ($C_2^5:S_3^2$) x 3, $C_2^7$ ($S_3\times C_2^3:S_4$), $C_2^5\times C_4$ ($S_3\times C_2^3:S_4$)

Order 96: $C_2^3\times D_6$ ($C_2\times C_2^5.S_4$) x 7, $C_2^4\times C_6$ ($C_2^6:S_4$) x 6, $C_2^3\times D_6$ ($C_2^6:S_4$) x 4, $C_2^3\times D_6$ ($C_2^6:S_4$) x 4, $C_{12}:C_2^3$ ($C_2\times C_2^5.S_4$) x 4, $C_2^3:D_6$ ($C_2\times C_2^5.S_4$) x 4, $C_2^3:D_6$ ($C_2^6:S_4$) x 4, $C_2^2\times D_{12}$ ($C_2^6:S_4$) x 4, $C_{12}:C_2^3$ ($C_2^6:S_4$) x 4, $C_2^4\times C_6$ ($C_2^6:S_4$), $C_2^4\times C_6$ ($C_2\times C_2^5.S_4$), $C_2^3\times C_{12}$ ($C_2\times C_2^5.S_4$), $C_6.C_2^4$ ($C_2\times C_2^5.S_4$)

Order 64: $C_2^6$ ($S_3\times C_2^4:S_4$) x 3, $D_4\times C_2^3$ ($C_2^6:S_3^2$)

Order 48: $C_2^2\times D_6$ ($C_2^4.C_2^4.D_6$) x 4, $C_2^2\times D_6$ ($C_2^4.C_2^4.D_6$) x 4, $C_2^3\times C_6$ ($C_2^5.S_4.C_2^2$) x 3, $C_2^3\times C_6$ ($C_2^4.C_2^4.D_6$) x 2, $C_2^3\times C_6$ ($C_2^5.S_4.C_2^2$) x 2, $C_2^2\times D_6$ ($C_2^6.S_4.C_2$) x 2, $C_2^2\times C_{12}$ ($C_2^6.A_4.C_2^2$) x 2, $C_6.C_2^3$ ($C_2^4.C_2^4.D_6$) x 2, $C_2^2\times D_6$ ($C_2^8.D_6$)

Order 32: $C_2^5$ ($S_3\times C_2^5:S_4$) x 6, $C_2^2\times D_4$ ($S_3\times C_2^5:S_4$) x 4, $C_2^5$ ($C_2^6.S_3^2.C_2$), $C_2^5$ ($S_3\times C_2^5:S_4$), $C_2^3\times C_4$ ($S_3\times C_2^5:S_4$)

Order 24: $C_2\times D_6$ ($C_2^7.S_4.C_2$) x 4, $C_2^2\times C_6$ ($C_2^7.(C_2\times S_4)$) x 2, $C_2\times D_6$ ($C_2^7.A_4.C_2^2$) x 2, $C_2^2\times C_6$ ($C_2^6.A_4.C_2^3$), $C_2^2\times C_6$ ($C_2^8.D_6.C_2$)

Order 16: $C_2^4$ ($C_2^5.A_4.D_6.C_2$) x 2, $C_2^4$ ($C_2.C_2^6.C_3^2.C_2^3$) x 2, $C_2^4$ ($S_3\times C_2^6:S_4$) x 2, $C_2^2\times C_4$ ($S_3\times C_2^6:S_4$) x 2, $C_2^4$ ($C_2.C_2^6.S_3^2.C_2$)

Order 12: $D_6$ ($C_2^4.C_2^6.D_6$) x 4, $C_2\times C_6$ ($C_2^3.C_2^6.D_6.C_2$) x 2, $D_6$ ($C_2^6.C_2^4.D_6$) x 2, $D_6$ ($C_2^4.C_2^6.D_6$) x 2, $C_2\times C_6$ ($C_2^7.S_4.C_2^2$)

Order 8: $C_2^3$ ($C_2^6.(D_6\times S_4)$) x 2, $C_2^3$ ($C_2^8.S_3^2.C_2$), $C_2^3$ ($C_2^6.(D_6\times S_4)$)

Order 6: $S_3$ ($C_2^9.S_4.C_2$) x 4, $C_6$ ($C_2^6.C_2^4.D_6.C_2$) x 2, $C_6$ ($C_2^4.C_2^6.C_6.C_2^2$)

Order 4: $C_2^2$ ($C_2^7.A_4.D_6.C_2$) x 2, $C_2^2$ ($C_2^7.(S_3\times S_4).C_2$)

Order 3: $C_3$ ($C_2^9.A_4.C_2^3$)

Order 2: $C_2$ ($C_2^8.A_4.D_6.C_2$) x 2, $C_2$ ($C_2^6.C_2^4.C_3.D_6.C_2$)

Order 1: $C_1$ ($C_2^9.(D_6\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 147456: $C_2^9.(D_6\times S_4)$ ($C_1$)

Order 73728: $C_2^9.(S_3\times S_4)$ ($C_2$), $C_2^7.C_2^4.S_3^2$ ($C_2$), $C_2^7.C_2^4.S_3^2$ ($C_2$), $C_2^7.C_2^4.(C_6\times S_3)$ ($C_2$), $C_2^9.A_4.D_6$ ($C_2$), $(C_2^8\times C_6).A_4.C_2^2$ ($C_2$), $C_2^5.C_2^6.C_3.D_6$ ($C_2$)

Order 36864: $(C_2^8\times C_6).A_4.C_2$ ($C_2^2$), $C_2^7.C_2^4.C_3.C_6$ ($C_2^2$), $C_2^9.(C_6\times A_4)$ ($C_2^2$), $(C_2^8\times C_6).A_4.C_2$ ($C_2^2$), $C_2^9.A_4.S_3$ ($C_2^2$), $C_2^9.A_4.C_6$ ($C_2^2$), $C_2^9.A_4.S_3$ ($C_2^2$)

Order 24576: $C_2^9.A_4.C_2^2$ ($S_3$), $C_3:(C_2^9.C_2^4)$ ($S_3$)

Order 18432: $C_2^9.A_4.C_3$ ($C_2^3$)

Order 12288: $C_3:(C_2^9.C_2^3)$ ($D_6$), $C_3:(C_2^9.C_2^3)$ ($D_6$), $C_2^9.S_4$ ($D_6$), $C_2^9.S_4$ ($D_6$), $C_3:(C_2^9.C_2^3)$ ($D_6$), $C_2^9.A_4.C_2$ ($D_6$)

Order 6144: $C_2^9.A_4$ ($C_2\times D_6$), $C_3:(C_2^7.C_2^4)$ ($S_4$), $C_3:(C_2^7.C_2^4)$ ($S_4$), $C_3:(C_2^7.C_2^4)$ ($S_4$), $C_3:(C_2^9.C_2^2)$ ($C_2\times D_6$), $C_3:(C_2^8.C_2^3)$ ($S_4$)

Order 4096: $C_2^5.C_2^6.C_2$ ($S_3^2$)

Order 3072: $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$) x 3, $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^8.C_2^2)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_3:(C_2^9.C_2)$ ($C_2\times S_4$), $C_3:(C_2^7.C_2^3)$ ($C_2\times S_4$), $C_2^9\times S_3$ ($C_2\times S_4$)

Order 2048: $C_2^9.C_2^2$ ($S_3\times D_6$)

Order 1536: $C_6\times C_2^6.C_2^2$ ($C_2^2\times S_4$) x 3, $C_2\times D_4.C_2^4\times S_3$ ($C_2^2:S_4$), $C_2^5\times S_3\times D_4$ ($C_2^2:S_4$), $C_2\times C_2^4.C_2^3\times S_3$ ($C_2^2:S_4$), $C_2^7:D_6$ ($C_2^2:S_4$), $C_2^8\times C_6$ ($C_2^2\times S_4$)

Order 1024: $C_2^6.C_2^4$ ($S_3\times S_4$), $C_2^6.C_2^4$ ($S_3\times S_4$), $C_2^4.C_2^6$ ($S_3\times S_4$), $C_2^6:C_2^4$ ($S_3\times S_4$)

Order 768: $D_6\times C_2^6$ ($C_2^3:S_4$) x 2, $D_{12}:C_2^5$ ($C_2^3:S_4$) x 2, $C_6\times C_2.C_2^6$ ($C_2^3:S_4$), $C_2^4\times C_6\times D_4$ ($C_2^3:S_4$), $C_6\times C_2^4.C_2^3$ ($C_2^3:S_4$), $C_2\times C_3:(C_2^6.C_2)$ ($C_2^3:S_4$), $C_2^6:D_6$ ($C_2^3:S_4$), $C_2^7:C_6$ ($C_2^3:S_4$), $D_6\times C_2^6$ ($C_2^3:S_4$), $C_2\times C_3:(C_2.C_2^6)$ ($C_2^3:S_4$), $D_{12}:C_2^5$ ($C_2^3:S_4$), $C_2^5\times D_{12}$ ($C_2^3:S_4$), $C_2^5\times C_3:D_4$ ($C_2^3:S_4$), $C_2^5\times C_3:D_4$ ($C_2^3:S_4$), $C_2^5\times C_4\times S_3$ ($C_2^3:S_4$), $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2^3:S_4$)

Order 512: $C_4^2:C_2^5$ ($D_6\times S_4$) x 3, $C_2^9$ ($D_6\times S_4$)

Order 384: $D_6\times C_2^5$ ($C_2^4:S_4$) x 4, $C_2^6\times C_6$ ($C_2^2\wr S_3$) x 2, $C_{12}.C_2^5$ ($C_2^2\wr S_3$) x 2, $C_2^6\times C_6$ ($C_2^4:S_4$), $C_{12}.C_2^5$ ($C_2^4:S_4$), $C_2^5\times C_{12}$ ($C_2^4:S_4$), $C_6.C_2^6$ ($C_2^4:S_4$), $C_{12}:C_2^5$ ($C_2^4:S_4$), $C_2^6:C_6$ ($C_2^4:S_4$), $C_2^6:S_3$ ($C_2^4:S_4$)

Order 256: $C_2^4:C_2^4$ ($C_2^4:S_3^2$), $D_4\times C_2^5$ ($C_2^4:S_3^2$), $C_2^5:D_4$ ($C_2^4:S_3^2$), $C_2^5:D_4$ ($C_2^4:S_3^2$)

Order 192: $C_2^5\times C_6$ ($C_2^5:S_4$) x 2, $D_6\times C_2^4$ ($C_2^5:S_4$) x 2, $D_6\times C_2^4$ ($C_2^5:S_4$), $D_6\times C_2^4$ ($C_2^5:S_4$), $C_{12}:C_2^4$ ($C_2^5:S_4$), $D_6:C_2^4$ ($C_2^5:S_4$), $D_6:C_2^4$ ($C_2^5:S_4$), $C_{12}:C_2^4$ ($C_2^5:S_4$), $C_2^3\times D_{12}$ ($C_2^5:S_4$), $C_{12}:C_2^4$ ($C_2^5:S_4$)

Order 128: $C_2^7$ ($C_2^5:S_3^2$) x 2, $D_4:C_2^4$ ($C_2^5:S_3^2$) x 2, $C_2^7$ ($S_3\times C_2^3:S_4$), $D_4:C_2^4$ ($S_3\times C_2^3:S_4$), $C_2^5\times C_4$ ($S_3\times C_2^3:S_4$), $C_2^3\wr C_2$ ($S_3\times C_2^3:S_4$)

Order 96: $C_2^3\times D_6$ ($C_2\times C_2^5.S_4$) x 4, $C_2^4\times C_6$ ($C_2^6:S_4$) x 2, $C_2^3\times D_6$ ($C_2^6:S_4$) x 2, $C_2^4\times C_6$ ($C_2^6:S_4$), $C_2^4\times C_6$ ($C_2\times C_2^5.S_4$), $C_2^3\times D_6$ ($C_2^6:S_4$), $C_{12}:C_2^3$ ($C_2\times C_2^5.S_4$), $C_2^3\times C_{12}$ ($C_2\times C_2^5.S_4$), $C_2^3:D_6$ ($C_2\times C_2^5.S_4$), $C_2^3:D_6$ ($C_2^6:S_4$), $C_6.C_2^4$ ($C_2\times C_2^5.S_4$), $C_2^2\times D_{12}$ ($C_2^6:S_4$), $C_{12}:C_2^3$ ($C_2^6:S_4$)

Order 64: $C_2^6$ ($S_3\times C_2^4:S_4$) x 2, $D_4\times C_2^3$ ($C_2^6:S_3^2$)

Order 48: $C_2^3\times C_6$ ($C_2^5.S_4.C_2^2$) x 2, $C_2^2\times D_6$ ($C_2^4.C_2^4.D_6$) x 2, $C_2^3\times C_6$ ($C_2^4.C_2^4.D_6$), $C_2^3\times C_6$ ($C_2^5.S_4.C_2^2$), $C_2^2\times D_6$ ($C_2^8.D_6$), $C_2^2\times D_6$ ($C_2^4.C_2^4.D_6$), $C_2^2\times D_6$ ($C_2^6.S_4.C_2$), $C_2^2\times C_{12}$ ($C_2^6.A_4.C_2^2$), $C_6.C_2^3$ ($C_2^4.C_2^4.D_6$)

Order 32: $C_2^5$ ($S_3\times C_2^5:S_4$) x 2, $C_2^5$ ($C_2^6.S_3^2.C_2$), $C_2^5$ ($S_3\times C_2^5:S_4$), $C_2^2\times D_4$ ($S_3\times C_2^5:S_4$), $C_2^3\times C_4$ ($S_3\times C_2^5:S_4$)

Order 24: $C_2\times D_6$ ($C_2^7.A_4.C_2^2$) x 2, $C_2^2\times C_6$ ($C_2^6.A_4.C_2^3$), $C_2^2\times C_6$ ($C_2^8.D_6.C_2$), $C_2^2\times C_6$ ($C_2^7.(C_2\times S_4)$), $C_2\times D_6$ ($C_2^7.S_4.C_2$)

Order 16: $C_2^4$ ($C_2^5.A_4.D_6.C_2$), $C_2^4$ ($C_2.C_2^6.S_3^2.C_2$), $C_2^4$ ($C_2.C_2^6.C_3^2.C_2^3$), $C_2^4$ ($S_3\times C_2^6:S_4$), $C_2^2\times C_4$ ($S_3\times C_2^6:S_4$)

Order 12: $C_2\times C_6$ ($C_2^7.S_4.C_2^2$), $C_2\times C_6$ ($C_2^3.C_2^6.D_6.C_2$), $D_6$ ($C_2^4.C_2^6.D_6$), $D_6$ ($C_2^6.C_2^4.D_6$), $D_6$ ($C_2^4.C_2^6.D_6$)

Order 8: $C_2^3$ ($C_2^8.S_3^2.C_2$), $C_2^3$ ($C_2^6.(D_6\times S_4)$), $C_2^3$ ($C_2^6.(D_6\times S_4)$)

Order 6: $C_6$ ($C_2^6.C_2^4.D_6.C_2$), $C_6$ ($C_2^4.C_2^6.C_6.C_2^2$), $S_3$ ($C_2^9.S_4.C_2$)

Order 4: $C_2^2$ ($C_2^7.A_4.D_6.C_2$), $C_2^2$ ($C_2^7.(S_3\times S_4).C_2$)

Order 3: $C_3$ ($C_2^9.A_4.C_2^3$)

Order 2: $C_2$ ($C_2^8.A_4.D_6.C_2$), $C_2$ ($C_2^6.C_2^4.C_3.D_6.C_2$)

Order 1: $C_1$ ($C_2^9.(D_6\times S_4)$)

Series

Derived series

$C_2^9.(D_6\times S_4)$

$\rhd$

$C_2^9.A_4.C_3$

$\rhd$

$C_2^9.C_2^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^9.(D_6\times S_4)$

$\rhd$

$(C_2^8\times C_6).A_4.C_2^2$

$\rhd$

$C_2^9.(C_6\times A_4)$

$\rhd$

$C_2^9.A_4.C_3$

$\rhd$

$C_2^9.A_4$

$\rhd$

$C_2^9.C_2^2$

$\rhd$

$C_2^9$

$\rhd$

$C_2^7$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^9.(D_6\times S_4)$

$\rhd$

$C_2^9.A_4.C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $432 \times 432$ rational character table is not available for this group.