Abstract group 294912.sf: $S_3\times C_2^5.C_2^6:S_4$

• Label: 294912.sf
• Order: \( 2^{15} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $35$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$S_3\times C_2^5.C_2^6:S_4$
Order:	\(294912\)\(\medspace = 2^{15} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^6.C_2^4.C_6^2.C_2^6.C_2$, of order \(4718592\)\(\medspace = 2^{19} \cdot 3^{2} \)
Composition factors:	$C_2$ x 15, $C_3$ x 2
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	7807	1538	92544	63806	30720	83136	15360	294912
Conjugacy classes	1	95	3	136	78	12	77	6	408
Divisions	1	95	3	136	78	10	77	5	405
Autjugacy classes	1	39	3	62	39	6	38	3	191

Minimal presentations

Permutation degree:	$35$
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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid a^{2}=d^{2}=e^{2}=f^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $35$ $\langle(1,2)(3,5)(4,6)(7,10)(8,11)(9,12)(13,16)(14,15)(18,19), (1,3)(2,5)(4,9) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $(C_2^5.C_2^6:S_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^5.C_2^6:S_4)$ . $S_3$	$C_2^8$ . $(S_3\times C_2^3:S_4)$	$C_2^6$ . $(S_3\times C_2^5:S_4)$ (2)	$S_3$ . $(C_2^5.C_2^6:S_4)$	all 64
Aut. group:	$\Aut(C_2^4.D_6)$	$\Aut(C_2^4.D_6)$

Elements of the group are displayed as permutations of degree 35.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 186 normal subgroups (78 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^6.C_2^4.S_3^2.C_2$
Commutator:	$G' \simeq$ $C_2^6.C_2^6.C_3^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $D_4:C_2^5$	$G/\Phi \simeq$ $C_2^5:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_2^6.C_2^6.C_2$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $S_3\times C_2^5.C_2^6:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^6.C_2^4.D_6.C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^6.C_2^2$
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 16384) are shown, as well as all normal subgroups of any index.

Order 294912: $S_3\times C_2^5.C_2^6:S_4$

Order 147456: $(C_2^7\times C_6).C_2^4.C_6.C_2$, $C_2^8.C_2^4.C_3.D_6$, $C_3\times C_2^6.C_2^6.D_6$, $C_2^8.C_2^4.S_3^2$, $C_2^6.C_2^6.S_3^2$, $C_2^6.C_2^6.S_3^2$, $C_2^8.C_2^4.S_3^2$

Order 98304: $C_3:(C_2^7.C_2^5.C_2^3)$, $C_2^6.C_2^6:S_4$

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

Order 49152: $C_2^5.C_2^6:S_4$ x 5, $C_3:(C_2^7.C_2.D_4^2)$ x 2, $C_3:(C_2^7.C_2^4.C_2^3)$ x 2, $C_3:(C_2^6.D_4^2.C_2^2)$ x 2, $C_3:(C_2^7.C_2.D_4^2)$ x 2, $C_3:(C_2^6.C_2^5.C_2^3)$ x 2, $C_3:(C_2^7.C_2^4.C_2^3)$ x 2, $C_3:(C_2^6.C_2^5.C_2^3)$ x 2, $C_2^6.D_4^2.D_6$ x 2, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_2^8.C_2^4.D_6$, $C_3:(C_2^8.C_2^4.C_2^2)$, $C_3:(C_2^8.C_2^6)$, $C_2^8.C_2^4.D_6$, $C_2^8.C_2^4.C_6.C_2$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^8.C_2^4.C_2^2)$, $C_3\times C_2^7.C_2^5.C_2^2$, $C_3:(C_2^7.C_2^4.C_2^3)$, $C_3:(C_2^8.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2^3)$

Order 36864: $C_2^5.C_2^4.S_3^2.C_2$ x 2, $C_2^5.C_2^4.C_3^2.C_2^3$ x 2, $C_2^6.(C_6\times A_4).C_2^3$ x 2, $C_2^6.C_2^4.S_3^2$ x 2, $C_3\times C_2^6.(D_4\times S_4)$ x 2, $C_2^4.C_2^4.C_3.C_6.C_2^3$ x 2, $(C_2^6\times C_6).A_4.C_2^3$ x 2, $C_2^4.C_2^4.D_6:D_6$ x 2, $C_2^6.A_4.D_6.C_2^2$ x 2, $C_2^4.C_2^4.(C_4\times S_3^2)$ x 2, $C_2^6.A_4.D_6.C_2^2$ x 2, $C_2^4.C_2^6.S_3^2$ x 2, $C_2^6.(C_6\times A_4).C_2^3$ x 2, $C_2^4.C_2^4.C_6^2.C_2^2$ x 2, $C_2^6.C_2^2.D_6:D_6$ x 2, $C_2^6.C_2^2.D_6^2$ x 2, $C_2^4.C_2^4.D_6^2$ x 2, $C_2^4.C_2^4.C_6^2.C_2^2$ x 2, $C_2^6.C_2^4.S_3^2$ x 2, $C_2^6.A_4.D_6.C_2^2$ x 2, $C_2^7.(D_6\times S_4)$ x 2, $C_2^6.C_2^6.C_3^2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_3\times C_2^5.C_2^4.D_6.C_2$, $(C_2^3\times C_6).C_2^6.C_6.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.C_3^2.C_2^3$, $C_2^5.C_2^4.S_3^2.C_2$, $(C_2^4\times C_6).C_2^4.C_6.C_2^2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2.C_2^6.A_4.D_6.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.S_3^2.C_2$

Order 32768: $C_2^7.C_2^6.C_2^2$

Order 24576: $C_3:(C_2^8.C_2^5)$ x 5, $C_3:(C_2^8.C_2^5)$ x 5, $C_3:(C_2^8.C_2^5)$ x 5, $C_3:(C_2^8.C_2^5)$ x 4, $C_3:(C_2^8.C_2^5)$ x 4, $C_3:(C_2^8.C_2^5)$ x 4, $C_3:(C_2^7.C_2^6)$ x 4, $C_3:(C_2^8.C_2^5)$ x 3, $C_2^8.C_2^4.C_6$ x 3, $C_2^8.C_2^4.S_3$ x 3, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2^6)$ x 2, $C_3:(C_2^5.C_2^6.C_2^2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2\times C_2^6.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2^5.C_2)$ x 2, $C_3:(C_2^5.C_2^6.C_2^2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2.C_2^6.C_2^6)$ x 2, $C_3\times C_2^7.C_2^4.C_2^2$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2^4.C_2^2)$ x 2, $C_3:(C_2^5.C_2^6.C_2^2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_3:(C_2^7.C_2^5.C_2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3\times C_2^7.C_2^4.C_2^2$ x 2, $C_3:(C_2^5.C_2^5.C_2^3)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^5.C_2^4.C_2^4)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3\times C_2^8.C_2^4.C_2$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3\times C_2^7.C_2.C_2^5$ x 2, $C_3:(C_2^5.C_2^4.C_2^4)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2.C_2^6.C_2^6)$ x 2, $C_3:(C_2^5.C_2^3.C_2^5)$ x 2, $C_3:(C_2^7.C_2^4.C_2^2)$ x 2, $C_3:(C_2^7.C_2^4.C_2^2)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_3:(C_2^7.C_2^5.C_2)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_2^4.C_2^4.C_6.C_2^4$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^8.C_2^3.C_2^2)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^8.C_2^3.C_2^2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^7.C_2^4.C_2^2)$ x 2, $C_3:(C_2^7.C_2^4.C_2^2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_2^6.C_2^6.S_3$ x 2, $C_3:(C_2^5.C_2^4.C_2^4)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^8.C_2^3.C_2^2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^5.C_2^6.C_2^2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^5.C_2^5.C_2^3)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^8.C_2^3.C_2^2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^5.C_2^6.C_2^2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^3.C_2^4)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2.C_2^6)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^5.C_2^3.C_2^5)$ x 2, $C_3:(C_2^6.C_2^3.C_2^4)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^5.C_2^4.C_2^4)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_3:(C_2^7.C_2^3.C_2^3)$ x 2, $C_3:(C_2^6.C_2.C_2^6)$ x 2, $C_3:(C_2^6.C_2^5.C_2^2)$ x 2, $C_3:(C_2^7.C_2^4.C_2^2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $(C_2^4\times D_{12}).D_4^2$ x 2, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2^2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_3:(C_2^8.C_2^5)$, $C_2^6.C_2^6.C_6$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3\times C_2^7.C_2^4.C_2^2$, $C_3\times C_2^6.C_2^5.C_2^2$, $C_3:(C_2^8.C_2^4.C_2)$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_2^5.C_2^4.C_6.C_2^3$, $C_2^8.C_2^4.S_3$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3\times C_2^6.C_2^6.C_2$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3\times C_2^6.C_2^5.C_2^2$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3\times C_2^8.C_2^4.C_2$, $C_3\times C_2^6.C_2^6.C_2$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.D_4^2.C_2)$

Order 18432: $C_2^6.(D_6\times S_4)$ x 40, $C_2^5.C_2^4.S_3^2$ x 17, $C_2^6.(D_6\times S_4)$ x 8, $(C_2^3\times C_6).C_2^6.C_6$ x 6, $(C_2^4\times C_6).C_2^4.C_6.C_2$ x 5, $(C_2^3\times C_6).C_2^6.C_6$ x 4, $(C_2^5\times C_6).C_2^4.C_6$ x 4, $C_2^5.C_2^4.C_3.D_6$ x 3, $C_2.C_2^6.A_4.D_6$ x 3, $(C_2^3\times C_6).C_2^6.C_6$ x 3, $C_2^5.C_2^4.(C_6\times S_3)$ x 3, $C_2^6.A_4.D_6.C_2$ x 2, $C_2^6.A_4.C_6.C_2^2$ x 2, $C_3\times C_2^6.A_4.D_4$ x 2, $C_3\times C_2^6.C_2^4.S_3$ x 2, $C_2.C_2^6.A_4.D_6$ x 2, $C_2^4.C_2^4.C_3^2.C_2^3$ x 2, $C_2^4.C_2^4.C_6^2.C_2$ x 2, $C_2^6.A_4.D_6.C_2$ 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, $C_2^4.C_2^4.C_3^2.D_4$ x 2, $C_2^6.A_4.D_6.C_2$ x 2, $C_2^6.A_4.C_6.C_2^2$ x 2, $C_2^4.C_2^4.C_3^2.C_2^3$ x 2, $C_2^6.C_2^4.C_3.S_3$ x 2, $Q_8:C_2^2.C_6^2.C_2^4$ x 2, $C_2^3:A_4.C_6.C_2^5$ x 2, $C_2^6.D_6:S_4$ x 2, $C_2^6.(C_6\times S_4).C_2$ x 2, $(C_2^7\times C_6).A_4.C_2$ x 2, $C_2^4.C_2^4.C_3.D_6.C_2$ x 2, $C_2^6.A_4.C_3.D_4$ x 2, $(C_2^5\times C_6).A_4.C_2^3$ x 2, $C_2^4.C_2^6.C_3.S_3$ x 2, $C_2^6.A_4.D_6.C_2$ x 2, $C_2^6.(C_6\times S_4).C_2$ x 2, $C_2^6.A_4.D_6.C_2$ x 2, $(C_2^5\times C_6).C_2^4.C_6$ x 2, $C_3\times C_2^6.A_4.D_4$ x 2, $C_3\times C_2^6.(C_4\times S_4)$ x 2, $C_3\times C_2^7.A_4.C_2^2$ x 2, $C_2^6.C_2^4.C_3.S_3$ x 2, $C_3\times C_2^6.C_2^4.S_3$ x 2, $C_2^6.C_3:(C_4\times S_4)$ x 2, $C_2^6.A_4.D_{12}$ x 2, $C_3\times C_2^6.A_4.D_4$ x 2, $(C_2^6\times C_6).A_4.C_2^2$ x 2, $C_2.C_2^6.(C_6\times S_4)$ x 2, $C_3\times C_2^6.C_2^4.S_3$ x 2, $C_2^6.A_4.D_6.C_2$ x 2, $C_2^6.A_4.D_6.C_2$ x 2, $(C_2^5\times C_6).A_4.C_2^2.C_2$ x 2, $(C_2^5\times C_6).A_4.C_2^3$ x 2, $(C_2\times C_6).C_2^6.C_6.C_2^2$ x 2, $C_2^6.D_6:S_4$ x 2, $C_2^7.D_6^2$ x 2, $C_2^7.(C_6\times S_4)$ x 2, $C_3\times C_2^5.C_2^4.D_6$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2^5.C_2^4.S_3^2$, $C_2.C_2^6.A_4.D_6$, $C_3\times C_2^4.C_2^6.C_6$, $C_2.C_2^6.A_4.D_6$, $C_2^5.C_2^4.S_3^2$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2^5.C_2^4.S_3^2$, $C_3\times C_2^5.C_2^4.D_6$, $C_2.C_2^6.A_4.D_6$, $C_3\times C_2^3.C_2^6.D_6$, $C_3\times C_2^5.C_2^4.D_6$, $C_2^5.C_2^4.S_3^2$, $C_3\times C_2^5.C_2^4.D_6$, $Q_8:C_2^2.C_6^2.C_2^4$, $C_2^5.C_2^4.S_3^2$, $C_2^5.C_2^4.C_3.D_6$, $C_2.C_2^6.A_4.D_6$, $C_2.C_2^6.A_4.D_6$, $C_2.C_2^6.A_4.D_6$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2.C_2^6.A_4.D_6$, $(C_2^5\times C_6).C_2^4.C_6$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2^5.C_2^4.S_3^2$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2^5.C_2^4.S_3^2$, $C_2^5.C_2^4.C_3.D_6$, $C_3\times C_2^5.C_2^4.C_6.C_2$, $C_2^5.C_2^4.C_3.D_6$

Order 16384: $C_2^7.C_2^5.C_2^2$ x 10, $C_2^8.C_2^5.C_2$ x 6, $C_2^7.C_2^5.C_2^2$ x 2, $C_2^5.C_2^6.C_2^3$ x 2, $C_2^8.C_2\wr C_2^2$ x 2, $C_2^{10}.(C_2\times D_4)$ x 2, $C_2^7.C_2^5.C_2^2$, $C_2^6.C_2^6.C_2^2$, $C_2^9.C_2^5$, $C_2^7.C_2^5.C_2^2$, $C_2^6.C_2^6.C_2^2$, $C_2^6.C_2^5.C_2^3$, $C_2^6.C_2^6.C_2^2$

Order 12288: $C_3:(C_2^8.C_2^4)$ x 2, $C_2^6.C_2^6.C_3$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^6)$

Order 8192: $C_2^6.C_2^6.C_2$

Order 6144: $C_3:(C_2^5.C_2^6)$ x 2, $C_3:(D_4^2.C_2^5)$ x 2, $C_3\times C_2^8.C_2^3$ x 2, $C_3\times C_2^6.C_2^5$, $C_3:(C_2^6.C_2^5)$, $C_3:(C_2^6.C_2^5)$

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

Order 3072: $C_3:(C_2^6.C_2^4)$ x 2, $C_3:(C_2^5.C_2^5)$, $C_3:(C_2^6.C_2^4)$

Order 2048: $C_2^8.C_2^3$ x 2, $C_2^6.C_2^5$

Order 1536: $C_2^3\times Q_8:C_2^2\times S_3$ x 5, $C_2^8\times S_3$ x 2, $C_2\times D_4.C_2^4\times S_3$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_2^5.C_2^3\times S_3$, $C_3\times C_2^5.C_2^4$

Order 1024: $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$

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

Order 512: $C_2^3.C_2^6$

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

Order 256: $D_4:C_2^5$ x 5, $C_2^8$ x 2, $C_2^4:C_2^4$, $C_4^2.C_2^4$

Order 192: $D_6\times C_2^4$ x 6, $C_2^5\times C_6$ x 3

Order 128: $D_4:C_2^4$

Order 96: $C_2^3\times D_6$ x 5, $C_2^4\times C_6$ x 3

Order 64: $C_2^6$ x 3

Order 48: $C_2^2\times D_6$ x 6, $C_2^3\times C_6$ x 3

Order 32: $C_2^5$ x 3

Order 24: $C_2\times D_6$ x 16, $C_2^2\times C_6$ x 3

Order 16: $C_2^4$ x 3

Order 12: $D_6$ x 12, $C_2\times C_6$ x 7

Order 9: $C_3^2$

Order 8: $C_2^3$ x 3

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

Order 4: $C_2^2$ x 7

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 16384) are shown, as well as all normal subgroups of any index.

Order 294912: $S_3\times C_2^5.C_2^6:S_4$

Order 147456: $(C_2^7\times C_6).C_2^4.C_6.C_2$, $C_2^8.C_2^4.C_3.D_6$, $C_3\times C_2^6.C_2^6.D_6$, $C_2^8.C_2^4.S_3^2$, $C_2^6.C_2^6.S_3^2$, $C_2^6.C_2^6.S_3^2$, $C_2^8.C_2^4.S_3^2$

Order 98304: $C_3:(C_2^7.C_2^5.C_2^3)$, $C_2^6.C_2^6:S_4$

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

Order 49152: $C_2^5.C_2^6:S_4$ x 5, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2.D_4^2)$, $C_3:(C_2^7.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_2^8.C_2^4.D_6$, $C_3:(C_2^6.D_4^2.C_2^2)$, $C_3:(C_2^7.C_2.D_4^2)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^8.C_2^4.C_2^2)$, $C_3:(C_2^8.C_2^6)$, $C_2^8.C_2^4.D_6$, $C_3:(C_2^7.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_2^8.C_2^4.C_6.C_2$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^8.C_2^4.C_2^2)$, $C_3\times C_2^7.C_2^5.C_2^2$, $C_3:(C_2^7.C_2^4.C_2^3)$, $C_3:(C_2^8.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2^3)$, $C_2^6.D_4^2.D_6$

Order 36864: $C_2^6.C_2^6.C_3^2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.C_3^2.C_2^3$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^6.(C_6\times A_4).C_2^3$, $C_2^5.C_2^4.S_3^2.C_2$, $C_3\times C_2^5.C_2^4.D_6.C_2$, $(C_2^3\times C_6).C_2^6.C_6.C_2$, $C_2^6.C_2^4.S_3^2$, $C_3\times C_2^6.(D_4\times S_4)$, $C_2^4.C_2^4.C_3.C_6.C_2^3$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.C_3^2.C_2^3$, $(C_2^6\times C_6).A_4.C_2^3$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^4.C_2^4.D_6:D_6$, $C_2^6.A_4.D_6.C_2^2$, $(C_2^4\times C_6).C_2^4.C_6.C_2^2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2.C_2^6.A_4.D_6.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^4.C_2^4.(C_4\times S_3^2)$, $C_2^6.A_4.D_6.C_2^2$, $C_2^4.C_2^6.S_3^2$, $C_2^6.(C_6\times A_4).C_2^3$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^4.C_2^4.C_6^2.C_2^2$, $C_2^6.C_2^2.D_6:D_6$, $C_2^6.C_2^2.D_6^2$, $C_2^4.C_2^4.D_6^2$, $C_2^4.C_2^4.C_6^2.C_2^2$, $C_2^6.C_2^4.S_3^2$, $C_2^6.A_4.D_6.C_2^2$, $C_2^7.(D_6\times S_4)$

Order 32768: $C_2^7.C_2^6.C_2^2$

Order 24576: $C_3:(C_2^8.C_2^5)$ x 3, $C_3:(C_2^8.C_2^5)$ x 3, $C_3:(C_2^8.C_2^5)$ x 3, $C_2^8.C_2^4.C_6$ x 3, $C_2^8.C_2^4.S_3$ x 3, $C_3:(C_2^8.C_2^5)$ x 2, $C_3:(C_2^8.C_2^5)$ x 2, $C_3:(C_2^8.C_2^5)$ x 2, $C_3:(C_2^8.C_2^5)$ x 2, $C_2^6.C_2^6.S_3$ x 2, $C_3:(C_2^7.C_2^6)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2^2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2\times C_2^6.C_2.C_2^5)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3\times C_2^7.C_2^4.C_2^2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_2^6.C_2^6.C_6$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3\times C_2^7.C_2^4.C_2^2$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3\times C_2^8.C_2^4.C_2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3\times C_2^7.C_2.C_2^5$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^6.C_2^2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^5.C_2^3.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^5.C_2)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_2^4.C_2^4.C_6.C_2^4$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3\times C_2^7.C_2^4.C_2^2$, $C_3\times C_2^6.C_2^5.C_2^2$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_3:(C_2^8.C_2^4.C_2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^5.C_2^4.C_6.C_2^3$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_2^8.C_2^4.S_3$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3\times C_2^6.C_2^6.C_2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3\times C_2^6.C_2^5.C_2^2$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^8.C_2^3.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3\times C_2^8.C_2^4.C_2$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3\times C_2^6.C_2^6.C_2$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^3.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.D_4^2.C_2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $(C_2^4\times D_{12}).D_4^2$

Order 18432: $C_2^6.(D_6\times S_4)$ x 7, $C_2^5.C_2^4.S_3^2$ x 3, $C_2.C_2^6.A_4.D_6$ x 3, $(C_2^3\times C_6).C_2^6.C_6$ x 3, $(C_2^4\times C_6).C_2^4.C_6.C_2$ x 3, $C_2^5.C_2^4.C_3.D_6$ x 2, $(C_2^3\times C_6).C_2^6.C_6$ x 2, $(C_2^3\times C_6).C_2^6.C_6$ x 2, $C_2^5.C_2^4.(C_6\times S_3)$ x 2, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.C_6.C_2^2$, $C_3\times C_2^6.A_4.D_4$, $C_3\times C_2^5.C_2^4.D_6$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_3\times C_2^6.C_2^4.S_3$, $C_2.C_2^6.A_4.D_6$, $C_2^5.C_2^4.S_3^2$, $C_2.C_2^6.A_4.D_6$, $C_2^4.C_2^4.C_3^2.C_2^3$, $C_2^4.C_2^4.C_6^2.C_2$, $C_2^6.A_4.D_6.C_2$, $C_3\times C_2^4.C_2^6.C_6$, $C_2.C_2^6.A_4.D_6$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^5.C_2^4.S_3^2$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^4.C_2^4.C_3^2.D_4$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.C_6.C_2^2$, $C_2^4.C_2^4.C_3^2.C_2^3$, $C_2^6.C_2^4.C_3.S_3$, $Q_8:C_2^2.C_6^2.C_2^4$, $C_2^3:A_4.C_6.C_2^5$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2^6.D_6:S_4$, $C_2^6.(C_6\times S_4).C_2$, $C_2^5.C_2^4.S_3^2$, $C_3\times C_2^5.C_2^4.D_6$, $C_2.C_2^6.A_4.D_6$, $(C_2^7\times C_6).A_4.C_2$, $C_2^4.C_2^4.C_3.D_6.C_2$, $(C_2^5\times C_6).C_2^4.C_6$, $C_3\times C_2^3.C_2^6.D_6$, $C_2^6.A_4.C_3.D_4$, $(C_2^5\times C_6).A_4.C_2^3$, $C_2^4.C_2^6.C_3.S_3$, $C_3\times C_2^5.C_2^4.D_6$, $C_2^6.A_4.D_6.C_2$, $C_2^6.(C_6\times S_4).C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^5.C_2^4.S_3^2$, $(C_2^5\times C_6).C_2^4.C_6$, $C_3\times C_2^6.A_4.D_4$, $C_3\times C_2^6.(C_4\times S_4)$, $C_3\times C_2^5.C_2^4.D_6$, $Q_8:C_2^2.C_6^2.C_2^4$, $C_2^5.C_2^4.S_3^2$, $C_2^5.C_2^4.C_3.D_6$, $C_3\times C_2^7.A_4.C_2^2$, $C_2.C_2^6.A_4.D_6$, $C_2^6.C_2^4.C_3.S_3$, $C_2.C_2^6.A_4.D_6$, $C_2.C_2^6.A_4.D_6$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_3\times C_2^6.C_2^4.S_3$, $C_2.C_2^6.A_4.D_6$, $(C_2^5\times C_6).C_2^4.C_6$, $C_2^6.C_3:(C_4\times S_4)$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2^6.A_4.D_{12}$, $C_3\times C_2^6.A_4.D_4$, $(C_2^6\times C_6).A_4.C_2^2$, $C_2^5.C_2^4.S_3^2$, $C_2.C_2^6.(C_6\times S_4)$, $C_3\times C_2^6.C_2^4.S_3$, $(C_2^4\times C_6).C_2^4.C_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^6.A_4.D_6.C_2$, $C_2^5.C_2^4.S_3^2$, $(C_2^5\times C_6).A_4.C_2^2.C_2$, $C_2^5.C_2^4.C_3.D_6$, $(C_2^5\times C_6).A_4.C_2^3$, $(C_2\times C_6).C_2^6.C_6.C_2^2$, $C_3\times C_2^5.C_2^4.C_6.C_2$, $C_2^5.C_2^4.C_3.D_6$, $C_2^6.D_6:S_4$, $C_2^6.(D_6\times S_4)$, $C_2^7.D_6^2$, $C_2^7.(C_6\times S_4)$

Order 16384: $C_2^8.C_2^5.C_2$ x 6, $C_2^7.C_2^5.C_2^2$ x 6, $C_2^7.C_2^5.C_2^2$, $C_2^6.C_2^6.C_2^2$, $C_2^9.C_2^5$, $C_2^7.C_2^5.C_2^2$, $C_2^7.C_2^5.C_2^2$, $C_2^6.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^3$, $C_2^6.C_2^5.C_2^3$, $C_2^6.C_2^6.C_2^2$, $C_2^8.C_2\wr C_2^2$, $C_2^{10}.(C_2\times D_4)$

Order 12288: $C_2^6.C_2^6.C_3$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^6)$, $C_3:(C_2^8.C_2^4)$

Order 8192: $C_2^6.C_2^6.C_2$

Order 6144: $C_3\times C_2^6.C_2^5$, $C_3:(C_2^5.C_2^6)$, $C_3:(C_2^6.C_2^5)$, $C_3:(D_4^2.C_2^5)$, $C_3\times C_2^8.C_2^3$, $C_3:(C_2^6.C_2^5)$

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

Order 3072: $C_3:(C_2^5.C_2^5)$, $C_3:(C_2^6.C_2^4)$, $C_3:(C_2^6.C_2^4)$

Order 2048: $C_2^6.C_2^5$, $C_2^8.C_2^3$

Order 1536: $C_2^3\times Q_8:C_2^2\times S_3$ x 3, $C_2^8\times S_3$, $C_2\times D_4.C_2^4\times S_3$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_2^5.C_2^3\times S_3$, $C_3\times C_2^5.C_2^4$

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

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

Order 512: $C_2^3.C_2^6$

Order 384: $D_6\times C_2^5$ x 2, $C_{12}.C_2^5$

Order 256: $D_4:C_2^5$ x 3, $C_2^8$, $C_2^4:C_2^4$, $C_4^2.C_2^4$

Order 192: $D_6\times C_2^4$ x 4, $C_2^5\times C_6$ x 2

Order 128: $D_4:C_2^4$

Order 96: $C_2^4\times C_6$ x 2, $C_2^3\times D_6$ x 2

Order 64: $C_2^6$ x 2

Order 48: $C_2^2\times D_6$ x 4, $C_2^3\times C_6$ x 2

Order 32: $C_2^5$ x 2

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

Order 16: $C_2^4$ x 2

Order 12: $C_2\times C_6$ x 4, $D_6$ x 4

Order 9: $C_3^2$

Order 8: $C_2^3$ x 2

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

Order 4: $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 294912: $S_3\times C_2^5.C_2^6:S_4$ ($C_1$)

Order 147456: $(C_2^7\times C_6).C_2^4.C_6.C_2$ ($C_2$), $C_2^8.C_2^4.C_3.D_6$ ($C_2$), $C_3\times C_2^6.C_2^6.D_6$ ($C_2$), $C_2^8.C_2^4.S_3^2$ ($C_2$), $C_2^6.C_2^6.S_3^2$ ($C_2$), $C_2^6.C_2^6.S_3^2$ ($C_2$), $C_2^8.C_2^4.S_3^2$ ($C_2$)

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

Order 49152: $C_3:(C_2^6.C_2^6.C_2^2)$ ($S_3$), $C_2^5.C_2^6:S_4$ ($S_3$)

Order 36864: $C_2^6.C_2^6.C_3^2$ ($C_2^3$)

Order 24576: $C_2^8.C_2^4.S_3$ ($D_6$), $C_3:(C_2^6.C_2^6.C_2)$ ($D_6$), $C_3\times C_2^6.C_2^6.C_2$ ($D_6$), $C_3:(C_2^6.C_2^6.C_2)$ ($D_6$), $C_2^8.C_2^4.C_6$ ($D_6$), $C_2^8.C_2^4.S_3$ ($D_6$)

Order 12288: $C_3:(C_2^8.C_2^4)$ ($S_4$) x 2, $C_2^6.C_2^6.C_3$ ($C_2\times D_6$), $C_3:(C_2^6.C_2^6)$ ($C_2\times D_6$), $C_3:(C_2^6.C_2^6)$ ($S_4$)

Order 8192: $C_2^6.C_2^6.C_2$ ($S_3^2$)

Order 6144: $C_3:(C_2^5.C_2^6)$ ($C_2\times S_4$) x 2, $C_3:(D_4^2.C_2^5)$ ($C_2\times S_4$) x 2, $C_3\times C_2^8.C_2^3$ ($C_2\times S_4$) x 2, $C_3\times C_2^6.C_2^5$ ($C_2\times S_4$), $C_3:(C_2^6.C_2^5)$ ($C_2\times S_4$), $C_3:(C_2^6.C_2^5)$ ($C_2\times S_4$)

Order 4096: $C_2^6.C_2^6$ ($S_3\times D_6$)

Order 3072: $C_3:(C_2^6.C_2^4)$ ($C_2^2\times S_4$) x 2, $C_3:(C_2^5.C_2^5)$ ($C_2^2:S_4$), $C_3:(C_2^6.C_2^4)$ ($C_2^2\times S_4$)

Order 2048: $C_2^8.C_2^3$ ($S_3\times S_4$) x 2, $C_2^6.C_2^5$ ($S_3\times S_4$)

Order 1536: $C_2^3\times Q_8:C_2^2\times S_3$ ($C_2^3:S_4$) x 4, $C_2^8\times S_3$ ($C_2^3:S_4$) x 2, $C_2\times D_4.C_2^4\times S_3$ ($C_2^3:S_4$), $C_2^3\times Q_8:C_2^2\times S_3$ ($C_2^3:S_4$), $C_3:(C_2^5.C_2^4)$ ($C_2^3:S_4$), $C_3:(C_2^5.C_2^4)$ ($C_2^3:S_4$), $C_3:(C_2^5.C_2^4)$ ($C_2^3:S_4$), $C_2^5.C_2^3\times S_3$ ($C_2^3:S_4$), $C_3\times C_2^5.C_2^4$ ($C_2^3:S_4$)

Order 1024: $C_2^6.C_2^4$ ($D_6\times S_4$) x 2, $C_2^6.C_2^4$ ($D_6\times S_4$)

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

Order 512: $C_2^3.C_2^6$ ($C_2^4:S_3^2$)

Order 384: $D_6\times C_2^5$ ($C_2^5:S_4$) x 3, $C_{12}.C_2^5$ ($C_2^5:S_4$)

Order 256: $D_4:C_2^5$ ($S_3\times C_2^3:S_4$) x 4, $C_2^8$ ($S_3\times C_2^3:S_4$) x 2, $C_2^4:C_2^4$ ($S_3\times C_2^3:S_4$), $D_4:C_2^5$ ($C_2^5:S_3^2$), $C_4^2.C_2^4$ ($S_3\times C_2^3:S_4$)

Order 192: $D_6\times C_2^4$ ($C_2^6:S_4$) x 6, $C_2^5\times C_6$ ($C_2^6:S_4$) x 3

Order 128: $D_4:C_2^4$ ($S_3\times C_2^4:S_4$)

Order 96: $C_2^3\times D_6$ ($C_2^6:(C_2\times S_4)$) x 4, $C_2^4\times C_6$ ($C_2^5.S_4.C_2^2$) x 3, $C_2^3\times D_6$ ($C_2^8:D_6$)

Order 64: $C_2^6$ ($S_3\times C_2^5:S_4$) x 3

Order 48: $C_2^2\times D_6$ ($C_2^5.C_2^4.D_6$) x 4, $C_2^3\times C_6$ ($C_2^4.C_2^4.D_6.C_2$) x 2, $C_2^2\times D_6$ ($C_2^7.S_4.C_2$) x 2, $C_2^3\times C_6$ ($C_2^9.D_6$)

Order 32: $C_2^5$ ($C_2.C_2^6.S_3^2.C_2$) x 3

Order 24: $C_2\times D_6$ ($C_2^8.S_4.C_2$) x 4, $C_2\times D_6$ ($C_2^6.C_2^4.D_6$) x 4, $C_2\times D_6$ ($C_2^4.C_2^6.D_6$) x 4, $C_2\times D_6$ ($C_2^4.C_2^5:S_4$) x 4, $C_2^2\times C_6$ ($C_2^5.C_2^4.D_6.C_2$) x 2, $C_2^2\times C_6$ ($C_2^7.A_4.C_2^3$)

Order 16: $C_2^4$ ($C_2^6.(D_6\times S_4)$) x 2, $C_2^4$ ($C_2^8.S_3^2.C_2$)

Order 12: $D_6$ ($C_2^7.C_2^4.D_6$) x 4, $D_6$ ($C_2^7.C_2^4.D_6$) x 4, $D_6$ ($C_2^5.C_2^6.D_6$) x 4, $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$) x 2, $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$) x 2, $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$) x 2, $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$)

Order 8: $C_2^3$ ($C_2.C_2^6.A_4.D_6.C_2$) x 2, $C_2^3$ ($C_2^7.A_4.D_6.C_2$)

Order 6: $S_3$ ($C_2^5.C_2^6:S_4$) x 4, $C_6$ ($C_2^5.C_2^6.D_6.C_2$) x 2, $C_6$ ($C_2^7.C_2^4.D_6.C_2$)

Order 4: $C_2^2$ ($C_2^4.C_2^6.S_3^2.C_2$) x 2, $C_2^2$ ($C_2^6.C_2^4.S_3^2.C_2$) x 2, $C_2^2$ ($C_2^6.C_2^4.S_3^2.C_2$) x 2, $C_2^2$ ($C_2^6.C_2^4.S_3^2.C_2$)

Order 3: $C_3$ ($C_2^6.C_2^6:S_4$)

Order 2: $C_2$ ($C_2^5.C_2^6.S_3^2.C_2$) x 2, $C_2$ ($C_2^7.C_2^4.S_3^2.C_2$)

Order 1: $C_1$ ($S_3\times C_2^5.C_2^6:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 294912: $S_3\times C_2^5.C_2^6:S_4$ ($C_1$)

Order 147456: $(C_2^7\times C_6).C_2^4.C_6.C_2$ ($C_2$), $C_2^8.C_2^4.C_3.D_6$ ($C_2$), $C_3\times C_2^6.C_2^6.D_6$ ($C_2$), $C_2^8.C_2^4.S_3^2$ ($C_2$), $C_2^6.C_2^6.S_3^2$ ($C_2$), $C_2^6.C_2^6.S_3^2$ ($C_2$), $C_2^8.C_2^4.S_3^2$ ($C_2$)

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

Order 49152: $C_3:(C_2^6.C_2^6.C_2^2)$ ($S_3$), $C_2^5.C_2^6:S_4$ ($S_3$)

Order 36864: $C_2^6.C_2^6.C_3^2$ ($C_2^3$)

Order 24576: $C_2^8.C_2^4.S_3$ ($D_6$), $C_3:(C_2^6.C_2^6.C_2)$ ($D_6$), $C_3\times C_2^6.C_2^6.C_2$ ($D_6$), $C_3:(C_2^6.C_2^6.C_2)$ ($D_6$), $C_2^8.C_2^4.C_6$ ($D_6$), $C_2^8.C_2^4.S_3$ ($D_6$)

Order 12288: $C_2^6.C_2^6.C_3$ ($C_2\times D_6$), $C_3:(C_2^6.C_2^6)$ ($C_2\times D_6$), $C_3:(C_2^6.C_2^6)$ ($S_4$), $C_3:(C_2^8.C_2^4)$ ($S_4$)

Order 8192: $C_2^6.C_2^6.C_2$ ($S_3^2$)

Order 6144: $C_3\times C_2^6.C_2^5$ ($C_2\times S_4$), $C_3:(C_2^5.C_2^6)$ ($C_2\times S_4$), $C_3:(C_2^6.C_2^5)$ ($C_2\times S_4$), $C_3:(D_4^2.C_2^5)$ ($C_2\times S_4$), $C_3\times C_2^8.C_2^3$ ($C_2\times S_4$), $C_3:(C_2^6.C_2^5)$ ($C_2\times S_4$)

Order 4096: $C_2^6.C_2^6$ ($S_3\times D_6$)

Order 3072: $C_3:(C_2^5.C_2^5)$ ($C_2^2:S_4$), $C_3:(C_2^6.C_2^4)$ ($C_2^2\times S_4$), $C_3:(C_2^6.C_2^4)$ ($C_2^2\times S_4$)

Order 2048: $C_2^6.C_2^5$ ($S_3\times S_4$), $C_2^8.C_2^3$ ($S_3\times S_4$)

Order 1536: $C_2^3\times Q_8:C_2^2\times S_3$ ($C_2^3:S_4$) x 2, $C_2^8\times S_3$ ($C_2^3:S_4$), $C_2\times D_4.C_2^4\times S_3$ ($C_2^3:S_4$), $C_2^3\times Q_8:C_2^2\times S_3$ ($C_2^3:S_4$), $C_3:(C_2^5.C_2^4)$ ($C_2^3:S_4$), $C_3:(C_2^5.C_2^4)$ ($C_2^3:S_4$), $C_3:(C_2^5.C_2^4)$ ($C_2^3:S_4$), $C_2^5.C_2^3\times S_3$ ($C_2^3:S_4$), $C_3\times C_2^5.C_2^4$ ($C_2^3:S_4$)

Order 1024: $C_2^6.C_2^4$ ($D_6\times S_4$), $C_2^6.C_2^4$ ($D_6\times S_4$)

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

Order 512: $C_2^3.C_2^6$ ($C_2^4:S_3^2$)

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

Order 256: $D_4:C_2^5$ ($S_3\times C_2^3:S_4$) x 2, $C_2^8$ ($S_3\times C_2^3:S_4$), $C_2^4:C_2^4$ ($S_3\times C_2^3:S_4$), $D_4:C_2^5$ ($C_2^5:S_3^2$), $C_4^2.C_2^4$ ($S_3\times C_2^3:S_4$)

Order 192: $D_6\times C_2^4$ ($C_2^6:S_4$) x 4, $C_2^5\times C_6$ ($C_2^6:S_4$) x 2

Order 128: $D_4:C_2^4$ ($S_3\times C_2^4:S_4$)

Order 96: $C_2^4\times C_6$ ($C_2^5.S_4.C_2^2$) x 2, $C_2^3\times D_6$ ($C_2^8:D_6$), $C_2^3\times D_6$ ($C_2^6:(C_2\times S_4)$)

Order 64: $C_2^6$ ($S_3\times C_2^5:S_4$) x 2

Order 48: $C_2^2\times D_6$ ($C_2^5.C_2^4.D_6$) x 2, $C_2^2\times D_6$ ($C_2^7.S_4.C_2$) x 2, $C_2^3\times C_6$ ($C_2^4.C_2^4.D_6.C_2$), $C_2^3\times C_6$ ($C_2^9.D_6$)

Order 32: $C_2^5$ ($C_2.C_2^6.S_3^2.C_2$) x 2

Order 24: $C_2\times D_6$ ($C_2^6.C_2^4.D_6$) x 3, $C_2^2\times C_6$ ($C_2^5.C_2^4.D_6.C_2$), $C_2^2\times C_6$ ($C_2^7.A_4.C_2^3$), $C_2\times D_6$ ($C_2^8.S_4.C_2$), $C_2\times D_6$ ($C_2^4.C_2^6.D_6$), $C_2\times D_6$ ($C_2^4.C_2^5:S_4$)

Order 16: $C_2^4$ ($C_2^8.S_3^2.C_2$), $C_2^4$ ($C_2^6.(D_6\times S_4)$)

Order 12: $D_6$ ($C_2^7.C_2^4.D_6$) x 2, $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$), $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$), $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$), $C_2\times C_6$ ($C_2^6.C_2^4.D_6.C_2$), $D_6$ ($C_2^7.C_2^4.D_6$), $D_6$ ($C_2^5.C_2^6.D_6$)

Order 8: $C_2^3$ ($C_2^7.A_4.D_6.C_2$), $C_2^3$ ($C_2.C_2^6.A_4.D_6.C_2$)

Order 6: $C_6$ ($C_2^7.C_2^4.D_6.C_2$), $C_6$ ($C_2^5.C_2^6.D_6.C_2$), $S_3$ ($C_2^5.C_2^6:S_4$)

Order 4: $C_2^2$ ($C_2^6.C_2^4.S_3^2.C_2$), $C_2^2$ ($C_2^4.C_2^6.S_3^2.C_2$), $C_2^2$ ($C_2^6.C_2^4.S_3^2.C_2$), $C_2^2$ ($C_2^6.C_2^4.S_3^2.C_2$)

Order 3: $C_3$ ($C_2^6.C_2^6:S_4$)

Order 2: $C_2$ ($C_2^5.C_2^6.S_3^2.C_2$), $C_2$ ($C_2^7.C_2^4.S_3^2.C_2$)

Order 1: $C_1$ ($S_3\times C_2^5.C_2^6:S_4$)

Series

Derived series

$S_3\times C_2^5.C_2^6:S_4$

$\rhd$

$C_2^6.C_2^6.C_3^2$

$\rhd$

$C_2^6.C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$S_3\times C_2^5.C_2^6:S_4$

$\rhd$

$C_2^6.C_2^6.S_3^2$

$\rhd$

$(C_2^5\times C_6).C_2^6.C_6$

$\rhd$

$C_2^6.C_2^6.C_3^2$

$\rhd$

$C_2^6.C_2^6.C_3$

$\rhd$

$C_2^6.C_2^6$

$\rhd$

$C_2^6.C_2^4$

$\rhd$

$D_4:C_2^5$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$S_3\times C_2^5.C_2^6:S_4$

$\rhd$

$C_2^6.C_2^6.C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^4$

Character theory

Complex character table

The $408 \times 408$ character table is not available for this group.

Rational character table

The $405 \times 405$ rational character table is not available for this group.