Abstract group 6144.bkl: $C_2^6.(D_4\times D_6)$

• Label: 6144.bkl
• Order: \( 2^{11} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{25} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{17} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $6$

Group information

Description:	$C_2^6.(D_4\times D_6)$
Order:	\(6144\)\(\medspace = 2^{11} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_5:F_5^2$, of order \(301989888\)\(\medspace = 2^{25} \cdot 3^{2} \)
Composition factors:	$C_2$ x 11, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.

Group statistics

Order	1	2	3	4	6	12
Elements	1	1151	2	2944	574	1472	6144
Conjugacy classes	1	123	1	112	61	56	354
Divisions	1	123	1	112	61	56	354
Autjugacy classes	1	18	1	8	11	4	43

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$6$
Inequivalent generating 6-tuples:	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:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=b^{6}=c^{4}=d^{4}=e^{2}=f^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $64$ $\langle(1,19)(3,9)(5,60)(6,12)(8,61)(11,58)(13,59)(15,21)(17,64)(18,24)(23,62) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_2^6:D_4)$
Semidirect product:	$C_3$ $\,\rtimes\,$ $(C_2^8.D_4)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^7:D_4)$ . $S_3$	$C_2^7$ . $(S_3\times D_4)$	$S_3$ . $(C_2^7:D_4)$	$(D_4^2:D_6)$ . $D_4$	all 330
Aut. group:	$\Aut(C_4^2.D_6)$	$\Aut(C_{12}.C_4^2)$	$\Aut(C_{12}.C_4^2)$	$\Aut(C_2\times C_{12}.D_4)$	all 9

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	$C_{2}^{18}$
Commutator length:	$1$

Subgroups

There are 7805 normal subgroups (49 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^6:D_6$
Commutator:	$G' \simeq$ $C_2^4\times C_6$	$G/G' \simeq$ $C_2^6$
Frattini:	$\Phi \simeq$ $C_2^5$	$G/\Phi \simeq$ $D_6\times C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_2^6.C_2^4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2^6.(D_4\times D_6)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^5:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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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 12 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 6144: $C_2^6.(D_4\times D_6)$

Order 3072: $S_3\times C_2^5.C_2^4$ x 32, $C_3:(C_2^6.C_2^4)$ x 6, $C_3:(C_2^6.C_2^4)$ x 6, $C_3:(C_2^6.C_2^4)$ x 6, $S_3\times C_2^5.C_2^4$ x 6, $C_3:(C_2^6.C_2^4)$, $C_3:(C_2^6.C_2^4)$, $C_3:(C_2^6.C_2^4)$, $C_3\times C_2^6.C_2^4$, $C_3:(C_2^4.C_2^6)$, $C_2\times C_3:(C_2^4.C_2^5)$, $C_3:(C_2^6.C_2^4)$

Order 2048: $C_2^8.D_4$

Order 1536: $S_3\times C_2^5:D_4$ x 144, $C_3:(C_2^5.C_2^4)$ x 96, $C_3:(C_2^5.C_2^4)$ x 96, $C_2^5.C_2^3\times S_3$ x 96, $S_3\times C_4^2:C_2^4$ x 72, $C_2^6:C_2^2\times S_3$ x 24, $C_2\times C_3:(C_2^5.C_2^3)$ x 24, $C_3\times C_2^5.C_2^4$ x 16, $C_3:(C_2^5.C_2^4)$ x 16, $C_3:(C_2^5.C_2^4)$ x 16, $C_3:(C_2^5.C_2^4)$ x 16, $S_3\times C_2^5.C_2^3$ x 16, $C_2\times C_3:(C_2^6.C_2^2)$ x 12, $C_2\times C_3:(C_2^6.C_2^2)$ x 12, $C_2\times C_2^5.C_2^2\times S_3$ x 12, $C_2\times D_4^2:D_6$ x 12, $C_2\times C_3:(C_2^5.C_2^3)$ x 12, $C_3:(C_2^4.C_2^5)$ x 12, $C_2\times C_3:(C_2^5.C_2^3)$ x 12, $S_3\times C_2^5.C_2^3$ x 12, $C_2\times C_2^5.C_2^2\times S_3$ x 12, $C_2\times C_3:(C_2^6.C_2^2)$ x 9, $C_2\times C_2^4.C_2^3\times S_3$ x 9, $C_6\times C_2^5.C_2^3$ x 6, $C_6\times C_2^5:D_4$ x 6, $C_6.(D_4\times C_2^5)$ x 6, $C_2^4:D_4\times D_6$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_2^4.C_2^3\times S_3$ x 6, $C_2\times C_2^5.C_2^2\times S_3$ x 6, $C_2\times C_3:(C_2^6.C_2^2)$ x 6, $C_2\times C_2^5.C_2^2\times S_3$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^6.C_2^2)$ x 5, $C_2^7:D_6$ x 4, $C_2\times C_3:(C_2^6.C_2^2)$ x 2, $C_6\times C_2^5.C_2^3$, $C_6\times C_2^4.C_2^4$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_6\times C_2^6.C_2^2$

Order 1024: $C_2^7:D_4$ x 48, $C_2^6.C_2^4$ x 6, $C_2^7:D_4$ x 6, $C_2^4.C_2^6$, $C_2\times C_2^4.C_2^5$, $C_2^6.C_2^4$

Order 768: $C_3:(C_2^6.C_2^2)$ x 768, $C_2^5.C_2^2\times S_3$ x 768, $D_4^2:D_6$ x 624, $C_3:(C_2^6.C_2^2)$ x 600, $C_3:(C_2^6.C_2^2)$ x 576, $C_2^4.C_2^3\times S_3$ x 576, $C_2^6:D_6$ x 392, $C_3:(C_2^5.C_2^3)$ x 192, $(C_2^2\times D_{12}):D_4$ x 192, $S_3\times C_2^4:D_4$ x 192, $C_2\wr C_2^2\times D_6$ x 192, $C_2^4:(C_2\times D_{12})$ x 144, $C_2^6:D_6$ x 144, $C_2^4:(C_4\times D_6)$ x 144, $C_3:(C_2^6.C_2^2)$ x 100, $C_2^4:(C_2\times D_{12})$ x 96, $(C_2^2\times D_{12}):D_4$ x 96, $C_2^4:(C_4\times D_6)$ x 96, $C_2^5.C_2^2\times S_3$ x 84, $C_3:(C_2^6.C_2^2)$ x 84, $C_2^6:D_6$ x 80, $C_2^5:D_{12}$ x 72, $(C_2^3\times C_{12}):D_4$ x 72, $C_2^6:D_6$ x 72, $C_3\times C_2^5:D_4$ x 72, $C_2^5:C_4\times S_3$ x 72, $C_2\times C_3:(C_2^4.C_2^3)$ x 72, $C_2\times C_3:(C_2^4.C_2^3)$ x 72, $C_2\times C_3:(C_2^4.C_2^3)$ x 72, $C_2\times C_3:(C_2^4.C_2^3)$ x 72, $C_2\times C_3:(C_2^5.C_2^2)$ x 72, $C_2\times C_3:(C_2^5.C_2^2)$ x 72, $C_2\times (C_2\times D_4):C_2^2\times S_3$ x 72, $C_3:(C_2^6.C_2^2)$ x 64, $C_3:(C_2^5.C_2^3)$ x 48, $C_3:(C_2^5.C_2^3)$ x 48, $C_3:(C_2^5.C_2^3)$ x 48, $C_3:(C_2^5.C_2^3)$ x 48, $C_3\times C_2^5.C_2^3$ x 48, $C_2\times C_3:(C_2^4.C_2^3)$ x 42, $C_3:(C_2^3.C_2^5)$ x 42, $C_3\times C_2^6.C_2^2$ x 36, $C_2\times C_3:(C_2^4.C_2^3)$ x 36, $C_2\times C_3:(C_2^4.C_2^3)$ x 36, $C_2\times C_3:(C_2^5.C_2^2)$ x 36, $C_2\times C_3:(C_2^5.C_2^2)$ x 36, $C_2\times C_3:(C_2^5.C_2^2)$ x 36, $C_2\times C_3:(C_2^5.C_2^2)$ x 36, $C_2\times C_3:(C_2^5.C_2^2)$ x 36, $C_2\times C_3:(C_2^5.C_2^2)$ x 36, $C_2\times C_2^2\wr C_2.C_2\times S_3$ x 36, $D_6\times D_4^2$ x 36, $C_2\times C_2.Q_8:C_2^2\times S_3$ x 36, $C_2\times C_2^4:C_2^2\times S_3$ x 33, $C_2\times C_3:(C_2^4.C_2^3)$ x 24, $C_2\times C_3:(C_2^4.C_2^3)$ x 24, $C_2^4:C_4\times D_6$ x 24, $C_2\times C_3:(C_2^5.C_2^2)$ x 21, $C_2\times C_3:(C_2^4.C_2^3)$ x 18, $C_3:(C_2^3.C_2^5)$ x 18, $C_2\times C_3:(C_2^4.C_2^3)$ x 18, $C_2\times C_3:(C_2^4.C_2^3)$ x 18, $C_2\times (C_2^2\times D_4):C_2\times S_3$ x 18, $C_4^2:C_2^2\times D_6$ x 18, $C_2^5:D_{12}$ x 18, $C_2^6:D_6$ x 18, $C_2\times C_3:(C_2^6.C_2)$ x 16, $C_3:(C_2^7.C_2)$ x 16, $C_2^6.C_2\times S_3$ x 16, $C_2^3\times C_2^2:D_{12}$ x 15, $C_2^3\times C_2^3:D_6$ x 15, $C_3:(C_2^7.C_2)$ x 15, $C_3\times C_2^4.C_2^4$ x 12, $C_2\times D_4^2:C_6$ x 12, $C_6\times C_2^5.C_2^2$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_3:(C_2^4.C_2^4)$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 12, $C_2\times C_2^2:C_4:C_4\times S_3$ x 12, $C_6\times C_2^4:D_4$ x 12, $C_6\times C_2^5.C_2^2$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_2\times C_3:(C_2^4.C_2^3)$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 12, $C_2\times (C_2\times C_4).C_2^3\times S_3$ x 12, $C_2\times C_2^3:(C_2\times C_4)\times S_3$ x 12, $C_2\times C_3:(C_2^5.C_2^2)$ x 9, $C_6\times C_2^4.C_2^3$ x 9, $C_{12}:C_2^6$ x 9, $C_2\times C_3:(C_2^4.C_2^3)$ x 9, $C_2\times C_3:(C_2^4.C_2^3)$ x 9, $C_2\times C_3:(C_2^4.C_2^3)$ x 9, $C_3:(C_2^5.C_2^3)$ x 8, $C_3\times C_2^5.C_2^3$ x 8, $C_2\times C_3:(C_2^5.C_2^2)$ x 6, $C_6\times C_2^5.C_2^2$ x 6, $C_2^7:C_6$ x 6, $C_6\times C_2^4:D_4$ x 6, $C_3:(C_2^4.C_2^4)$ x 6, $C_2\times C_3:(C_2^4.C_2^3)$ x 6, $C_2\times C_3:(C_2^4.C_2^3)$ x 6, $C_3:(C_2^4.C_2^4)$ x 6, $C_3\times C_2^4.C_2^4$ x 6, $C_3\times C_2^4.C_2^4$ x 6, $D_{12}:C_2^5$ x 6, $C_2^3\times C_2^4:S_3$ x 6, $(C_2^3\times C_{12}):D_4$ x 6, $C_2\times C_3:(C_2^6.C_2)$ x 4, $C_2^7:C_6$ x 4, $C_2\times C_2^2\wr C_2:C_2\times S_3$ x 4, $C_2^2\times C_2.C_4^2\times S_3$ x 3, $C_2\times C_2^3:Q_8\times S_3$ x 3, $D_6\times C_2^6$ x 2

Order 512: $C_2^6:D_4$ x 256, $C_2^6:D_4$ x 216, $C_2^5.C_2^4$ x 144, $C_4^2:C_2^5$ x 108, $C_2^4.C_2^5$ x 36, $C_2^5.C_2^4$ x 24, $C_2^4.C_2^5$ x 12, $C_2^2\times C_2^5.C_2^2$ x 12, $D_4^2.C_2^3$ x 12, $C_4^2:C_2^5$ x 12, $C_2^2\times C_2^4.C_2^3$ x 9, $C_2^4.C_2^5$ x 6, $C_2^4.C_2^5$ x 6, $C_2^4.C_2^5$ x 6, $C_2^6:D_4$ x 6, $C_2^6:D_4$ x 6, $C_2^8.C_2$ x 4

Order 384: $C_2^3:(S_3\times D_4)$ x 96, $S_3\times C_2^3:D_4$ x 96, $C_2^5:D_6$ x 96, $C_{12}:C_2^5$ x 84, $C_2^5:D_6$ x 64, $C_2^4:(C_2\times C_{12})$ x 48, $C_2^5.D_6$ x 48, $C_2^4.D_{12}$ x 48, $C_2^4.D_{12}$ x 48, $C_2^4.D_{12}$ x 48, $D_{12}:C_2^4$ x 48, $D_4^2:C_6$ x 48, $(C_6\times D_4):D_4$ x 48, $C_2^5:D_6$ x 48, $C_2^5:D_6$ x 48, $(C_2\times D_4):D_{12}$ x 48, $C_4^2:C_2^2\times S_3$ x 48, $C_2^4:D_{12}$ x 48, $C_2^5.D_6$ x 48, $C_2^5:C_{12}$ x 24, $C_2^5.D_6$ x 24, $C_2^4.D_{12}$ x 24, $C_2^6:C_6$ x 24, $C_2^5:D_6$ x 24, $C_2^5:D_6$ x 24, $D_6\times C_2^5$ x 20, $D_{12}:C_2^4$ x 18, $C_2^4:D_{12}$ x 16, $C_2^5.D_6$ x 16, $C_2^6:C_6$ x 16, $C_2^6:S_3$ x 16, $C_2^5.D_6$ x 16, $(C_2^2\times C_{12}):D_4$ x 12, $C_2^5.D_6$ x 12, $C_6\times C_2^3:D_4$ x 12, $C_2^4:D_{12}$ x 12, $C_2^5.D_6$ x 12, $C_{12}.C_2^5$ x 6, $D_6:C_2^5$ x 6, $C_4^2:C_2^2\times C_6$ x 6, $C_2^5:C_{12}$ x 6, $C_2^5.D_6$ x 6, $C_2^5.D_6$ x 6, $C_2^4.D_{12}$ x 6, $C_2^5:C_{12}$ x 4, $C_2^5.D_6$ x 4, $C_{12}:C_2^5$ x 3, $C_2^4\times D_{12}$ x 3, $C_{12}:C_2^5$ x 3, $C_2^6\times C_6$ x 2

Order 256: $C_2^5:D_4$ x 48, $C_2^5.D_4$ x 48, $C_2^5.D_4$ x 12, $C_2^6:C_2^2$ x 8, $C_4^2:C_2^4$ x 8, $C_2^6:C_4$ x 8, $C_4^2:C_2^4$ x 6, $C_2^6:C_4$ x 6, $C_2^5:D_4$ x 4, $C_2^5:D_4$ x 4, $C_2^5.D_4$ x 3

Order 192: $C_{12}:C_2^4$ x 288, $D_6:C_2^4$ x 96, $C_2^4:D_6$ x 96, $C_2^4:D_6$ x 96, $C_2^3:D_{12}$ x 96, $C_2^4.D_6$ x 96, $D_6\times C_2^4$ x 87, $D_{12}:C_2^3$ x 72, $C_{12}:C_2^4$ x 48, $C_2^3\times D_{12}$ x 48, $C_{12}:C_2^4$ x 48, $C_3\times C_2^3:D_4$ x 48, $C_2^3.D_{12}$ x 48, $C_2^3:D_{12}$ x 48, $C_2^4.D_6$ x 48, $C_2^5:C_6$ x 32, $C_2^4:D_6$ x 32, $C_2^4:C_{12}$ x 30, $C_2^4.D_6$ x 30, $C_{12}.C_2^4$ x 24, $C_3\times C_4^2:C_2^2$ x 24, $C_2^4.D_6$ x 24, $C_2^5\times C_6$ x 10, $C_2^4:C_{12}$ x 8, $C_2^4.D_6$ x 8, $C_2^4\times C_{12}$ x 3, $C_6.C_2^5$ x 3

Order 128: $C_2^4.D_4$ x 48, $D_4^2:C_2$ x 48, $C_2^5:C_4$ x 24, $C_2^4:D_4$ x 24, $C_2^3\wr C_2$ x 16, $C_2^4:D_4$ x 12, $C_2^4.D_4$ x 12, $D_4:C_2^4$ x 6, $C_4^2:C_2^3$ x 6, $C_2^5:C_4$ x 6, $C_2^5:C_4$ x 4, $D_4\times C_2^4$ x 3, $C_2^7$ x 2

Order 96: $C_2^3\times D_6$ x 250, $C_2^3:D_6$ x 192, $C_{12}:C_2^3$ x 96, $C_2^2\times D_{12}$ x 96, $C_{12}:C_2^3$ x 96, $C_2^2.D_{12}$ x 96, $C_2^3:C_{12}$ x 48, $C_2^3.D_6$ x 48, $C_2^4\times C_6$ x 37, $C_2^3\times C_{12}$ x 30, $C_6.C_2^4$ x 30

Order 64: $D_4\times C_2^3$ x 48, $C_2^3:D_4$ x 48, $C_2^3:D_4$ x 32, $C_2^4:C_4$ x 30, $D_4:C_2^3$ x 24, $C_4^2:C_2^2$ x 24, $C_2^6$ x 10, $C_2^4:C_4$ x 8, $C_2^4\times C_4$ x 3

Order 48: $C_2^2\times D_6$ x 224, $C_2^3\times C_6$ x 67, $C_2^2\times C_{12}$ x 36, $C_6.C_2^3$ x 36

Order 32: $C_2^2\times D_4$ x 96, $C_2^3:C_4$ x 48, $C_2^5$ x 37, $C_2^3\times C_4$ x 30

Order 24: $C_2\times D_6$ x 100, $C_2^2\times C_6$ x 39

Order 16: $C_2^4$ x 67, $C_2^2\times C_4$ x 36

Order 12: $D_6$ x 40, $C_2\times C_6$ x 19

Order 8: $C_2^3$ x 39

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$ x 19

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 12 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 6144: $C_2^6.(D_4\times D_6)$

Order 3072: $C_3:(C_2^6.C_2^4)$, $C_3:(C_2^6.C_2^4)$, $C_3:(C_2^6.C_2^4)$, $C_3:(C_2^6.C_2^4)$, $C_3:(C_2^6.C_2^4)$, $C_3\times C_2^6.C_2^4$, $C_3:(C_2^4.C_2^6)$, $C_3:(C_2^6.C_2^4)$, $C_2\times C_3:(C_2^4.C_2^5)$, $C_3:(C_2^6.C_2^4)$, $S_3\times C_2^5.C_2^4$, $S_3\times C_2^5.C_2^4$

Order 2048: $C_2^8.D_4$

Order 1536: $C_6.(D_4\times C_2^5)$ x 4, $S_3\times C_4^2:C_2^4$ x 3, $C_2\times C_3:(C_2^6.C_2^2)$ x 3, $C_2\times C_3:(C_2^6.C_2^2)$ x 2, $C_2\times C_3:(C_2^6.C_2^2)$ x 2, $C_2\times C_3:(C_2^6.C_2^2)$ x 2, $C_2\times C_2^4.C_2^3\times S_3$ x 2, $C_2\times D_4^2:D_6$ x 2, $S_3\times C_2^5:D_4$ x 2, $C_2\times C_3:(C_2^6.C_2^2)$ x 2, $C_2^6:C_2^2\times S_3$ x 2, $C_2\times C_2^5.C_2^2\times S_3$ x 2, $C_6\times C_2^5.C_2^3$, $C_6\times C_2^5.C_2^3$, $C_6\times C_2^5:D_4$, $C_6\times C_2^4.C_2^4$, $C_2\times C_3:(C_2^6.C_2^2)$, $C_2\times C_2^5.C_2^2\times S_3$, $C_2^4:D_4\times D_6$, $C_3\times C_2^5.C_2^4$, $C_2^7:D_6$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_3:(C_2^4.C_2^5)$, $C_2\times C_3:(C_2^5.C_2^3)$, $S_3\times C_2^5.C_2^3$, $C_2\times C_2^5.C_2^2\times S_3$, $C_2\times C_2^4.C_2^3\times S_3$, $C_2\times C_2^5.C_2^2\times S_3$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $S_3\times C_2^5.C_2^3$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2\times C_3:(C_2^5.C_2^3)$, $C_2^5.C_2^3\times S_3$, $C_6\times C_2^6.C_2^2$

Order 1024: $C_2^7:D_4$ x 7, $C_2^4.C_2^6$, $C_2\times C_2^4.C_2^5$, $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^7:D_4$

Order 768: $C_2^6:D_6$ x 28, $C_3:(C_2^6.C_2^2)$ x 8, $D_4^2:D_6$ x 8, $C_3:(C_2^6.C_2^2)$ x 6, $C_2^4.C_2^3\times S_3$ x 6, $C_2\times C_3:(C_2^5.C_2^2)$ x 5, $C_3:(C_2^6.C_2^2)$ x 5, $C_3:(C_2^6.C_2^2)$ x 5, $C_2\times C_3:(C_2^4.C_2^3)$ x 5, $C_3:(C_2^3.C_2^5)$ x 5, $C_2^3\times C_2^2:D_{12}$ x 5, $C_2^3\times C_2^3:D_6$ x 5, $C_3:(C_2^7.C_2)$ x 5, $C_2^7:C_6$ x 4, $C_3:(C_2^6.C_2^2)$ x 4, $C_2^5.C_2^2\times S_3$ x 4, $C_2\times C_2^4:C_2^2\times S_3$ x 4, $C_2^3\times C_2^4:S_3$ x 4, $C_2^5.C_2^2\times S_3$ x 3, $C_3\times C_2^6.C_2^2$ x 3, $C_3:(C_2^6.C_2^2)$ x 3, $C_{12}:C_2^6$ x 3, $C_2\times C_3:(C_2^4.C_2^3)$ x 3, $C_3:(C_2^3.C_2^5)$ x 3, $C_2\times C_3:(C_2^4.C_2^3)$ x 3, $C_2\times C_3:(C_2^4.C_2^3)$ x 3, $C_2\times C_3:(C_2^4.C_2^3)$ x 3, $C_2\times C_3:(C_2^4.C_2^3)$ x 3, $C_2\times C_3:(C_2^5.C_2^2)$ x 3, $C_2\times C_3:(C_2^5.C_2^2)$ x 3, $C_2\times (C_2\times D_4):C_2^2\times S_3$ x 3, $C_2^4:C_4\times D_6$ x 3, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_3\times C_2^4.C_2^4$ x 2, $C_6\times C_2^5.C_2^2$ x 2, $C_6\times C_2^4.C_2^3$ x 2, $C_2\times D_4^2:C_6$ x 2, $C_2\times C_3:(C_2^6.C_2)$ x 2, $C_3:(C_2^7.C_2)$ x 2, $C_2^6:D_6$ x 2, $C_2^6.C_2\times S_3$ x 2, $C_2^4:(C_2\times D_{12})$ x 2, $C_2^5:D_{12}$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_2^6:D_6$ x 2, $C_2^6:D_6$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2^4:(C_4\times D_6)$ x 2, $C_3\times C_2^5:D_4$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2^5:C_4\times S_3$ x 2, $D_6\times C_2^6$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2\times C_3:(C_2^5.C_2^2)$ x 2, $C_2\times C_2^2\wr C_2.C_2\times S_3$ x 2, $D_6\times D_4^2$ x 2, $C_2\times (C_2^2\times D_4):C_2\times S_3$ x 2, $C_2\times C_2.Q_8:C_2^2\times S_3$ x 2, $C_4^2:C_2^2\times D_6$ x 2, $C_2\times C_2^3:(C_2\times C_4)\times S_3$ x 2, $C_2^5:D_{12}$ x 2, $C_2^6:D_6$ x 2, $C_2\wr C_2^2\times D_6$ x 2, $C_3:(C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^3)$, $C_6\times C_2^5.C_2^2$, $C_6\times C_2^4:D_4$, $C_2\times C_3:(C_2^6.C_2)$, $C_2^2\times C_2.C_4^2\times S_3$, $C_2^7:C_6$, $C_2^4:(C_2\times D_{12})$, $(C_2^2\times D_{12}):D_4$, $(C_2^2\times D_{12}):D_4$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $S_3\times C_2^4:D_4$, $C_3:(C_2^4.C_2^4)$, $C_2\times C_3:(C_2^5.C_2^2)$, $C_2\times C_3:(C_2^5.C_2^2)$, $C_2\times C_3:(C_2^5.C_2^2)$, $C_2\times C_3:(C_2^5.C_2^2)$, $C_3:(C_2^4.C_2^4)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2\times C_3:(C_2^4.C_2^3)$, $C_2^4:(C_4\times D_6)$, $C_3:(C_2^4.C_2^4)$, $C_2\times C_3:(C_2^5.C_2^2)$, $C_2\times C_2^2:C_4:C_4\times S_3$, $C_6\times C_2^4:D_4$, $C_3\times C_2^5.C_2^3$, $C_3\times C_2^5.C_2^3$, $C_6\times C_2^5.C_2^2$, $C_3\times C_2^4.C_2^4$, $C_3\times C_2^4.C_2^4$, $D_{12}:C_2^5$, $C_2\times C_3:(C_2^5.C_2^2)$, $C_2\times C_2^2\wr C_2:C_2\times S_3$, $C_2\times (C_2\times C_4).C_2^3\times S_3$, $C_2\times C_2^3:Q_8\times S_3$, $(C_2^3\times C_{12}):D_4$

Order 512: $C_2^6:D_4$ x 11, $C_4^2:C_2^5$ x 11, $C_2^4.C_2^5$ x 8, $C_2^5.C_2^4$ x 6, $C_2^6:D_4$ x 6, $C_2^5.C_2^4$ x 4, $C_2^6:D_4$ x 4, $C_2^4.C_2^5$ x 2, $C_2^2\times C_2^4.C_2^3$ x 2, $C_4^2:C_2^5$ x 2, $C_2^8.C_2$, $C_2^4.C_2^5$, $C_2^2\times C_2^5.C_2^2$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $D_4^2.C_2^3$, $C_2^6:D_4$

Order 384: $C_{12}:C_2^5$ x 8, $D_6\times C_2^5$ x 6, $C_2^6:C_6$ x 4, $C_2^5:D_6$ x 4, $C_2^5:D_6$ x 3, $C_2^4:D_{12}$ x 3, $C_2^5.D_6$ x 3, $C_2^6\times C_6$ x 2, $D_6:C_2^5$ x 2, $D_{12}:C_2^4$ x 2, $D_4^2:C_6$ x 2, $C_2^5:C_{12}$ x 2, $C_2^5.D_6$ x 2, $C_2^5:D_6$ x 2, $C_2^5:D_6$ x 2, $C_2^5:D_6$ x 2, $C_2^5:D_6$ x 2, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^4:(C_2\times C_{12})$, $C_2^5:C_{12}$, $(C_2^2\times C_{12}):D_4$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^4.D_{12}$, $C_2^4.D_{12}$, $C_2^4.D_{12}$, $C_2^4.D_{12}$, $C_{12}.C_2^5$, $C_{12}:C_2^5$, $D_{12}:C_2^4$, $C_2^4\times D_{12}$, $C_{12}:C_2^5$, $C_6\times C_2^3:D_4$, $C_4^2:C_2^2\times C_6$, $(C_6\times D_4):D_4$, $C_2^5.D_6$, $C_2^4.D_{12}$, $C_2^3:(S_3\times D_4)$, $S_3\times C_2^3:D_4$, $(C_2\times D_4):D_{12}$, $C_4^2:C_2^2\times S_3$, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^6:C_6$, $C_2^5:C_{12}$, $C_2^6:S_3$, $C_2^5.D_6$, $C_2^5.D_6$

Order 256: $C_2^5:D_4$ x 2, $C_2^6:C_2^2$, $C_2^5.D_4$, $C_4^2:C_2^4$, $C_4^2:C_2^4$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^6:C_4$, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^6:C_4$

Order 192: $D_6\times C_2^4$ x 17, $D_6:C_2^4$ x 11, $C_{12}:C_2^4$ x 10, $C_{12}:C_2^4$ x 6, $C_2^3\times D_{12}$ x 5, $C_{12}:C_2^4$ x 5, $C_2^5\times C_6$ x 4, $C_2^4:C_{12}$ x 3, $C_2^4.D_6$ x 3, $D_{12}:C_2^3$ x 2, $C_2^5:C_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^3.D_{12}$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_{12}.C_2^4$, $C_2^4\times C_{12}$, $C_6.C_2^5$, $C_3\times C_2^3:D_4$, $C_3\times C_4^2:C_2^2$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 128: $C_2^4:D_4$ x 4, $C_2^7$ x 2, $D_4^2:C_2$ x 2, $C_2^5:C_4$ x 2, $C_2^4.D_4$, $C_2^5:C_4$, $D_4:C_2^4$, $D_4\times C_2^4$, $C_2^4:D_4$, $C_4^2:C_2^3$, $C_2^4.D_4$, $C_2^3\wr C_2$, $C_2^5:C_4$

Order 96: $C_2^3\times D_6$ x 26, $C_2^4\times C_6$ x 10, $C_2^3:D_6$ x 9, $C_{12}:C_2^3$ x 6, $C_2^3\times C_{12}$ x 4, $C_6.C_2^4$ x 4, $C_2^2\times D_{12}$ x 3, $C_{12}:C_2^3$ x 3, $C_2^3:C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^2.D_{12}$

Order 64: $D_4\times C_2^3$ x 6, $C_2^6$ x 4, $C_2^4:C_4$ x 3, $C_2^3:D_4$ x 2, $C_2^4:C_4$, $D_4:C_2^3$, $C_2^4\times C_4$, $C_2^3:D_4$, $C_4^2:C_2^2$

Order 48: $C_2^2\times D_6$ x 20, $C_2^3\times C_6$ x 11, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2

Order 32: $C_2^5$ x 10, $C_2^2\times D_4$ x 6, $C_2^3\times C_4$ x 4, $C_2^3:C_4$ x 2

Order 24: $C_2\times D_6$ x 9, $C_2^2\times C_6$ x 8

Order 16: $C_2^4$ x 11, $C_2^2\times C_4$ x 2

Order 12: $C_2\times C_6$ x 5, $D_6$ x 3

Order 8: $C_2^3$ x 8

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

Order 4: $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6144: $C_2^6.(D_4\times D_6)$ ($C_1$)

Order 3072: $S_3\times C_2^5.C_2^4$ ($C_2$) x 32, $C_3:(C_2^6.C_2^4)$ ($C_2$) x 6, $C_3:(C_2^6.C_2^4)$ ($C_2$) x 6, $C_3:(C_2^6.C_2^4)$ ($C_2$) x 6, $S_3\times C_2^5.C_2^4$ ($C_2$) x 6, $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3\times C_2^6.C_2^4$ ($C_2$), $C_3:(C_2^4.C_2^6)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^5)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$)

Order 1536: $S_3\times C_2^5:D_4$ ($C_2^2$) x 96, $C_3:(C_2^5.C_2^4)$ ($C_2^2$) x 96, $C_3:(C_2^5.C_2^4)$ ($C_2^2$) x 96, $C_2^5.C_2^3\times S_3$ ($C_2^2$) x 96, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 24, $S_3\times C_4^2:C_2^4$ ($C_2^2$) x 16, $C_3\times C_2^5.C_2^4$ ($C_2^2$) x 16, $C_2^6:C_2^2\times S_3$ ($C_2^2$) x 16, $C_3:(C_2^5.C_2^4)$ ($C_2^2$) x 16, $C_3:(C_2^5.C_2^4)$ ($C_2^2$) x 16, $C_3:(C_2^5.C_2^4)$ ($C_2^2$) x 16, $S_3\times C_2^5.C_2^3$ ($C_2^2$) x 16, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 12, $C_3:(C_2^4.C_2^5)$ ($C_2^2$) x 12, $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^2$) x 12, $C_6\times C_2^5.C_2^3$ ($C_2^2$) x 6, $C_6\times C_2^5:D_4$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$) x 6, $C_2\times D_4^2:D_6$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 6, $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$) x 6, $C_6.(D_4\times C_2^5)$ ($C_2^2$) x 4, $C_2^7:D_6$ ($C_2^2$) x 4, $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$) x 4, $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$) x 3, $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^2$) x 3, $C_6\times C_2^5.C_2^3$ ($C_2^2$), $C_6\times C_2^4.C_2^4$ ($C_2^2$), $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_6\times C_2^6.C_2^2$ ($C_2^2$)

Order 1024: $C_2^7:D_4$ ($S_3$)

Order 768: $C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 192, $C_2^4:(C_2\times D_{12})$ ($C_2^3$) x 96, $C_2^6:D_6$ ($C_2^3$) x 96, $C_2^4:(C_4\times D_6)$ ($C_2^3$) x 96, $C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 48, $C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 48, $C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 48, $C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 48, $C_2^5:D_{12}$ ($C_2^3$) x 48, $(C_2^3\times C_{12}):D_4$ ($C_2^3$) x 48, $C_2^6:D_6$ ($C_2^3$) x 48, $C_3\times C_2^5:D_4$ ($C_2^3$) x 48, $C_2^5:C_4\times S_3$ ($C_2^3$) x 48, $C_3\times C_2^5.C_2^3$ ($C_2^3$) x 48, $C_3:(C_2^6.C_2^2)$ ($C_2^3$) x 48, $D_4^2:D_6$ ($C_2^3$) x 48, $D_4^2:D_6$ ($D_4$) x 48, $C_2^6:D_6$ ($C_2^3$) x 36, $C_2^6:D_6$ ($C_2^3$) x 32, $C_3:(C_2^6.C_2^2)$ ($C_2^3$) x 32, $C_2^5.C_2^2\times S_3$ ($C_2^3$) x 24, $C_3:(C_2^6.C_2^2)$ ($C_2^3$) x 24, $C_3:(C_2^4.C_2^4)$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 12, $C_6\times C_2^5.C_2^2$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^4.C_2^3)$ ($D_4$) x 12, $C_3:(C_2^3.C_2^5)$ ($C_2^3$) x 12, $C_2\times C_2^4:C_2^2\times S_3$ ($D_4$) x 12, $C_3\times C_2^4.C_2^4$ ($C_2^3$) x 8, $C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 8, $C_3\times C_2^6.C_2^2$ ($C_2^3$) x 8, $C_3\times C_2^5.C_2^3$ ($C_2^3$) x 8, $C_3:(C_2^6.C_2^2)$ ($C_2^3$) x 8, $C_2^6:D_6$ ($D_4$) x 8, $C_2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 6, $C_2\times D_4^2:C_6$ ($C_2^3$) x 6, $C_3:(C_2^4.C_2^4)$ ($C_2^3$) x 6, $C_3\times C_2^4.C_2^4$ ($C_2^3$) x 6, $D_{12}:C_2^5$ ($D_4$) x 6, $C_2\times C_3:(C_2^4.C_2^3)$ ($D_4$) x 6, $C_2\times C_3:(C_2^4.C_2^3)$ ($C_2^3$) x 6, $C_2\times C_3:(C_2^4.C_2^3)$ ($D_4$) x 6, $C_2^3\times C_2^3:D_6$ ($C_2^3$) x 6, $C_2\times C_2^3:(C_2\times C_4)\times S_3$ ($D_4$) x 6, $C_2^7:C_6$ ($C_2^3$) x 4, $C_2\times C_3:(C_2^6.C_2)$ ($C_2^3$) x 4, $C_2\times C_3:(C_2^6.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^7.C_2)$ ($C_2^3$) x 4, $C_2^6.C_2\times S_3$ ($C_2^3$) x 4, $C_2^7:C_6$ ($C_2^3$) x 4, $C_2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 3, $C_6\times C_2^5.C_2^2$ ($C_2^3$) x 3, $C_{12}:C_2^6$ ($C_2^3$) x 3, $C_2^3\times C_2^2:D_{12}$ ($C_2^3$) x 3, $C_2^3\times C_2^2:D_{12}$ ($D_4$) x 3, $C_2^3\times C_2^4:S_3$ ($C_2^3$) x 3, $C_3:(C_2^7.C_2)$ ($C_2^3$) x 3, $C_3:(C_2^7.C_2)$ ($D_4$) x 3, $D_6\times C_2^6$ ($C_2^3$), $D_6\times C_2^6$ ($D_4$), $C_2^3\times C_2^4:S_3$ ($D_4$)

Order 512: $C_2^6:D_4$ ($D_6$) x 16, $C_2^5.C_2^4$ ($D_6$) x 6, $C_2^6:D_4$ ($D_6$) x 6, $C_2^5.C_2^4$ ($D_6$), $C_2^4.C_2^5$ ($D_6$), $C_4^2:C_2^5$ ($D_6$)

Order 384: $C_2^3:(S_3\times D_4)$ ($C_2\times D_4$) x 96, $S_3\times C_2^3:D_4$ ($C_2\times D_4$) x 96, $C_{12}:C_2^5$ ($C_2\times D_4$) x 72, $C_2^4:(C_2\times C_{12})$ ($C_2^4$) x 48, $C_2^5.D_6$ ($C_2^4$) x 48, $C_2^4.D_{12}$ ($C_2^4$) x 48, $C_2^4.D_{12}$ ($C_2^4$) x 48, $C_2^4.D_{12}$ ($C_2^4$) x 48, $D_{12}:C_2^4$ ($C_2\times D_4$) x 48, $(C_6\times D_4):D_4$ ($C_2\times D_4$) x 48, $C_2^5:D_6$ ($C_2^4$) x 48, $C_2^5:D_6$ ($C_2\times D_4$) x 48, $C_2^5:D_6$ ($C_2\times D_4$) x 48, $(C_2\times D_4):D_{12}$ ($C_2\times D_4$) x 48, $C_4^2:C_2^2\times S_3$ ($C_2\times D_4$) x 48, $C_2^4:D_{12}$ ($C_2\times D_4$) x 36, $C_2^5.D_6$ ($C_2\times D_4$) x 36, $C_2^5:C_{12}$ ($C_2^4$) x 24, $C_2^5.D_6$ ($C_2^4$) x 24, $C_2^4.D_{12}$ ($C_2^4$) x 24, $D_4^2:C_6$ ($C_2^4$) x 24, $D_4^2:C_6$ ($C_2\times D_4$) x 24, $C_2^5:D_6$ ($C_2^4$) x 24, $C_2^5:D_6$ ($C_2\times D_4$) x 24, $C_2^5:D_6$ ($C_2\times D_4$) x 24, $C_2^5:D_6$ ($C_2^4$) x 24, $C_2^5:D_6$ ($C_2\times D_4$) x 24, $C_2^6:C_6$ ($C_2^4$) x 20, $D_{12}:C_2^4$ ($C_2\times D_4$) x 18, $C_2^4:D_{12}$ ($C_2^4$) x 16, $C_2^5.D_6$ ($C_2^4$) x 16, $C_2^5:D_6$ ($C_2^4$) x 16, $C_2^5:D_6$ ($C_2^4$) x 16, $C_2^6:C_6$ ($C_2^4$) x 16, $C_2^6:S_3$ ($C_2^4$) x 16, $C_2^5.D_6$ ($C_2^4$) x 16, $D_6\times C_2^5$ ($C_2\times D_4$) x 14, $(C_2^2\times C_{12}):D_4$ ($C_2^4$) x 12, $C_2^5.D_6$ ($C_2^4$) x 12, $C_{12}:C_2^5$ ($C_2^4$) x 12, $C_6\times C_2^3:D_4$ ($C_2\times D_4$) x 12, $C_2^4:D_{12}$ ($C_2\times D_4$) x 12, $C_2^4:D_{12}$ ($C_2^4$) x 12, $C_2^5.D_6$ ($C_2\times D_4$) x 12, $C_2^5.D_6$ ($C_2^4$) x 12, $C_2^5:D_6$ ($C_2\times D_4$) x 8, $D_6\times C_2^5$ ($C_2^4$) x 6, $C_{12}.C_2^5$ ($C_2\times D_4$) x 6, $C_4^2:C_2^2\times C_6$ ($C_2\times D_4$) x 6, $C_2^5.D_6$ ($C_2\times D_4$) x 6, $C_2^4.D_{12}$ ($C_2\times D_4$) x 6, $C_2^6:C_6$ ($C_2\times D_4$) x 4, $C_2^5:C_{12}$ ($C_2^4$) x 4, $C_2^5.D_6$ ($C_2^4$) x 4, $C_{12}:C_2^5$ ($C_2^4$) x 3, $D_6:C_2^5$ ($C_2^4$) x 3, $D_6:C_2^5$ ($C_2\times D_4$) x 3, $C_2^4\times D_{12}$ ($C_2\times D_4$) x 3, $C_{12}:C_2^5$ ($C_2\times D_4$) x 3, $C_2^5:C_{12}$ ($C_2^4$) x 3, $C_2^5:C_{12}$ ($C_2\times D_4$) x 3, $C_2^5.D_6$ ($C_2^4$) x 3, $C_2^5.D_6$ ($C_2\times D_4$) x 3, $C_2^6\times C_6$ ($C_2^4$), $C_2^6\times C_6$ ($C_2\times D_4$)

Order 256: $C_2^5:D_4$ ($C_2\times D_6$) x 48, $C_2^5.D_4$ ($C_2\times D_6$) x 48, $C_2^5.D_4$ ($C_2\times D_6$) x 12, $C_2^6:C_2^2$ ($C_2\times D_6$) x 8, $C_4^2:C_2^4$ ($C_2\times D_6$) x 8, $C_2^6:C_4$ ($C_2\times D_6$) x 8, $C_4^2:C_2^4$ ($C_2\times D_6$) x 6, $C_2^6:C_4$ ($C_2\times D_6$) x 6, $C_2^5:D_4$ ($C_2\times D_6$) x 4, $C_2^5:D_4$ ($C_2\times D_6$) x 4, $C_2^5.D_4$ ($C_2\times D_6$) x 3

Order 192: $C_{12}:C_2^4$ ($C_2^2\times D_4$) x 144, $C_{12}:C_2^4$ ($C_2^2\wr C_2$) x 144, $C_2^4:D_6$ ($C_2^2\times D_4$) x 96, $D_{12}:C_2^3$ ($C_2^2\times D_4$) x 72, $D_6\times C_2^4$ ($C_2^2\times D_4$) x 55, $D_6:C_2^4$ ($C_2^2\times D_4$) x 54, $C_3\times C_2^3:D_4$ ($C_2^2\times D_4$) x 48, $C_2^4:D_6$ ($C_2^2\times D_4$) x 48, $C_2^4:D_6$ ($C_2^2\wr C_2$) x 48, $C_2^3:D_{12}$ ($C_2^2\times D_4$) x 48, $C_2^3:D_{12}$ ($C_2^2\times D_4$) x 48, $C_2^3:D_{12}$ ($C_2^2\wr C_2$) x 48, $C_2^4.D_6$ ($C_2^2\times D_4$) x 48, $C_2^4.D_6$ ($C_2^2\times D_4$) x 48, $C_2^4.D_6$ ($C_2^2\wr C_2$) x 48, $C_{12}:C_2^4$ ($C_2^2\times D_4$) x 42, $D_6:C_2^4$ ($C_2^2\wr C_2$) x 36, $C_2^3\times D_{12}$ ($C_2^2\wr C_2$) x 36, $C_{12}:C_2^4$ ($C_2^2\wr C_2$) x 36, $D_6\times C_2^4$ ($C_2^2\wr C_2$) x 28, $C_{12}.C_2^4$ ($C_2^2\times D_4$) x 24, $C_2^5:C_6$ ($C_2^2\times D_4$) x 24, $C_3\times C_4^2:C_2^2$ ($C_2^2\times D_4$) x 24, $C_2^4:C_{12}$ ($C_2^2\times D_4$) x 24, $C_2^4:D_6$ ($C_2^2\times D_4$) x 24, $C_2^4.D_6$ ($C_2^2\times D_4$) x 24, $C_2^4.D_6$ ($C_2^2\times D_4$) x 24, $C_2^3.D_{12}$ ($C_2^2\times D_4$) x 24, $C_2^3.D_{12}$ ($C_2^2\wr C_2$) x 24, $C_2^3\times D_{12}$ ($C_2^2\times D_4$) x 12, $C_{12}:C_2^4$ ($C_2^2\times D_4$) x 12, $C_2^4:C_{12}$ ($C_2^5$) x 8, $C_2^4.D_6$ ($C_2^5$) x 8, $C_2^5:C_6$ ($C_2^5$) x 8, $C_2^4:D_6$ ($C_2^5$) x 8, $C_2^5\times C_6$ ($C_2^2\times D_4$) x 7, $C_{12}:C_2^4$ ($C_2^5$) x 6, $D_6:C_2^4$ ($C_2^5$) x 6, $C_2^4:C_{12}$ ($C_2^5$) x 6, $C_2^4.D_6$ ($C_2^5$) x 6, $D_6\times C_2^4$ ($C_2^5$) x 4, $C_2^5\times C_6$ ($C_2^5$) x 3, $C_2^4\times C_{12}$ ($C_2^2\times D_4$) x 3, $C_6.C_2^5$ ($C_2^2\times D_4$) x 3

Order 128: $C_2^4.D_4$ ($C_2^2\times D_6$) x 48, $C_2^5:C_4$ ($C_2^2\times D_6$) x 24, $D_4^2:C_2$ ($C_2^2\times D_6$) x 24, $D_4^2:C_2$ ($S_3\times D_4$) x 24, $C_2^4:D_4$ ($C_2^2\times D_6$) x 20, $C_2^3\wr C_2$ ($C_2^2\times D_6$) x 16, $C_2^4:D_4$ ($S_3\times D_4$) x 12, $C_2^4.D_4$ ($C_2^2\times D_6$) x 12, $D_4:C_2^4$ ($S_3\times D_4$) x 6, $C_4^2:C_2^3$ ($S_3\times D_4$) x 6, $C_2^4:D_4$ ($S_3\times D_4$) x 4, $C_2^5:C_4$ ($C_2^2\times D_6$) x 4, $D_4\times C_2^4$ ($C_2^2\times D_6$) x 3, $C_2^5:C_4$ ($C_2^2\times D_6$) x 3, $C_2^5:C_4$ ($S_3\times D_4$) x 3, $C_2^7$ ($C_2^2\times D_6$), $C_2^7$ ($S_3\times D_4$)

Order 96: $C_2^3\times D_6$ ($C_2^3:D_4$) x 192, $C_2^3:D_6$ ($C_2^3:D_4$) x 144, $C_2^2\times D_{12}$ ($C_2^3:D_4$) x 96, $C_{12}:C_2^3$ ($C_2^3:D_4$) x 96, $C_2^2.D_{12}$ ($C_2^3:D_4$) x 96, $C_{12}:C_2^3$ ($D_4\times C_2^3$) x 48, $C_{12}:C_2^3$ ($C_2^3:D_4$) x 48, $C_2^3:D_6$ ($D_4\times C_2^3$) x 48, $C_2^3\times D_6$ ($D_4\times C_2^3$) x 34, $C_2^3\times D_6$ ($C_2\wr C_2^2$) x 24, $C_2^3\times C_{12}$ ($C_2^3:D_4$) x 24, $C_6.C_2^4$ ($C_2^3:D_4$) x 24, $C_2^3:C_{12}$ ($D_4\times C_2^3$) x 24, $C_2^3:C_{12}$ ($C_2^3:D_4$) x 24, $C_2^3.D_6$ ($D_4\times C_2^3$) x 24, $C_2^3.D_6$ ($C_2^3:D_4$) x 24, $C_2^4\times C_6$ ($D_4\times C_2^3$) x 20, $C_2^4\times C_6$ ($C_2^3:D_4$) x 16, $C_2^3\times C_{12}$ ($D_4\times C_2^3$) x 6, $C_6.C_2^4$ ($D_4\times C_2^3$) x 6, $C_2^4\times C_6$ ($C_2^6$)

Order 64: $C_2^3:D_4$ ($D_4\times D_6$) x 48, $D_4\times C_2^3$ ($D_4\times D_6$) x 42, $D_4:C_2^3$ ($D_4\times D_6$) x 24, $C_2^3:D_4$ ($D_4\times D_6$) x 24, $C_4^2:C_2^2$ ($D_4\times D_6$) x 24, $C_2^4:C_4$ ($D_4\times D_6$) x 24, $C_2^4:C_4$ ($C_2^3\times D_6$) x 8, $C_2^3:D_4$ ($C_2^3\times D_6$) x 8, $C_2^6$ ($D_4\times D_6$) x 7, $D_4\times C_2^3$ ($C_2^3\times D_6$) x 6, $C_2^4:C_4$ ($C_2^3\times D_6$) x 6, $C_2^6$ ($C_2^3\times D_6$) x 3, $C_2^4\times C_4$ ($D_4\times D_6$) x 3

Order 48: $C_2^2\times D_6$ ($C_2^4:D_4$) x 76, $C_2^2\times D_6$ ($C_2^4:D_4$) x 72, $C_2^2\times D_6$ ($C_2^3\wr C_2$) x 64, $C_2^3\times C_6$ ($C_2^4:D_4$) x 48, $C_2^2\times C_{12}$ ($C_2^4:D_4$) x 36, $C_6.C_2^3$ ($C_2^4:D_4$) x 36, $C_2^3\times C_6$ ($C_2^4:D_4$) x 12, $C_2^2\times D_6$ ($C_2^4:D_4$) x 12, $C_2^3\times C_6$ ($D_4\times C_2^4$) x 7

Order 32: $C_2^2\times D_4$ ($C_{12}:C_2^4$) x 48, $C_2^2\times D_4$ ($C_2^4:D_6$) x 48, $C_2^3\times C_4$ ($C_2^4:D_6$) x 24, $C_2^3:C_4$ ($C_{12}:C_2^4$) x 24, $C_2^3:C_4$ ($C_2^4:D_6$) x 24, $C_2^5$ ($C_{12}:C_2^4$) x 20, $C_2^5$ ($C_2^4:D_6$) x 16, $C_2^3\times C_4$ ($C_{12}:C_2^4$) x 6, $C_2^5$ ($D_6\times C_2^4$)

Order 24: $C_2\times D_6$ ($C_2^5:D_4$) x 48, $C_2\times D_6$ ($C_2^5:D_4$) x 24, $C_2^2\times C_6$ ($C_2^5:D_4$) x 18, $C_2\times D_6$ ($C_2^5:D_4$) x 16, $C_2\times D_6$ ($C_2^5:D_4$) x 12, $C_2^2\times C_6$ ($C_2^5:D_4$) x 8, $C_2^2\times C_6$ ($C_2^5:D_4$) x 7, $C_2^2\times C_6$ ($C_2^5:D_4$) x 6

Order 16: $C_2^4$ ($C_2^5:D_6$) x 48, $C_2^2\times C_4$ ($C_2^5:D_6$) x 36, $C_2^4$ ($C_2\wr C_2^2\times S_3$) x 12, $C_2^4$ ($C_{12}:C_2^5$) x 7

Order 12: $D_6$ ($C_2^6:D_4$) x 24, $D_6$ ($C_2^6:D_4$) x 16, $C_2\times C_6$ ($C_2^6:D_4$) x 12, $C_2\times C_6$ ($C_2^3.C_2^6$) x 3, $C_2\times C_6$ ($C_2^6:D_4$) x 3, $C_2\times C_6$ ($C_2^8.C_2$)

Order 8: $C_2^3$ ($C_2\wr C_2^2\times D_6$) x 18, $C_2^3$ ($C_2^6:D_6$) x 8, $C_2^3$ ($C_2^6:D_6$) x 7, $C_2^3$ ($C_2^4.C_2^3\times S_3$) x 6

Order 6: $S_3$ ($C_2^7:D_4$) x 8, $C_6$ ($C_2^7:D_4$) x 4, $C_6$ ($C_2^6.C_2^4$) x 3

Order 4: $C_2^2$ ($S_3\times C_2^5:D_4$) x 12, $C_2^2$ ($C_2\times C_2^4.C_2^3\times S_3$) x 3, $C_2^2$ ($C_2^4:D_4\times D_6$) x 3, $C_2^2$ ($C_2^7:D_6$)

Order 3: $C_3$ ($C_2^8.D_4$)

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

Order 1: $C_1$ ($C_2^6.(D_4\times D_6)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6144: $C_2^6.(D_4\times D_6)$ ($C_1$)

Order 3072: $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_3\times C_2^6.C_2^4$ ($C_2$), $C_3:(C_2^4.C_2^6)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $C_2\times C_3:(C_2^4.C_2^5)$ ($C_2$), $C_3:(C_2^6.C_2^4)$ ($C_2$), $S_3\times C_2^5.C_2^4$ ($C_2$), $S_3\times C_2^5.C_2^4$ ($C_2$)

Order 1536: $C_6.(D_4\times C_2^5)$ ($C_2^2$) x 2, $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$) x 2, $C_6\times C_2^5.C_2^3$ ($C_2^2$), $C_6\times C_2^5.C_2^3$ ($C_2^2$), $C_6\times C_2^5:D_4$ ($C_2^2$), $C_6\times C_2^4.C_2^4$ ($C_2^2$), $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$), $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$), $S_3\times C_4^2:C_2^4$ ($C_2^2$), $C_2\times D_4^2:D_6$ ($C_2^2$), $C_3\times C_2^5.C_2^4$ ($C_2^2$), $C_2^7:D_6$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_3:(C_2^4.C_2^5)$ ($C_2^2$), $S_3\times C_2^5:D_4$ ($C_2^2$), $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^2$), $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^2$), $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^2$), $C_2^6:C_2^2\times S_3$ ($C_2^2$), $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^2$), $C_3:(C_2^5.C_2^4)$ ($C_2^2$), $C_3:(C_2^5.C_2^4)$ ($C_2^2$), $C_3:(C_2^5.C_2^4)$ ($C_2^2$), $C_3:(C_2^5.C_2^4)$ ($C_2^2$), $C_3:(C_2^5.C_2^4)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $S_3\times C_2^5.C_2^3$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^2$), $C_2^5.C_2^3\times S_3$ ($C_2^2$), $C_6\times C_2^6.C_2^2$ ($C_2^2$)

Order 1024: $C_2^7:D_4$ ($S_3$)

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

Order 512: $C_2^5.C_2^4$ ($D_6$), $C_2^5.C_2^4$ ($D_6$), $C_2^6:D_4$ ($D_6$), $C_2^4.C_2^5$ ($D_6$), $C_2^6:D_4$ ($D_6$), $C_4^2:C_2^5$ ($D_6$)

Order 384: $C_{12}:C_2^5$ ($C_2\times D_4$) x 7, $D_6\times C_2^5$ ($C_2^4$) x 3, $D_6\times C_2^5$ ($C_2\times D_4$) x 3, $C_2^6:C_6$ ($C_2^4$) x 3, $D_{12}:C_2^4$ ($C_2\times D_4$) x 2, $C_2^5:D_6$ ($C_2^4$) x 2, $C_2^5:D_6$ ($C_2\times D_4$) x 2, $C_2^5:D_6$ ($C_2^4$) x 2, $C_2^5:D_6$ ($C_2\times D_4$) x 2, $C_2^4:D_{12}$ ($C_2\times D_4$) x 2, $C_2^5.D_6$ ($C_2\times D_4$) x 2, $C_2^4:D_{12}$ ($C_2^4$), $C_2^5.D_6$ ($C_2^4$), $C_2^4:(C_2\times C_{12})$ ($C_2^4$), $C_2^5:C_{12}$ ($C_2^4$), $(C_2^2\times C_{12}):D_4$ ($C_2^4$), $C_2^5.D_6$ ($C_2^4$), $C_2^5.D_6$ ($C_2^4$), $C_2^5.D_6$ ($C_2^4$), $C_2^4.D_{12}$ ($C_2^4$), $C_2^4.D_{12}$ ($C_2^4$), $C_2^4.D_{12}$ ($C_2^4$), $C_2^4.D_{12}$ ($C_2^4$), $C_2^6\times C_6$ ($C_2^4$), $C_2^6\times C_6$ ($C_2\times D_4$), $C_{12}.C_2^5$ ($C_2\times D_4$), $C_{12}:C_2^5$ ($C_2^4$), $D_6:C_2^5$ ($C_2^4$), $D_6:C_2^5$ ($C_2\times D_4$), $D_{12}:C_2^4$ ($C_2\times D_4$), $C_{12}:C_2^5$ ($C_2^4$), $C_2^4\times D_{12}$ ($C_2\times D_4$), $C_{12}:C_2^5$ ($C_2\times D_4$), $D_4^2:C_6$ ($C_2^4$), $D_4^2:C_6$ ($C_2\times D_4$), $C_6\times C_2^3:D_4$ ($C_2\times D_4$), $C_2^6:C_6$ ($C_2\times D_4$), $C_4^2:C_2^2\times C_6$ ($C_2\times D_4$), $C_2^5:C_{12}$ ($C_2^4$), $C_2^5:C_{12}$ ($C_2\times D_4$), $C_2^5:D_6$ ($C_2\times D_4$), $C_2^5.D_6$ ($C_2^4$), $C_2^5.D_6$ ($C_2\times D_4$), $(C_6\times D_4):D_4$ ($C_2\times D_4$), $C_2^5:D_6$ ($C_2^4$), $C_2^5:D_6$ ($C_2\times D_4$), $C_2^5.D_6$ ($C_2\times D_4$), $C_2^4.D_{12}$ ($C_2\times D_4$), $C_2^3:(S_3\times D_4)$ ($C_2\times D_4$), $S_3\times C_2^3:D_4$ ($C_2\times D_4$), $C_2^5:D_6$ ($C_2^4$), $C_2^5:D_6$ ($C_2\times D_4$), $C_2^5:D_6$ ($C_2^4$), $C_2^5:D_6$ ($C_2\times D_4$), $(C_2\times D_4):D_{12}$ ($C_2\times D_4$), $C_4^2:C_2^2\times S_3$ ($C_2\times D_4$), $C_2^4:D_{12}$ ($C_2\times D_4$), $C_2^4:D_{12}$ ($C_2^4$), $C_2^5.D_6$ ($C_2\times D_4$), $C_2^5.D_6$ ($C_2^4$), $C_2^6:C_6$ ($C_2^4$), $C_2^5:C_{12}$ ($C_2^4$), $C_2^6:S_3$ ($C_2^4$), $C_2^5.D_6$ ($C_2^4$), $C_2^5.D_6$ ($C_2^4$)

Order 256: $C_2^5:D_4$ ($C_2\times D_6$) x 2, $C_2^6:C_2^2$ ($C_2\times D_6$), $C_2^5.D_4$ ($C_2\times D_6$), $C_4^2:C_2^4$ ($C_2\times D_6$), $C_4^2:C_2^4$ ($C_2\times D_6$), $C_2^5:D_4$ ($C_2\times D_6$), $C_2^5:D_4$ ($C_2\times D_6$), $C_2^6:C_4$ ($C_2\times D_6$), $C_2^5.D_4$ ($C_2\times D_6$), $C_2^5.D_4$ ($C_2\times D_6$), $C_2^6:C_4$ ($C_2\times D_6$)

Order 192: $D_6\times C_2^4$ ($C_2^2\times D_4$) x 9, $D_6\times C_2^4$ ($C_2^2\wr C_2$) x 6, $D_6:C_2^4$ ($C_2^2\times D_4$) x 6, $C_{12}:C_2^4$ ($C_2^2\times D_4$) x 5, $C_{12}:C_2^4$ ($C_2^2\times D_4$) x 5, $C_{12}:C_2^4$ ($C_2^2\wr C_2$) x 5, $D_6:C_2^4$ ($C_2^2\wr C_2$) x 4, $C_2^3\times D_{12}$ ($C_2^2\wr C_2$) x 4, $C_{12}:C_2^4$ ($C_2^2\wr C_2$) x 4, $C_2^5\times C_6$ ($C_2^5$) x 2, $C_2^5\times C_6$ ($C_2^2\times D_4$) x 2, $D_6\times C_2^4$ ($C_2^5$) x 2, $D_{12}:C_2^3$ ($C_2^2\times D_4$) x 2, $C_2^4:C_{12}$ ($C_2^2\times D_4$) x 2, $C_2^4.D_6$ ($C_2^2\times D_4$) x 2, $C_2^4:D_6$ ($C_2^2\times D_4$) x 2, $C_2^4:C_{12}$ ($C_2^5$), $C_2^4.D_6$ ($C_2^5$), $C_{12}.C_2^4$ ($C_2^2\times D_4$), $C_{12}:C_2^4$ ($C_2^5$), $C_2^4\times C_{12}$ ($C_2^2\times D_4$), $D_6:C_2^4$ ($C_2^5$), $C_6.C_2^5$ ($C_2^2\times D_4$), $C_2^3\times D_{12}$ ($C_2^2\times D_4$), $C_{12}:C_2^4$ ($C_2^2\times D_4$), $C_3\times C_2^3:D_4$ ($C_2^2\times D_4$), $C_2^5:C_6$ ($C_2^5$), $C_2^5:C_6$ ($C_2^2\times D_4$), $C_3\times C_4^2:C_2^2$ ($C_2^2\times D_4$), $C_2^4:C_{12}$ ($C_2^5$), $C_2^4:D_6$ ($C_2^5$), $C_2^4:D_6$ ($C_2^2\times D_4$), $C_2^4.D_6$ ($C_2^5$), $C_2^4:D_6$ ($C_2^2\times D_4$), $C_2^4:D_6$ ($C_2^2\wr C_2$), $C_2^4.D_6$ ($C_2^2\times D_4$), $C_2^3.D_{12}$ ($C_2^2\times D_4$), $C_2^3.D_{12}$ ($C_2^2\wr C_2$), $C_2^3:D_{12}$ ($C_2^2\times D_4$), $C_2^3:D_{12}$ ($C_2^2\times D_4$), $C_2^3:D_{12}$ ($C_2^2\wr C_2$), $C_2^4.D_6$ ($C_2^2\times D_4$), $C_2^4.D_6$ ($C_2^2\times D_4$), $C_2^4.D_6$ ($C_2^2\wr C_2$)

Order 128: $C_2^4:D_4$ ($C_2^2\times D_6$) x 3, $C_2^4.D_4$ ($C_2^2\times D_6$), $C_2^5:C_4$ ($C_2^2\times D_6$), $C_2^7$ ($C_2^2\times D_6$), $C_2^7$ ($S_3\times D_4$), $D_4:C_2^4$ ($S_3\times D_4$), $D_4\times C_2^4$ ($C_2^2\times D_6$), $D_4^2:C_2$ ($C_2^2\times D_6$), $D_4^2:C_2$ ($S_3\times D_4$), $C_2^4:D_4$ ($S_3\times D_4$), $C_2^4:D_4$ ($S_3\times D_4$), $C_4^2:C_2^3$ ($S_3\times D_4$), $C_2^5:C_4$ ($C_2^2\times D_6$), $C_2^5:C_4$ ($S_3\times D_4$), $C_2^4.D_4$ ($C_2^2\times D_6$), $C_2^3\wr C_2$ ($C_2^2\times D_6$), $C_2^5:C_4$ ($C_2^2\times D_6$)

Order 96: $C_2^3\times D_6$ ($C_2^3:D_4$) x 20, $C_2^3:D_6$ ($C_2^3:D_4$) x 6, $C_2^4\times C_6$ ($D_4\times C_2^3$) x 5, $C_2^3\times D_6$ ($D_4\times C_2^3$) x 5, $C_2^4\times C_6$ ($C_2^3:D_4$) x 4, $C_{12}:C_2^3$ ($D_4\times C_2^3$) x 3, $C_{12}:C_2^3$ ($C_2^3:D_4$) x 3, $C_2^3\times C_{12}$ ($C_2^3:D_4$) x 3, $C_2^3:D_6$ ($D_4\times C_2^3$) x 3, $C_6.C_2^4$ ($C_2^3:D_4$) x 3, $C_2^2\times D_{12}$ ($C_2^3:D_4$) x 3, $C_{12}:C_2^3$ ($C_2^3:D_4$) x 3, $C_2^4\times C_6$ ($C_2^6$), $C_2^3\times D_6$ ($C_2\wr C_2^2$), $C_2^3\times C_{12}$ ($D_4\times C_2^3$), $C_6.C_2^4$ ($D_4\times C_2^3$), $C_2^3:C_{12}$ ($D_4\times C_2^3$), $C_2^3:C_{12}$ ($C_2^3:D_4$), $C_2^3.D_6$ ($D_4\times C_2^3$), $C_2^3.D_6$ ($C_2^3:D_4$), $C_2^2.D_{12}$ ($C_2^3:D_4$)

Order 64: $D_4\times C_2^3$ ($D_4\times D_6$) x 5, $C_2^6$ ($C_2^3\times D_6$) x 2, $C_2^6$ ($D_4\times D_6$) x 2, $C_2^4:C_4$ ($D_4\times D_6$) x 2, $C_2^4:C_4$ ($C_2^3\times D_6$), $D_4:C_2^3$ ($D_4\times D_6$), $D_4\times C_2^3$ ($C_2^3\times D_6$), $C_2^4\times C_4$ ($D_4\times D_6$), $C_2^3:D_4$ ($D_4\times D_6$), $C_2^3:D_4$ ($C_2^3\times D_6$), $C_2^3:D_4$ ($D_4\times D_6$), $C_4^2:C_2^2$ ($D_4\times D_6$), $C_2^4:C_4$ ($C_2^3\times D_6$)

Order 48: $C_2^3\times C_6$ ($C_2^4:D_4$) x 8, $C_2^2\times D_6$ ($C_2^4:D_4$) x 8, $C_2^2\times D_6$ ($C_2^3\wr C_2$) x 8, $C_2^2\times D_6$ ($C_2^4:D_4$) x 3, $C_2^3\times C_6$ ($D_4\times C_2^4$) x 2, $C_2^2\times C_{12}$ ($C_2^4:D_4$) x 2, $C_6.C_2^3$ ($C_2^4:D_4$) x 2, $C_2^3\times C_6$ ($C_2^4:D_4$), $C_2^2\times D_6$ ($C_2^4:D_4$)

Order 32: $C_2^5$ ($C_{12}:C_2^4$) x 5, $C_2^5$ ($C_2^4:D_6$) x 4, $C_2^2\times D_4$ ($C_{12}:C_2^4$) x 3, $C_2^2\times D_4$ ($C_2^4:D_6$) x 3, $C_2^3\times C_4$ ($C_2^4:D_6$) x 3, $C_2^5$ ($D_6\times C_2^4$), $C_2^3\times C_4$ ($C_{12}:C_2^4$), $C_2^3:C_4$ ($C_{12}:C_2^4$), $C_2^3:C_4$ ($C_2^4:D_6$)

Order 24: $C_2^2\times C_6$ ($C_2^5:D_4$) x 3, $C_2\times D_6$ ($C_2^5:D_4$) x 3, $C_2\times D_6$ ($C_2^5:D_4$) x 3, $C_2^2\times C_6$ ($C_2^5:D_4$) x 2, $C_2^2\times C_6$ ($C_2^5:D_4$) x 2, $C_2\times D_6$ ($C_2^5:D_4$) x 2, $C_2^2\times C_6$ ($C_2^5:D_4$), $C_2\times D_6$ ($C_2^5:D_4$)

Order 16: $C_2^4$ ($C_2^5:D_6$) x 8, $C_2^4$ ($C_{12}:C_2^5$) x 2, $C_2^2\times C_4$ ($C_2^5:D_6$) x 2, $C_2^4$ ($C_2\wr C_2^2\times S_3$)

Order 12: $C_2\times C_6$ ($C_2^6:D_4$) x 2, $D_6$ ($C_2^6:D_4$) x 2, $C_2\times C_6$ ($C_2^8.C_2$), $C_2\times C_6$ ($C_2^3.C_2^6$), $C_2\times C_6$ ($C_2^6:D_4$), $D_6$ ($C_2^6:D_4$)

Order 8: $C_2^3$ ($C_2^6:D_6$) x 3, $C_2^3$ ($C_2^6:D_6$) x 2, $C_2^3$ ($C_2\wr C_2^2\times D_6$) x 2, $C_2^3$ ($C_2^4.C_2^3\times S_3$)

Order 6: $C_6$ ($C_2^6.C_2^4$), $C_6$ ($C_2^7:D_4$), $S_3$ ($C_2^7:D_4$)

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

Order 3: $C_3$ ($C_2^8.D_4$)

Order 2: $C_2$ ($C_3:(C_2^5.C_2^5)$), $C_2$ ($S_3\times C_2^5.C_2^4$)

Order 1: $C_1$ ($C_2^6.(D_4\times D_6)$)

Series

Derived series

$C_2^6.(D_4\times D_6)$

$\rhd$

$C_2^4\times C_6$

$\rhd$

$C_1$

Chief series

$C_2^6.(D_4\times D_6)$

$\rhd$

$C_3:(C_2^6.C_2^4)$

$\rhd$

$C_6\times C_2^5.C_2^3$

$\rhd$

$C_2^5.C_2^4$

$\rhd$

$C_4^2:C_2^4$

$\rhd$

$C_4^2:C_2^3$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6.(D_4\times D_6)$

$\rhd$

$C_2^4\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^6$

$\lhd$

$C_2^7:D_4$

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 $354 \times 354$ rational character table is not available for this group.