Abstract group 49152.nz: $C_2^8.C_2^3:S_4$

• Label: 49152.nz
• Order: \( 2^{14} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{21} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^8.C_2^3:S_4$
Order:	\(49152\)\(\medspace = 2^{14} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^5.C_2^6.C_6.C_2^6.C_2^3$, of order \(6291456\)\(\medspace = 2^{21} \cdot 3 \)
Composition factors:	$C_2$ x 14, $C_3$
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
Elements	1	1983	2048	15424	6144	15360	8192	49152
Conjugacy classes	1	42	1	97	3	16	4	164
Divisions	1	42	1	94	2	16	2	158
Autjugacy classes	1	24	1	29	2	5	2	64

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{2}=d^{4}=f^{4}=g^{4}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,13)(2,14)(3,16)(4,15)(5,10,7,11)(6,9,8,12)(17,21,19,23)(18,22,20,24) \!\cdots\! \rangle$
Transitive group:	24T15166				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^6.C_2^5)$ . $S_4$ (5)	$C_2^8$ . $(C_2^3:S_4)$	$C_2^6$ . $(C_2^5:S_4)$	$C_2^3$ . $(C_2^8.S_4)$	all 25

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$2$

Subgroups

There are 58 normal subgroups (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^7.C_2^4.D_6$
Commutator:	$G' \simeq$ $C_2^8.C_2^4.C_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^6$	$G/\Phi \simeq$ $C_2^5:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6.C_2^6.C_2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^8.C_2^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2\times C_2^5.S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^6.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 32 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 49152: $C_2^8.C_2^3:S_4$

Order 24576: $C_2^4.Q_8^2:S_4$ x 2, $C_2^8.C_2^4.C_6$

Order 16384: $C_2^7.C_2^6.C_2$

Order 12288: $(C_2^3\times C_4^3).S_4$ x 2, $C_2^8.(C_2\times S_4)$ x 2, $C_2^8.C_2^4.C_3$, $(C_2^3\times C_4^3).S_4$, $C_2^2.C_4^4:D_6$, $C_2^3.C_2\wr S_4$

Order 8192: $C_2^7.C_2^5.C_2$ x 4, $C_2^7.C_2^5.C_2$ x 4, $C_2^7.C_2^5.C_2$ x 2, $C_2^7.C_2^5.C_2$ x 2, $C_2^7.C_2^5.C_2$ x 2, $C_2^6.C_2^6.C_2$ x 2, $C_2^7.C_2^5.C_2$ x 2, $C_2^7.C_2^5.C_2$ x 2, $C_2^7.C_2^5.C_2$ x 2, $C_2^7.C_2^5.C_2$ x 2, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^5.C_2^2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^7.C_2^5.C_2$

Order 6144: $C_2^7.A_4.C_2^2$ x 4, $C_2^8.S_4$ x 4, $C_2^8.S_4$ x 4, $C_2^5.C_2^4.D_6$ x 2, $C_2^3.C_2^6.C_{12}$ x 2, $C_2^8.S_4$ x 2, $(C_2^2\times C_4^3).S_4$ x 2, $C_2^7.(C_2\times S_4)$ x 2, $(C_2^4\times C_4^2).S_4$ x 2, $C_2^4.C_4^2:S_4$ x 2, $C_2^8.(C_2\times A_4)$ x 2, $C_2^3.C_2\wr A_4$ x 2, $(C_2^3\times C_4^3).A_4$ x 2, $C_2^5.C_2^4.C_{12}$, $C_2^6.C_2^4.C_6$, $C_2^6.C_2^4.C_6$, $C_2^6.C_2^4.C_6$, $(C_2^2\times C_4^3).S_4$, $(C_2^2\times C_4^3).S_4$, $(C_2^3\times C_4^3).A_4$, $C_2^3.C_2\wr A_4$

Order 4096: $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^6.C_2^5.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^6.C_2^4.C_2^2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^6.C_2^5.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^6.C_2^5.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^6.C_2^5.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^7.C_2^4.C_2$ x 4, $C_2^5.C_2^4.C_2^3$ x 3, $C_2^6.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2^2$ x 2, $C_2^5.C_2^6.C_2$ x 2, $C_2^6.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^7.C_2^5$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.D_4^2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^5.C_2^4.C_2^3$ x 2, $C_2^7.C_2^5$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^7.C_2^5$ x 2, $C_2^6.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2^2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^6$ x 2, $C_2^6.C_2^4.C_2^2$ x 2, $C_2^6.C_2^6$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^7.C_2^5$ x 2, $C_2^5.C_2^4.C_2^3$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^4.C_2^2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^6$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^3.C_2^3$ x 2, $C_2^7.C_2^5$ x 2, $C_2^6.C_2^4.C_2^2$ x 2, $C_2^5.C_2^6.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^2.C_2^4$ x 2, $C_2^6.D_4^2$ x 2, $C_2^7.C_2^5$, $C_2^6.C_2^6$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^6$, $C_2^5.C_2^5.C_2^2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^5$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^5$, $C_2^5.C_2^6.C_2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^5.C_2$, $C_2^4.C_2^6.C_2^2$, $C_2^6.C_2^6$, $C_2.C_2^6.C_2^5$, $C_2^6.C_2^6$, $C_2^4.C_2^6.C_2^2$, $C_2^7.C_2^5$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^6.C_2$, $C_2^7.C_2^5$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^3.C_2^4$, $C_2^6.C_2^6$, $C_2^7.C_2^5$, $C_2^6.C_2^6$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^3.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^7.C_2^5$, $C_2^6.C_2^5.C_2$

Order 3072: $(C_2\times C_4^3).S_4$ x 4, $C_2^7.S_4$ x 4, $C_2^7.S_4$ x 4, $(C_2^5\times C_4).S_4$ x 4, $C_2^7.(C_2\times A_4)$ x 4, $C_2^8.A_4$ x 4, $C_2^8:A_4$ x 3, $(C_2^3\times C_4^2).S_4$ x 2, $(C_2^3\times C_4^2).S_4$ x 2, $C_2^8.D_6$ x 2, $(C_2^2\times C_4^3):A_4$ x 2, $C_2^8.A_4$ x 2, $C_2^6.C_2^4.C_3$, $C_2^5.C_2^4.C_6$, $C_2^6.C_2^4.C_3$, $C_2^6.C_2^4.C_3$, $(C_2\times C_4^3):S_4$, $C_2^8:A_4$

Order 2048: $C_2^5.C_2^6$ x 64, $C_2^6.(C_4\times D_4)$ x 16, $C_2^6.C_2^4.C_2$ x 16, $C_2^6.C_2^4.C_2$ x 12, $C_2^6.C_2^4.C_2$ x 12, $C_2^6.(C_4\times D_4)$ x 12, $C_2^6.C_2^4.C_2$ x 12, $C_2^6.C_2^4.C_2$ x 10, $C_2^5.C_2^4.C_2^2$ x 10, $C_2^5.C_2^5.C_2$ x 10, $C_2^6.C_2^4.C_2$ x 10, $C_2^6.C_2^4.C_2$ x 8, $C_2^5.C_2^5.C_2$ x 8, $C_2^5.C_2^5.C_2$ x 8, $C_2^5.C_2^5.C_2$ x 8, $C_2^6.C_2^5$ x 8, $C_2^7.C_2^3.C_2$ x 8, $C_2^5.C_2^3.C_2^3$ x 8, $C_2^6.C_2^3.C_2^2$ x 8, $C_2^4.C_2^6.C_2$ x 8, $C_2^6.C_2^4.C_2$ x 8, $C_2^7.C_2^3.C_2$ x 8, $C_2^6.C_2^4.C_2$ x 8, $C_2^6.C_2^4.C_2$ x 8, $C_2^6.C_2^4.C_2$ x 7, $C_2^5.C_2^5.C_2$ x 6, $C_2^5.C_2^5.C_2$ x 6, $C_2^5.C_2^5.C_2$ x 6, $C_2^5.C_2^3.C_2^3$ x 6, $C_2^6.C_2^4.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^4.C_2^6.C_2$ x 4, $C_2^7.C_2^4$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^7.C_2.C_2^3$ x 4, $C_2^5.C_2^3.C_2^3$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^4.C_2^4.C_2^3$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^7.C_2^4$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^7.C_2^3.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^6.C_2^3.C_2^2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^3.C_2^6.C_2^2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^8.C_2^3$ x 4, $C_2^5.C_2^4.C_2^2$ x 4, $C_2^5.C_2^3.C_2^3$ x 4, $C_2^4.C_2^6.C_2$ x 4, $C_2^6.C_2^3.C_2^2$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^6.C_2^5$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^5.C_2^3.C_2^3$ x 4, $C_2^4.C_2^5.C_2^2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^4.C_2^2$ x 4, $C_2^6.C_2^3.C_2^2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^6.(C_4\times D_4)$ x 4, $C_2^4.C_2^6.C_2$ x 4, $C_2.C_2^6.C_2^4$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^6.C_2^3.C_2^2$ x 4, $C_2^6.C_2^5$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^4.C_2^2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^6.C_2^5$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^5.C_2^5.C_2$ x 4, $C_2^5.C_2^4.C_2^2$ x 4, $C_2^5.C_2^3.C_2^3$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^5.C_2^3.C_2^3$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^5.C_2^3.C_2^3$ x 4, $C_2^4.C_2^5.C_2^2$ x 4, $C_2^7.C_2^4$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^3.C_2^6.C_2^2$ x 3, $C_2^7.C_2^4$ x 3, $C_2^7.C_2^4$ x 3, $C_2^6.C_2^5$ x 3, $C_2^6.C_2^4.C_2$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^3.C_2^6.C_2^2$ x 2, $C_2^3.C_2^6.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^8.C_2^3$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^8.C_2^3$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^6$ x 2, $C_2^6.C_2^5$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^7.C_2.C_2^3$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^3.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^8.C_2^3$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^8.C_2^3$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^6$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^6$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^7.C_2^3.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^8.C_2^3$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^3.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^3.C_2^2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^6$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^6$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^6$ x 2, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2^6.C_2^5$, $(C_2^5\times C_4).C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $(C_2^5\times C_4).C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^2.C_2^6.C_2^3$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^7.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^7.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2^3.C_2^5.C_2^3$, $(C_2^5\times C_4).C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$

Order 1536: $C_2\times C_4^3:A_4$ x 10, $C_4^3.S_4$ x 8, $(C_2^2\times C_4^2).S_4$ x 8, $C_2^5.(C_2\times S_4)$ x 8, $C_2^6.S_4$ x 8, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 4, $C_4^3.S_4$ x 4, $C_4^3.S_4$ x 4, $C_2^5.(C_2\times S_4)$ x 4, $C_2^5.(C_2\times S_4)$ x 4, $C_2^7.D_6$ x 4, $C_2^6:S_4$ x 4, $(C_2^5\times C_4).A_4$ x 4, $C_2^7:A_4$ x 3, $C_2^7:C_{12}$ x 3, $C_2^5.(C_4\times A_4)$ x 3, $C_4^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:D_{12}$ x 2, $C_2^7.D_6$ x 2, $C_4^3:S_4$ x 2, $C_2^7.A_4$ x 2, $(C_2^3\times C_4^2):C_{12}$ x 2, $C_4^3:S_4$, $C_4^3.S_4$, $(C_2\times C_4)\wr C_3$, $C_2^6.C_2^3.C_3$, $C_2^5.C_2^4.C_3$

Order 1024: $C_2^6.C_2^4$ x 2, $C_2^7.C_2^3$ x 2, $C_2^7.C_2^3$, $C_2^7.C_2^3$, $C_2^6.C_2^4$

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

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

Order 128: $C_2^3.C_4^2$, $C_2^5\times C_4$

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

Order 32: $C_2^5$

Order 16: $C_2^4$ x 4

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 32 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 49152: $C_2^8.C_2^3:S_4$

Order 24576: $C_2^8.C_2^4.C_6$, $C_2^4.Q_8^2:S_4$

Order 16384: $C_2^7.C_2^6.C_2$

Order 12288: $C_2^8.C_2^4.C_3$, $(C_2^3\times C_4^3).S_4$, $C_2^2.C_4^4:D_6$, $(C_2^3\times C_4^3).S_4$, $C_2^3.C_2\wr S_4$, $C_2^8.(C_2\times S_4)$

Order 8192: $C_2^7.C_2^5.C_2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2^5.C_2$, $C_2^6.C_2^5.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2^5.C_2$, $C_2^5.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2$, $C_2^6.C_2^6.C_2$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2^5.C_2$, $C_2^7.C_2^5.C_2$

Order 6144: $C_2^5.C_2^4.C_{12}$, $C_2^5.C_2^4.D_6$, $C_2^3.C_2^6.C_{12}$, $C_2^6.C_2^4.C_6$, $C_2^6.C_2^4.C_6$, $C_2^8.S_4$, $C_2^6.C_2^4.C_6$, $(C_2^2\times C_4^3).S_4$, $(C_2^2\times C_4^3).S_4$, $(C_2^2\times C_4^3).S_4$, $C_2^7.(C_2\times S_4)$, $C_2^7.A_4.C_2^2$, $(C_2^4\times C_4^2).S_4$, $C_2^4.C_4^2:S_4$, $C_2^8.S_4$, $C_2^8.S_4$, $C_2^8.(C_2\times A_4)$, $(C_2^3\times C_4^3).A_4$, $C_2^3.C_2\wr A_4$, $C_2^3.C_2\wr A_4$, $(C_2^3\times C_4^3).A_4$

Order 4096: $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2^5.C_2$ x 2, $C_2^5.C_2^4.C_2^3$ x 2, $C_2^5.C_2^4.C_2^3$ x 2, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^6.C_2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^5$, $C_2^7.C_2^5$, $C_2^6.C_2^6$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^6.D_4^2$, $C_2^6.C_2^6$, $C_2^5.C_2^5.C_2^2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^5$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^5$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^5.C_2^4.C_2^3$, $C_2^7.C_2^5$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^6.C_2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^5$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^6$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^6$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^4.C_2^6.C_2^2$, $C_2^6.C_2^6$, $C_2.C_2^6.C_2^5$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^6$, $C_2^7.C_2^5$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^4.C_2^6.C_2^2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^5$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^4.C_2^2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^5.C_2^6.C_2$, $C_2^6.C_2^6$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^6.C_2$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^5$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^3.C_2^4$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^6$, $C_2^7.C_2^5$, $C_2^6.C_2^3.C_2^3$, $C_2^7.C_2^5$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^4.C_2^2$, $C_2^7.C_2^4.C_2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^6$, $C_2^7.C_2^4.C_2$, $C_2^5.C_2^6.C_2$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$, $C_2^7.C_2^4.C_2$, $C_2^5.C_2^4.C_2^3$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^3.C_2^4$, $C_2^6.C_2^5.C_2$, $C_2^4.C_2^4.C_2^4$, $C_2^7.C_2^5$, $C_2^6.C_2^2.C_2^4$, $C_2^6.C_2^5.C_2$, $C_2^6.D_4^2$

Order 3072: $(C_2^5\times C_4).S_4$ x 3, $(C_2^2\times C_4^3):A_4$ x 2, $C_2^7.(C_2\times A_4)$ x 2, $C_2^8.A_4$ x 2, $C_2^8:A_4$ x 2, $C_2^6.C_2^4.C_3$, $C_2^5.C_2^4.C_6$, $C_2^6.C_2^4.C_3$, $C_2^6.C_2^4.C_3$, $(C_2\times C_4^3).S_4$, $(C_2^3\times C_4^2).S_4$, $(C_2^3\times C_4^2).S_4$, $C_2^7.S_4$, $C_2^7.S_4$, $(C_2\times C_4^3):S_4$, $C_2^8.D_6$, $C_2^8:A_4$, $C_2^8.A_4$

Order 2048: $C_2^6.C_2^4.C_2$ x 6, $C_2^6.C_2^4.C_2$ x 5, $C_2^5.C_2^5.C_2$ x 5, $C_2^6.(C_4\times D_4)$ x 4, $C_2^7.C_2^4$ x 4, $C_2^5.C_2^3.C_2^3$ x 4, $C_2^5.C_2^4.C_2^2$ x 4, $C_2^6.C_2^4.C_2$ x 4, $C_2^6.C_2^4.C_2$ x 3, $C_2^6.C_2^4.C_2$ x 3, $C_2^7.C_2^4$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^6.C_2^4.C_2$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^6.(C_4\times D_4)$ x 3, $C_2^7.C_2^4$ x 3, $C_2^6.C_2^4.C_2$ x 3, $C_2^4.C_2^6.C_2$ x 3, $C_2^4.C_2^6.C_2$ x 3, $C_2^6.C_2^4.C_2$ x 3, $C_2^4.C_2^5.C_2^2$ x 3, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^3.C_2^6.C_2^2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^7.C_2^3.C_2$ x 2, $C_2^4.C_2^6.C_2$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^6.C_2^3.C_2^2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^6.C_2^5$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^7.C_2^3.C_2$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^6.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^6$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^3.C_2^3$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^4.C_2$, $C_2^7.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^5.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2.C_2^3$, $C_2^5.C_2^4.C_2^2$, $C_2^8.C_2^3$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^3.C_2^3$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^6.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $(C_2^5\times C_4).C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^2$, $C_2^7.C_2^4$, $C_2^7.C_2^3.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^8.C_2^3$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^3.C_2^2$, $C_2^5.C_2^6$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^7.C_2.C_2^3$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^3.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^6$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^8.C_2^3$, $C_2^7.C_2^4$, $C_2^3.C_2^6.C_2^2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $(C_2^5\times C_4).C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^6$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^8.C_2^3$, $C_2^8.C_2^3$, $C_2^7.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^3.C_2^2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^4.C_2$, $C_2^5.C_2^3.C_2^3$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^5.C_2^6$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^3.C_2^3$, $C_2^7.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^5.C_2^6$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^7.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^4.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2^7.C_2^3.C_2$, $C_2.C_2^6.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2.C_2^6.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^3.C_2^2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^2$, $C_2^8.C_2^3$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.(C_4\times D_4)$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^4.C_2^6.C_2$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^3.C_2^2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^5.C_2^2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2^3.C_2^5.C_2^3$, $(C_2^5\times C_4).C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^7.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^3.C_2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^4.C_2^6.C_2$, $C_2^4.C_2^5.C_2^2$, $C_2^7.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^4.C_2^2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^2$, $C_2^7.C_2^4$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^4.C_2$, $C_2^6.C_2^3.C_2^2$, $C_2^5.C_2^4.C_2^2$, $C_2^5.C_2^6$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^5.C_2^3.C_2^3$, $C_2^7.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^4.C_2^2$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^5.C_2^3.C_2^3$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^6.C_2^5$, $C_2^5.C_2^6$, $C_2^5.C_2^4.C_2^2$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^6.C_2^5$, $C_2^5.C_2^6$

Order 1536: $C_2\times C_4^3:A_4$ x 4, $C_2^5.(C_2\times S_4)$ x 3, $C_2^5.(C_2\times S_4)$ x 3, $C_2^5.(C_2\times S_4)$ x 3, $C_2^6.S_4$ x 3, $(C_2^5\times C_4).A_4$ x 3, $C_2^7:A_4$ x 2, $C_2^7:C_{12}$ x 2, $C_2^5.(C_4\times A_4)$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^3.S_4$, $C_4^3.S_4$, $C_4^3.S_4$, $(C_2^2\times C_4^2).S_4$, $C_2^6:D_{12}$, $C_2^7.D_6$, $C_2^7.D_6$, $C_4^3:S_4$, $C_4^3:S_4$, $C_4^3.S_4$, $C_2^6:S_4$, $(C_2\times C_4)\wr C_3$, $C_2^6.C_2^3.C_3$, $C_2^5.C_2^4.C_3$, $C_2^7.A_4$, $(C_2^3\times C_4^2):C_{12}$

Order 1024: $C_2^6.C_2^4$, $C_2^7.C_2^3$, $C_2^7.C_2^3$, $C_2^6.C_2^4$, $C_2^7.C_2^3$

Order 512: $C_2^3.C_2^6$ x 3, $C_2^8.C_2$, $C_2^3\times C_4^3$

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

Order 128: $C_2^3.C_4^2$, $C_2^5\times C_4$

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

Order 32: $C_2^5$

Order 16: $C_2^4$ x 3

Order 8: $C_2^3$ x 2

Order 4: $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 49152: $C_2^8.C_2^3:S_4$ ($C_1$)

Order 24576: $C_2^4.Q_8^2:S_4$ ($C_2$) x 2, $C_2^8.C_2^4.C_6$ ($C_2$)

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

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

Order 4096: $C_2^7.C_2^5$ ($D_6$)

Order 2048: $C_2^6.C_2^5$ ($S_4$) x 2, $C_2^7.C_2^4$ ($S_4$) x 2, $C_2^7.C_2^4$ ($S_4$), $C_2^7.C_2^4$ ($S_4$), $C_2^7.C_2^4$ ($S_4$)

Order 1024: $C_2^6.C_2^4$ ($C_2\times S_4$) x 2, $C_2^7.C_2^3$ ($C_2\times S_4$) x 2, $C_2^7.C_2^3$ ($C_2\times S_4$), $C_2^7.C_2^3$ ($C_2\times S_4$), $C_2^6.C_2^4$ ($C_2\times S_4$)

Order 512: $C_2^3.C_2^6$ ($C_2^2:S_4$) x 4, $C_2^8.C_2$ ($C_2^2:S_4$) x 2, $C_2^3\times C_4^3$ ($C_2^2:S_4$)

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

Order 128: $C_2^3.C_4^2$ ($C_2^4:S_4$), $C_2^5\times C_4$ ($C_2^4:S_4$)

Order 64: $C_2^2.C_4^2$ ($C_2^5:S_4$) x 2, $C_2^4\times C_4$ ($C_2^5:S_4$) x 2, $C_2^6$ ($C_2^5:S_4$)

Order 32: $C_2^5$ ($C_2\times C_2^5.S_4$)

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

Order 8: $C_2^3$ ($C_4^2.C_2^4:S_4$) x 2, $C_2^3$ ($C_2^8.S_4$)

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

Order 2: $C_2$ ($C_2^7.C_2^4.D_6$)

Order 1: $C_1$ ($C_2^8.C_2^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 49152: $C_2^8.C_2^3:S_4$ ($C_1$)

Order 24576: $C_2^8.C_2^4.C_6$ ($C_2$), $C_2^4.Q_8^2:S_4$ ($C_2$)

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

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

Order 4096: $C_2^7.C_2^5$ ($D_6$)

Order 2048: $C_2^6.C_2^5$ ($S_4$), $C_2^7.C_2^4$ ($S_4$), $C_2^7.C_2^4$ ($S_4$), $C_2^7.C_2^4$ ($S_4$), $C_2^7.C_2^4$ ($S_4$)

Order 1024: $C_2^6.C_2^4$ ($C_2\times S_4$), $C_2^7.C_2^3$ ($C_2\times S_4$), $C_2^7.C_2^3$ ($C_2\times S_4$), $C_2^6.C_2^4$ ($C_2\times S_4$), $C_2^7.C_2^3$ ($C_2\times S_4$)

Order 512: $C_2^3.C_2^6$ ($C_2^2:S_4$) x 3, $C_2^8.C_2$ ($C_2^2:S_4$), $C_2^3\times C_4^3$ ($C_2^2:S_4$)

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

Order 128: $C_2^3.C_4^2$ ($C_2^4:S_4$), $C_2^5\times C_4$ ($C_2^4:S_4$)

Order 64: $C_2^4\times C_4$ ($C_2^5:S_4$) x 2, $C_2^2.C_4^2$ ($C_2^5:S_4$), $C_2^6$ ($C_2^5:S_4$)

Order 32: $C_2^5$ ($C_2\times C_2^5.S_4$)

Order 16: $C_2^4$ ($C_2^4.C_2^4.D_6$), $C_2^4$ ($C_2^4.C_2^4.D_6$), $C_2^4$ ($C_2^6.S_4.C_2$)

Order 8: $C_2^3$ ($C_4^2.C_2^4:S_4$), $C_2^3$ ($C_2^8.S_4$)

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

Order 2: $C_2$ ($C_2^7.C_2^4.D_6$)

Order 1: $C_1$ ($C_2^8.C_2^3:S_4$)

Series

Derived series

$C_2^8.C_2^3:S_4$

$\rhd$

$C_2^8.C_2^4.C_3$

$\rhd$

$C_2^7.C_2^5$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$C_2^8.C_2^3:S_4$

$\rhd$

$C_2^8.C_2^4.C_6$

$\rhd$

$C_2^8.C_2^4.C_3$

$\rhd$

$C_2^7.C_2^5$

$\rhd$

$C_2^6.C_2^4$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^8.C_2^3:S_4$

$\rhd$

$C_2^8.C_2^4.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

The $164 \times 164$ character table is not available for this group.

Rational character table

The $158 \times 158$ rational character table is not available for this group.