Abstract group 5160960.b: $C_2^7.S_8$

• Label: 5160960.b
• Order: \( 2^{14} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$C_2^7.S_8$
Order:	\(5160960\)\(\medspace = 2^{14} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^8.S_8$, of order \(10321920\)\(\medspace = 2^{15} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 8, $A_8$
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	20	24	30
Elements	1	17039	18368	430192	21504	766528	368640	994560	408576	949760	368640	172032	258048	215040	172032	5160960
Conjugacy classes	1	15	2	25	1	21	1	9	5	13	1	1	2	2	1	100
Divisions	1	15	2	25	1	21	1	9	5	13	1	1	2	2	1	100
Autjugacy classes	1	11	2	20	1	16	1	7	4	10	1	1	1	1	1	78

Dimension	1	7	8	14	20	21	28	35	42	48	56	64	70	90	112	120	140	160	168	210	224	252	280	315	336	420	448	560	630	672
Irr. complex chars.	2	2	2	2	2	2	6	6	1	2	6	2	3	1	6	2	12	1	2	4	4	4	10	4	2	2	4	2	1	1	100
Irr. rational chars.	2	2	2	2	2	2	6	6	1	2	6	2	3	1	6	2	12	1	2	4	4	4	10	4	2	2	4	2	1	1	100

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $16$ $\langle(1,16,2,15)(3,8,9,13,6)(4,7,10,14,5)(11,12), (1,10,8,15,13)(2,9,7,16,14)(3,11,5)(4,12,6)\rangle$
Transitive group:	16T1945				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:	$C_2^7$ . $S_8$	$(C_2^7.A_8)$ . $C_2$	$C_2$ . $(C_2^6.A_8.C_2)$		more information

Elements of the group are displayed as permutations of degree 16.

Homology

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

Subgroups

There are 5 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^6.A_8.C_2$
Commutator:	$G' \simeq$ $C_2^7.A_8$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^6.A_8.C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^7$	$G/\operatorname{Fit} \simeq$ $S_8$
Radical:	$R \simeq$ $C_2^7$	$G/R \simeq$ $S_8$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^6.A_8.C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2\wr D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 2240 (order at least 2304) are shown, as well as all normal subgroups of any index.

Order 5160960: $C_2^7.S_8$

Order 2580480: $C_2^7.A_8$

Order 645120: $C_2^7.S_7$

Order 322560: $C_2^6.S_7$ x 2, $C_2^7.A_7$

Order 184320: $C_2^7.C_2.A_6.C_2$

Order 172032: $C_2^4.C_2^6:\GL(3,2)$

Order 161280: $C_2^6:A_7$

Order 147456: $C_2^7.S_4\wr C_2$

Order 92160: $C_2^6.C_2^2.A_6$, $C_2^7.A_6.C_2$, $C_2^7.A_6.C_2$, $(C_2^5\times C_4).A_6.C_2$, $C_2^7.A_6.C_2$, $(C_2^5\times C_4).A_6.C_2$, $C_2\wr S_5.S_4$, $C_2^7.S_6$

Order 80640: $C_2\times S_8$ x 2

Order 73728: $C_2^6.S_4\wr C_2$ x 2, $C_2^6.S_4\wr C_2$ x 2, $C_2^6.A_4^2.C_2^3$, $C_2^7.\POPlus(4,3)$, $C_2^7.A_4^2:C_4$

Order 49152: $C_2\wr C_2^3.S_4$

Order 46080: $S_4\times C_2^4:S_5$ x 4, $C_2^6.S_6$ x 4, $C_2^6.S_6$ x 2, $C_2^6.A_6.C_2$, $C_2^7.A_6$, $(C_2^5\times C_4).A_6$, $C_2^6.S_6$, $C_2^7:A_6$, $C_2^5.(S_4\times A_5)$, $C_2\wr S_5\times A_4$, $A_4.C_2\wr S_5$

Order 43008: $C_2^7:\PGL(2,7)$

Order 40320: $S_8$ x 4, $C_2\times A_8$ x 2

Order 36864: $C_2^6.\POPlus(4,3)$ x 2, $C_2^6.A_4^2:C_4$ x 2, $C_2^6.A_4^2.C_2^2$, $C_2^6.A_4^2.C_2^2$, $C_2^4.C_2^6.S_3^2$, $C_2^6.A_4^2.C_2^2$, $C_2^3.(C_2^3\times A_4).A_4.C_2^2$, $Q_8^2.\POPlus(4,3)$, $C_2^7.A_4\wr C_2$

Order 30720: $C_2^7.C_2.S_5$, $C_2^5.(D_4\times S_5)$

Order 24576: $Q_8:C_2^2.(C_6\times D_4).C_2^4$, $C_2^7.Q_8:S_4$, $C_2^4.C_2^6:S_4$, $C_2\wr Q_8.A_4$, $C_2^4.C_2^6:S_4$

Order 23040: $C_2^5:S_6$ x 3, $C_2^6:A_6$ x 2, $(A_4\times C_2^4):S_5$ x 2, $S_4\times C_2^4:A_5$ x 2, $A_4\times C_2^4:S_5$ x 2, $C_2\wr S_5\times S_3$, $C_2\wr A_6$, $C_2^5.S_6$, $C_2^7:\GL(2,4)$

Order 21504: $C_2.C_2^6.\PSL(2,7)$ x 2, $C_2^4:C_2^3:\PSL(2,7)$ x 2, $C_2^7.F_8.C_3$, $C_2^7:\PSL(2,7)$, $C_2^7.\PSL(2,7)$

Order 20160: $A_8$ x 2

Order 18432: $Q_8^2.A_4\wr C_2$ x 2, $C_2^6.A_4^2.C_2$, $C_2^4.(C_2^4\times A_4).C_6$, $A_4^2.C_2^5.C_2^2$, $C_2^4.C_2^6.C_3.S_3$, $C_2^4.(C_2^4\times A_4).C_6$, $C_2^6.A_4^2.C_2$, $C_2^5.S_4^2$

Order 16384: $C_2^7.C_2\wr D_4$

Order 15360: $D_4\times C_2^4:S_5$ x 8, $C_2^7.S_5$ x 2, $C_2^7.S_5$ x 2, $C_2^7.S_5$, $C_2^7.S_5$, $(C_2^5\times C_4).S_5$, $C_2^7.S_5$, $(C_2^5\times C_4).S_5$, $C_2^7.C_2.A_5$, $C_4:C_2\wr S_5$, $C_2^5.(C_4\times S_5)$, $C_2^5.(D_4\times A_5)$, $C_2\wr F_5.S_4$, $C_2^7:S_5$

Order 12288: $C_2^6.(D_4\times S_4)$ x 2, $C_2^4.C_2^5:S_4$ x 2, $Q_8^2.C_2^3:S_4$ x 2, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^5.C_2^5.A_4$, $C_2^3:A_4.C_2^5.C_2^2$, $C_2^3:A_4.C_2^5.C_2^2$, $C_2^3:A_4.C_4:D_4.C_2^2$, $C_2^3:A_4.C_4:D_4.C_2^2$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.C_2^5.C_6.C_2^2$, $C_2^3:A_4.C_2^3.C_2^4$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^3:A_4.C_2^3.C_2^3.C_2$, $C_2^5.A_4.C_2.C_2^4$, $C_2^6.C_2^4.D_6$, $C_2^3:A_4.C_2^3.C_2^3.C_2$, $Q_8:C_2^2.(C_6\times D_4).C_2^3$, $C_2^4.C_2^5:S_4$, $C_2^4.C_2^6:A_4$

Order 11520: $S_3\times C_2^4:S_5$ x 4, $C_2\wr S_5\times C_3$, $(C_2^4\times C_6):S_5$, $C_2\wr A_5\times S_3$, $C_2^6:\GL(2,4)$, $C_2^5:A_6$

Order 10752: $C_2^6:\GL(3,2)$

Order 10080: $C_2\times S_7$

Order 9216: $C_2^4.S_4^2$ x 8, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $C_2^4.S_4^2$ x 2, $C_2\times S_4^2:D_4$ x 2, $C_2^4.C_2^6.C_3^2$, $A_4^2.C_2^4.C_2^2$, $C_2^2.C_2^6.S_3^2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^4.C_2^2$, $C_2^4.S_4^2$, $C_2^4.S_4^2$, $(A_4\times C_2^5).S_4$, $A_4\times C_2^5:S_4$, $C_2^6.D_6^2$, $C_2^7:\SOPlus(4,2)$, $S_4^2.C_2^4$, $C_2^5:A_4\times S_4$, $C_2^4.S_4^2$, $(C_2\times S_4^2).D_4$

Order 8192: $C_2^6.C_2\wr D_4$ x 2, $C_2^6.C_2\wr D_4$ x 2, $C_2^2\wr C_2^2.D_4$ x 2, $C_2\wr C_2^3.C_4$ x 2, $C_2^6.C_2^5.C_2^2$, $C_2^7.C_2\wr C_4$, $C_2^2\wr C_2^2.D_4$, $C_2^7.C_2\wr C_2^2$, $C_2^7.C_2\wr C_2^2$, $C_2^7.C_2\wr C_2^2$, $C_2\wr D_4.C_2^2$

Order 7680: $C_2^6:S_5$ x 15, $C_2^6:S_5$ x 8, $C_4\times C_2^4:S_5$ x 4, $C_2^4:F_5\times S_4$ x 4, $C_2^6:S_5$ x 4, $(C_2^4\times C_4):S_5$ x 4, $D_4\times C_2^4:A_5$ x 4, $C_2\wr S_5$ x 2, $C_2^7:A_5$ x 2, $(C_2^5\times C_4).A_5$, $C_2^7.A_5$, $C_2^6.S_5$, $C_2\wr A_5\times C_4$, $C_2^4.(D_{10}\times S_4)$, $A_4:C_2\wr F_5$, $A_4\times C_2\wr F_5$, $C_2^6.S_5$, $C_2^6.S_5$, $C_2^7:A_5$

Order 7168: $C_2^7.F_8$

Order 6144: $C_2^6.A_4.C_2^3$ x 2, $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_2^3:A_4.C_2^3.C_2^3$ x 2, $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_4^2:C_2^4.S_4$ x 2, $C_2^6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^7.(C_2\times S_4)$ x 2, $Q_8^2.C_2^3:A_4$ x 2, $C_2^6.(C_4\times S_4)$ x 2, $C_2^7.(C_2\times S_4)$ x 2, $C_2\wr C_2^3:C_3$ x 2, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_4^2:C_4$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.C_4.C_2^4$, $C_2^3:A_4.C_2^5.C_2$, $C_2^5.C_2\wr A_4$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^5.(D_4\times S_4)$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^6.A_4.C_2^3$, $C_2^5.A_4.C_2^4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^4.C_2^6.S_3$, $C_2^3:A_4.C_2^3.C_2^3$, $C_2^4.C_2^4.C_6.C_2^2$, $C_2^5.A_4.C_2.C_2^3$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.D_4^2$, $C_2^6.A_4.C_2^3$, $C_2^6.A_4.C_2^2.C_2$, $C_2^2.C_2^6.C_6.C_2^2$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2\wr C_2^2$, $Q_8:C_2^2.D_6.C_2^4$, $C_2^3:A_4.C_2^2.C_2^3.C_2$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.C_4^2:C_4$, $C_2^5.A_4.C_2^4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4:D_4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2.C_4^2:A_4.D_4.C_2$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_2^5.C_2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_2.C_2^3$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_4^2.C_2^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_2^3.C_2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3.C_2^5.C_6.C_2^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^6.C_2^4.C_6$, $C_2.C_4^2:A_4.D_4.C_2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_4^2:C_4$, $C_2^4.C_2^6.S_3$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.D_4:D_4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2^3.C_2^3$, $(C_2\times C_4^3):\GL(2,3)$, $C_2^7:\GL(2,3)$, $C_4^3.(C_2^2\times S_4)$, $C_2^5.(D_4\times S_4)$, $D_4^2.(C_2^2\times S_4)$, $D_4^3.D_6$, $C_2^6.\GL(2,\mathbb{Z}/4)$

Order 5760: $S_3\times C_2^4:A_5$ x 2, $(C_2^3\times C_6):S_5$ x 2, $C_3\times C_2^4:S_5$ x 2, $C_2^3\times S_6$, $C_2\times S_4\times S_5$, $C_2^5:\GL(2,4)$

Order 5376: $C_2\wr F_7$

Order 5120: $C_2\wr F_5\times D_4$

Order 5040: $S_7$ x 2, $C_2\times A_7$

Order 4608: $S_4^2:C_2^3$ x 18, $S_4^2:D_4$ x 10, $D_4\times S_4^2$ x 8, $C_2^3:S_4^2$ x 8, $C_2\times S_4^2:C_4$ x 6, $C_2^3.S_4^2$ x 4, $Q_8:S_4^2$ x 4, $C_2^3.S_4^2$ x 4, $C_2\wr A_4:S_4$ x 4, $C_2\wr A_4\times S_4$ x 4, $A_4\times C_2\wr S_4$ x 4, $C_2^6:\SOPlus(4,2)$ x 4, $C_2^3:(A_4^2:C_4)$ x 3, $A_4^2.C_2^2.C_2^3$ x 2, $A_4^2.C_2^3.C_2^2$ x 2, $A_4^2.C_2^3.C_2^2$ x 2, $C_2^6.\SOPlus(4,2)$ x 2, $C_2.C_2^6.C_3.D_6$ x 2, $A_4\times C_2^4:S_4$ x 2, $C_2^3:S_4^2$ x 2, $C_2^3:S_4^2$ x 2, $C_2^4:A_4\times S_4$ x 2, $C_2^3\times S_4^2$ x 2, $C_2^7:S_3^2$ x 2, $C_2^4.C_6^2.C_2^3$, $A_4^2.C_2^2\wr C_2$, $C_2^3.(Q_8\times A_4).C_6$, $C_2^4.D_6^2:C_2$, $C_2^2.(C_2^4\times A_4).C_6$, $A_4^2.C_2^2\wr C_2$, $C_2^4.A_4^2.C_2$, $C_2^5.D_6^2$, $(A_4\times C_2^4).S_4$, $(C_2^4\times S_4):A_4$, $C_2^5:A_4^2$, $A_4\times C_2^4.S_4$, $C_2^5:D_6^2$, $A_4^2:C_4\times C_2^3$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $A_4^2:C_2^2\times D_4$, $A_4\times \GL(2,\mathbb{Z}/4):C_2^2$, $C_2^7.S_3^2$, $C_2\wr S_4\times D_6$

Order 4096: $C_2^5.C_2\wr D_4$ x 4, $C_2^2\wr C_2^2.C_4$ x 4, $C_2^5.C_2\wr D_4$ x 4, $C_2^2.C_2^2\wr C_2^2$ x 3, $C_2^6.C_2\wr C_2^2$ x 3, $C_2^5.C_2^5.C_2^2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2^2$ x 2, $C_2^6.C_2\wr C_4$ x 2, $C_2^6.C_4^2:C_4$ x 2, $C_2^6.D_4^2$ x 2, $C_2^6.C_2\wr C_4$ x 2, $C_2^2.C_2^2\wr C_4$ x 2, $C_2^5.C_2^5.C_2^2$, $C_2^6.D_4^2$, $C_2^6.C_2^4.C_2^2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^7.C_2^5$, $C_2^4.C_2^4.C_2^4$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^6.C_2.C_2^4.C_2$, $C_2^5.C_2^4.C_2^3$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^5.C_2^5.C_2^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^2.C_2^2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2\wr D_4.C_2$, $C_2^6.C_2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^5.C_2^5:C_4$

Order 3840: $C_2\wr S_5$ x 21, $C_2^6:A_5$ x 7, $C_2^4:D_5\times S_4$ x 4, $C_2^5:S_5$ x 3, $A_4\times C_2^4:F_5$ x 2, $(A_4\times C_2^4):F_5$ x 2, $C_2^6.A_5$ x 2, $C_4\times C_2^4:A_5$ x 2, $C_2^5.S_5$ x 2, $C_2^6:D_{30}$, $A_4.C_2^4:C_5.C_2^2$, $C_2\wr F_5\times S_3$, $C_2\times C_2^4:D_5\times A_4$, $C_2\wr A_5$, $C_2^5.S_5$

Order 3072: $(D_4\times C_2^4):S_4$ x 17, $C_4^2.(D_4\times S_4)$ x 16, $C_2^7:S_4$ x 9, $D_4\times C_2\wr S_4$ x 8, $C_2^7:S_4$ x 5, $C_2^4.C_2^4.D_6$ x 4, $C_2^7.S_4$ x 4, $C_2^3\wr C_2:S_4$ x 4, $D_4^3:S_3$ x 4, $D_4\times C_2^4:S_4$ x 4, $C_2^4:D_4\times S_4$ x 3, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5.A_4.C_2^3$ x 2, $C_2^3:A_4.C_2^4.C_2$ x 2, $C_2^3:A_4.C_2^2\wr C_2$ x 2, $C_2^5.A_4.C_2^3$ x 2, $C_2^3:A_4.C_2^2\wr C_2$ x 2, $Q_8:C_2^2.C_2^3:D_6$ x 2, $C_2^3:A_4.C_2^4.C_2$ x 2, $C_2^3:A_4.C_2^4.C_2$ x 2, $C_2^3:A_4.C_2^3.C_2^2$ x 2, $C_2^6.S_4.C_2$ x 2, $C_2^3:A_4.C_2^3.C_2^2$ x 2, $C_2^3:A_4.C_4:D_4$ x 2, $C_2^5:Q_8:A_4$ x 2, $C_2^7:\SL(2,3)$ x 2, $C_4^3.(C_2\times S_4)$ x 2, $C_4^3:(C_2\times S_4)$ x 2, $Q_8^2:(C_2\times S_4)$ x 2, $C_2^7:S_4$ x 2, $C_2^7.S_4$ x 2, $C_2^2\wr A_4$ x 2, $C_2^6.(C_4\times A_4)$ x 2, $C_2^7:(C_2\times A_4)$ x 2, $C_2^3.(D_4\times S_4).C_2$ x 2, $C_2^7:S_4$ x 2, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times D_4\times A_4).C_2^4$ x 2, $C_2^5.(C_4\times S_4)$ x 2, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 2, $(D_4\times C_2^4):S_4$ x 2, $C_2^7:S_4$ x 2, $C_2^5.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $Q_8:C_2^2.(C_{12}.D_4)$, $C_2^3.(D_4\times S_4).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^5.A_4.C_2^2.C_2$, $C_2^5.(C_4\times S_4)$, $C_2.C_4^2:A_4.D_4$, $C_2^3.(D_4\times S_4).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^6.A_4.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.C_2^3$, $C_2^4.(C_2^3\times A_4).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_4\wr C_2$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3.(D_4\times S_4).C_2$, $Q_8:C_2^2.C_8:D_6$, $C_2^4.(C_2^4\times A_4)$, $C_4^2.(C_6\times D_4).C_2^2$, $C_2^6.A_4.C_2^2$, $C_2^5.A_4.C_4.C_2$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.C_2^3$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.D_4$, $C_2.C_4^2:A_4.D_4$, $(C_2^3\times A_4).C_2^5$, $C_2^5.(C_4\times S_4)$, $C_2^5.A_4.D_4$, $C_2^3:A_4.C_2^2\wr C_2$, $C_2^4.(C_2^4\times A_4)$, $C_2^3.(C_2^5.S_3).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.Q_8:C_2^2$, $C_2^3.(D_4\times S_4).C_2$, $C_2^5.A_4.C_2^3$, $C_2^3:A_4.(C_4\times D_4)$, $Q_8:C_2^2.D_8:S_3$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^6.C_2^4.C_3$, $C_2^3:A_4.C_2^2:D_4$, $C_2^3:A_4.C_2^4.C_2$, $Q_8.(C_2\times D_4\times A_4).C_2$, $Q_8:C_2^2.C_2^3:D_6$, $C_2^5.A_4.D_4$, $C_2^5.(C_4\times S_4)$, $C_2^5.A_4.C_2^3$, $C_2^5.A_4.C_2^2.C_2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.C_4.C_2$, $C_2^5.A_4.C_2^2.C_2$, $C_2^4.(C_6\times D_4).C_2^2$, $Q_8:C_2^2.D_{24}:C_2$, $C_2^5.C_2^3.A_4$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^2.C_2^5.C_{12}.C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:S_4.C_2^4$, $Q_8:C_2^2.D_6:D_4$, $C_2^3:A_4.C_4.C_2^3$, $C_2^3:A_4.C_4:D_4$, $C_2^3.(D_4\times S_4).C_2$, $Q_8:C_2^2.D_6:D_4$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.(C_4\times D_4)$, $Q_8:C_2^2.C_4:D_{12}$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3:A_4.C_4\wr C_2$, $C_2^5.A_4.D_4$, $Q_8:C_2^2.C_2^3:D_6$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3.(C_2^5.S_3).C_2$, $Q_8:C_2^2.C_4:D_{12}$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3:A_4.C_4.C_2^3$, $C_2^5.A_4.C_2^3$, $Q_8:C_2^2.Q_8:D_6$, $C_2^5.A_4.C_2^3$, $C_2^6.A_4.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.(C_4\times S_4)$, $C_2^3:A_4.C_4:D_4$, $C_2^6.A_4.C_2^2$, $C_2^6.A_4.C_2^2$, $C_2^2.C_2^5.C_6.C_2^2$, $C_2^3:A_4.C_4:D_4$, $Q_8:C_2^2.C_2^4:S_3$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3.(C_2^5.S_3).C_2$, $Q_8.C_2^3:D_6.C_2^2$, $C_2^5.A_4.D_4$, $C_2^3:A_4.C_4\wr C_2$, $C_2^3:A_4.C_2^2:D_4$, $C_4^3.A_4.C_2^2$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^6.A_4.C_2^2$, $(C_2\times C_4^3):S_4$, $(C_2\times C_4^3):S_4$, $D_4\times C_2^4.S_4$, $(C_2^5\times C_4):S_4$, $D_4\times C_2^5:A_4$, $C_4\times C_2^5:S_4$, $(D_4\times C_2^4):S_4$, $A_4\times D_4^2:C_2^2$, $C_2^7:S_4$, $C_2^4.D_4\times S_4$, $C_2^4.(D_4\times S_4)$, $C_2^5:\GL(2,\mathbb{Z}/4)$, $(C_2^4\times S_4):D_4$, $C_2\times C_4^2:\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^3:S_4$, $C_2^7.S_4$, $D_4^3:C_6$, $C_2\times C_4^3:S_4$, $C_2^8:D_6$

Order 2880: $C_2^2\times S_6$ x 14, $S_4\times S_5$ x 4, $C_2^3\times A_6$, $C_2\times S_4\times A_5$, $C_2\times A_4\times S_5$, $C_2\times A_4:S_5$, $C_2^4:\GL(2,4)$

Order 2688: $C_2^4:\PSL(2,7)$ x 4, $C_2^4:\GL(3,2)$ x 3, $C_2^4.F_8.C_3$ x 2, $C_2^6:F_7$ x 2, $C_2\wr C_7:C_3$, $C_2^4.\PSL(2,7)$

Order 2560: $D_4\times C_2^4:F_5$ x 8, $C_2^7:F_5$ x 2, $C_2^7:F_5$ x 2, $C_2\wr F_5\times C_4$, $C_4:C_2\wr F_5$, $C_2^7:D_{10}$

Order 2520: $A_7$

Order 2304: $S_4^2:C_2^2$ x 72, $C_2^2\times S_4^2$ x 30, $S_4\times \GL(2,\mathbb{Z}/4)$ x 16, $C_2^4:D_6^2$ x 13, $\GL(2,\mathbb{Z}/4):S_4$ x 10, $A_4^2:C_4\times C_2^2$ x 10, $S_4^2:C_4$ x 10, $C_4:S_4^2$ x 8, $D_4\times A_4\times S_4$ x 8, $S_3\times C_2\wr S_4$ x 8, $C_2\wr S_3^2$ x 8, $(C_2^2\times A_4^2):C_4$ x 6, $A_4^2.C_2^3.C_2$ x 4, $C_4:S_4^2$ x 4, $C_4\times S_4^2$ x 4, $D_4\times \PSOPlus(4,3)$ x 4, $A_4\times C_2^3:S_4$ x 4, $Q_8:A_4\times S_4$ x 4, $(C_2^3\times A_4):S_4$ x 4, $\POPlus(4,3):C_4$ x 4, $A_4^2.C_2^2:C_4$ x 2, $A_4^2.C_2^3.C_2$ x 2, $A_4^2.C_2^3.C_2$ x 2, $C_2^4.C_6^2.C_2^2$ x 2, $C_2^4:A_4^2$ x 2, $C_2^2.S_4^2$ x 2, $A_4^2:C_2^4$ x 2, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 2, $A_4^2:C_2^4$ x 2, $C_2\times A_4^2:D_4$ x 2, $C_2^6.S_3^2$ x 2, $C_2^6.S_3^2$ x 2, $(Q_8\times A_4):S_4$ x 2, $A_4\times Q_8:S_4$ x 2, $(C_2^3\times S_4):A_4$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4.A_4^2$, $C_2^2.S_4^2$, $C_2\times A_4^2:D_4$, $C_4\times A_4^2:C_2^2$, $A_4^2:C_2^4$, $C_2^4:D_6^2$, $C_2^2.S_4^2$, $A_4\times C_2^3:D_{12}$, $C_2^2.S_4^2$, $C_2\times C_4\times S_4\times A_4$, $C_2^6:C_6^2$, $(C_2^2\times A_4^2):C_4$, $C_2^6.S_3^2$, $C_2\wr A_4\times D_6$, $D_6\times Q_8:S_4$, $C_2\wr S_4\times C_6$, $S_3\times C_2^4:S_4$, $(C_2^3\times D_6):S_4$, $C_6:C_2\wr S_4$, $(C_2^3\times D_6):S_4$, $Q_8^2:S_3^2$, $C_2^2.S_4^2$

Order 128: $C_2^7$

Order 9: $C_3^2$

Order 7: $C_7$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 2240 (order at least 2304) are shown, as well as all normal subgroups of any index.

Order 5160960: $C_2^7.S_8$

Order 2580480: $C_2^7.A_8$

Order 645120: $C_2^7.S_7$

Order 322560: $C_2^7.A_7$, $C_2^6.S_7$

Order 184320: $C_2^7.C_2.A_6.C_2$

Order 172032: $C_2^4.C_2^6:\GL(3,2)$

Order 161280: $C_2^6:A_7$

Order 147456: $C_2^7.S_4\wr C_2$

Order 92160: $C_2^6.C_2^2.A_6$, $C_2^7.A_6.C_2$, $C_2^7.A_6.C_2$, $(C_2^5\times C_4).A_6.C_2$, $C_2^7.A_6.C_2$, $(C_2^5\times C_4).A_6.C_2$, $C_2\wr S_5.S_4$, $C_2^7.S_6$

Order 80640: $C_2\times S_8$

Order 73728: $C_2^6.A_4^2.C_2^3$, $C_2^6.S_4\wr C_2$, $C_2^6.S_4\wr C_2$, $C_2^7.\POPlus(4,3)$, $C_2^7.A_4^2:C_4$

Order 49152: $C_2\wr C_2^3.S_4$

Order 46080: $C_2^6.S_6$ x 3, $S_4\times C_2^4:S_5$ x 2, $C_2^6.A_6.C_2$, $C_2^7.A_6$, $(C_2^5\times C_4).A_6$, $C_2^6.S_6$, $C_2^7:A_6$, $C_2^6.S_6$, $C_2^5.(S_4\times A_5)$, $C_2\wr S_5\times A_4$, $A_4.C_2\wr S_5$

Order 43008: $C_2^7:\PGL(2,7)$

Order 40320: $C_2\times A_8$, $S_8$

Order 36864: $C_2^6.A_4^2.C_2^2$, $C_2^6.A_4^2.C_2^2$, $C_2^4.C_2^6.S_3^2$, $C_2^6.A_4^2.C_2^2$, $C_2^3.(C_2^3\times A_4).A_4.C_2^2$, $Q_8^2.\POPlus(4,3)$, $C_2^6.\POPlus(4,3)$, $C_2^6.A_4^2:C_4$, $C_2^7.A_4\wr C_2$

Order 30720: $C_2^7.C_2.S_5$, $C_2^5.(D_4\times S_5)$

Order 24576: $Q_8:C_2^2.(C_6\times D_4).C_2^4$, $C_2^7.Q_8:S_4$, $C_2^4.C_2^6:S_4$, $C_2\wr Q_8.A_4$, $C_2^4.C_2^6:S_4$

Order 23040: $C_2^6:A_6$ x 2, $C_2^5:S_6$ x 2, $(A_4\times C_2^4):S_5$ x 2, $C_2\wr S_5\times S_3$, $C_2\wr A_6$, $C_2^5.S_6$, $C_2^7:\GL(2,4)$, $S_4\times C_2^4:A_5$, $A_4\times C_2^4:S_5$

Order 21504: $C_2^7.F_8.C_3$, $C_2.C_2^6.\PSL(2,7)$, $C_2^4:C_2^3:\PSL(2,7)$, $C_2^7:\PSL(2,7)$, $C_2^7.\PSL(2,7)$

Order 20160: $A_8$

Order 18432: $C_2^6.A_4^2.C_2$, $C_2^4.(C_2^4\times A_4).C_6$, $A_4^2.C_2^5.C_2^2$, $C_2^4.C_2^6.C_3.S_3$, $C_2^4.(C_2^4\times A_4).C_6$, $C_2^6.A_4^2.C_2$, $Q_8^2.A_4\wr C_2$, $C_2^5.S_4^2$

Order 16384: $C_2^7.C_2\wr D_4$

Order 15360: $D_4\times C_2^4:S_5$ x 4, $C_2^7.S_5$ x 2, $C_2^7.S_5$ x 2, $C_2^7.S_5$, $C_2^7.S_5$, $(C_2^5\times C_4).S_5$, $C_2^7.S_5$, $(C_2^5\times C_4).S_5$, $C_2^7.C_2.A_5$, $C_4:C_2\wr S_5$, $C_2^5.(C_4\times S_5)$, $C_2^5.(D_4\times A_5)$, $C_2\wr F_5.S_4$, $C_2^7:S_5$

Order 12288: $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^5.C_2^5.A_4$, $C_2^3:A_4.C_2^5.C_2^2$, $C_2^3:A_4.C_2^5.C_2^2$, $C_2^3:A_4.C_4:D_4.C_2^2$, $C_2^3:A_4.C_4:D_4.C_2^2$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.C_2^5.C_6.C_2^2$, $C_2^3:A_4.C_2^3.C_2^4$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^3:A_4.C_2^3.C_2^3.C_2$, $C_2^5.A_4.C_2.C_2^4$, $C_2^6.C_2^4.D_6$, $C_2^3:A_4.C_2^3.C_2^3.C_2$, $Q_8:C_2^2.(C_6\times D_4).C_2^3$, $C_2^6.(D_4\times S_4)$, $C_2^4.C_2^5:S_4$, $Q_8^2.C_2^3:S_4$, $C_2^4.C_2^5:S_4$, $C_2^4.C_2^6:A_4$

Order 11520: $S_3\times C_2^4:S_5$ x 2, $C_2\wr S_5\times C_3$, $(C_2^4\times C_6):S_5$, $C_2\wr A_5\times S_3$, $C_2^6:\GL(2,4)$, $C_2^5:A_6$

Order 10752: $C_2^6:\GL(3,2)$

Order 10080: $C_2\times S_7$

Order 9216: $C_2^4.S_4^2$ x 4, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $C_2^4.S_4^2$ x 2, $C_2\times S_4^2:D_4$ x 2, $C_2^4.C_2^6.C_3^2$, $A_4^2.C_2^4.C_2^2$, $C_2^2.C_2^6.S_3^2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^4.C_2^2$, $C_2^4.S_4^2$, $C_2^4.S_4^2$, $(A_4\times C_2^5).S_4$, $A_4\times C_2^5:S_4$, $C_2^6.D_6^2$, $C_2^7:\SOPlus(4,2)$, $S_4^2.C_2^4$, $C_2^5:A_4\times S_4$, $C_2^4.S_4^2$, $(C_2\times S_4^2).D_4$

Order 8192: $C_2^6.C_2^5.C_2^2$, $C_2^6.C_2\wr D_4$, $C_2^6.C_2\wr D_4$, $C_2^7.C_2\wr C_4$, $C_2^2\wr C_2^2.D_4$, $C_2^2\wr C_2^2.D_4$, $C_2^7.C_2\wr C_2^2$, $C_2^7.C_2\wr C_2^2$, $C_2^7.C_2\wr C_2^2$, $C_2\wr C_2^3.C_4$, $C_2\wr D_4.C_2^2$

Order 7680: $C_2^6:S_5$ x 10, $C_2^6:S_5$ x 6, $C_2^6:S_5$ x 3, $C_4\times C_2^4:S_5$ x 2, $C_2^7:A_5$ x 2, $C_2^4:F_5\times S_4$ x 2, $(C_2^4\times C_4):S_5$ x 2, $D_4\times C_2^4:A_5$ x 2, $(C_2^5\times C_4).A_5$, $C_2^7.A_5$, $C_2\wr S_5$, $C_2^6.S_5$, $C_2\wr A_5\times C_4$, $C_2^4.(D_{10}\times S_4)$, $A_4:C_2\wr F_5$, $A_4\times C_2\wr F_5$, $C_2^6.S_5$, $C_2^6.S_5$, $C_2^7:A_5$

Order 7168: $C_2^7.F_8$

Order 6144: $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_2^3:A_4.C_2^3.C_2^3$ x 2, $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_4^2:C_4$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.C_4.C_2^4$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.C_2^5.C_2$, $C_2^5.C_2\wr A_4$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^5.(D_4\times S_4)$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^6.A_4.C_2^3$, $C_2^5.A_4.C_2^4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^4.C_2^6.S_3$, $C_2^3:A_4.C_2^3.C_2^3$, $C_2^4.C_2^4.C_6.C_2^2$, $C_2^5.A_4.C_2.C_2^3$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.D_4^2$, $C_2^6.A_4.C_2^3$, $C_2^6.A_4.C_2^2.C_2$, $C_2^2.C_2^6.C_6.C_2^2$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2\wr C_2^2$, $Q_8:C_2^2.D_6.C_2^4$, $C_2^3:A_4.C_2^2.C_2^3.C_2$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.C_4^2:C_4$, $C_2^5.A_4.C_2^4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4:D_4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2.C_4^2:A_4.D_4.C_2$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_2^5.C_2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_2.C_2^3$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_4^2.C_2^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_2^3.C_2$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3.C_2^5.C_6.C_2^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^6.C_2^4.C_6$, $C_2.C_4^2:A_4.D_4.C_2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_4^2:C_4$, $C_2^4.C_2^6.S_3$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.D_4:D_4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2^3.C_2^3$, $(C_2\times C_4^3):\GL(2,3)$, $C_4^2:C_2^4.S_4$, $C_2^7:\GL(2,3)$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^7.(C_2\times S_4)$, $Q_8^2.C_2^3:A_4$, $C_4^3.(C_2^2\times S_4)$, $C_2^6.(C_4\times S_4)$, $C_2^7.(C_2\times S_4)$, $C_2\wr C_2^3:C_3$, $C_2^5.(D_4\times S_4)$, $D_4^2.(C_2^2\times S_4)$, $D_4^3.D_6$, $C_2^6.\GL(2,\mathbb{Z}/4)$

Order 5760: $(C_2^3\times C_6):S_5$ x 2, $C_2^3\times S_6$, $C_2\times S_4\times S_5$, $S_3\times C_2^4:A_5$, $C_2^5:\GL(2,4)$, $C_3\times C_2^4:S_5$

Order 5376: $C_2\wr F_7$

Order 5120: $C_2\wr F_5\times D_4$

Order 5040: $S_7$, $C_2\times A_7$

Order 4608: $S_4^2:C_2^3$ x 12, $C_2\times S_4^2:C_4$ x 6, $S_4^2:D_4$ x 5, $D_4\times S_4^2$ x 4, $C_2^3:S_4^2$ x 4, $C_2\wr A_4:S_4$ x 4, $C_2^3:(A_4^2:C_4)$ x 3, $A_4^2.C_2^2.C_2^3$ x 2, $A_4^2.C_2^3.C_2^2$ x 2, $A_4^2.C_2^3.C_2^2$ x 2, $C_2^6.\SOPlus(4,2)$ x 2, $C_2.C_2^6.C_3.D_6$ x 2, $A_4\times C_2^4:S_4$ x 2, $C_2^3:S_4^2$ x 2, $C_2^3:S_4^2$ x 2, $C_2^4:A_4\times S_4$ x 2, $C_2^3\times S_4^2$ x 2, $C_2^7:S_3^2$ x 2, $C_2^3.S_4^2$ x 2, $Q_8:S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $C_2\wr A_4\times S_4$ x 2, $A_4\times C_2\wr S_4$ x 2, $C_2^6:\SOPlus(4,2)$ x 2, $C_2^4.C_6^2.C_2^3$, $A_4^2.C_2^2\wr C_2$, $C_2^3.(Q_8\times A_4).C_6$, $C_2^4.D_6^2:C_2$, $C_2^2.(C_2^4\times A_4).C_6$, $A_4^2.C_2^2\wr C_2$, $C_2^4.A_4^2.C_2$, $C_2^5.D_6^2$, $(A_4\times C_2^4).S_4$, $(C_2^4\times S_4):A_4$, $C_2^5:A_4^2$, $A_4\times C_2^4.S_4$, $C_2^5:D_6^2$, $A_4^2:C_4\times C_2^3$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $A_4^2:C_2^2\times D_4$, $A_4\times \GL(2,\mathbb{Z}/4):C_2^2$, $C_2^7.S_3^2$, $C_2\wr S_4\times D_6$

Order 4096: $C_2^2.C_2^2\wr C_2^2$ x 2, $C_2^5.C_2\wr D_4$ x 2, $C_2^6.C_2\wr C_2^2$ x 2, $C_2^2\wr C_2^2.C_4$ x 2, $C_2^5.C_2\wr D_4$ x 2, $C_2^5.C_2^5.C_2^2$, $C_2^6.D_4^2$, $C_2^6.C_2^4.C_2^2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^5.C_2^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^7.C_2^5$, $C_2^4.C_2^4.C_2^4$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^6.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^6.C_2.C_2^4.C_2$, $C_2^5.C_2^4.C_2^3$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^5.C_2^5.C_2^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.D_4^2$, $C_2^7.C_2^4.C_2$, $C_2^5.C_2^5.C_2^2$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^2.C_2^2\wr C_2^2$, $C_2^6.C_2\wr C_4$, $C_2^6.C_4^2:C_4$, $C_2^6.C_2\wr C_2^2$, $C_2^6.D_4^2$, $C_2^6.C_2\wr C_4$, $C_2\wr D_4.C_2$, $C_2^2.C_2^2\wr C_4$, $C_2^6.C_2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^5.C_2^5:C_4$

Order 3840: $C_2\wr S_5$ x 14, $C_2^6:A_5$ x 6, $C_2^4:D_5\times S_4$ x 2, $(A_4\times C_2^4):F_5$ x 2, $C_2^6.A_5$ x 2, $C_2^5.S_5$ x 2, $C_2^5:S_5$ x 2, $A_4\times C_2^4:F_5$, $C_2^6:D_{30}$, $A_4.C_2^4:C_5.C_2^2$, $C_2\wr F_5\times S_3$, $C_2\times C_2^4:D_5\times A_4$, $C_4\times C_2^4:A_5$, $C_2\wr A_5$, $C_2^5.S_5$

Order 3072: $(D_4\times C_2^4):S_4$ x 13, $C_4^2.(D_4\times S_4)$ x 8, $C_2^7:S_4$ x 7, $D_4\times C_2\wr S_4$ x 4, $D_4\times C_2^4:S_4$ x 4, $C_2^7:S_4$ x 3, $C_2^4:D_4\times S_4$ x 3, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5.A_4.C_2^3$ x 2, $C_2^3:A_4.C_2^4.C_2$ x 2, $C_2^3:A_4.C_2^2\wr C_2$ x 2, $C_2^5.A_4.C_2^3$ x 2, $C_2^3:A_4.C_2^2\wr C_2$ x 2, $Q_8:C_2^2.C_2^3:D_6$ x 2, $C_2^3:A_4.C_2^4.C_2$ x 2, $C_2^3:A_4.C_2^4.C_2$ x 2, $C_2^3:A_4.C_2^3.C_2^2$ x 2, $C_2^4.C_2^4.D_6$ x 2, $C_2^3:A_4.C_2^3.C_2^2$ x 2, $C_2^3:A_4.C_4:D_4$ x 2, $C_2^7:\SL(2,3)$ x 2, $C_2^7.S_4$ x 2, $C_2^3\wr C_2:S_4$ x 2, $D_4^3:S_3$ x 2, $C_2^3.(D_4\times S_4).C_2$ x 2, $C_2^7:S_4$ x 2, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times D_4\times A_4).C_2^4$ x 2, $C_2^5.(C_4\times S_4)$ x 2, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 2, $(D_4\times C_2^4):S_4$ x 2, $C_2^7:S_4$ x 2, $C_2^5.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $Q_8:C_2^2.(C_{12}.D_4)$, $C_2^3.(D_4\times S_4).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^5.A_4.C_2^2.C_2$, $C_2^5.(C_4\times S_4)$, $C_2.C_4^2:A_4.D_4$, $C_2^3.(D_4\times S_4).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^6.A_4.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.C_2^3$, $C_2^4.(C_2^3\times A_4).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_4\wr C_2$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3.(D_4\times S_4).C_2$, $Q_8:C_2^2.C_8:D_6$, $C_2^4.(C_2^4\times A_4)$, $C_4^2.(C_6\times D_4).C_2^2$, $C_2^6.A_4.C_2^2$, $C_2^5.A_4.C_4.C_2$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.C_2^3$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.D_4$, $C_2.C_4^2:A_4.D_4$, $(C_2^3\times A_4).C_2^5$, $C_2^5.(C_4\times S_4)$, $C_2^5.A_4.D_4$, $C_2^3:A_4.C_2^2\wr C_2$, $C_2^4.(C_2^4\times A_4)$, $C_2^3.(C_2^5.S_3).C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.Q_8:C_2^2$, $C_2^3.(D_4\times S_4).C_2$, $C_2^5.A_4.C_2^3$, $C_2^3:A_4.(C_4\times D_4)$, $Q_8:C_2^2.D_8:S_3$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^6.C_2^4.C_3$, $C_2^3:A_4.C_2^2:D_4$, $C_2^3:A_4.C_2^4.C_2$, $Q_8.(C_2\times D_4\times A_4).C_2$, $Q_8:C_2^2.C_2^3:D_6$, $C_2^5.A_4.D_4$, $C_2^5.(C_4\times S_4)$, $C_2^5.A_4.C_2^3$, $C_2^5.A_4.C_2^2.C_2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.A_4.C_4.C_2$, $C_2^5.A_4.C_2^2.C_2$, $C_2^4.(C_6\times D_4).C_2^2$, $Q_8:C_2^2.D_{24}:C_2$, $C_2^5.C_2^3.A_4$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^2.C_2^5.C_{12}.C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:S_4.C_2^4$, $Q_8:C_2^2.D_6:D_4$, $C_2^3:A_4.C_4.C_2^3$, $C_2^3:A_4.C_4:D_4$, $C_2^3.(D_4\times S_4).C_2$, $Q_8:C_2^2.D_6:D_4$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.(C_4\times D_4)$, $Q_8:C_2^2.C_4:D_{12}$, $C_2^3:A_4.(C_4\times D_4)$, $C_2^3:A_4.C_4\wr C_2$, $C_2^5.A_4.D_4$, $Q_8:C_2^2.C_2^3:D_6$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3.(C_2^5.S_3).C_2$, $C_2^6.S_4.C_2$, $Q_8:C_2^2.C_4:D_{12}$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^3:A_4.C_4.C_2^3$, $C_2^5.A_4.C_2^3$, $Q_8:C_2^2.Q_8:D_6$, $C_2^5.A_4.C_2^3$, $C_2^6.A_4.C_2^2$, $C_2^3:A_4.C_2^2:D_4$, $C_2^3:A_4.C_2^2:D_4$, $C_2^5.(C_4\times S_4)$, $C_2^3:A_4.C_4:D_4$, $C_2^6.A_4.C_2^2$, $C_2^6.A_4.C_2^2$, $C_2^2.C_2^5.C_6.C_2^2$, $C_2^3:A_4.C_4:D_4$, $Q_8:C_2^2.C_2^4:S_3$, $C_2^3:A_4.C_2^4.C_2$, $C_2^3.(C_2^5.S_3).C_2$, $Q_8.C_2^3:D_6.C_2^2$, $C_2^5.A_4.D_4$, $C_2^3:A_4.C_4\wr C_2$, $C_2^3:A_4.C_2^2:D_4$, $C_4^3.A_4.C_2^2$, $C_2^3:A_4.C_2^3.C_2^2$, $C_2^6.A_4.C_2^2$, $C_2^5:Q_8:A_4$, $(C_2\times C_4^3):S_4$, $C_4^3.(C_2\times S_4)$, $C_4^3:(C_2\times S_4)$, $Q_8^2:(C_2\times S_4)$, $C_2^7:S_4$, $C_2^7.S_4$, $(C_2\times C_4^3):S_4$, $C_2^2\wr A_4$, $C_2^6.(C_4\times A_4)$, $C_2^7:(C_2\times A_4)$, $D_4\times C_2^4.S_4$, $(C_2^5\times C_4):S_4$, $D_4\times C_2^5:A_4$, $C_4\times C_2^5:S_4$, $(D_4\times C_2^4):S_4$, $A_4\times D_4^2:C_2^2$, $C_2^7:S_4$, $C_2^4.D_4\times S_4$, $C_2^4.(D_4\times S_4)$, $C_2^5:\GL(2,\mathbb{Z}/4)$, $(C_2^4\times S_4):D_4$, $C_2\times C_4^2:\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^3:S_4$, $C_2^7.S_4$, $D_4^3:C_6$, $C_2\times C_4^3:S_4$, $C_2^8:D_6$

Order 2880: $C_2^2\times S_6$ x 7, $S_4\times S_5$ x 2, $C_2^3\times A_6$, $C_2\times S_4\times A_5$, $C_2\times A_4\times S_5$, $C_2\times A_4:S_5$, $C_2^4:\GL(2,4)$

Order 2688: $C_2^4:\PSL(2,7)$ x 2, $C_2^4:\GL(3,2)$ x 2, $C_2^4.F_8.C_3$, $C_2\wr C_7:C_3$, $C_2^4.\PSL(2,7)$, $C_2^6:F_7$

Order 2560: $D_4\times C_2^4:F_5$ x 4, $C_2^7:F_5$ x 2, $C_2^7:F_5$ x 2, $C_2\wr F_5\times C_4$, $C_4:C_2\wr F_5$, $C_2^7:D_{10}$

Order 2520: $A_7$

Order 2304: $S_4^2:C_2^2$ x 33, $C_2^2\times S_4^2$ x 22, $C_2^4:D_6^2$ x 9, $S_4\times \GL(2,\mathbb{Z}/4)$ x 8, $A_4^2:C_4\times C_2^2$ x 8, $S_4^2:C_4$ x 8, $\GL(2,\mathbb{Z}/4):S_4$ x 7, $C_2\wr S_3^2$ x 5, $C_4:S_4^2$ x 4, $D_4\times A_4\times S_4$ x 4, $(C_2^3\times A_4):S_4$ x 4, $\POPlus(4,3):C_4$ x 4, $S_3\times C_2\wr S_4$ x 4, $(C_2^2\times A_4^2):C_4$ x 4, $C_4:S_4^2$ x 3, $C_4\times S_4^2$ x 3, $A_4^2.C_2^3.C_2$ x 2, $A_4^2.C_2^2:C_4$ x 2, $C_2^4.C_6^2.C_2^2$ x 2, $C_2^4:A_4^2$ x 2, $C_2^2.S_4^2$ x 2, $A_4^2:C_2^4$ x 2, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 2, $A_4^2:C_2^4$ x 2, $D_4\times \PSOPlus(4,3)$ x 2, $C_2\times A_4^2:D_4$ x 2, $C_2^6.S_3^2$ x 2, $C_2^6.S_3^2$ x 2, $(Q_8\times A_4):S_4$ x 2, $A_4\times C_2^3:S_4$ x 2, $Q_8:A_4\times S_4$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4.A_4^2$, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^3.C_2$, $C_2^2.S_4^2$, $C_2\times A_4^2:D_4$, $C_4\times A_4^2:C_2^2$, $A_4^2:C_2^4$, $C_2^4:D_6^2$, $C_2^2.S_4^2$, $A_4\times C_2^3:D_{12}$, $C_2^2.S_4^2$, $C_2\times C_4\times S_4\times A_4$, $C_2^6:C_6^2$, $(C_2^2\times A_4^2):C_4$, $C_2^6.S_3^2$, $A_4\times Q_8:S_4$, $(C_2^3\times S_4):A_4$, $C_2\wr A_4\times D_6$, $D_6\times Q_8:S_4$, $C_2\wr S_4\times C_6$, $S_3\times C_2^4:S_4$, $(C_2^3\times D_6):S_4$, $C_6:C_2\wr S_4$, $(C_2^3\times D_6):S_4$, $Q_8^2:S_3^2$, $C_2^2.S_4^2$

Order 128: $C_2^7$

Order 9: $C_3^2$

Order 7: $C_7$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5160960: $C_2^7.S_8$ ($C_1$)

Order 2580480: $C_2^7.A_8$ ($C_2$)

Order 128: $C_2^7$ ($S_8$)

Order 2: $C_2$ ($C_2^6.A_8.C_2$)

Order 1: $C_1$ ($C_2^7.S_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5160960: $C_2^7.S_8$ ($C_1$)

Order 2580480: $C_2^7.A_8$ ($C_2$)

Order 128: $C_2^7$ ($S_8$)

Order 2: $C_2$ ($C_2^6.A_8.C_2$)

Order 1: $C_1$ ($C_2^7.S_8$)

Series

Derived series	$C_2^7.S_8$	$\rhd$	$C_2^7.A_8$
Chief series	$C_2^7.S_8$	$\rhd$	$C_2^7.A_8$	$\rhd$	$C_2^7$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2^7.S_8$	$\rhd$	$C_2^7.A_8$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 3 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

See the $100 \times 100$ rational character table. Alternatively, you may search for characters of this group with desired properties.