Abstract group 46448640.a: $C_2^8.A_9$

• Label: 46448640.a
• Order: \( 2^{14} \cdot 3^{4} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{4} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $18$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_2^8.A_9$
Order:	\(46448640\)\(\medspace = 2^{14} \cdot 3^{4} \cdot 5 \cdot 7 \)
Exponent:	\(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^8.S_9$, of order \(92897280\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 8, $A_9$
Derived length:	$0$

This group is nonabelian and perfect (hence nonsolvable).

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	14	15	20	24	30
Elements	1	39567	197792	1992816	48384	3698016	1658880	3144960	10321920	1306368	9192960	4976640	1548288	1741824	1935360	4644864	46448640
Conjugacy classes	1	8	3	15	1	16	1	6	2	5	11	3	2	2	2	6	84
Divisions	1	8	3	15	1	16	1	6	2	5	11	2	1	2	2	3	79
Autjugacy classes	1	8	3	15	1	16	1	6	1	5	11	2	1	2	2	3	78

Dimension	1	8	9	21	27	28	35	36	42	48	56	63	84	105	120	126	162	168	180	189	216	252	315	378	405	420	504	540	576	630	720	756	810	840	1008	1260	1344	1512	1680	1890	2268	2688
Irr. complex chars.	1	1	1	2	1	1	2	2	1	1	1	1	3	1	1	3	1	2	1	4	3	2	1	2	2	4	7	2	1	3	1	7	0	4	1	3	3	3	1	2	1	0	84
Irr. rational chars.	1	1	1	0	1	1	2	2	2	1	1	1	3	1	1	3	1	2	1	2	3	2	1	3	0	4	7	2	1	3	1	5	1	4	1	3	1	4	1	2	1	1	79

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	9	9
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $18$ $\langle(1,8,6,11,4,17,14,10,15)(2,7,5,12,3,18,13,9,16), (1,9,11,3,5,15,14,2,10,12,4,6,16,13)(7,8)\rangle$
Transitive group:	18T963				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:	$C_2^8$ . $A_9$				more information

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 3 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^8.A_9$
Commutator:	$G' \simeq$ $C_2^8.A_9$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^8.A_9$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8$	$G/\operatorname{Fit} \simeq$ $A_9$
Radical:	$R \simeq$ $C_2^8$	$G/R \simeq$ $A_9$
Socle:	$\operatorname{soc} \simeq$ $C_2^8$	$G/\operatorname{soc} \simeq$ $A_9$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2\wr D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
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 13440 (order at least 3456) are shown, as well as all normal subgroups of any index.

Order 46448640: $C_2^8.A_9$

Order 5160960: $C_2^8.A_8$

Order 2580480: $C_2^7.A_8$

Order 1290240: $C_2^8.S_7$

Order 645120: $C_2^8.A_7$, $C_2^6.A_7:C_4$, $C_2^7.S_7$

Order 552960: $C_2\wr A_6.S_4$

Order 387072: $C_2^8:{}^2G(2,3)$ x 2

Order 368640: $C_2^8.A_4:S_5$

Order 344064: $C_2\wr C_2^3.\PSL(2,7)$ x 2

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

Order 276480: $(A_4\times C_2^5).S_6$, $(A_4\times C_2^5).S_6$, $A_4\times C_2\wr A_6$

Order 184320: $C_2^8.A_6.C_2$, $C_2^7.A_4:S_5$, $C_2^7.A_4:S_5$, $C_2^8.(A_4\times A_5)$

Order 181440: $A_9$

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

Order 165888: $A_4:S_4^2.S_4$

Order 161280: $C_2^6:A_7$

Order 147456: $C_2^8.\POPlus(4,3)$

Order 138240: $(C_2^5\times C_6).S_6$, $A_4\times C_2^5:A_6$

Order 129024: $C_2^8:\SL(2,8)$ x 2

Order 122880: $C_2^7.(D_4\times S_5)$

Order 92160: $C_2^7.A_6.C_2$, $C_2^7.A_6.C_2$, $C_2^7.A_6.C_2$, $C_2^8.A_6$, $C_2^7.A_6.C_2$, $A_4.C_2^6.S_5$, $C_2^7.A_6.C_2$, $C_2^7.(A_4\times A_5)$, $C_2\wr A_5.S_4$, $C_2^7.S_6$

Order 82944: $C_2^6.C_3^3.C_2^4.C_3$

Order 73728: $C_2^7.\POPlus(4,3)$ x 2, $C_2^8.A_4\wr C_2$ x 2, $C_2^4.C_2^6.C_3.D_6.C_2$, $C_2^7.\POPlus(4,3)$, $C_2^7.\POPlus(4,3)$

Order 69120: $(C_2^4\times C_6):S_6$, $(C_2^4\times C_6).S_6$, $C_3\times C_2\wr A_6$

Order 61440: $C_2^7.C_2^2.S_5$, $C_2^7.C_4:S_5$, $C_2^7.(C_4\times S_5)$, $D_4.C_2^6:S_5$, $C_2^7.(C_2^2\times S_5)$, $C_2^6:D_4.S_5$, $C_2^7.(D_4\times A_5)$, $C_2^8.A_4:F_5$

Order 55296: $C_2^8:\PU(3,2)$, $A_4^3.C_2^2\wr C_2$

Order 49152: $C_2\wr C_2^3.S_4$ x 2, $C_2\wr C_2^3.S_4$

Order 46080: $A_4.C_2\wr S_5$ x 7, $C_2^7:A_6$ x 2, $C_2^6.S_6$ x 2, $C_2^6.A_6.C_2$, $C_2^7.A_6$, $A_4.C_2^6.A_5$, $C_2^6.A_6.C_2$, $C_2^6.S_6$, $C_2^6.S_6$, $(C_2^4\times \SL(2,3)):S_5$, $(A_4\times C_2^5).S_5$, $(A_4\times C_2^5):S_5$, $C_2^8.\GL(2,4)$

Order 43008: $C_2^7.F_8:C_6$ x 2, $C_2^8.\GL(3,2)$, $C_2^8.\PSL(2,7)$

Order 41472: $A_4^3:S_4$ x 2, $A_4^3.S_4$

Order 40320: $C_2\times A_8$

Order 36864: $C_2^7.A_4\wr C_2$ x 2, $C_2^7.A_4\wr C_2$ x 2, $C_2.C_2^6.C_6^2.C_2^3$, $C_2^6.A_4^2.C_2^2$, $C_2^4.C_2^6.C_3.D_6$, $C_2^4.C_2^6.C_3.D_6$, $C_2^4.C_2^6.C_3.D_6$, $C_2^6.A_4^2.C_2^2$, $Q_8^2.\POPlus(4,3)$, $C_2^6.\POPlus(4,3)$, $C_2^6.S_4^2$

Order 34560: $C_3\times C_2^5:A_6$

Order 30720: $C_2^5.(D_4\times S_5)$ x 2, $C_2^7.(C_2\times S_5)$ x 2, $C_2^7.C_2.S_5$, $C_2^8.S_5$, $C_2^6.C_2^2.S_5$, $C_2^5.C_2^3.S_5$, $C_2^8.S_5$, $C_2^2.C_2^6.S_5$, $C_2^5.(D_4\times S_5)$, $C_2^6.(D_4\times A_5)$, $C_4.C_2^6:S_5$, $D_4:C_2^5.S_5$, $C_2^5.(D_4\times S_5)$, $(D_4\times C_2^5).S_5$, $C_2^5.(C_2^4\times A_5)$, $C_2^7.(C_4\times A_5)$, $C_2^6:D_{10}.S_4$, $C_2^6:D_{10}.S_4$, $C_2^7.(A_4\times D_{10})$

Order 27648: $(C_2\times A_4^3).D_4$ x 2, $C_2\times A_4^3:D_4$ x 2, $C_2^6.C_3^3.C_2^4$, $A_4^2.(C_2\times \GL(2,\mathbb{Z}/4))$, $A_4^3:C_2^2:C_4$

Order 24576: $C_2^4.C_2^6.D_6.C_2$ x 2, $C_2^2\wr C_2^2.S_4$ x 2, $C_2^7.C_2^3:S_4$ x 2, $C_4^2:C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.C_2^6:S_4$ x 2, $C_2^7.C_2^3:S_4$ x 2, $C_2\wr C_2^3.A_4$ x 2, $C_2^4.(C_2^4\times A_4).C_2^3$, $C_2^7.Q_8:S_4$, $C_2^4.C_2^6:S_4$, $C_2\wr Q_8.A_4$

Order 23040: $(A_4\times C_2^4):S_5$ x 6, $C_2^7:\GL(2,4)$ x 4, $C_2\wr A_6$ x 2, $C_2^6:A_6$, $C_2^5.S_6$, $C_2^5:S_6$, $C_6:C_2\wr S_5$, $C_2\wr A_4:S_5$, $C_2^4:A_5\times \SL(2,3)$, $C_3:C_2\wr S_5$, $C_2^7:\GL(2,4)$

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

Order 20736: $A_4^3:A_4$ x 3, $A_4^3.A_4$

Order 20480: $C_2^7.(D_4\times F_5)$

Order 20160: $C_2^2:S_7$, $A_8$

Order 18432: $Q_8^2.A_4\wr C_2$ x 2, $C_2.C_2^6.D_6:D_6$, $C_2^2.C_2^6.C_3:S_4$, $Q_8.A_4^2.D_4.C_2$, $C_2^3.A_4^2.C_4.C_2^2$, $(C_2^4\times A_4).A_4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $C_2^3:A_4.(D_4\times A_4).C_2$, $C_2^5.(C_6\times S_4).C_2^2$, $C_2^6.A_4^2.C_2$, $C_2^5.(C_6\times A_4).C_2^3$, $C_2^6.A_4^2.C_2$, $C_2.C_2^6.D_6:D_6$, $C_2.C_2^6.D_6:D_6$, $C_2.C_2^6.D_6:D_6$, $C_2^6.A_4^2.C_2$, $C_2.C_2^6.D_6:D_6$, $C_2^5.A_4.C_6.C_2^3$, $C_2.C_2^6.D_6:D_6$, $C_2^4.C_2^6.C_3.C_6$, $C_2.C_2^6.D_6:D_6$, $C_2^8.\PSU(3,2)$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2\wr A_4.S_4$, $A_4\times C_2\wr S_4$, $C_2\wr A_4.S_4$

Order 17280: $C_2\times A_4:S_6$

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

Order 15360: $C_2^7.S_5$ x 10, $D_4\times C_2^4:S_5$ x 4, $C_2^5.(C_4\times S_5)$ x 3, $C_2^5.(D_4\times A_5)$ x 3, $C_2^7.S_5$ x 2, $C_4:C_2\wr S_5$ x 2, $C_2^7.S_5$ x 2, $C_2^6:(C_4\times A_5)$ x 2, $C_2^7.S_5$, $C_2^7.S_5$, $C_2^7.S_5$, $C_2^8.A_5$, $C_2^7.S_5$, $C_2^6.C_{15}.C_2^3.C_2$, $C_2^8.A_5$, $(C_2^5\times C_4).S_5$, $C_2^7.S_5$, $(C_2^5\times C_4).S_5$, $(Q_8\times C_2^4):S_5$, $C_2^5.(C_4\times S_5)$, $(D_4\times C_2^4):S_5$, $C_4.C_2\wr S_5$, $D_4:C_2\wr A_5$, $C_2^7.(D_5\times A_4)$, $C_2^7.(D_5\times A_4)$, $C_2^7.(C_{10}\times A_4)$, $C_2^7:S_5$, $C_2^7.S_5$

Order 14336: $C_2^8.F_8$ x 2

Order 13824: $A_4^3:D_4$ x 4, $A_4^3.C_2^3$ x 3, $C_2^2:A_4^2.S_4$ x 2, $C_2\times A_4^3:C_4$ x 2, $A_4^2.A_4.C_2^3$, $C_2^2:A_4^2.S_4$, $(C_2^2\times A_4^2):D_{12}$, $A_4^2:\GL(2,\mathbb{Z}/4)$, $A_4^3:C_2^3$, $A_4^3:(C_2\times C_4)$, $A_4^2.\GL(2,\mathbb{Z}/4)$

Order 12288: $C_2^6.C_2^4.D_6$ x 2, $C_2^4.C_2^6.D_6$ x 2, $C_2^4.C_2^6.C_6.C_2$ x 2, $C_2^6.C_2^4.D_6$ x 2, $C_2^6.C_2^4.D_6$ x 2, $C_2^7.C_2^2:S_4$ x 2, $C_2^4.C_2^5:S_4$ x 2, $Q_8^2.C_2^3:S_4$ x 2, $C_2^4.C_2^5:S_4$ x 2, $C_2^6.(D_4\times S_4)$ x 2, $C_2^4.C_2^6:A_4$ x 2, $C_2^2.C_2^8:C_{12}$ x 2, $C_2^3:A_4.C_2^3.C_2.C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^3:A_4.C_2^3.C_2^3.C_2$, $C_2^3:A_4.(C_2\times D_4).C_2^3$, $Q_8:C_2^2.(C_6\times D_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2.C_2^6.C_6.C_2^4$, $C_2^3:A_4.C_2^3.C_2^4$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^5.A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^5.C_2^2$, $Q_8:C_2^2.(C_6\times D_4).C_2^3$, $C_2^3:A_4.C_2^2.C_2^4.C_2$, $(C_2\times A_4).C_2^4.C_2^5$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^7:Q_8:A_4$

Order 11520: $(C_2^4\times C_6):S_5$ x 6, $C_2^6:\GL(2,4)$ x 2, $C_2^6:\GL(2,4)$, $(C_2^3\times A_5):S_4$, $(Q_8\times A_5):S_4$, $A_5\times C_2\wr A_4$, $(C_2^4\times C_6).S_5$, $(C_2^4\times C_6):S_5$, $C_2^6:\GL(2,4)$, $C_2^5:A_6$

Order 10752: $C_2^8.F_7$, $C_2^6:\GL(3,2)$

Order 10368: $A_4^3:S_3$

Order 10240: $C_2^4:C_5.C_2^5.C_2^2$, $C_2^7.D_{10}:C_4$, $C_2^7.(C_4\times F_5)$, $C_2^7:C_{20}:C_4$, $C_2^7.(C_2^2\times F_5)$, $C_2^8.(C_2\times F_5)$, $C_2^6.(D_4\times D_{10})$

Order 10080: $C_2^2\times A_7$, $A_7:C_4$, $C_2\times S_7$

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

Order 8640: $A_4:S_6$ x 2, $C_2\times A_4\times A_6$

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 14, $C_2^6:S_5$ x 10, $A_4:C_2\wr F_5$ x 7, $C_2^7:A_5$ x 6, $C_2^6.S_5$ x 5, $D_4\times C_2^4:A_5$ x 4, $C_2\wr A_5\times C_4$ x 3, $(C_2^4\times C_4):S_5$ x 3, $C_4\times C_2^4:S_5$ x 2, $C_2\wr S_5$ x 2, $C_2^6.S_5$ x 2, $C_2^7:A_5$ x 2, $C_2^4:C_5.A_4.C_2^3$, $C_2^7.A_5$, $(C_2^3\times S_5):D_4$, $(C_2^4\times C_4).S_5$, $Q_8\times C_2^4:A_5$, $(C_2^4\times \SL(2,3)):F_5$, $C_2^7.(C_5\times A_4)$, $C_2^6:S_5$, $C_2^6.S_5$

Order 7168: $C_2^7:F_8$ x 2

Order 6912: $A_4:S_4^2$ x 6, $A_4^3:C_2^2$ x 4, $C_2^6.C_6^2.C_3$ x 2, $(C_2\times A_4^2).D_{12}$ x 2, $C_2\times A_4^2:D_{12}$ x 2, $A_4^3:C_4$ x 2, $A_4^3:C_2^2$ x 2, $C_2^6.C_3^3.C_2^2$, $C_2\times C_6:S_4^2$, $(C_2^2\times A_4^2):D_6$, $(C_2^2\times A_4^2):C_{12}$, $A_4^3:C_4$, $A_4^3:C_2^2$, $C_2^8.\He_3$

Order 6144: $C_2^7.(C_2\times S_4)$ x 3, $C_2^2.C_2^6.S_4$ x 2, $C_2^3:A_4.C_2^3.C_2^3$ x 2, $C_2^4.C_2^6.C_6$ x 2, $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_2^4.C_2^6.S_3$ x 2, $C_2^4.C_2^6.C_6$ x 2, $C_2^4.C_2^6.S_3$ 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, $C_2^8.S_4$ x 2, $C_2^6.(C_4\times S_4)$ x 2, $C_2\wr C_2^3:C_3$ x 2, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_4^2:C_4$, $C_2^3:A_4.(C_2^3.C_4).C_2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^4\times A_4).C_2^4.C_2$, $A_4.C_2^6.C_2^3$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^5.A_4.C_2^4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4:D_4$, $C_2^5.(D_4\times S_4)$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4:D_4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_2^3.C_2^3$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.C_2^3.C_2^3$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_2^5.C_2$, $C_2^6.A_4.C_2^3$, $C_2.C_4^2:A_4.C_2^4$, $C_2^3:A_4.C_2^5.C_2$, $C_2^6.C_6.C_2^4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2^3.C_2^3$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^3\times A_4).D_4^2$, $C_2^5.A_4.C_4.C_2^2$, $C_2^3:A_4.D_4:D_4$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.D_4^2$, $C_2.C_4^2:A_4.D_4.C_2$, $C_2^4.(C_2^4\times A_4).C_2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2\times D_4\times A_4).C_2^5$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_2^4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4:D_4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^6.A_4.C_2^3$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.C_4.C_2^4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_4$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^3\times A_4).D_4^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_4.C_2^2$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^3:A_4.C_4^2:C_4$, $A_4.C_2^6.C_2^3$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_2^3.C_2^3$, $C_2^3:A_4.C_4^2:C_4$, $(C_2\times A_4).C_2^4.C_2^4$, $C_2^3.C_2^4.C_6.C_2^3$, $(C_2^3\times A_4).D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2^3.C_2^3$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_4\wr C_2.C_2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_4^2:C_4$, $C_2^5.A_4.C_2.C_2^3$, $(C_2^3\times A_4).C_2^5.C_2$, $A_4.C_2^5.C_2^4$, $(C_2^3\times A_4).C_2^5.C_2$, $C_2^5.(D_4\times S_4)$, $C_2^8.S_4$, $(C_2\times C_4^3):\GL(2,3)$, $C_2^7:\GL(2,3)$, $C_2\wr \SL(2,3)$, $C_4^3.(C_2^2\times S_4)$, $C_2^5.(D_4\times S_4)$, $D_4^2.(C_2^2\times S_4)$, $D_4^3.D_6$, $C_2^6.\GL(2,\mathbb{Z}/4)$

Order 5760: $(C_2^3\times C_6):S_5$ x 4, $C_2^5:\GL(2,4)$ x 3, $C_2^2\times A_4:S_5$, $C_2^3:S_6$, $A_5\times Q_8:A_4$, $C_2^5:\GL(2,4)$

Order 5376: $C_2\wr C_7.C_6$, $C_2^5:\GL(3,2)$, $C_2^5.\GL(3,2)$, $C_2^7.F_7$, $C_2\wr F_7$

Order 5184: $A_4\wr C_3$

Order 5120: $C_2\wr F_5:D_4$ x 3, $(D_4\times C_2^5):F_5$ x 2, $C_2^4:C_5.C_2^5.C_2$, $C_2^4:C_5.C_2^4.C_2^2$, $C_2^4.C_2^2:C_4:F_5$, $C_2^8.D_{10}$, $C_2\wr D_5:C_4^2$, $D_4:C_2\wr F_5$, $D_4:C_2\wr F_5$, $C_2^5.(D_4\times F_5)$, $C_2\wr F_5\times D_4$, $C_2^7.(C_2\times F_5)$, $(D_4\times C_2^5).F_5$, $C_2^4:C_5.C_2\wr C_2^2$, $(D_4\times C_2^5).D_{10}$, $(D_4\times C_2^5).D_{10}$, $C_2^7:(C_4\times D_5)$, $C_2^8.F_5$, $C_2^8.D_{10}$, $C_2^6.(D_4\times C_{10})$

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

Order 4608: $C_2^3:S_4^2$ x 15, $C_2\wr A_4:S_4$ x 14, $C_2^6:\SOPlus(4,2)$ x 8, $C_2\times S_4^2:C_4$ x 6, $C_2.C_2^6.C_3.D_6$ x 5, $(A_4\times C_2^4).S_4$ x 5, $C_2^5:A_4^2$ x 5, $A_4^2.C_2^3.C_2^2$ x 4, $C_2^3:(A_4^2:C_4)$ x 4, $S_4^2:D_4$ x 4, $C_2^7:S_3^2$ x 3, $C_2^7.S_3^2$ x 3, $C_2^7.S_3^2$ x 3, $C_2^6.\SOPlus(4,2)$ x 2, $C_2^6.\SOPlus(4,2)$ x 2, $A_4^2:C_4\times C_2^3$ x 2, $C_2^3.S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $S_4^2:C_2^3$ 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, $C_2^3.S_4^2$ x 2, $C_2\wr A_4\times S_4$ x 2, $A_4\times C_2\wr S_4$ x 2, $C_2^8.D_9$ x 2, $A_4^2.C_2^3.C_2^2$, $C_2^4.A_4^2.C_2$, $Q_8.(C_6\times A_4).C_2^3$, $(C_2^3\times A_4).A_4.C_4$, $A_4^2.C_2^5$, $C_2.C_2^6.C_6^2$, $(C_2\times A_4^2).C_2^4$, $A_4^2.C_2^2.C_2^3$, $(C_2^3\times A_4).A_4.C_4$, $C_2^3.A_4^2.C_4$, $C_2^3.A_4^2.C_4$, $C_2^4.A_4^2.C_2$, $C_2^2.C_2^6.C_3.C_6$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $D_6:C_2\wr S_4$, $C_2^5:(S_3\times S_4)$, $(C_2^3\times C_6):\GL(2,\mathbb{Z}/4)$, $C_2^4.(D_6\times S_4)$, $C_3\times C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^5:(S_3\times S_4)$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $A_4\times C_2^4:S_4$, $(C_2^4\times S_4):A_4$, $C_2^4:A_4\times S_4$, $A_4\times C_2^4.S_4$, $C_2^3\times S_4^2$, $C_2^6:(S_3\times A_4)$, $C_2^4.(D_6\times S_4)$, $(C_2^5\times C_6):S_4$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^5.(S_3\times S_4)$, $(D_6\times C_2^4):S_4$, $C_2^5.(S_3\times S_4)$, $C_2^5:(S_3\times S_4)$, $C_2^4.(A_4\times S_4)$, $(C_2^2\times A_4).\GL(2,\mathbb{Z}/4)$, $C_3:C_2\wr S_4$, $A_4\times C_2^4.S_4$, $C_3\times C_2\wr S_4$, $C_2\wr A_4:S_3$, $C_2^4.(A_4\times S_4)$, $C_4^2:(C_3\times \GL(2,\mathbb{Z}/4))$, $(A_4\times C_2^4):S_4$, $C_2^3.S_4^2$, $C_2^3:S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^5:A_4^2$, $C_2^4:(A_4\times S_4)$, $A_4\times C_2^4:S_4$, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$, $(C_2^2\times \GL(2,\mathbb{Z}/4)):A_4$, $C_2^6.C_6^2.C_2$, $(C_2^2\times A_4)\wr C_2$, $C_2^4.\PSOPlus(4,3)$, $C_2\wr S_4\times D_6$

Order 4320: $C_6:S_6$, $A_4\times A_6$

Order 4096: $C_2^2.C_2^2\wr C_2^2$ x 3, $C_2^6.C_2\wr C_2^2$ x 3, $C_2^6.C_2\wr C_4$ x 2, $C_2^6.C_4^2:C_4$ x 2, $C_2^5.C_2\wr D_4$ x 2, $C_2^6.D_4^2$ x 2, $C_2^6.C_2\wr C_4$ x 2, $C_2^6.D_4:D_4$ x 2, $C_2^2.C_2^2\wr C_4$ x 2, $C_2^2\wr C_2^2.C_4$ x 2, $C_2^5.C_2\wr D_4$ x 2, $C_2^6.D_4^2$ x 2, $C_2^4.C_2^6.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^5.C_2^3$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4.C_2$, $C_2^4.C_2^5.C_2^3$, $C_2^8.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^6.C_2^4.C_2^2$, $C_2.C_2^6.C_2^5$, $C_2^3.C_2^6.C_2^3$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^6.C_2^2$, $C_2^6.C_2^4.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^6.C_2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^7:C_4\wr C_2$, $C_2^2.C_2^2\wr C_2^2$, $C_2^4.C_2^6:C_4$, $C_2^6.C_2\wr C_2^2$, $C_2^5.C_2^5:C_4$, $C_2\wr D_4.C_2$, $C_2^6.C_2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^5.C_2^5:C_4$, $(C_2^2\times D_4^2).C_2^4$

Order 3840: $C_2\wr S_5$ x 14, $C_2^6:A_5$ x 9, $C_2\times C_2^4:D_5\times A_4$ x 7, $(A_4\times C_2^4):F_5$ x 6, $C_2^5.S_5$ x 4, $C_4\times C_2^4:A_5$ x 2, $C_2\wr A_5$ x 2, $C_2^6.C_{30}.C_2$, $C_6:C_2\wr F_5$, $A_5.D_4^2$, $(C_2^3\times A_5).D_4$, $(D_4\times A_5):D_4$, $(C_2^3\times S_5):C_4$, $C_2^3.(C_2^2\times S_5)$, $(C_2^4\times A_5):C_4$, $C_2^3.(D_4\times A_5)$, $C_2\wr A_4:F_5$, $C_2^4:D_5\times \SL(2,3)$, $C_2^6.A_5$, $C_2^5:S_5$, $C_2^5.S_5$

Order 3584: $C_2^8:D_7$

Order 3456: $C_6:S_4^2$ x 7, $A_4^2:D_{12}$ x 4, $A_4^2:S_4$ x 3, $A_4^2:C_4\times C_6$ x 2, $C_2^6:C_9:C_6$ x 2, $C_2\times A_4^3$, $A_4^2:C_2^2\times C_6$, $C_2\times A_4^2:D_6$, $(A_4\times C_6^2):D_4$, $(C_3\times A_4^2):D_4$, $C_2\times A_4^2:D_6$, $C_6.\POPlus(4,3)$, $(C_3\times A_4^2):D_4$, $A_4^2:S_4$, $C_2^2:A_4^2:C_6$

Order 256: $C_2^8$

Order 81: $C_3\wr C_3$

Order 7: $C_7$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 13440 (order at least 3456) are shown, as well as all normal subgroups of any index.

Order 46448640: $C_2^8.A_9$

Order 5160960: $C_2^8.A_8$

Order 2580480: $C_2^7.A_8$

Order 1290240: $C_2^8.S_7$

Order 645120: $C_2^8.A_7$, $C_2^6.A_7:C_4$, $C_2^7.S_7$

Order 552960: $C_2\wr A_6.S_4$

Order 387072: $C_2^8:{}^2G(2,3)$

Order 368640: $C_2^8.A_4:S_5$

Order 344064: $C_2\wr C_2^3.\PSL(2,7)$

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

Order 276480: $(A_4\times C_2^5).S_6$, $(A_4\times C_2^5).S_6$, $A_4\times C_2\wr A_6$

Order 184320: $C_2^8.A_6.C_2$, $C_2^7.A_4:S_5$, $C_2^7.A_4:S_5$, $C_2^8.(A_4\times A_5)$

Order 181440: $A_9$

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

Order 165888: $A_4:S_4^2.S_4$

Order 161280: $C_2^6:A_7$

Order 147456: $C_2^8.\POPlus(4,3)$

Order 138240: $(C_2^5\times C_6).S_6$, $A_4\times C_2^5:A_6$

Order 129024: $C_2^8:\SL(2,8)$

Order 122880: $C_2^7.(D_4\times S_5)$

Order 92160: $C_2^7.A_6.C_2$, $C_2^7.A_6.C_2$, $C_2^7.A_6.C_2$, $C_2^8.A_6$, $C_2^7.A_6.C_2$, $A_4.C_2^6.S_5$, $C_2^7.A_6.C_2$, $C_2^7.(A_4\times A_5)$, $C_2\wr A_5.S_4$, $C_2^7.S_6$

Order 82944: $C_2^6.C_3^3.C_2^4.C_3$

Order 73728: $C_2^4.C_2^6.C_3.D_6.C_2$, $C_2^7.\POPlus(4,3)$, $C_2^7.\POPlus(4,3)$, $C_2^7.\POPlus(4,3)$, $C_2^8.A_4\wr C_2$

Order 69120: $(C_2^4\times C_6):S_6$, $(C_2^4\times C_6).S_6$, $C_3\times C_2\wr A_6$

Order 61440: $C_2^7.C_2^2.S_5$, $C_2^7.C_4:S_5$, $C_2^7.(C_4\times S_5)$, $D_4.C_2^6:S_5$, $C_2^7.(C_2^2\times S_5)$, $C_2^6:D_4.S_5$, $C_2^7.(D_4\times A_5)$, $C_2^8.A_4:F_5$

Order 55296: $C_2^8:\PU(3,2)$, $A_4^3.C_2^2\wr C_2$

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

Order 46080: $A_4.C_2\wr S_5$ x 7, $C_2^7:A_6$ x 2, $C_2^6.S_6$ x 2, $C_2^6.A_6.C_2$, $C_2^7.A_6$, $A_4.C_2^6.A_5$, $C_2^6.A_6.C_2$, $C_2^6.S_6$, $C_2^6.S_6$, $(C_2^4\times \SL(2,3)):S_5$, $(A_4\times C_2^5).S_5$, $(A_4\times C_2^5):S_5$, $C_2^8.\GL(2,4)$

Order 43008: $C_2^7.F_8:C_6$, $C_2^8.\GL(3,2)$, $C_2^8.\PSL(2,7)$

Order 41472: $A_4^3:S_4$ x 2, $A_4^3.S_4$

Order 40320: $C_2\times A_8$

Order 36864: $C_2.C_2^6.C_6^2.C_2^3$, $C_2^6.A_4^2.C_2^2$, $C_2^4.C_2^6.C_3.D_6$, $C_2^4.C_2^6.C_3.D_6$, $C_2^4.C_2^6.C_3.D_6$, $C_2^6.A_4^2.C_2^2$, $Q_8^2.\POPlus(4,3)$, $C_2^6.\POPlus(4,3)$, $C_2^6.S_4^2$, $C_2^7.A_4\wr C_2$, $C_2^7.A_4\wr C_2$

Order 34560: $C_3\times C_2^5:A_6$

Order 30720: $C_2^5.(D_4\times S_5)$ x 2, $C_2^7.(C_2\times S_5)$ x 2, $C_2^7.C_2.S_5$, $C_2^8.S_5$, $C_2^6.C_2^2.S_5$, $C_2^5.C_2^3.S_5$, $C_2^8.S_5$, $C_2^2.C_2^6.S_5$, $C_2^5.(D_4\times S_5)$, $C_2^6.(D_4\times A_5)$, $C_4.C_2^6:S_5$, $D_4:C_2^5.S_5$, $C_2^5.(D_4\times S_5)$, $(D_4\times C_2^5).S_5$, $C_2^5.(C_2^4\times A_5)$, $C_2^7.(C_4\times A_5)$, $C_2^6:D_{10}.S_4$, $C_2^6:D_{10}.S_4$, $C_2^7.(A_4\times D_{10})$

Order 27648: $(C_2\times A_4^3).D_4$ x 2, $C_2\times A_4^3:D_4$ x 2, $C_2^6.C_3^3.C_2^4$, $A_4^2.(C_2\times \GL(2,\mathbb{Z}/4))$, $A_4^3:C_2^2:C_4$

Order 24576: $C_2^4.(C_2^4\times A_4).C_2^3$, $C_2^4.C_2^6.D_6.C_2$, $C_2^7.Q_8:S_4$, $C_2^4.C_2^6:S_4$, $C_2^2\wr C_2^2.S_4$, $C_2^7.C_2^3:S_4$, $C_4^2:C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2\wr Q_8.A_4$, $C_2^4.C_2^6:S_4$, $C_2^7.C_2^3:S_4$, $C_2\wr C_2^3.A_4$

Order 23040: $(A_4\times C_2^4):S_5$ x 6, $C_2^7:\GL(2,4)$ x 4, $C_2\wr A_6$ x 2, $C_2^6:A_6$, $C_2^5.S_6$, $C_2^5:S_6$, $C_6:C_2\wr S_5$, $C_2\wr A_4:S_5$, $C_2^4:A_5\times \SL(2,3)$, $C_3:C_2\wr S_5$, $C_2^7:\GL(2,4)$

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

Order 20736: $A_4^3:A_4$ x 3, $A_4^3.A_4$

Order 20480: $C_2^7.(D_4\times F_5)$

Order 20160: $C_2^2:S_7$, $A_8$

Order 18432: $C_2.C_2^6.D_6:D_6$, $C_2^2.C_2^6.C_3:S_4$, $Q_8.A_4^2.D_4.C_2$, $C_2^3.A_4^2.C_4.C_2^2$, $(C_2^4\times A_4).A_4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $C_2^3:A_4.(D_4\times A_4).C_2$, $C_2^5.(C_6\times S_4).C_2^2$, $C_2^6.A_4^2.C_2$, $C_2^5.(C_6\times A_4).C_2^3$, $C_2^6.A_4^2.C_2$, $C_2.C_2^6.D_6:D_6$, $C_2.C_2^6.D_6:D_6$, $C_2.C_2^6.D_6:D_6$, $C_2^6.A_4^2.C_2$, $C_2.C_2^6.D_6:D_6$, $C_2^5.A_4.C_6.C_2^3$, $C_2.C_2^6.D_6:D_6$, $C_2^4.C_2^6.C_3.C_6$, $C_2.C_2^6.D_6:D_6$, $C_2^8.\PSU(3,2)$, $Q_8^2.A_4\wr C_2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2\wr A_4.S_4$, $A_4\times C_2\wr S_4$, $C_2\wr A_4.S_4$

Order 17280: $C_2\times A_4:S_6$

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

Order 15360: $C_2^7.S_5$ x 10, $D_4\times C_2^4:S_5$ x 4, $C_2^5.(C_4\times S_5)$ x 3, $C_2^5.(D_4\times A_5)$ x 3, $C_2^7.S_5$ x 2, $C_4:C_2\wr S_5$ x 2, $C_2^7.S_5$ x 2, $C_2^6:(C_4\times A_5)$ x 2, $C_2^7.S_5$, $C_2^7.S_5$, $C_2^7.S_5$, $C_2^8.A_5$, $C_2^7.S_5$, $C_2^6.C_{15}.C_2^3.C_2$, $C_2^8.A_5$, $(C_2^5\times C_4).S_5$, $C_2^7.S_5$, $(C_2^5\times C_4).S_5$, $(Q_8\times C_2^4):S_5$, $C_2^5.(C_4\times S_5)$, $(D_4\times C_2^4):S_5$, $C_4.C_2\wr S_5$, $D_4:C_2\wr A_5$, $C_2^7.(D_5\times A_4)$, $C_2^7.(D_5\times A_4)$, $C_2^7.(C_{10}\times A_4)$, $C_2^7:S_5$, $C_2^7.S_5$

Order 14336: $C_2^8.F_8$

Order 13824: $A_4^3:D_4$ x 4, $A_4^3.C_2^3$ x 3, $C_2\times A_4^3:C_4$ x 2, $A_4^2.A_4.C_2^3$, $C_2^2:A_4^2.S_4$, $C_2^2:A_4^2.S_4$, $(C_2^2\times A_4^2):D_{12}$, $A_4^2:\GL(2,\mathbb{Z}/4)$, $A_4^3:C_2^3$, $A_4^3:(C_2\times C_4)$, $A_4^2.\GL(2,\mathbb{Z}/4)$

Order 12288: $C_2^3:A_4.C_2^3.C_2.C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^3:A_4.C_2^3.C_2^3.C_2$, $C_2^3:A_4.(C_2\times D_4).C_2^3$, $C_2^6.C_2^4.D_6$, $Q_8:C_2^2.(C_6\times D_4).C_2^3$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^4.C_2^6.D_6$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2.C_2^6.C_6.C_2^4$, $C_2^3:A_4.C_2^3.C_2^4$, $C_2^4.C_2^6.C_6.C_2$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^6.C_2^4.D_6$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^5.A_4.C_2^4.C_2$, $C_2^3:A_4.C_2^5.C_2^2$, $Q_8:C_2^2.(C_6\times D_4).C_2^3$, $C_2^6.C_2^4.D_6$, $C_2^3:A_4.C_2^2.C_2^4.C_2$, $(C_2\times A_4).C_2^4.C_2^5$, $C_2^4.(C_2^3\times A_4).C_2^3$, $C_2^7:Q_8:A_4$, $C_2^7.C_2^2:S_4$, $C_2^4.C_2^5:S_4$, $Q_8^2.C_2^3:S_4$, $C_2^4.C_2^5:S_4$, $C_2^6.(D_4\times S_4)$, $C_2^4.C_2^6:A_4$, $C_2^2.C_2^8:C_{12}$

Order 11520: $(C_2^4\times C_6):S_5$ x 6, $C_2^6:\GL(2,4)$ x 2, $C_2^6:\GL(2,4)$, $(C_2^3\times A_5):S_4$, $(Q_8\times A_5):S_4$, $A_5\times C_2\wr A_4$, $(C_2^4\times C_6).S_5$, $(C_2^4\times C_6):S_5$, $C_2^6:\GL(2,4)$, $C_2^5:A_6$

Order 10752: $C_2^8.F_7$, $C_2^6:\GL(3,2)$

Order 10368: $A_4^3:S_3$

Order 10240: $C_2^4:C_5.C_2^5.C_2^2$, $C_2^7.D_{10}:C_4$, $C_2^7.(C_4\times F_5)$, $C_2^7:C_{20}:C_4$, $C_2^7.(C_2^2\times F_5)$, $C_2^8.(C_2\times F_5)$, $C_2^6.(D_4\times D_{10})$

Order 10080: $C_2^2\times A_7$, $A_7:C_4$, $C_2\times S_7$

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

Order 8640: $A_4:S_6$ x 2, $C_2\times A_4\times A_6$

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 14, $C_2^6:S_5$ x 10, $A_4:C_2\wr F_5$ x 7, $C_2^7:A_5$ x 6, $C_2^6.S_5$ x 5, $D_4\times C_2^4:A_5$ x 4, $C_2\wr A_5\times C_4$ x 3, $(C_2^4\times C_4):S_5$ x 3, $C_4\times C_2^4:S_5$ x 2, $C_2\wr S_5$ x 2, $C_2^6.S_5$ x 2, $C_2^7:A_5$ x 2, $C_2^4:C_5.A_4.C_2^3$, $C_2^7.A_5$, $(C_2^3\times S_5):D_4$, $(C_2^4\times C_4).S_5$, $Q_8\times C_2^4:A_5$, $(C_2^4\times \SL(2,3)):F_5$, $C_2^7.(C_5\times A_4)$, $C_2^6:S_5$, $C_2^6.S_5$

Order 7168: $C_2^7:F_8$

Order 6912: $A_4:S_4^2$ x 6, $A_4^3:C_2^2$ x 4, $(C_2\times A_4^2).D_{12}$ x 2, $C_2\times A_4^2:D_{12}$ x 2, $A_4^3:C_4$ x 2, $A_4^3:C_2^2$ x 2, $C_2^6.C_6^2.C_3$, $C_2^6.C_3^3.C_2^2$, $C_2\times C_6:S_4^2$, $(C_2^2\times A_4^2):D_6$, $(C_2^2\times A_4^2):C_{12}$, $A_4^3:C_4$, $A_4^3:C_2^2$, $C_2^8.\He_3$

Order 6144: $C_2^3:A_4.C_2^3.C_2^3$ x 2, $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_2^3:A_4.C_2\wr C_2^2$ x 2, $C_2^7.(C_2\times S_4)$ x 2, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_4^2:C_4$, $C_2^3:A_4.(C_2^3.C_4).C_2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^4\times A_4).C_2^4.C_2$, $A_4.C_2^6.C_2^3$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^5.A_4.C_2^4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4:D_4$, $C_2^5.(D_4\times S_4)$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4:D_4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_2^3.C_2^3$, $C_2^3:A_4.C_2^4.C_2^2$, $C_2^3:A_4.C_2^3.C_2^3$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_2^5.C_2$, $C_2^6.A_4.C_2^3$, $C_2.C_4^2:A_4.C_2^4$, $C_2^3:A_4.C_2^5.C_2$, $C_2^6.C_6.C_2^4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2^3.C_2^3$, $C_2^2.C_2^6.S_4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^3\times A_4).D_4^2$, $C_2^5.A_4.C_4.C_2^2$, $C_2^3:A_4.D_4:D_4$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.D_4^2$, $C_2.C_4^2:A_4.D_4.C_2$, $C_2^4.(C_2^4\times A_4).C_2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2\times D_4\times A_4).C_2^5$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_2^4$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.D_4:D_4$, $C_2^4.C_2^6.C_6$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^6.A_4.C_2^3$, $C_2^6.A_4.C_2^3$, $C_2^3:A_4.C_4.C_2^4$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_2\wr C_4$, $(C_2^4\times A_4).C_2^4.C_2$, $(C_2^3\times A_4).D_4^2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^3:A_4.C_2\wr C_4$, $C_2^5.A_4.C_4.C_2^2$, $C_2^3:A_4.D_4:D_4$, $C_2^3:A_4.C_2\wr C_2^2$, $C_2^3:A_4.C_4^2:C_4$, $A_4.C_2^6.C_2^3$, $C_2^4.C_2^6.S_3$, $C_2^3:A_4.D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_2^3.C_2^3$, $C_2^3:A_4.C_4^2:C_4$, $(C_2\times A_4).C_2^4.C_2^4$, $C_2^3.C_2^4.C_6.C_2^3$, $(C_2^3\times A_4).D_4^2$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.D_4^2$, $C_2^3:A_4.C_2^3.C_2^3$, $(C_2^4\times A_4).C_2^4.C_2$, $C_2^3:A_4.C_4\wr C_2.C_2$, $C_2^3:A_4.C_2\wr C_4$, $C_2^4.C_2^6.C_6$, $C_2^5.(D_4\times S_4)$, $C_2^3:A_4.C_4^2:C_4$, $C_2^5.A_4.C_2.C_2^3$, $(C_2^3\times A_4).C_2^5.C_2$, $A_4.C_2^5.C_2^4$, $C_2^4.C_2^6.S_3$, $(C_2^3\times A_4).C_2^5.C_2$, $C_2^5.(D_4\times S_4)$, $C_2^8.S_4$, $(C_2\times C_4^3):\GL(2,3)$, $C_4^2:C_2^4.S_4$, $C_2^7:\GL(2,3)$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^7.(C_2\times S_4)$, $Q_8^2.C_2^3:A_4$, $C_2\wr \SL(2,3)$, $C_2^8.S_4$, $C_4^3.(C_2^2\times S_4)$, $C_2^6.(C_4\times S_4)$, $C_2\wr C_2^3:C_3$, $C_2^5.(D_4\times S_4)$, $D_4^2.(C_2^2\times S_4)$, $D_4^3.D_6$, $C_2^6.\GL(2,\mathbb{Z}/4)$

Order 5760: $(C_2^3\times C_6):S_5$ x 4, $C_2^5:\GL(2,4)$ x 3, $C_2^2\times A_4:S_5$, $C_2^3:S_6$, $A_5\times Q_8:A_4$, $C_2^5:\GL(2,4)$

Order 5376: $C_2\wr C_7.C_6$, $C_2^5:\GL(3,2)$, $C_2^5.\GL(3,2)$, $C_2^7.F_7$, $C_2\wr F_7$

Order 5184: $A_4\wr C_3$

Order 5120: $C_2\wr F_5:D_4$ x 3, $(D_4\times C_2^5):F_5$ x 2, $C_2^4:C_5.C_2^5.C_2$, $C_2^4:C_5.C_2^4.C_2^2$, $C_2^4.C_2^2:C_4:F_5$, $C_2^8.D_{10}$, $C_2\wr D_5:C_4^2$, $D_4:C_2\wr F_5$, $D_4:C_2\wr F_5$, $C_2^5.(D_4\times F_5)$, $C_2\wr F_5\times D_4$, $C_2^7.(C_2\times F_5)$, $(D_4\times C_2^5).F_5$, $C_2^4:C_5.C_2\wr C_2^2$, $(D_4\times C_2^5).D_{10}$, $(D_4\times C_2^5).D_{10}$, $C_2^7:(C_4\times D_5)$, $C_2^8.F_5$, $C_2^8.D_{10}$, $C_2^6.(D_4\times C_{10})$

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

Order 4608: $C_2^3:S_4^2$ x 15, $C_2\wr A_4:S_4$ x 14, $C_2^6:\SOPlus(4,2)$ x 8, $C_2\times S_4^2:C_4$ x 6, $C_2.C_2^6.C_3.D_6$ x 5, $(A_4\times C_2^4).S_4$ x 5, $C_2^5:A_4^2$ x 5, $A_4^2.C_2^3.C_2^2$ x 4, $C_2^3:(A_4^2:C_4)$ x 4, $S_4^2:D_4$ x 4, $C_2^7:S_3^2$ x 3, $C_2^7.S_3^2$ x 3, $C_2^7.S_3^2$ x 3, $C_2^6.\SOPlus(4,2)$ x 2, $C_2^6.\SOPlus(4,2)$ x 2, $A_4^2:C_4\times C_2^3$ x 2, $C_2^3.S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $S_4^2:C_2^3$ 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, $C_2^3.S_4^2$ x 2, $C_2\wr A_4\times S_4$ x 2, $A_4\times C_2\wr S_4$ x 2, $A_4^2.C_2^3.C_2^2$, $C_2^4.A_4^2.C_2$, $Q_8.(C_6\times A_4).C_2^3$, $(C_2^3\times A_4).A_4.C_4$, $A_4^2.C_2^5$, $C_2.C_2^6.C_6^2$, $(C_2\times A_4^2).C_2^4$, $A_4^2.C_2^2.C_2^3$, $(C_2^3\times A_4).A_4.C_4$, $C_2^3.A_4^2.C_4$, $C_2^3.A_4^2.C_4$, $C_2^4.A_4^2.C_2$, $C_2^2.C_2^6.C_3.C_6$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $D_6:C_2\wr S_4$, $C_2^5:(S_3\times S_4)$, $(C_2^3\times C_6):\GL(2,\mathbb{Z}/4)$, $C_2^4.(D_6\times S_4)$, $C_3\times C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^5:(S_3\times S_4)$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $A_4\times C_2^4:S_4$, $(C_2^4\times S_4):A_4$, $C_2^4:A_4\times S_4$, $A_4\times C_2^4.S_4$, $C_2^3\times S_4^2$, $C_2^6:(S_3\times A_4)$, $C_2^4.(D_6\times S_4)$, $(C_2^5\times C_6):S_4$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^5.(S_3\times S_4)$, $(D_6\times C_2^4):S_4$, $C_2^5.(S_3\times S_4)$, $C_2^5:(S_3\times S_4)$, $C_2^4.(A_4\times S_4)$, $(C_2^2\times A_4).\GL(2,\mathbb{Z}/4)$, $C_3:C_2\wr S_4$, $A_4\times C_2^4.S_4$, $C_3\times C_2\wr S_4$, $C_2\wr A_4:S_3$, $C_2^4.(A_4\times S_4)$, $C_4^2:(C_3\times \GL(2,\mathbb{Z}/4))$, $(A_4\times C_2^4):S_4$, $C_2^3.S_4^2$, $C_2^3:S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^5:A_4^2$, $C_2^4:(A_4\times S_4)$, $A_4\times C_2^4:S_4$, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$, $(C_2^2\times \GL(2,\mathbb{Z}/4)):A_4$, $C_2^6.C_6^2.C_2$, $(C_2^2\times A_4)\wr C_2$, $C_2^8.D_9$, $C_2^4.\PSOPlus(4,3)$, $C_2\wr S_4\times D_6$

Order 4320: $C_6:S_6$, $A_4\times A_6$

Order 4096: $C_2^2.C_2^2\wr C_2^2$ x 2, $C_2^6.C_2\wr C_2^2$ x 2, $C_2^4.C_2^6.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^5.C_2^3$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^4.C_2^2$, $C_2.C_2^6.C_2^4.C_2$, $C_2^4.C_2^5.C_2^3$, $C_2^8.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^6.C_2^4.C_2^2$, $C_2.C_2^6.C_2^5$, $C_2^3.C_2^6.C_2^3$, $C_2^6.C_2^4.C_2^2$, $C_2^6.C_2^5.C_2$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^6.C_2^2$, $C_2^6.C_2^4.C_2^2$, $C_2^4.C_2^4.C_2^4$, $C_2^4.C_2^4.C_2^4$, $C_2^6.C_2\wr C_2^2$, $C_2^6.C_2\wr C_2^2$, $C_2^7:C_4\wr C_2$, $C_2^2.C_2^2\wr C_2^2$, $C_2^6.C_2\wr C_4$, $C_2^6.C_4^2:C_4$, $C_2^4.C_2^6:C_4$, $C_2^6.C_2\wr C_2^2$, $C_2^5.C_2\wr D_4$, $C_2^6.D_4^2$, $C_2^6.C_2\wr C_4$, $C_2^5.C_2^5:C_4$, $C_2^6.D_4:D_4$, $C_2\wr D_4.C_2$, $C_2^2.C_2^2\wr C_4$, $C_2^6.C_2\wr C_2^2$, $C_2^2\wr C_2^2.C_4$, $C_2^6.C_2\wr C_2^2$, $C_2^5.C_2\wr D_4$, $C_2^6.D_4^2$, $C_2^5.C_2^5:C_4$, $(C_2^2\times D_4^2).C_2^4$

Order 3840: $C_2\wr S_5$ x 14, $C_2^6:A_5$ x 9, $C_2\times C_2^4:D_5\times A_4$ x 7, $(A_4\times C_2^4):F_5$ x 6, $C_2^5.S_5$ x 4, $C_4\times C_2^4:A_5$ x 2, $C_2\wr A_5$ x 2, $C_2^6.C_{30}.C_2$, $C_6:C_2\wr F_5$, $A_5.D_4^2$, $(C_2^3\times A_5).D_4$, $(D_4\times A_5):D_4$, $(C_2^3\times S_5):C_4$, $C_2^3.(C_2^2\times S_5)$, $(C_2^4\times A_5):C_4$, $C_2^3.(D_4\times A_5)$, $C_2\wr A_4:F_5$, $C_2^4:D_5\times \SL(2,3)$, $C_2^6.A_5$, $C_2^5:S_5$, $C_2^5.S_5$

Order 3584: $C_2^8:D_7$

Order 3456: $C_6:S_4^2$ x 7, $A_4^2:D_{12}$ x 4, $A_4^2:S_4$ x 3, $A_4^2:C_4\times C_6$ x 2, $C_2\times A_4^3$, $A_4^2:C_2^2\times C_6$, $C_2\times A_4^2:D_6$, $(A_4\times C_6^2):D_4$, $(C_3\times A_4^2):D_4$, $C_2\times A_4^2:D_6$, $C_6.\POPlus(4,3)$, $(C_3\times A_4^2):D_4$, $A_4^2:S_4$, $C_2^6:C_9:C_6$, $C_2^2:A_4^2:C_6$

Order 256: $C_2^8$

Order 81: $C_3\wr C_3$

Order 7: $C_7$

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 46448640: $C_2^8.A_9$ ($C_1$)

Order 256: $C_2^8$ ($A_9$)

Order 1: $C_1$ ($C_2^8.A_9$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 46448640: $C_2^8.A_9$ ($C_1$)

Order 256: $C_2^8$ ($A_9$)

Order 1: $C_1$ ($C_2^8.A_9$)

Series

Derived series	$C_2^8.A_9$
Chief series	$C_2^8.A_9$	$\rhd$	$C_2^8$	$\rhd$	$C_1$
Lower central series	$C_2^8.A_9$
Upper central series	$C_1$

Supergroups

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

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $79 \times 79$ rational character table.