Abstract group 594542592.a: $C_2^{16}.\SL(2,8).(C_3\times S_3)$

• Label: 594542592.a
• Order: \( 2^{20} \cdot 3^{4} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{20} \cdot 3^{4} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2^{16}.\SL(2,8).(C_3\times S_3)$
Order:	\(594542592\)\(\medspace = 2^{20} \cdot 3^{4} \cdot 7 \)
Exponent:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Automorphism group:	Group of order \(594542592\)\(\medspace = 2^{20} \cdot 3^{4} \cdot 7 \)
Composition factors:	$C_2$ x 17, $C_3$ x 2, $\SL(2,8)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	9	12	14	18	21	28
Elements	1	130815	1894400	16646400	80123904	884736	49545216	212631552	23887872	148635648	28311552	31850496	594542592
Conjugacy classes	1	29	7	35	49	1	6	39	5	6	1	3	182
Divisions	1	29	5	35	28	1	4	20	4	4	1	2	134
Autjugacy classes	1	29	7	35	49	1	6	39	5	6	1	3	182

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	27	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $36$ $\langle(1,20,27,14,11,35,22,32,8,4,19,25,15,12,33,23,31,6)(2,18,26,13,9,34,21,30,5,3,17,28,16,10,36,24,29,7) \!\cdots\! \rangle$
Transitive group:	36T89207				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:	$(C_2^{16}.S_3)$ . ${}^2G(2,3)$	$C_2^{16}$ . $(S_3\times {}^2G(2,3))$	$(C_2^{16}.S_3.\SL(2,8))$ . $C_3$	$(C_2^{16}.\SL(2,8).C_3)$ . $S_3$	all 8
Aut. group:	$\Aut(C_2^{16}.\SL(2,8))$	$\Aut(C_2^{17}.\SL(2,8))$	$\Aut(C_2^{16}.\SL(2,8).C_3)$	$\Aut(C_2^{16}.\SL(2,8).C_6)$	all 5

Elements of the group are displayed as permutations of degree 36.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 10 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_2^{16}.C_3.\SL(2,8)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{15}.C_2^5$

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 4096 (order at least 145152) are shown, as well as all normal subgroups of any index.

Order 594542592: $C_2^{16}.\SL(2,8).(C_3\times S_3)$

Order 297271296: $C_2^{16}.\SL(2,8).C_3^2$

Order 198180864: $C_2^{16}.S_3.\SL(2,8)$, $C_2^{16}.\SL(2,8).C_6$

Order 99090432: $C_2^{16}.\SL(2,8).C_3$, $C_2^{16}.C_3.\SL(2,8)$, $C_2^{16}.\SL(2,8).C_3$

Order 66060288: $C_2^{16}.F_8.C_3.C_6$, $C_2^9.C_2.\SL(2,8)$

Order 33030144: $C_2^{14}.F_8.(C_3\times A_4)$, $C_2^{16}.\SL(2,8)$

Order 22020096: $C_2^{16}.(S_3\times F_8)$, $C_2^{14}.F_8.C_6.C_2^2$

Order 21233664: $C_2^{16}.(S_3\times D_9:C_3)$

Order 16515072: $C_2^{16}.(S_3\times F_7)$, $C_2^{14}.F_8.C_3.C_6$

Order 11010048: $C_2^{14}.F_8.C_6.C_2$ x 2, $C_2^{14}.F_8.C_{12}$, $C_2^{14}.F_8.A_4$, $C_2^{14}.C_2^5.C_{21}$

Order 10616832: $C_2^{12}:C_3^2.\PSOPlus(4,3)$, $C_2^{16}.D_9.C_3^2$, $C_2^{16}.C_3^3.C_6$

Order 9437184: $C_2^{14}.A_4^2.C_2^2$

Order 8257536: $C_2^{16}.C_{21}.C_6$, $C_2^{14}.F_8.C_3^2$, $C_2^{14}.A_4:F_7$, $C_2^{12}.C_7.C_2^4.C_3.C_6$

Order 7340032: $C_2^{15}.F_8.C_2^2$

Order 7077888: $C_2^{16}.C_3^3.C_2^2$, $C_2^{16}.C_9.D_6$, $C_2^{16}.C_{18}.C_6$

Order 5505024: $C_2^{16}.C_7.D_6$, $C_2^{14}.F_8.C_6$, $C_2^{14}.(S_3\times F_8)$, $C_2^{12}.C_{14}.(C_6\times D_4).C_2$, $C_2^8.C_2^6.F_8.C_6$

Order 5308416: $C_2^{12}:C_3^2.A_4^2$

Order 4718592: $C_2^{14}.A_4^2.C_2$ x 3

Order 4128768: $C_2^{12}.C_7.C_2^4.C_3^2$, $C_2^{12}.(S_4\times F_7)$

Order 3670016: $C_2^{14}.F_8.C_2^2$ x 2, $C_2^{14}.F_8.C_4$

Order 3538944: $C_2^{16}.C_3^3.C_2$ x 2, $C_2^{10}.A_4^2.S_4$ x 2, $C_2^{16}.C_3.C_{18}$ x 2, $C_2^{12}.C_3.C_2^4.C_3.C_6$, $C_2^{12}:C_3^2.\GL(2,\mathbb{Z}/4)$, $C_2^{16}.C_9.C_6$, $C_2^{12}.A_4^2.S_3$, $C_2^{16}.C_3:D_9$, $C_2^{16}.C_9.C_6$

Order 3145728: $C_2^{16}.D_6.C_2^2$, $C_2^{15}.A_4.C_2^3$

Order 2752512: $C_2^{12}.C_{14}.C_6.C_2^3$ x 3, $C_2^{14}.F_8.C_3$ x 2, $C_2^{12}.C_{14}.C_6.C_4.C_2$ x 2, $C_2^{12}.C_7.C_2^4.C_6$ x 2, $C_2^{16}.D_{21}$, $C_2^{16}.C_{21}.C_2$, $C_2^{12}.C_7.C_6.C_2^4$, $C_2^{12}.C_7.(C_6\times D_4).C_2$, $C_2^8.C_2^3.C_2^6.C_{21}$

Order 2654208: $C_2^8.A_4^3.S_3$

Order 2359296: $C_2^{16}.S_3^2$, $C_2^{16}.(C_6\times S_3)$, $C_2^{14}.A_4^2$, $C_2^{12}.C_6.(C_6\times D_4).C_2$, $C_2^{12}.A_4^2.C_2^2$, $C_2^{16}.D_{18}$

Order 2064384: $C_2^{14}.C_{21}.C_6$ x 2, $C_2^{12}.A_4:F_7$ x 2, $C_2^{12}.(A_4\times F_7)$

Order 1835008: $C_2^{14}.F_8.C_2$ x 2, $C_2^{12}.C_7.C_2\wr C_2^2$

Order 1769472: $C_2^{12}.C_3.A_4^2$ x 2, $C_2^{14}.C_9.A_4$ x 2, $C_2^{16}.C_3^3$, $C_2^{12}.C_6^2:D_6$, $C_2^{10}.A_4^2.A_4$, $C_2^{14}.C_{18}.C_6$, $C_2^{14}.C_{18}.C_6$, $C_2^9.A_4^2.S_4$, $C_2^9.A_4^2.S_4$

Order 1572864: $C_2^{14}.A_4.C_2^3$ x 3, $C_2^{16}.D_6.C_2$ x 2, $C_2^{14}.A_4.C_4.C_2$ x 2, $C_2^{16}.C_6.C_2^2$, $C_2^{16}.A_4.C_2$, $C_2^{14}.A_4.C_2^2.C_2$, $C_2^{13}.A_4.C_2^4$

Order 1376256: $C_2^{12}.C_7.C_6.C_2^3$ x 3, $C_2^{12}.(C_2^3.F_7)$ x 3, $C_2^{12}.D_4:F_7$ x 2, $C_2^{12}.C_2^2:F_7.C_2$ x 2, $C_2^8.C_2\wr F_7$ x 2, $C_2^{16}.C_{21}$, $C_2^{12}.C_{14}.C_6.C_2^2$, $C_2^9.F_8.C_6.C_4.C_2$, $C_2^9.F_8.C_6.C_2^3$, $C_2^9.F_8.C_2^4.C_3$, $C_2^8.C_2^6.C_7.C_6.C_2$, $C_2^2\times C_2^9.F_8.C_6.C_2$, $C_2\times C_2^{12}.C_{14}.C_6.C_2$, $C_2^{12}.(D_7\times S_4)$

Order 1327104: $C_2^8.A_4^3.C_3$, $C_2^6.A_4^3.D_6$

Order 1179648: $C_2^{16}.C_3.C_6$ x 4, $C_2^{14}.(S_3\times A_4)$ x 4, $C_2^{12}.C_6^2.C_2^3$ x 3, $C_2^{12}.C_6^2.C_4.C_2$ x 2, $C_2^{16}.D_9$ x 2, $C_2^{16}.C_{18}$ x 2, $C_2^{16}.C_3.S_3$, $C_2^{14}.C_6^2.C_2$, $C_2^{12}.C_6^2.C_2^2.C_2$, $C_2^{12}.C_3.C_2^4.C_6$, $C_2^8.C_2^6.A_4.C_6$, $C_2^8.A_4.C_2^5.C_6.C_2$, $C_2^8.C_2^8:D_9$, $C_2^{12}.(A_4\times S_4)$

Order 1048576: $C_2^{15}.C_2^5$

Order 1032192: $C_2^{14}.C_{21}.C_3$ x 3, $C_2^{12}.(S_3\times F_7)$

Order 917504: $C_2^{12}.C_{14}.C_2^2:C_4$ x 2, $C_2^{12}.C_2^4:D_7$ x 2, $C_2^{14}.F_8$, $C_2^{14}.C_7.C_2^3$, $C_2^{12}.C_{14}.C_2^4$, $C_2^{12}.C_{14}.C_2^3.C_2$

Order 884736: $C_2^{12}:C_3^2.S_4$ x 4, $C_2^8.A_4^2:S_4$ x 2, $C_2^{14}.C_3^3.C_2$, $C_2^{10}.A_4^2:C_6$, $C_2^{12}.C_9:S_4$, $C_2^{12}.(C_3^2\times S_4)$, $C_2^9.A_4^2.A_4$, $C_2^{12}.D_{18}:C_6$, $C_2^{10}.A_4^2.C_6$

Order 786432: $C_2^{13}.A_4.C_2^3$ x 11, $C_2^{13}.A_4.C_2^2.C_2$ x 4, $C_2^{16}.D_6$ x 2, $C_2^{16}.C_6.C_2$, $C_2^{16}.A_4$, $C_2^{14}.D_6.C_2^2$, $C_2^{14}.C_6.C_2^3$, $C_2^{13}.A_4.C_4.C_2$, $C_2^{12}.C_6.C_2.D_4.C_2$, $C_2^{12}.A_4.C_2^4$

Order 774144: $C_2^9.{}^2G(2,3)$

Order 688128: $C_2^7.C_2\wr F_7$ x 6, $C_2^{12}.C_{28}:C_6$ x 4, $C_2^{12}.C_7.C_6.C_2^2$ x 3, $C_2^{11}.F_8.C_6$ x 3, $C_2^9.F_8.C_6.C_2^2$ x 2, $C_2\times C_2^9.F_8.C_6.C_2$ x 2, $C_2^2\times C_2^{12}.F_7$ x 2, $C_2^7.C_2\wr F_7$ x 2, $C_2^7.C_2\wr F_7$ x 2, $C_2^{12}.(C_7\times S_4)$ x 2, $C_2^{14}.D_{21}$ x 2, $C_2^{12}.C_7.C_{12}.C_2$, $C_2^{12}.(C_4.F_7)$, $C_2^9.F_8.C_{12}.C_2$, $C_2^2\times C_2^9.F_8.C_6$, $C_2^7.C_2\wr F_7$, $C_2^7.C_2\wr F_7$, $C_2^{12}.(A_4\times D_7)$, $C_2^{12}.D_7:A_4$

Order 663552: $C_2^6.A_4^3.S_3$, $C_2^6.A_4^3.S_3$, $C_2^6.A_4^3.C_6$

Order 589824: $C_2^{12}.C_6^2.C_2^2$ x 10, $C_2^{14}.(C_6\times S_3)$ x 3, $C_2^{16}.C_3^2$ x 2, $C_2^{13}.C_6^2.C_2$ x 2, $C_2^{12}.A_4^2$ x 2, $C_2^{16}.C_9$ x 2, $C_2^7.C_2\wr C_9$ x 2, $C_2^{12}.(C_6\times S_4)$ x 2, $C_2^{14}.S_3^2$, $C_2^{14}.C_6^2$, $C_2^{12}.(S_3\times A_4).C_2$, $C_2^{10}.C_6.(C_6\times D_4).C_2$, $C_2^{10}.C_3.C_2^4.C_{12}$, $C_2^8.C_2^4.A_4^2$, $C_2^8.A_4^2.C_2^4$, $C_2^8.A_4.C_2^4.C_6.C_2$, $C_2^{12}.A_4^2$, $C_2^{14}.D_{18}$, $C_2^9.C_2^6:D_9$, $C_2^{14}.D_{18}$

Order 524288: $C_2^{15}.C_2^4$ x 3, $C_2^{16}.C_2^3$ x 2, $C_2^{14}.C_2^5$, $C_2^{10}.D_4.C_2^6$

Order 516096: $C_2^{12}.C_{21}:C_6$, $C_2^{12}.C_{21}:C_6$, $C_2^{12}.(C_3\times F_7)$

Order 458752: $C_2^{12}.C_{14}.C_2^3$ x 5, $C_2^{12}.(C_{14}.D_4)$ x 3, $C_2^{12}.D_4:D_7$ x 2, $C_2^{11}.F_8.C_2^2$ x 2, $C_2^8.C_2\wr D_7$ x 2, $C_2^{13}.C_{28}.C_2$, $C_2^{13}.C_{14}.C_2^2$, $C_2^{13}.C_7.C_2^3$, $C_2^2\times C_2^{12}.C_{14}.C_2$, $C_2\times C_2^{13}.C_{14}.C_2$

Order 442368: $C_2^{12}.C_9:A_4$ x 3, $C_2^{12}:C_3^2.A_4$ x 3, $C_2^8.A_4^3$ x 2, $C_2^{12}.(C_9\times A_4)$ x 2, $C_2^{14}.C_3^3$, $C_2^{12}.C_{18}:C_6$, $C_2^{12}.C_9:C_{12}$, $C_2^{12}.C_{18}:C_6$, $C_2^{12}.(C_3\times S_3^2)$, $C_2^{12}.(C_3\times S_3^2)$, $C_2^{12}.(S_3\times D_9)$, $C_2^{12}.C_{18}:C_6$

Order 393216: $C_2^{13}.A_4.C_2^2$ x 15, $C_2^{13}.C_6.C_2^3$ x 9, $C_2^{12}.A_4.C_2^3$ x 8, $C_2^{12}.C_6.C_2^4$ x 7, $C_2^{12}.A_4.C_2^2.C_2$ x 5, $C_2^{14}.D_6.C_2$ x 4, $C_2^{14}.C_6.C_2^2$ x 3, $C_2^{12}.A_4.C_4.C_2$ x 3, $C_2^{16}.S_3$ x 2, $C_2^{12}.C_6.C_2^2:C_4$ x 2, $C_2^{12}.C_2^4:S_3$ x 2, $C_2^{16}.C_6$, $C_2^{12}.C_6.C_2^3.C_2$, $C_2^{11}.A_4.C_2^4$, $C_2^{10}.A_4.C_2^5$, $C_2^8.A_4.C_2^6.C_2$

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

Order 344064: $C_2^{12}.C_{14}:C_6$ x 8, $C_2^{11}.F_8.C_3$ x 3, $C_2^{12}.C_7:C_{12}$ x 3, $C_2^{14}.C_{21}$ x 3, $C_2^{12}.C_7:A_4$ x 3, $C_2\times C_2^{12}.F_7$ x 3, $C_2^{12}.C_{14}:C_6$ x 2, $C_2^{12}.C_7:C_{12}$ x 2, $C_2^9.C_{14}.C_6.C_2^3$, $C_2^6.C_2\wr F_7$, $C_2^{12}.(S_3\times D_7)$

Order 331776: $C_2^{12}:(C_9:C_3^2)$

Order 294912: $C_2^{14}.C_3.C_6$ x 21, $C_2^{12}.C_6^2.C_2$ x 9, $C_2^8.(A_4\times \GL(2,\mathbb{Z}/4))$ x 7, $C_2^{10}.(C_3\times \GL(2,\mathbb{Z}/4))$ x 6, $C_2^{14}.C_3.S_3$ x 5, $C_2^{10}.C_6^2.C_2^3$ x 4, $C_2^{14}.D_9$ x 4, $C_2^{12}.(S_3\times A_4)$ x 3, $C_2^{10}.C_6^2.C_2^2.C_2$ x 3, $C_2^{12}.(C_3\times S_4)$ x 2, $C_2^{14}.C_{18}$ x 2, $C_2^{10}.(C_{12}\times S_4)$ x 2, $C_2^{10}.(C_3\times \GL(2,\mathbb{Z}/4))$ x 2, $C_2^{13}.C_6^2$ x 2, $C_2^{14}.C_{18}$ x 2, $C_2^{12}.C_6.C_6.C_2$, $C_2^{10}.C_6.A_4.C_2^2$, $C_2^{10}.A_4^2.C_2$, $C_2^8.A_4.C_2^4.C_6$, $C_2^6.C_2^6.A_4.C_6$, $C_2^{12}.(C_6\times D_6)$, $C_2^{12}.(C_3\times D_{12})$, $C_2^{12}.C_9:D_4$, $C_2^{10}.(A_4\times S_4)$, $C_2^{10}.(A_4\times S_4)$

Order 262144: $C_2^{15}.C_2^3$ x 12, $C_2^{11}.C_2^6.C_2$ x 10, $C_2^{14}.C_2^4$ x 4, $C_2^{16}.C_2^2$ x 2, $C_2^{13}.C_2^5$ x 2, $C_2^{12}.C_2^6$

Order 258048: $C_2^8.F_8.C_3.C_6$, $C_2^{12}:C_7:C_3^2$, $C_2^9.\SL(2,8)$

Order 229376: $C_2^{12}.C_{14}.C_2^2$ x 6, $C_2^{11}.F_8.C_2$ x 6, $C_2^7.C_2\wr D_7$ x 6, $\ASigmaL(1,16384)$ x 4, $C_2\times C_2^{12}.C_{14}.C_2$ x 3, $C_2\times C_2^{12}.D_{14}$ x 2, $C_2^7.C_2\wr D_7$ x 2, $C_2^7.C_2\wr D_7$ x 2, $C_2^{12}.C_7.Q_8$, $C_2^2\times C_2^{12}.C_{14}$, $C_2^{12}.D_{28}$, $C_2\times C_2^{12}.D_{14}$

Order 221184: $C_2^{12}.C_9:C_6$ x 4, $C_2^8.A_4^2:C_6$ x 3, $C_2^{12}.C_3^3.C_2$ x 2, $C_2^{12}.(S_3\times C_3^2)$ x 2, $C_2^{12}.(S_3\times C_9)$ x 2, $C_2^8.C_3.C_2^4.C_3.C_6$, $C_2^8.A_4^2:C_6$, $C_2^{12}.C_3:D_9$, $C_2^{12}.C_9:C_6$, $C_2^{12}.(C_3\times D_9)$, $C_2^{12}.(S_3\times C_3^2)$, $C_2^8.C_6^2:S_4$, $C_2^{12}.C_9:C_6$

Order 196608: $C_2^{12}.C_6.C_2^3$ x 45, $C_2^{10}.A_4.C_2^4$ x 32, $C_2^{11}.C_6.C_2^4$ x 28, $C_2^{12}.A_4.C_2^2$ x 19, $C_2^{11}.A_4.C_2^3$ x 17, $C_2^9.A_4.C_2^5$ x 16, $C_2^{14}.D_6$ x 10, $C_2^8.A_4.C_2^5.C_2$ x 6, $C_2^{13}.A_4.C_2$ x 5, $C_2^{12}.A_4.C_4$ x 5, $C_2^{13}.C_6.C_2^2$ x 4, $C_2^{14}.C_6.C_2$ x 3, $C_2^{12}.(C_6.D_4)$ x 3, $C_2^{11}.C_6.C_2^3.C_2$ x 3, $C_2^8.A_4.C_2\wr C_2^2$ x 3, $C_2^{12}.D_4:S_3$ x 2, $C_2^{11}.C_6.C_2.C_2^3$ x 2, $C_2^8.A_4.C_2^6$ x 2, $C_2^{16}.C_3$, $C_2^{12}.D_{12}:C_2$, $C_2^{12}.D_6.C_2^2$, $C_2^{11}.C_6.C_4.C_2^2$, $C_2^9.(D_4\times A_4).C_2^2$, $C_2^2\times C_2^8.C_6.C_2^5$, $C_2\times C_2^9.A_4.C_2^4$

Order 172032: $C_2^9.F_8:C_6$ x 3, $C_2^{12}.F_7$ x 3, $C_2^9.C_{14}.C_{12}.C_2$ x 2, $C_2\times C_2^9.C_{14}.C_6.C_2$ x 2, $C_2^9.C_{14}.C_6.C_2^2$, $C_2^9.C_7.C_6.C_2^3$, $C_2^9.C_7.C_2^4.C_3$, $C_2^7.F_8.C_6.C_2^2$, $C_2^2\times C_2^9.C_7.C_6.C_2$, $C_2^9.F_8:C_6$, $C_2^{12}.(S_3\times C_7)$, $C_2^{12}.D_{21}$, $C_2^{12}.(C_3\times D_7)$

Order 165888: $C_2^{10}.(C_3^3.S_3)$

Order 147456: $C_2^{14}.C_3^2$ x 23, $C_2^{10}.C_6^2.C_2^2$ x 16, $C_2^{10}.A_4^2$ x 9, $C_2^9.(A_4\times S_4)$ x 7, $C_2^9.(A_4\times S_4)$ x 7, $C_2^{10}.(C_6\times S_4)$ x 7, $C_2^{14}.C_9$ x 6, $C_2^{10}.A_4^2$ x 5, $C_2^{10}.A_4^2$ x 4, $C_2^{12}.C_6^2$ x 3, $C_2^{11}.C_6^2.C_2$ x 3, $C_2^{10}.(C_6\times S_4)$ x 3, $C_2^9.A_4\wr C_2$ x 3, $C_2^{12}.S_3^2$ x 3, $C_2^{12}.(C_6\times S_3)$ x 2, $C_2^{10}.C_6.C_{12}.C_2$ x 2, $C_2^{10}.A_4^2$ x 2, $C_2^8.A_4^2.C_2^2$ x 2, $C_2^{12}.(C_2\times C_{18})$ x 2, $C_2^{10}.(C_6\times S_4)$ x 2, $C_2^{12}.S_3^2$, $C_2^{11}.(C_6\times C_{12})$, $C_2^{10}.C_6.C_6.C_2^2$, $C_2^8.C_6.(C_6\times D_4).C_2$, $C_2^8.C_2^6.C_3^2$, $C_2^8.A_4^2.C_4$, $C_2^6.C_2^4.A_4^2$, $C_2^6.A_4^2.C_2^4$, $C_2^{12}.D_{18}$, $C_2^{12}.C_9:C_4$, $C_2^{12}.(C_6\times S_3)$, $C_2^8.(C_4\times A_4^2)$, $C_2^{12}.C_6^2$, $C_2^{12}.D_{18}$

Order 65536: $C_2^{16}$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 594542592: $C_2^{16}.\SL(2,8).(C_3\times S_3)$ ($C_1$)

Order 297271296: $C_2^{16}.\SL(2,8).C_3^2$ ($C_2$)

Order 198180864: $C_2^{16}.S_3.\SL(2,8)$ ($C_3$)

Order 99090432: $C_2^{16}.C_3.\SL(2,8)$ ($C_6$), $C_2^{16}.\SL(2,8).C_3$ ($S_3$)

Order 33030144: $C_2^{16}.\SL(2,8)$ ($C_3\times S_3$)

Order 393216: $C_2^{16}.S_3$ (${}^2G(2,3)$)

Order 196608: $C_2^{16}.C_3$ ($\SL(2,8):C_6$)

Order 65536: $C_2^{16}$ ($S_3\times {}^2G(2,3)$)

Order 1: $C_1$ ($C_2^{16}.\SL(2,8).(C_3\times S_3)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $182 \times 182$ character table is not available for this group.

Rational character table

The $134 \times 134$ rational character table is not available for this group.