Abstract group 2654208.dk: $C_2^8.S_3^4:C_8$

• Label: 2654208.dk
• Order: \( 2^{15} \cdot 3^{4} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{17} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_2^8.S_3^4:C_8$
Order:	\(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	Group of order \(10616832\)\(\medspace = 2^{17} \cdot 3^{4} \)
Composition factors:	$C_2$ x 15, $C_3$ x 4
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	16	24
Elements	1	19999	6560	226272	183392	732672	600576	331776	552960	2654208
Conjugacy classes	1	47	5	104	87	20	106	4	6	380
Divisions	1	47	5	98	87	7	102	1	2	350
Autjugacy classes	1	33	5	58	50	7	49	1	2	206

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m \mid a^{8}=c^{6}=e^{6}=f^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,19,15,9)(2,20,16,10)(3,23,18,7)(4,24,17,8)(5,21,13,11,6,22,14,12), (1,16,6,18,4,14)(2,15,5,17,3,13)(7,24,11,22,8,23,12,21)(9,20)(10,19)\rangle$
Transitive group:	24T22665	24T22667	24T22669	24T22670	all 6
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:	$C_2^8$ . $(S_3^4:C_8)$	$(C_2^9.C_3:S_3^3)$ . $D_4$	$(C_2^9.S_3^2\wr C_2)$ . $C_2$	$C_2^9$ . $(C_3^4.C_2\wr C_4)$	all 22

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 26 normal subgroups (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_4^2.S_4\wr C_2.C_2$
Commutator:	$G' \simeq$ $C_2^8.C_3.S_3^3$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_4^2.S_4\wr C_2.C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $C_3^4.C_2\wr C_4$
Radical:	$R \simeq$ $C_2^8.S_3^4:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_3^4.C_2\wr C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^4.C_2^3.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 64 (order at least 41472) are shown, as well as all normal subgroups of any index.

Order 2654208: $C_2^8.S_3^4:C_8$

Order 1327104: $C_2^9.S_3^2\wr C_2$, $C_2^8.C_3:S_3^3:C_8$, $C_2^8.C_3:S_3^3:C_8$

Order 663552: $C_2^8.C_3^4.C_2^5$, $C_2\times S_4^2.A_4^2:C_4$, $A_4^2.(C_2\times S_4^2:C_4)$, $C_2^5.S_4^2:S_3^2$, $A_4^2.(C_2\times S_4^2:C_4)$, $C_2^9.C_3^4:\OD_{16}$, $C_2.A_4^2\wr C_2.D_4$

Order 331776: $C_2^8.C_3^4.C_2^4$ x 6, $C_2^8.C_3^4.C_2^4$ x 2, $S_4^2.A_4^2:C_4$ x 2, $C_2^6.C_3^3.A_4.C_2^4$, $C_2^8.C_3^4.C_2^4$, $C_2^8.C_3^4.C_2^4$, $C_2^9.C_3^4:D_4$, $A_4^2.S_4^2:C_4$, $A_4^2.(C_2^6.S_3^2)$, $C_2^9.C_3:S_3^3$, $A_4^2.(A_4^2:C_4\times C_2^2)$, $C_2^9.C_3\wr D_4$, $\PSOPlus(4,3).S_4\wr C_2$, $C_2^9.C_3^2:F_9$, $C_2.A_4\wr C_4.C_2$

Order 221184: $C_2^6.C_3^3.C_2.C_2^6$, $C_2^5.S_4^2:D_6$

Order 165888: $C_2^8.C_3^4.C_2^3$ x 10, $C_2^6.C_3^3.A_4.C_2^3$ x 8, $C_2^8.C_3.S_3^3$ x 6, $C_2^8.C_3.S_3^3$ x 4, $C_2^8.C_3.S_3^3$ x 4, $C_2^6.C_3^3.A_4.C_2^3$ x 3, $C_2^8.C_3^4.C_2^3$ x 2, $A_4^2.A_4^2.C_2^3$ x 2, $C_2^8.C_3^4.C_2^3$ x 2, $C_2^8.C_3^4.C_2^3$ x 2, $A_4^2.(C_2^5.S_3^2)$ x 2, $C_2^8.C_3^4.C_2^3$, $C_2^6.C_3^3.A_4.C_2^3$, $C_2^6.C_3^4.C_2^5$, $C_2.A_4\wr C_4$, $C_2^8.C_3^2:F_9$, $C_2^9.C_3^2:S_3^2$, $C_2^9.C_3^4:C_4$, $C_2^9.C_3\wr C_4$, $C_2^9.C_3\wr C_2^2$, $A_4^2.(C_2\times A_4^2:C_4)$

Order 147456: $A_4^2.C_2^3.C_2^6.C_2$

Order 110592: $C_2^6.C_3^3.C_2.C_2^5$ x 24, $C_2^6.C_3^3.C_2^5.C_2$ x 4, $C_2^6.C_3^2.D_6.C_2^4$ x 4, $C_2^6.C_3^3.C_2.C_2^5$ x 4, $C_2^6.C_3^3.C_2.C_2^5$ x 4, $A_4^2.C_2^3.C_6.C_2^4$ x 3, $C_2^6.C_3^3.C_2.C_2^5$ x 3, $C_2^6.C_3^3.C_2.C_2^5$ x 3, $C_2^6.C_3^3.C_2.C_2^5$ x 2, $C_2^8.C_3.D_6^2$ x 2, $C_2^6.C_3^2.D_6.C_2^4$ x 2, $C_2^6.C_3^2.D_6.C_2^4$ x 2, $C_2^6.C_6^3.C_2^3$ x 2, $C_2^6.C_3^3.C_2.C_2^5$, $C_2^6.S_3^3.C_2^3$, $C_2^6.C_3^3.C_2^6$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2.C_2^5$, $C_2^6.C_3^3.C_2.C_2^5$, $(C_2^3\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2:C_2^3.\GL(2,\mathbb{Z}/4)$, $(C_2^3\times S_4^2).S_4$, $C_2^9.S_3^3$, $A_4^2.(C_2^6.D_6)$, $C_2^4.S_4^2:C_{12}$, $C_2^4:A_4^2.(S_3\times C_8)$

Order 82944: $S_3\times S_4^3$ x 16, $C_2^6.C_3^3.A_4.C_2^2$ x 12, $A_4^2.A_4^2.C_2^2$ x 7, $C_2^8.C_3^3.D_6$ x 7, $C_2^8.C_3^3.D_6$ x 6, $C_2^6.C_3^3.A_4.C_2^2$ x 4, $C_2^8.C_3^3.D_6$ x 4, $C_2^6.C_3^4.C_2^4$ x 3, $C_2^6.C_3^4.C_2^4$ x 3, $C_2^6.C_3^4.C_2^4$ x 3, $(C_3\times A_4^2).A_4.C_2^4$ x 3, $A_4^2.A_4^2.C_2^2$ x 2, $C_2^6.C_3^3.A_4.C_2^2$, $A_4.C_2^6.C_3.C_6^2$, $C_2^6.C_3^4.C_2^4$, $C_2^8.C_3^3.D_6$, $C_2^6.C_3^4.C_2^4$, $C_2^6.C_3^4.C_2^4$, $S_4^2.S_3^2:C_4$, $C_2.A_4^2\wr C_2$, $C_2^8.C_3\wr C_4$, $A_4^2:(A_4^2:C_4)$

Order 73728: $A_4^2.C_2^4.C_2^4.C_2$ x 2, $C_2^5.C_6^2.C_2^5.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^5.C_2^4$, $A_4^2.C_2^3.C_2^5.C_2$, $C_2^5.C_6^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2^5.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^3.C_2^5.C_2$, $A_4^2.C_2^3.C_2^5.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^4.C_2^5$, $C_2^4.S_4^2:C_8$

Order 55296: $C_2^6.C_3^3.C_2.C_2^4$ x 62, $C_2^6.C_3^3.C_2^5$ x 48, $C_2^6.C_3^3.C_2.C_2^4$ x 38, $C_2^6.C_3^2.D_6.C_2^3$ x 36, $C_2^6.C_3^3.C_2.C_2^4$ x 34, $C_2^6.C_3^3.D_4.C_2^2$ x 28, $A_4^2.C_2^3.C_6.C_2^3$ x 22, $C_2^6.C_3^2.D_6.C_2^3$ x 22, $C_2^6.C_3^3.C_2.C_2^4$ x 16, $C_2^6.C_3^3.C_2.C_2^4$ x 16, $C_2^6.C_3^3.C_2.C_2^4$ x 16, $C_2^6.C_3^3.C_2^4.C_2$ x 16, $C_2^6.S_3^3.C_2^2$ x 15, $C_2^6.C_3^3.C_2.C_2^4$ x 15, $C_2^6.C_3^2.D_6.C_2^3$ x 15, $A_4^2.(D_4\times A_4).C_2^2$ x 14, $C_2^8.S_3^3$ x 14, $C_2^6.C_3^2.D_6.C_2^3$ x 14, $A_4^2.(C_2^4\times A_4).C_2$ x 12, $C_2^6.C_3^3.C_2.C_2^4$ x 12, $C_2^6.C_3^3.C_2^4.C_2$ x 12, $C_2^6.C_3^3.C_2.C_2^4$ x 10, $C_2^6.C_3^3.C_2.C_2^4$ x 10, $C_2^6.C_3^2.D_6.C_2^3$ x 8, $A_4^2.C_2^3.C_6.C_2^3$ x 8, $C_2^6.C_3^3.D_4.C_2^2$ x 8, $A_4^2.C_2^3.C_6.C_2^3$ x 6, $C_2^6.C_3^3.D_4.C_2^2$ x 6, $A_4^2.A_4.C_2^5$ x 6, $C_2^6.C_3^3.D_4.C_2^2$ x 6, $C_2^6.C_3^3.C_2.C_2^4$ x 6, $A_4^2.C_2^3.C_6.C_2^3$ x 4, $C_2^6.C_3^2.D_6.C_2^3$ x 4, $C_2^6.C_3^3.C_2^4.C_2$ x 4, $C_2^6.C_3^2.D_6.C_2^3$ x 4, $C_2^6.C_3^2.D_6.C_2^3$ x 4, $C_2^6.C_3^3.C_2^4.C_2$ x 4, $A_4^2.(D_4\times A_4).C_2^2$ x 4, $C_2^6.C_6^3.C_2^2$ x 4, $C_2^6.C_3^3.C_2^4.C_2$ x 4, $C_2^6.C_3^3.C_2^4.C_2$ x 4, $C_2^6.C_6^3.C_2^2$ x 4, $C_2^6.C_3^3.C_2^5$ x 4, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_3^2.D_6.C_2^3$ x 3, $C_2^8.C_3^3.C_2^3$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_6^2.C_6.C_2^2$ x 3, $A_4^2.A_4.C_2.C_2^4$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $A_4.C_2^5.C_6^2.C_2^2$ x 3, $(C_3\times A_4^2).C_2.C_2^6$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $A_4^2.(D_4\times A_4).C_2^2$ x 2, $C_2^6.C_3^3.D_4.C_2^2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $C_2^6.C_6^3.C_2^2$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $A_4^2.C_2^4.C_6.C_2^2$ x 2, $C_2^6.C_3^3.D_4.C_2^2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $A_4^2.C_2^3.C_6.C_2^3$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $A_4^2.C_2^3.C_6.C_2^3$ x 2, $C_2^8.C_3^3.C_2^3$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.C_2^5$ x 2, $C_2^6.C_6^3.C_2^2$ x 2, $C_2^8.C_3^3.C_2^3$ x 2, $A_4^2.(C_2^5.D_6)$ x 2, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.D_4.C_2^2$, $C_2^8.C_3^3.C_2^3$, $C_2^6.C_3^3.D_4.C_2^2$, $C_2^6.C_3^2.D_6.C_2^3$, $A_4^2.(D_4\times A_4).C_2^2$, $A_4^2.(D_4\times A_4).C_2^2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^4:A_4^2.C_{24}$, $C_2^2.A_4^3.C_8$, $S_4^3.C_4$, $C_2^5.A_4^2:D_6$, $C_2^3.A_4^3:C_2^2$, $A_4^2:C_2^4.S_4$, $A_4^2:C_2^4.S_4$, $C_2^9.(C_3\times S_3^2)$, $C_2^5.A_4^2:C_{12}$, $C_2^7.(A_4\times S_3^2)$

Order 41472: $C_3.S_4^3$ x 24, $C_3.S_4^3$ x 24, $S_3\times A_4\times S_4^2$ x 24, $(S_3\times A_4).S_4^2$ x 24, $C_3\times S_4^3$ x 8, $S_3\times A_4:S_4^2$ x 8, $C_3.S_4^3$ x 8, $(C_3\times A_4^2).A_4.C_2^3$ x 6, $A_4^2.A_4^2.C_2$ x 4, $(C_3\times A_4^2).A_4.C_2^3$ x 3, $(C_3\times A_4).A_4^2.C_2^3$ x 3, $C_2^6.C_3^4.C_2^3$ x 3, $C_2^6.C_3^4.C_2^3$ x 3, $C_2^6.C_3^4.C_2^3$ x 3, $A_4^2.C_2^2.C_6^2.C_2$ x 3, $C_2^6.C_3^4.C_2^3$ x 3, $C_2^6.C_3^4.C_2^3$ x 3, $C_2^8.C_3^4.C_2$ x 2, $A_4^2.(C_6\times A_4).C_2^2$ x 2, $C_2^6.C_3^3.A_4.C_2$ x 2, $A_4.C_2^6.C_3^3.C_2$ x 2, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $A_4^2.(S_3\times A_4).C_2^2$, $C_2^4.C_3^4.C_2^5$, $A_4^2.D_6^2.C_2$, $C_2^6.C_3^4.C_2^3$, $A_4^2:(C_6^2.D_4)$, $C_2\times (C_3^2\times S_4^2):C_4$, $A_4^2.D_6^2.C_2$, $(C_6\times S_4^2).D_6$, $(C_2^3\times D_6^2).S_3^2$, $A_4^2:(C_6^2.D_4)$

Order 32768: $C_2^7.C_2^4.C_2^3.C_2$

Order 20736: $C_2^8.C_3^4$

Order 10368: $S_3^4:C_8$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 64 (order at least 41472) are shown, as well as all normal subgroups of any index.

Order 2654208: $C_2^8.S_3^4:C_8$

Order 1327104: $C_2^9.S_3^2\wr C_2$, $C_2^8.C_3:S_3^3:C_8$, $C_2^8.C_3:S_3^3:C_8$

Order 663552: $C_2^8.C_3^4.C_2^5$, $C_2\times S_4^2.A_4^2:C_4$, $A_4^2.(C_2\times S_4^2:C_4)$, $C_2^5.S_4^2:S_3^2$, $A_4^2.(C_2\times S_4^2:C_4)$, $C_2^9.C_3^4:\OD_{16}$, $C_2.A_4^2\wr C_2.D_4$

Order 331776: $C_2^8.C_3^4.C_2^4$ x 4, $C_2^8.C_3^4.C_2^4$ x 2, $C_2^6.C_3^3.A_4.C_2^4$, $C_2^8.C_3^4.C_2^4$, $C_2^8.C_3^4.C_2^4$, $C_2^9.C_3^4:D_4$, $A_4^2.S_4^2:C_4$, $A_4^2.(C_2^6.S_3^2)$, $C_2^9.C_3:S_3^3$, $A_4^2.(A_4^2:C_4\times C_2^2)$, $C_2^9.C_3\wr D_4$, $\PSOPlus(4,3).S_4\wr C_2$, $S_4^2.A_4^2:C_4$, $C_2^9.C_3^2:F_9$, $C_2.A_4\wr C_4.C_2$

Order 221184: $C_2^6.C_3^3.C_2.C_2^6$, $C_2^5.S_4^2:D_6$

Order 165888: $C_2^8.C_3^4.C_2^3$ x 6, $C_2^8.C_3.S_3^3$ x 4, $C_2^6.C_3^3.A_4.C_2^3$ x 3, $C_2^6.C_3^3.A_4.C_2^3$ x 2, $C_2^8.C_3^4.C_2^3$ x 2, $A_4^2.A_4^2.C_2^3$ x 2, $C_2^8.C_3^4.C_2^3$ x 2, $C_2^8.C_3.S_3^3$ x 2, $C_2^8.C_3.S_3^3$ x 2, $A_4^2.(C_2^5.S_3^2)$ x 2, $C_2^8.C_3^4.C_2^3$, $C_2^6.C_3^3.A_4.C_2^3$, $C_2^6.C_3^4.C_2^5$, $C_2^8.C_3^4.C_2^3$, $C_2.A_4\wr C_4$, $C_2^8.C_3^2:F_9$, $C_2^9.C_3^2:S_3^2$, $C_2^9.C_3^4:C_4$, $C_2^9.C_3\wr C_4$, $C_2^9.C_3\wr C_2^2$, $A_4^2.(C_2\times A_4^2:C_4)$

Order 147456: $A_4^2.C_2^3.C_2^6.C_2$

Order 110592: $C_2^6.C_3^3.C_2.C_2^5$ x 9, $C_2^6.C_3^3.C_2.C_2^5$ x 3, $C_2^6.C_3^3.C_2.C_2^5$ x 3, $C_2^6.C_3^3.C_2.C_2^5$ x 2, $C_2^8.C_3.D_6^2$ x 2, $C_2^6.C_3^2.D_6.C_2^4$ x 2, $A_4^2.C_2^3.C_6.C_2^4$ x 2, $C_2^6.C_3^3.C_2.C_2^5$ x 2, $C_2^6.C_3^3.C_2.C_2^5$ x 2, $C_2^6.C_3^3.C_2.C_2^5$, $C_2^6.S_3^3.C_2^3$, $C_2^6.C_3^3.C_2^6$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2.C_2^5$, $C_2^6.C_3^3.C_2^5.C_2$, $C_2^6.C_3^2.D_6.C_2^4$, $C_2^6.C_3^2.D_6.C_2^4$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2.C_2^5$, $(C_2^3\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2:C_2^3.\GL(2,\mathbb{Z}/4)$, $(C_2^3\times S_4^2).S_4$, $C_2^9.S_3^3$, $A_4^2.(C_2^6.D_6)$, $C_2^4.S_4^2:C_{12}$, $C_2^4:A_4^2.(S_3\times C_8)$

Order 82944: $C_2^8.C_3^3.D_6$ x 6, $S_3\times S_4^3$ x 6, $A_4^2.A_4^2.C_2^2$ x 4, $C_2^6.C_3^3.A_4.C_2^2$ x 4, $C_2^8.C_3^3.D_6$ x 4, $C_2^6.C_3^3.A_4.C_2^2$ x 3, $C_2^6.C_3^4.C_2^4$ x 2, $A_4^2.A_4^2.C_2^2$ x 2, $C_2^8.C_3^3.D_6$ x 2, $C_2^6.C_3^4.C_2^4$ x 2, $C_2^6.C_3^4.C_2^4$ x 2, $(C_3\times A_4^2).A_4.C_2^4$ x 2, $C_2^6.C_3^3.A_4.C_2^2$, $A_4.C_2^6.C_3.C_6^2$, $C_2^6.C_3^4.C_2^4$, $C_2^8.C_3^3.D_6$, $C_2^6.C_3^4.C_2^4$, $C_2^6.C_3^4.C_2^4$, $S_4^2.S_3^2:C_4$, $C_2.A_4^2\wr C_2$, $C_2^8.C_3\wr C_4$, $A_4^2:(A_4^2:C_4)$

Order 73728: $A_4^2.C_2^4.C_2^4.C_2$ x 2, $C_2^5.C_6^2.C_2^5.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^5.C_2^4$, $A_4^2.C_2^3.C_2^5.C_2$, $C_2^5.C_6^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2^5.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^3.C_2^5.C_2$, $A_4^2.C_2^3.C_2^5.C_2$, $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^4.C_2^5$, $C_2^4.S_4^2:C_8$

Order 55296: $C_2^6.C_3^3.C_2^5$ x 25, $C_2^6.C_3^3.C_2.C_2^4$ x 18, $C_2^6.C_3^3.C_2.C_2^4$ x 13, $C_2^6.C_3^2.D_6.C_2^3$ x 11, $C_2^6.C_3^3.C_2.C_2^4$ x 10, $C_2^6.C_3^3.C_2.C_2^4$ x 9, $C_2^8.S_3^3$ x 8, $C_2^6.S_3^3.C_2^2$ x 7, $C_2^6.C_3^3.D_4.C_2^2$ x 7, $C_2^6.C_3^3.C_2.C_2^4$ x 7, $C_2^6.C_3^2.D_6.C_2^3$ x 7, $A_4^2.C_2^3.C_6.C_2^3$ x 6, $C_2^6.C_3^3.C_2.C_2^4$ x 6, $C_2^6.C_3^2.D_6.C_2^3$ x 6, $A_4^2.(D_4\times A_4).C_2^2$ x 5, $C_2^6.C_3^3.C_2.C_2^4$ x 5, $C_2^6.C_3^3.C_2.C_2^4$ x 5, $C_2^6.C_3^3.C_2.C_2^4$ x 4, $C_2^6.C_3^3.C_2.C_2^4$ x 4, $A_4^2.C_2^3.C_6.C_2^3$ x 4, $C_2^6.C_3^2.D_6.C_2^3$ x 4, $A_4^2.A_4.C_2^5$ x 4, $C_2^6.C_3^3.C_2^4.C_2$ x 4, $C_2^6.C_3^3.C_2.C_2^4$ x 4, $A_4^2.(C_2^4\times A_4).C_2$ x 3, $A_4^2.C_2^3.C_6.C_2^3$ x 3, $C_2^6.C_3^2.D_6.C_2^3$ x 3, $C_2^6.C_3^3.D_4.C_2^2$ x 3, $C_2^6.C_3^3.C_2^4.C_2$ x 3, $C_2^6.C_3^3.C_2^5$ x 3, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $A_4^2.C_2^3.C_6.C_2^3$ x 2, $A_4^2.(D_4\times A_4).C_2^2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $C_2^8.C_3^3.C_2^3$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $A_4^2.C_2^4.C_6.C_2^2$ x 2, $A_4^2.(D_4\times A_4).C_2^2$ x 2, $C_2^6.C_6^3.C_2^2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.D_4.C_2^2$ x 2, $C_2^6.C_6^2.C_6.C_2^2$ x 2, $A_4^2.A_4.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_6^3.C_2^2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^8.C_3^3.C_2^3$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.D_4.C_2^2$ x 2, $A_4.C_2^5.C_6^2.C_2^2$ x 2, $(C_3\times A_4^2).C_2.C_2^6$ x 2, $C_2^6.C_3^3.C_2^5$ x 2, $C_2^8.C_3^3.C_2^3$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $A_4^2.(C_2^5.D_6)$ x 2, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.D_4.C_2^2$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^8.C_3^3.C_2^3$, $C_2^6.C_3^3.D_4.C_2^2$, $C_2^6.C_3^3.D_4.C_2^2$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^3.D_4.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2^4.C_2$, $A_4^2.C_2^3.C_6.C_2^3$, $A_4^2.(D_4\times A_4).C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $A_4^2.C_2^3.C_6.C_2^3$, $A_4^2.(D_4\times A_4).C_2^2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_6^3.C_2^2$, $C_2^4:A_4^2.C_{24}$, $C_2^2.A_4^3.C_8$, $S_4^3.C_4$, $C_2^5.A_4^2:D_6$, $C_2^3.A_4^3:C_2^2$, $A_4^2:C_2^4.S_4$, $A_4^2:C_2^4.S_4$, $C_2^9.(C_3\times S_3^2)$, $C_2^5.A_4^2:C_{12}$, $C_2^7.(A_4\times S_3^2)$

Order 41472: $C_3.S_4^3$ x 8, $(S_3\times A_4).S_4^2$ x 8, $C_3.S_4^3$ x 7, $S_3\times A_4\times S_4^2$ x 7, $C_3.S_4^3$ x 6, $A_4^2.A_4^2.C_2$ x 4, $(C_3\times A_4^2).A_4.C_2^3$ x 3, $C_3\times S_4^3$ x 3, $S_3\times A_4:S_4^2$ x 3, $(C_3\times A_4^2).A_4.C_2^3$ x 2, $(C_3\times A_4).A_4^2.C_2^3$ x 2, $C_2^6.C_3^4.C_2^3$ x 2, $C_2^8.C_3^4.C_2$ x 2, $C_2^6.C_3^4.C_2^3$ x 2, $C_2^6.C_3^4.C_2^3$ x 2, $A_4^2.C_2^2.C_6^2.C_2$ x 2, $C_2^6.C_3^4.C_2^3$ x 2, $C_2^6.C_3^4.C_2^3$ x 2, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $A_4^2.(S_3\times A_4).C_2^2$, $A_4^2.(C_6\times A_4).C_2^2$, $C_2^6.C_3^3.A_4.C_2$, $A_4.C_2^6.C_3^3.C_2$, $C_2^4.C_3^4.C_2^5$, $A_4^2.D_6^2.C_2$, $C_2^6.C_3^4.C_2^3$, $A_4^2:(C_6^2.D_4)$, $C_2\times (C_3^2\times S_4^2):C_4$, $A_4^2.D_6^2.C_2$, $(C_6\times S_4^2).D_6$, $(C_2^3\times D_6^2).S_3^2$, $A_4^2:(C_6^2.D_4)$

Order 32768: $C_2^7.C_2^4.C_2^3.C_2$

Order 20736: $C_2^8.C_3^4$

Order 10368: $S_3^4:C_8$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2654208: $C_2^8.S_3^4:C_8$ ($C_1$)

Order 1327104: $C_2^9.S_3^2\wr C_2$ ($C_2$), $C_2^8.C_3:S_3^3:C_8$ ($C_2$), $C_2^8.C_3:S_3^3:C_8$ ($C_2$)

Order 663552: $C_2^8.C_3^4.C_2^5$ ($C_4$), $A_4^2.(C_2\times S_4^2:C_4)$ ($C_4$), $A_4^2.(C_2\times S_4^2:C_4)$ ($C_2^2$)

Order 331776: $C_2^8.C_3^4.C_2^4$ ($C_8$) x 2, $C_2^8.C_3^4.C_2^4$ ($C_2\times C_4$), $C_2^9.C_3:S_3^3$ ($D_4$), $A_4^2.(A_4^2:C_4\times C_2^2)$ ($D_4$)

Order 165888: $C_2^8.C_3.S_3^3$ ($\OD_{16}$), $C_2^8.C_3.S_3^3$ ($C_2\times C_8$), $C_2^8.C_3^4.C_2^3$ ($C_2^2:C_4$)

Order 82944: $C_2^8.C_3^3.D_6$ ($\OD_{16}:C_2$), $C_2^8.C_3^3.D_6$ ($C_2^2:C_8$), $C_2^8.C_3^3.D_6$ ($C_2^3:C_4$)

Order 41472: $C_2^8.C_3^4.C_2$ ($C_2\wr C_4$), $C_2^8.C_3^4.C_2$ ($C_2^3:C_8$), $C_2^8.C_3^4.C_2$ ($C_2\wr C_4$)

Order 20736: $C_2^8.C_3^4$ ($C_2^4:C_8$)

Order 512: $C_2^9$ ($C_3^4.C_2\wr C_4$)

Order 256: $C_2^8$ ($S_3^4:C_8$)

Order 2: $C_2$ ($S_4^2.S_4\wr C_2.C_2$)

Order 1: $C_1$ ($C_2^8.S_3^4:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2654208: $C_2^8.S_3^4:C_8$ ($C_1$)

Order 1327104: $C_2^9.S_3^2\wr C_2$ ($C_2$), $C_2^8.C_3:S_3^3:C_8$ ($C_2$), $C_2^8.C_3:S_3^3:C_8$ ($C_2$)

Order 663552: $C_2^8.C_3^4.C_2^5$ ($C_4$), $A_4^2.(C_2\times S_4^2:C_4)$ ($C_4$), $A_4^2.(C_2\times S_4^2:C_4)$ ($C_2^2$)

Order 331776: $C_2^8.C_3^4.C_2^4$ ($C_2\times C_4$), $C_2^8.C_3^4.C_2^4$ ($C_8$), $C_2^9.C_3:S_3^3$ ($D_4$), $A_4^2.(A_4^2:C_4\times C_2^2)$ ($D_4$)

Order 165888: $C_2^8.C_3.S_3^3$ ($\OD_{16}$), $C_2^8.C_3.S_3^3$ ($C_2\times C_8$), $C_2^8.C_3^4.C_2^3$ ($C_2^2:C_4$)

Order 82944: $C_2^8.C_3^3.D_6$ ($\OD_{16}:C_2$), $C_2^8.C_3^3.D_6$ ($C_2^2:C_8$), $C_2^8.C_3^3.D_6$ ($C_2^3:C_4$)

Order 41472: $C_2^8.C_3^4.C_2$ ($C_2\wr C_4$), $C_2^8.C_3^4.C_2$ ($C_2^3:C_8$), $C_2^8.C_3^4.C_2$ ($C_2\wr C_4$)

Order 20736: $C_2^8.C_3^4$ ($C_2^4:C_8$)

Order 512: $C_2^9$ ($C_3^4.C_2\wr C_4$)

Order 256: $C_2^8$ ($S_3^4:C_8$)

Order 2: $C_2$ ($S_4^2.S_4\wr C_2.C_2$)

Order 1: $C_1$ ($C_2^8.S_3^4:C_8$)

Series

Derived series

$C_2^8.S_3^4:C_8$

$\rhd$

$C_2^8.C_3.S_3^3$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^8.S_3^4:C_8$

$\rhd$

$C_2^9.S_3^2\wr C_2$

$\rhd$

$A_4^2.(C_2\times S_4^2:C_4)$

$\rhd$

$C_2^8.C_3^4.C_2^4$

$\rhd$

$C_2^8.C_3.S_3^3$

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Lower central series

$C_2^8.S_3^4:C_8$

$\rhd$

$C_2^8.C_3.S_3^3$

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

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

Character theory

Complex character table

The $380 \times 380$ character table is not available for this group.

Rational character table

The $350 \times 350$ rational character table is not available for this group.