Abstract group 663552.hk: $A_4^2.S_4^2:D_4$

• Label: 663552.hk
• Order: \( 2^{13} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$A_4^2.S_4^2:D_4$
Order:	\(663552\)\(\medspace = 2^{13} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^2\times A_4^2.A_4^2.C_2.C_2^5$, of order \(5308416\)\(\medspace = 2^{16} \cdot 3^{4} \)
Composition factors:	$C_2$ x 13, $C_3$ 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	24
Elements	1	10975	6560	117024	145760	27648	300288	55296	663552
Conjugacy classes	1	51	12	96	134	4	108	4	410
Divisions	1	51	12	96	132	4	102	2	400
Autjugacy classes	1	36	11	53	85	4	51	2	243

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid b^{12}=c^{6}=d^{6}=e^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(2,3,5,7)(4,6,8)(9,11)(10,12)(13,18,14,20)(15,16)(17,19), (1,2,4,3,6,7) \!\cdots\! \rangle$
Transitive group:	36T33414	36T33415			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:	$S_4^2$ . $(A_4^2:D_4)$	$A_4^2$ . $(S_4^2:D_4)$	$C_2^9$ . $(S_3^2:S_3^2)$	$C_2^5$ . $(S_4^2:S_3^2)$	all 61

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$2$

Subgroups

There are 86 normal subgroups (74 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_4^2.\POPlus(4,3)$
Commutator:	$G' \simeq$ $A_4^2.A_4^2.C_2^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_4^2.\POPlus(4,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $S_3^2:S_3^2$
Radical:	$R \simeq$ $A_4^2.S_4^2:D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $S_3^2:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^5.C_2$
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 36 (order at least 18432) are shown, as well as all normal subgroups of any index.

Order 663552: $A_4^2.S_4^2:D_4$

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

Order 221184: $(C_2^2\times S_4^2).\GL(2,\mathbb{Z}/4)$, $C_2^3.S_4^3:C_2$

Order 165888: $C_2^8.C_3^4.C_2^3$ x 2, $C_2^8.C_3.S_3^3$ x 2, $C_2^9.C_3\wr C_4$ x 2, $A_4^2.S_4\wr C_2$ x 2, $A_4^2.S_4\wr C_2$ x 2, $C_2^8.C_3^4.C_2^3$, $C_2^6.C_3^3.A_4.C_2^3$, $C_2^8.C_3^4.C_2^3$, $A_4^2.A_4^2.C_2^3$, $C_2^8.C_3^4.C_2^3$, $A_4^2.A_4^2:D_4$, $C_2^9.C_3\wr C_2^2$, $C_2^9.C_3\wr C_2^2$, $A_4^2\wr C_2.C_2^2$

Order 110592: $C_2^8.C_3.D_6^2$ x 2, $A_4^2:C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6.C_3^3.C_2.C_2^5$, $C_2^6.C_3^3.C_2^5.C_2$, $(C_2^3\times A_4^2).\GL(2,\mathbb{Z}/4)$, $(C_2^3\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2:C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^3.A_4^3:D_4$, $(C_2^3\times S_4^2).S_4$, $(C_2^3\times S_4^2).S_4$, $C_2^9.S_3^3$, $C_2^9.S_3^3$, $C_2^9.S_3^3$, $C_2^4.S_4^2:C_{12}$, $C_2^5.S_4^2:C_6$

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

Order 73728: $A_4^2.C_2^6.C_2^3$, $C_2^5.C_6^2.C_2^5.C_2$

Order 55296: $C_2^6.C_3^3.C_2^4.C_2$ x 16, $C_2^8.S_3^3$ x 8, $C_2^6.C_3^3.C_2^4.C_2$ x 8, $C_2^8.C_3^3.C_2^3$ x 4, $C_2^6.C_3^3.C_2.C_2^4$ x 4, $C_2^6.C_3^2.D_6.C_2^3$ x 4, $C_2^8.C_3^3.C_2^3$ x 3, $C_2^8.C_3^3.C_2^3$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ 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, $A_4^2.C_2^4.C_6.C_2^2$ x 2, $A_4^2.C_2^3.C_6.C_2^3$ x 2, $C_2^6.C_6^2.C_6.C_2^2$ x 2, $C_2^8.C_3^3.C_2^3$ x 2, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2.A_4^3:D_4$ x 2, $A_4^2:C_2^4.S_4$ x 2, $C_2^5.A_4^2:C_{12}$ x 2, $A_4^2.C_2\wr C_6$ x 2, $C_2^2.A_4^3:D_4$ x 2, $(C_2^2\times S_4^2).S_4$ x 2, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2^5$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2.C_2^4$, $A_4^2.A_4.C_2^5$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $A_4^2.(C_2^4\times A_4).C_2$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^2.D_6.C_2^3$, $A_4.C_2^5.C_6^2.C_2^2$, $C_2^2\times C_2^6.C_3^3.C_2^3$, $C_2^2\times A_4^2.C_2^2\times S_4$, $S_4^2.\GL(2,\mathbb{Z}/4)$, $(C_2\times S_4^3).C_2$, $C_2^4.A_4^2:D_{12}$, $C_2^4:A_4^2.D_{12}$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2:C_2^4.S_4$, $A_4^2.C_2^2\wr S_3$, $C_2^9.(C_3\times S_3^2)$, $C_2^8.(C_6\times S_3^2)$, $C_2^9.(C_3\times S_3^2)$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2.C_2\wr C_6$, $C_2^7.(A_4\times S_3^2)$, $C_2^7.(A_4\times S_3^2)$

Order 41472: $C_3.S_4^3$ x 8, $(S_3\times A_4).S_4^2$ x 8, $A_4^2.A_4^2.C_2$ x 3, $C_2^6.C_3^4.C_2^3$ x 3, $C_2^6.C_3^4.C_2^3$ x 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, $A_4^2.(C_6\times A_4).C_2^2$ x 2, $(C_3\times A_4).A_4^2.C_2^3$, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $(C_3\times A_4^2).A_4.C_2^3$, $A_4^2.C_2^2.C_6^2.C_2$, $C_2^6.C_3^4.C_2^3$, $C_2^6.C_3^4.C_2^3$, $(C_3\times S_4^2):D_{12}$, $(S_3^2\times A_4^2).D_4$, $A_4^2\wr C_2$

Order 36864: $C_2^5.C_6^2.C_2^4.C_2$ x 2, $C_2^5.C_6^2.C_2^4.C_2$ x 2, $A_4^2.C_2^5.C_2^3$ x 2, $C_2^6.C_6^2.C_2^4$ x 2, $A_4^2.C_2^4.C_2^3.C_2$ x 2, $A_4^2.C_2^6.C_2^2$, $C_2^5.C_6^2.C_2^4.C_2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^5.C_2^3$, $C_2^5.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^3.C_2^4.C_2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^3.C_2^4.C_2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $C_2^5.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^4.C_2^4$, $A_4^2.C_2^5.C_2^3$, $C_2^5.C_6^2.C_2^4.C_2$, $C_2^6.S_4^2$, $(C_2^4\times C_4).S_4^2$

Order 27648: $C_2^6.C_3^3.C_2^3.C_2$ x 32, $C_2^6.C_3^3.C_2^3.C_2$ x 20, $C_2\times S_4^3$ x 20, $C_2^8.C_3^2.D_6$ x 16, $C_2^6.C_3^3.C_2^3.C_2$ x 16, $A_4^2.(C_2\times D_4\times A_4)$ x 16, $C_2^6.C_3^3.C_2^3.C_2$ x 14, $A_4^2.A_4.C_2^4$ x 14, $C_2^2\times C_2^6.C_3^2.D_6$ x 12, $C_2\times C_2^6.C_3^3.C_2^3$ x 11, $C_2\times A_4^2.C_2^2\times S_4$ x 8, $C_2^6.(S_3\times C_3:D_{12})$ x 8, $C_2^6.C_3^3.C_2^3.C_2$ x 8, $A_4^2.(C_2\times D_4\times A_4)$ x 8, $C_2^6.(A_4\times S_3^2)$ x 8, $C_2^6.C_3^3.D_4.C_2$ x 8, $C_2^6.(S_3\times D_6:S_3)$ x 8, $C_2^6.(C_6\times S_3):D_6$ x 8, $C_2^8.C_3^2.D_6$ x 8, $C_2^6.C_3^3.C_2^3.C_2$ x 8, $A_4.C_2^5.C_6^2.C_2$ x 8, $C_2^6.(C_3:S_3\times D_{12})$ x 8, $C_2^8.C_3^2.D_6$ x 6, $C_2^8.C_3^2.D_6$ x 6, $A_4^2.C_2^4.C_6.C_2$ x 6, $C_2^8.C_3^2.D_6$ x 5, $C_2^6.C_3^3.D_4.C_2$ x 4, $C_2^6.C_3^3.C_2^3.C_2$ x 4, $C_2^6.C_3^3.D_4.C_2$ x 4, $C_2^6.C_3^3.C_2^3.C_2$ x 4, $C_2^6.C_3^3.D_4.C_2$ x 4, $C_2^6.C_3^3.C_2^3.C_2$ x 4, $C_2^6.C_3^2.D_6.C_2^2$ x 4, $C_2^6.C_3^3.D_4.C_2$ x 4, $C_2^6.C_3^3.C_2^3.C_2$ x 4, $C_2^6.C_3^3.C_2^4$ x 3, $A_4.C_2^5.C_6^2.C_2$ x 3, $C_2^8.C_3^2.D_6$ x 3, $C_2^6.C_3^3.D_4.C_2$ x 3, $C_2^4:C_3.A_4^2.C_2^2$ x 3, $C_2^6.C_3^2.C_2\times S_4$ x 2, $(C_2^4\times A_4).D_6:D_6$ x 2, $(C_3\times A_4^2).C_2^5.C_2$ x 2, $C_2^6.C_3^3.D_4.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $A_4.A_4^2.C_2^4$ x 2, $A_4^2.A_4.C_2.C_2^3$ x 2, $C_2^8.C_3^2.D_6$ x 2, $C_2^2\times C_2^6.C_3^2.D_6$ x 2, $A_4^2.(D_4\times A_4).C_2$ x 2, $C_2^6.C_3^2.C_2^2\times A_4$ x 2, $A_4^2.(D_4\times A_4).C_2$ x 2, $A_4^2.(C_2\times D_4\times A_4)$ x 2, $C_2^6.C_3^3.C_2^4$ x 2, $C_2^6.C_3^3.D_4.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $(C_3\times A_4^2).C_2.C_2^5$ x 2, $C_2^2\times A_4^2.A_4.C_2^2$ x 2, $A_4^2.C_2^4.C_6.C_2$ x 2, $C_2^6.C_3^2.D_6.C_2^2$ x 2, $C_2^6.C_3^2.C_2\times S_4$ x 2, $A_4.C_2^6.C_6^2$ x 2, $A_4^2.(D_4\times A_4).C_2$ x 2, $C_2^6.C_6^2.C_6.C_2$ x 2, $(C_2^4\times A_4).D_6\times A_4$ x 2, $C_2^8.(C_3\times S_3^2)$ x 2, $A_4^2:C_2^3.S_4$ x 2, $C_2^6.(A_4\times S_3^2)$ x 2, $A_4^2:(C_2^4:C_{12})$ x 2, $A_4^2.C_2^3:S_4$ x 2, $\PSOPlus(4,3).\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times A_4^2).S_4$ x 2, $(C_2^3\times A_4^2).S_4$ x 2, $C_2^8.(C_3^2\times D_6)$ x 2, $C_2^9.(S_3\times C_3^2)$ x 2, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^2\times C_2\times A_4^2\times S_4$, $C_2^2\times A_4.A_4^2.C_2^2$, $A_4.C_2^5.C_6^2.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2.C_2^3$, $A_4.A_4^2.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^4.C_6^3.C_2^3$, $A_4^2.A_4.C_2^3.C_2$, $A_4^2.(C_2\times D_4\times A_4)$, $C_2^6.C_6^3.C_2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.D_4.C_2$, $A_4^2.A_4.C_2^4$, $C_2^7.C_6^3$, $C_2^6.C_3^3.C_2^4$, $(C_3\times A_4^2).C_2.C_2^5$, $A_4^2.C_2^2\times C_2^4:C_3$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2.C_2^3$, $(C_2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $C_2\times A_4^2:\GL(2,\mathbb{Z}/4)$, $(C_2\times A_4^3).D_4$, $C_2\times A_4^3:D_4$, $C_2\times S_4^2:S_4$, $(A_4\times S_4^2):C_4$, $C_2^7.S_3^3$, $C_2^7.S_3^3$, $C_2^7.S_3^3$, $C_2\times A_4^3:D_4$, $A_4\times S_4^2:C_4$, $C_2^4:A_4^2.C_{12}$, $C_2^4.A_4^2:D_6$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 24576: $C_2^5.(C_2^3\times C_6).C_2^4$, $C_2^5.(C_2^3\times C_6).C_2^4$

Order 20736: $A_4^3:D_6$ x 12, $A_4^3:D_6$ x 8, $A_4^3.D_6$ x 8, $C_3\times A_4\times S_4^2$ x 4, $A_4^3:D_6$ x 4, $(C_3\times A_4).S_4^2$ x 4, $A_4^3.D_6$ x 4, $A_4^3:D_6$ x 4, $A_4^3:D_6$ x 4, $A_4^3:D_6$ x 4, $(C_3\times A_4^2).A_4.C_2^2$ x 3, $A_4.(C_3\times A_4^2).C_2^2$ x 3, $(C_3\times A_4).A_4^2.C_2^2$ x 2, $A_4^2.C_2^2.C_6^2$ x 2, $C_2^6.C_3^3.D_6$ x 2, $C_2^4.C_3^4.C_2^4$ x 2, $C_2^6.C_3^4.C_2^2$, $(C_2^5\times C_6).C_3.C_6^2$, $C_2^4.C_3^4.C_2^4$, $C_2^4.C_3^4.C_2^4$, $C_2^8.C_3^4$, $(C_6\times A_4^2):D_{12}$, $(C_6\times A_4^2).D_{12}$, $A_4^2:(S_3^2:C_4)$, $A_4^2:S_3^2:C_2^2$, $A_4^2:C_6.D_{12}$, $A_4^2:(C_6^2:C_4)$, $(C_2^4\times C_6).S_3^3$, $(C_2^4\times C_6).S_3^3$, $(S_3^2\times A_4^2):C_2^2$, $(S_3^2\times A_4^2):C_4$, $(C_3\times S_4^2):C_{12}$, $C_6\times S_4^2:S_3$

Order 18432: $C_2^6.D_6^2.C_2$ x 32, $A_4^2.C_2^5.C_2^2$ x 32, $A_4^2.C_2^4.C_2^3$ x 32, $A_4^2.C_2^5.C_2^2$ x 16, $A_4^2.C_2^4.C_2^3$ x 12, $A_4^2.C_2\wr D_4$ x 8, $A_4^2.C_2^6.C_2$ x 6, $A_4^2.C_2^4.C_2^3$ x 5, $(C_2^2\times S_4^2).D_4$ x 5, $A_4^2.C_2^5.C_2^2$ x 4, $A_4^2.C_2^3.C_2^3.C_2$ x 4, $A_4^2.C_2^4.C_2^3$ x 4, $A_4^2.C_2^4.C_2^3$ x 4, $A_4^2.C_2^4.C_2^3$ x 4, $A_4^2.C_2^4.C_2^3$ x 4, $C_2^6.C_6^2.C_2^3$ x 4, $A_4^2.C_2^5.C_2^2$ x 4, $A_4^2.C_2^4.C_2^3$ x 4, $A_4^2:C_2\wr D_4$ x 4, $(C_2^2\times S_4^2).D_4$ x 4, $(C_2^2\times S_4^2).D_4$ x 4, $(C_2^3\times S_4^2).C_4$ x 4, $A_4^2.C_2\wr D_4$ x 4, $(C_2^2\times S_4^2).D_4$ x 4, $A_4^2.C_2^5.C_2^2$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $A_4^2.C_2^4.C_2^3$ x 3, $A_4^2.C_2^4.C_2^3$ x 3, $A_4^2.C_2^6.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^2.C_2^4.C_2$ x 2, $A_4.(C_2^4\times A_4).C_2^3$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^6.D_6^2.C_2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $C_2^6.(C_6\times S_3).C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^2\times C_2^6.C_3^2.C_2^3$ x 2, $C_2^2\times C_2\times C_2^2:S_4\times S_4$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^6.C_2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^6.D_6^2.C_2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^8.C_3.D_6.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $(C_2\times A_4^2).C_2^6$ x 2, $C_2^4.(C_6\times A_4).C_2^4$ x 2, $A_4^2.C_2^6.C_2$ x 2, $C_2^2\times A_4^2.C_2^3.C_2^2$ x 2, $A_4^2.C_2^3.C_2^4$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^4.(C_6^2\times D_4).C_2^2$ x 2, $A_4^2.C_2^6.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^6.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4.C_2^5.C_6.C_2^3$ x 2, $A_4^2.C_2^6.C_2$ x 2, $C_2^4:C_3.C_2^3.C_6.C_2^3$ x 2, $C_2^4.(C_6^2\times D_4).C_2^2$ x 2, $C_2^5.S_4^2$ x 2, $C_2^9.S_3^2$ x 2, $C_2^9.S_3^2$ x 2, $C_2^5.S_4^2$ x 2, $C_2^9.S_3^2$ x 2, $A_4^2.C_2^4.C_2^3$, $C_2^4.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^6.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2^5.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^6.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $A_4^2.(C_2^2\times C_4^2).C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4.(C_2^4\times A_4).C_2^3$, $A_4^2.C_2^5.C_2^2$, $C_2^4:C_3.C_2^4.C_6.C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2^5.C_2$, $A_4^2.C_2^6.C_2$, $C_2^4.(C_6\times C_{12}).C_2^4$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $C_2^4.(C_6^2\times D_4).C_2^2$, $C_2^4.(C_2\times C_6^2).C_2^4$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2^6.C_6^2.C_2^3$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $C_2^3\times A_4^2.C_2^4$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2^4:C_3.C_2^3.C_6.C_2^3$, $C_2^4.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2^6.C_{12}:S_4$, $C_2^6.\PSOPlus(4,3)$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$, $C_2^6.(A_4\times S_4)$, $C_2^7.(C_6\times S_4)$, $C_2^6.(C_{12}\times S_4)$

Order 13824: $A_4^2.C_2^4.C_6$ x 4, $C_2^8.C_3^3.C_2$ x 2

Order 9216: $C_2^2\times A_4^2.C_2^4$ x 2, $C_2^4\times A_4^2.C_2^2$, $C_2^3\times A_4^2.C_2^3$

Order 8192: $C_2^7.C_2^5.C_2$

Order 6912: $C_2^8.C_3^3$ x 2

Order 4608: $C_2^2\times A_4^2.C_2^3$ x 2, $C_2^4\times C_2\times A_4^2$, $C_2^3\times A_4^2.C_2^2$

Order 2304: $C_2^4\times A_4^2$, $C_2^2\times A_4^2.C_2^2$

Order 1536: $C_2^9.C_3$ x 2

Order 1152: $C_2\times S_4^2$

Order 768: $C_2^8.C_3$ x 2

Order 576: $A_4^2:C_2^2$ x 2, $S_4^2$ x 2

Order 512: $C_2^9$

Order 288: $C_2\times A_4^2$ x 2, $\PSOPlus(4,3)$ x 2

Order 256: $C_2^8$

Order 144: $A_4^2$ x 2

Order 96: $C_2^3:A_4$ x 2

Order 81: $C_3^4$

Order 48: $C_2^2:A_4$ x 2

Order 32: $C_2^5$ x 2

Order 16: $C_2^4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 18432) are shown, as well as all normal subgroups of any index.

Order 663552: $A_4^2.S_4^2:D_4$

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

Order 221184: $(C_2^2\times S_4^2).\GL(2,\mathbb{Z}/4)$, $C_2^3.S_4^3:C_2$

Order 165888: $C_2^9.C_3\wr C_4$ x 2, $C_2^8.C_3^4.C_2^3$, $C_2^6.C_3^3.A_4.C_2^3$, $C_2^8.C_3^4.C_2^3$, $A_4^2.A_4^2.C_2^3$, $C_2^8.C_3^4.C_2^3$, $C_2^8.C_3^4.C_2^3$, $C_2^8.C_3.S_3^3$, $A_4^2.A_4^2:D_4$, $C_2^9.C_3\wr C_2^2$, $C_2^9.C_3\wr C_2^2$, $A_4^2\wr C_2.C_2^2$, $A_4^2.S_4\wr C_2$, $A_4^2.S_4\wr C_2$

Order 110592: $C_2^8.C_3.D_6^2$ x 2, $A_4^2:C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6.C_3^3.C_2.C_2^5$, $C_2^6.C_3^3.C_2^5.C_2$, $(C_2^3\times A_4^2).\GL(2,\mathbb{Z}/4)$, $(C_2^3\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2:C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^3.A_4^3:D_4$, $(C_2^3\times S_4^2).S_4$, $(C_2^3\times S_4^2).S_4$, $C_2^9.S_3^3$, $C_2^9.S_3^3$, $C_2^9.S_3^3$, $C_2^4.S_4^2:C_{12}$, $C_2^5.S_4^2:C_6$

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

Order 73728: $A_4^2.C_2^6.C_2^3$, $C_2^5.C_6^2.C_2^5.C_2$

Order 55296: $C_2^8.S_3^3$ x 4, $C_2^6.C_3^3.C_2^4.C_2$ x 4, $C_2^8.C_3^3.C_2^3$ x 3, $C_2^8.C_3^3.C_2^3$ x 3, $C_2^8.C_3^3.C_2^3$ x 2, $A_4^2.C_2^4.C_6.C_2^2$ x 2, $C_2^6.C_6^2.C_6.C_2^2$ 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, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2.A_4^3:D_4$ x 2, $A_4^2:C_2^4.S_4$ x 2, $C_2^5.A_4^2:C_{12}$ x 2, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2^5$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^3.C_2.C_2^4$, $A_4^2.C_2^3.C_6.C_2^3$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2.C_2^4$, $A_4^2.A_4.C_2^5$, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2$, $A_4^2.(C_2^4\times A_4).C_2$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^2.D_6.C_2^3$, $A_4.C_2^5.C_6^2.C_2^2$, $C_2^2\times C_2^6.C_3^3.C_2^3$, $C_2^2\times A_4^2.C_2^2\times S_4$, $S_4^2.\GL(2,\mathbb{Z}/4)$, $(C_2\times S_4^3).C_2$, $C_2^4.A_4^2:D_{12}$, $C_2^4:A_4^2.D_{12}$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2:C_2^4.S_4$, $A_4^2.C_2^2\wr S_3$, $C_2^9.(C_3\times S_3^2)$, $C_2^8.(C_6\times S_3^2)$, $C_2^9.(C_3\times S_3^2)$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2.C_2\wr C_6$, $C_2^7.(A_4\times S_3^2)$, $C_2^7.(A_4\times S_3^2)$, $A_4^2.C_2\wr C_6$, $C_2^2.A_4^3:D_4$, $(C_2^2\times S_4^2).S_4$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$

Order 41472: $A_4^2.A_4^2.C_2$ x 3, $C_2^6.C_3^4.C_2^3$ x 2, $C_2^6.C_3^4.C_2^3$ x 2, $C_3.S_4^3$ x 2, $(S_3\times A_4).S_4^2$ x 2, $(C_3\times A_4).A_4^2.C_2^3$, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $(C_3\times A_4^2).A_4.C_2^3$, $C_2^6.C_3^3.A_4.C_2$, $A_4.C_2^6.C_3^3.C_2$, $A_4^2.C_2^2.C_6^2.C_2$, $A_4^2.(C_6\times A_4).C_2^2$, $C_2^6.C_3^4.C_2^3$, $C_2^6.C_3^4.C_2^3$, $(C_3\times S_4^2):D_{12}$, $(S_3^2\times A_4^2).D_4$, $A_4^2\wr C_2$

Order 36864: $C_2^5.C_6^2.C_2^4.C_2$ x 2, $C_2^5.C_6^2.C_2^4.C_2$ x 2, $C_2^6.C_6^2.C_2^4$ x 2, $A_4^2.C_2^4.C_2^3.C_2$ x 2, $A_4^2.C_2^6.C_2^2$, $C_2^5.C_6^2.C_2^4.C_2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^5.C_2^3$, $C_2^5.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^3.C_2^4.C_2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^3.C_2^4.C_2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $C_2^5.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^4.C_2^4$, $A_4^2.C_2^5.C_2^3$, $C_2^5.C_6^2.C_2^4.C_2$, $C_2^6.S_4^2$, $(C_2^4\times C_4).S_4^2$

Order 27648: $C_2\times S_4^3$ x 11, $C_2\times C_2^6.C_3^3.C_2^3$ x 8, $A_4^2.A_4.C_2^4$ x 8, $C_2^6.C_3^3.C_2^3.C_2$ x 8, $C_2^6.C_3^3.C_2^3.C_2$ x 6, $C_2^8.C_3^2.D_6$ x 6, $C_2\times A_4^2.C_2^2\times S_4$ x 5, $C_2^6.C_3^3.C_2^3.C_2$ x 5, $C_2^8.C_3^2.D_6$ x 4, $A_4^2.C_2^4.C_6.C_2$ x 4, $C_2^8.C_3^2.D_6$ x 4, $A_4.C_2^5.C_6^2.C_2$ x 4, $C_2^2\times C_2^6.C_3^2.D_6$ x 3, $A_4.C_2^5.C_6^2.C_2$ x 3, $C_2^8.C_3^2.D_6$ x 3, $C_2^8.C_3^2.D_6$ x 3, $C_2^4:C_3.A_4^2.C_2^2$ x 3, $C_2^6.C_3^3.C_2^4$ x 2, $C_2^6.(S_3\times C_3:D_{12})$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^2.C_2\times S_4$ x 2, $A_4^2.(C_2\times D_4\times A_4)$ x 2, $C_2^6.C_3^3.D_4.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.(A_4\times S_3^2)$ x 2, $C_2^6.(C_6\times S_3):D_6$ x 2, $C_2^8.C_3^2.D_6$ x 2, $C_2^2\times C_2^6.C_3^2.D_6$ x 2, $C_2^6.C_3^2.C_2^2\times A_4$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $A_4^2.(C_2\times D_4\times A_4)$ x 2, $C_2^6.C_3^3.C_2^4$ x 2, $C_2^8.C_3^2.D_6$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^3.D_4.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^2\times A_4^2.A_4.C_2^2$ x 2, $C_2^6.C_3^3.D_4.C_2$ x 2, $A_4^2.C_2^4.C_6.C_2$ x 2, $A_4^2.(C_2\times D_4\times A_4)$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $A_4.C_2^6.C_6^2$ x 2, $C_2^6.C_6^2.C_6.C_2$ x 2, $C_2^6.(C_3:S_3\times D_{12})$ x 2, $C_2^8.(C_3\times S_3^2)$ x 2, $A_4^2:C_2^3.S_4$ x 2, $C_2^6.(A_4\times S_3^2)$ x 2, $A_4^2:(C_2^4:C_{12})$ x 2, $A_4^2.C_2^3:S_4$ x 2, $\PSOPlus(4,3).\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times A_4^2).S_4$ x 2, $(C_2^3\times A_4^2).S_4$ x 2, $C_2^8.(C_3^2\times D_6)$ x 2, $C_2^9.(S_3\times C_3^2)$ x 2, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $(C_2^4\times A_4).D_6:D_6$, $(C_3\times A_4^2).C_2^5.C_2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^2\times C_2\times A_4^2\times S_4$, $C_2^2\times A_4.A_4^2.C_2^2$, $A_4.A_4^2.C_2^4$, $A_4.C_2^5.C_6^2.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2.C_2^3$, $A_4.A_4^2.C_2^3.C_2$, $A_4^2.A_4.C_2.C_2^3$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.(S_3\times D_6:S_3)$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^4.C_6^3.C_2^3$, $A_4^2.(D_4\times A_4).C_2$, $A_4^2.A_4.C_2^3.C_2$, $A_4^2.(C_2\times D_4\times A_4)$, $A_4^2.(D_4\times A_4).C_2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_6^3.C_2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.D_4.C_2$, $(C_3\times A_4^2).C_2.C_2^5$, $A_4^2.A_4.C_2^4$, $C_2^7.C_6^3$, $C_2^6.C_3^3.C_2^4$, $(C_3\times A_4^2).C_2.C_2^5$, $A_4^2.C_2^2\times C_2^4:C_3$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^2.D_6.C_2^2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^2.D_6.C_2^2$, $C_2^6.C_3^2.C_2\times S_4$, $A_4^2.(D_4\times A_4).C_2$, $(C_2^4\times A_4).D_6\times A_4$, $C_2^6.C_3^3.C_2.C_2^3$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $(C_2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $C_2\times A_4^2:\GL(2,\mathbb{Z}/4)$, $(C_2\times A_4^3).D_4$, $C_2\times A_4^3:D_4$, $C_2\times S_4^2:S_4$, $(A_4\times S_4^2):C_4$, $C_2^7.S_3^3$, $C_2^7.S_3^3$, $C_2^7.S_3^3$, $C_2\times A_4^3:D_4$, $A_4\times S_4^2:C_4$, $C_2^4:A_4^2.C_{12}$, $C_2^4.A_4^2:D_6$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 24576: $C_2^5.(C_2^3\times C_6).C_2^4$, $C_2^5.(C_2^3\times C_6).C_2^4$

Order 20736: $A_4^3.D_6$ x 4, $(C_3\times A_4).A_4^2.C_2^2$ x 2, $A_4^2.C_2^2.C_6^2$ x 2, $(C_3\times A_4^2).A_4.C_2^2$ x 2, $C_2^6.C_3^3.D_6$ x 2, $A_4.(C_3\times A_4^2).C_2^2$ x 2, $C_3\times A_4\times S_4^2$ x 2, $A_4^3:D_6$ x 2, $A_4^3:D_6$ x 2, $A_4^3.D_6$ x 2, $A_4^3:D_6$ x 2, $C_2^6.C_3^4.C_2^2$, $(C_2^5\times C_6).C_3.C_6^2$, $C_2^4.C_3^4.C_2^4$, $C_2^4.C_3^4.C_2^4$, $C_2^4.C_3^4.C_2^4$, $C_2^8.C_3^4$, $A_4^3:D_6$, $(C_3\times A_4).S_4^2$, $A_4^3:D_6$, $A_4^3:D_6$, $(C_6\times A_4^2):D_{12}$, $(C_6\times A_4^2).D_{12}$, $A_4^2:(S_3^2:C_4)$, $A_4^2:S_3^2:C_2^2$, $A_4^2:C_6.D_{12}$, $A_4^2:(C_6^2:C_4)$, $(C_2^4\times C_6).S_3^3$, $(C_2^4\times C_6).S_3^3$, $(S_3^2\times A_4^2):C_2^2$, $(S_3^2\times A_4^2):C_4$, $(C_3\times S_4^2):C_{12}$, $C_6\times S_4^2:S_3$

Order 18432: $C_2^6.D_6^2.C_2$ x 8, $A_4^2.C_2^5.C_2^2$ x 6, $A_4^2.C_2^6.C_2$ x 6, $A_4^2.C_2^4.C_2^3$ x 5, $A_4^2.C_2^5.C_2^2$ x 4, $A_4^2.C_2^3.C_2^3.C_2$ x 4, $A_4^2.C_2^5.C_2^2$ x 4, $A_4^2.C_2^4.C_2^3$ x 4, $A_4^2.C_2^5.C_2^2$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $A_4^2.C_2^4.C_2^3$ x 3, $(C_2^2\times S_4^2).D_4$ x 3, $A_4^2.C_2\wr D_4$ x 3, $A_4^2.C_2^6.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^2.C_2^4.C_2$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^6.D_6^2.C_2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^2\times C_2^6.C_3^2.C_2^3$ x 2, $C_2^2\times C_2\times C_2^2:S_4\times S_4$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^6.C_2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^6.D_6^2.C_2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $C_2^8.C_3.D_6.C_2$ x 2, $C_2^4.(C_6\times A_4).C_2^4$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^4.(C_6^2\times D_4).C_2^2$ x 2, $A_4^2.C_2^6.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $A_4.C_2^5.C_6.C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^4:C_3.C_2^3.C_6.C_2^3$ x 2, $C_2^4.(C_6^2\times D_4).C_2^2$ x 2, $C_2^5.S_4^2$ x 2, $C_2^9.S_3^2$ x 2, $C_2^9.S_3^2$ x 2, $C_2^5.S_4^2$ x 2, $A_4^2.C_2\wr D_4$ x 2, $(C_2^2\times S_4^2).D_4$ x 2, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $C_2^4.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^6.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2^5.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^6.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4.(C_2^4\times A_4).C_2^3$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $C_2^6.(C_6\times S_3).C_2^3$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $A_4^2.(C_2^2\times C_4^2).C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4.(C_2^4\times A_4).C_2^3$, $A_4^2.C_2^5.C_2^2$, $C_2^4:C_3.C_2^4.C_6.C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2^5.C_2$, $A_4^2.C_2^6.C_2$, $C_2^4.(C_6\times C_{12}).C_2^4$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $C_2^4.(C_6^2\times D_4).C_2^2$, $C_2^4.(C_2\times C_6^2).C_2^4$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $C_2^6.C_6^2.C_2^3$, $A_4^2.C_2^4.C_2^3$, $(C_2\times A_4^2).C_2^6$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^6.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $C_2^2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_2^3.C_2^4$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $C_2^3\times A_4^2.C_2^4$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^6.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^6.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2^4:C_3.C_2^3.C_6.C_2^3$, $C_2^4.(C_6^2\times D_4).C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2^6.C_{12}:S_4$, $C_2^6.\PSOPlus(4,3)$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$, $C_2^6.(A_4\times S_4)$, $C_2^7.(C_6\times S_4)$, $C_2^6.(C_{12}\times S_4)$, $A_4^2:C_2\wr D_4$, $(C_2^2\times S_4^2).D_4$, $(C_2^2\times S_4^2).D_4$, $(C_2^3\times S_4^2).C_4$, $C_2^9.S_3^2$

Order 13824: $A_4^2.C_2^4.C_6$ x 4, $C_2^8.C_3^3.C_2$ x 2

Order 9216: $C_2^4\times A_4^2.C_2^2$, $C_2^3\times A_4^2.C_2^3$, $C_2^2\times A_4^2.C_2^4$

Order 8192: $C_2^7.C_2^5.C_2$

Order 6912: $C_2^8.C_3^3$ x 2

Order 4608: $C_2^2\times A_4^2.C_2^3$ x 2, $C_2^4\times C_2\times A_4^2$, $C_2^3\times A_4^2.C_2^2$

Order 2304: $C_2^4\times A_4^2$, $C_2^2\times A_4^2.C_2^2$

Order 1536: $C_2^9.C_3$ x 2

Order 1152: $C_2\times S_4^2$

Order 768: $C_2^8.C_3$ x 2

Order 576: $A_4^2:C_2^2$ x 2, $S_4^2$

Order 512: $C_2^9$

Order 288: $C_2\times A_4^2$ x 2, $\PSOPlus(4,3)$ x 2

Order 256: $C_2^8$

Order 144: $A_4^2$ x 2

Order 96: $C_2^3:A_4$ x 2

Order 81: $C_3^4$

Order 48: $C_2^2:A_4$ x 2

Order 32: $C_2^5$ x 2

Order 16: $C_2^4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 663552: $A_4^2.S_4^2:D_4$ ($C_1$)

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

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

Order 110592: $(C_2^3\times S_4^2).S_4$ ($S_3$), $(C_2^3\times S_4^2).S_4$ ($S_3$)

Order 82944: $A_4^2.A_4^2.C_2^2$ ($D_4$) x 2, $C_2^8.C_3^3.D_6$ ($D_4$) x 2, $A_4^2.A_4^2.C_2^2$ ($C_2^3$), $A_4^2.A_4^2.C_2^2$ ($D_4$), $C_2^8.C_3^3.D_6$ ($D_4$)

Order 55296: $A_4^2.C_2^4.C_6.C_2^2$ ($D_6$) x 2, $A_4^2:C_2^4.S_4$ ($D_6$) x 2, $A_4^2:C_2^4.S_4$ ($D_6$), $A_4^2.C_2^2\wr S_3$ ($D_6$)

Order 41472: $A_4^2.A_4^2.C_2$ ($C_2\times D_4$) x 2, $C_2^8.C_3^4.C_2$ ($C_2\times D_4$)

Order 27648: $A_4^2.C_2^4.C_6.C_2$ ($C_2\times D_6$) x 2, $A_4^2.C_2^4.C_6.C_2$ ($C_3:D_4$) x 2, $A_4^2.C_2^4.C_6.C_2$ ($D_{12}$) x 2

Order 20736: $C_2^8.C_3^4$ ($C_2^2\wr C_2$)

Order 18432: $C_2^3\times A_4^2.C_2^4$ ($S_3^2$)

Order 13824: $C_2^8.C_3^3.C_2$ ($S_3\times D_4$) x 2, $A_4^2.C_2^4.C_6$ ($S_3\times D_4$) x 2, $A_4^2.C_2^4.C_6$ ($C_6:D_4$), $A_4^2.C_2^4.C_6$ ($C_2\times D_{12}$)

Order 9216: $C_2^2\times A_4^2.C_2^4$ ($C_3:D_{12}$) x 2, $C_2^4\times A_4^2.C_2^2$ ($\SOPlus(4,2)$), $C_2^3\times A_4^2.C_2^3$ ($S_3\times D_6$)

Order 6912: $C_2^8.C_3^3$ ($C_2^2:D_{12}$), $C_2^8.C_3^3$ ($D_6:D_4$)

Order 4608: $C_2^4\times C_2\times A_4^2$ ($S_3^2:C_2^2$), $C_2^3\times A_4^2.C_2^2$ ($D_6:D_6$), $C_2^2\times A_4^2.C_2^3$ ($D_6:D_6$), $C_2^2\times A_4^2.C_2^3$ ($C_6:D_{12}$)

Order 2304: $C_2^4\times A_4^2$ ($D_6\wr C_2$), $C_2^2\times A_4^2.C_2^2$ ($C_6^2:D_4$)

Order 1536: $C_2^9.C_3$ ($S_3^3:C_2$) x 2

Order 1152: $C_2\times S_4^2$ ($\POPlus(4,3)$)

Order 768: $C_2^8.C_3$ ($S_3^2:D_{12}$), $C_2^8.C_3$ ($D_6:\SOPlus(4,2)$)

Order 576: $S_4^2$ ($A_4^2:D_4$) x 2, $A_4^2:C_2^2$ ($A_4^2:C_2^3$), $A_4^2:C_2^2$ ($S_4\wr C_2$)

Order 512: $C_2^9$ ($S_3^2:S_3^2$)

Order 288: $C_2\times A_4^2$ ($C_2\wr S_3^2$), $C_2\times A_4^2$ ($S_4^2:C_2^2$), $\PSOPlus(4,3)$ ($C_2^6.S_3^2$), $\PSOPlus(4,3)$ ($C_2\wr S_3^2$)

Order 256: $C_2^8$ ($C_3^4.C_2^4.C_2$)

Order 144: $A_4^2$ ($C_2^7.S_3^2$), $A_4^2$ ($S_4^2:D_4$)

Order 96: $C_2^3:A_4$ ($S_4^2:D_6$) x 2

Order 48: $C_2^2:A_4$ ($(C_3\times S_4^2):D_4$), $C_2^2:A_4$ ($S_4^2:D_{12}$)

Order 32: $C_2^5$ ($C_3^2.A_4^2.C_2^3.C_2$), $C_2^5$ ($S_4^2:S_3^2$)

Order 16: $C_2^4$ ($(C_3\times S_4^2):D_{12}$), $C_2^4$ ($(S_3^2\times A_4^2).D_4$)

Order 2: $C_2$ ($S_4^2.\POPlus(4,3)$)

Order 1: $C_1$ ($A_4^2.S_4^2:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 663552: $A_4^2.S_4^2:D_4$ ($C_1$)

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

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

Order 110592: $(C_2^3\times S_4^2).S_4$ ($S_3$), $(C_2^3\times S_4^2).S_4$ ($S_3$)

Order 82944: $A_4^2.A_4^2.C_2^2$ ($D_4$), $A_4^2.A_4^2.C_2^2$ ($C_2^3$), $A_4^2.A_4^2.C_2^2$ ($D_4$), $C_2^8.C_3^3.D_6$ ($D_4$), $C_2^8.C_3^3.D_6$ ($D_4$)

Order 55296: $A_4^2.C_2^4.C_6.C_2^2$ ($D_6$) x 2, $A_4^2:C_2^4.S_4$ ($D_6$) x 2, $A_4^2:C_2^4.S_4$ ($D_6$), $A_4^2.C_2^2\wr S_3$ ($D_6$)

Order 41472: $A_4^2.A_4^2.C_2$ ($C_2\times D_4$) x 2, $C_2^8.C_3^4.C_2$ ($C_2\times D_4$)

Order 27648: $A_4^2.C_2^4.C_6.C_2$ ($C_2\times D_6$) x 2, $A_4^2.C_2^4.C_6.C_2$ ($C_3:D_4$), $A_4^2.C_2^4.C_6.C_2$ ($D_{12}$)

Order 20736: $C_2^8.C_3^4$ ($C_2^2\wr C_2$)

Order 18432: $C_2^3\times A_4^2.C_2^4$ ($S_3^2$)

Order 13824: $C_2^8.C_3^3.C_2$ ($S_3\times D_4$) x 2, $A_4^2.C_2^4.C_6$ ($S_3\times D_4$) x 2, $A_4^2.C_2^4.C_6$ ($C_6:D_4$), $A_4^2.C_2^4.C_6$ ($C_2\times D_{12}$)

Order 9216: $C_2^4\times A_4^2.C_2^2$ ($\SOPlus(4,2)$), $C_2^3\times A_4^2.C_2^3$ ($S_3\times D_6$), $C_2^2\times A_4^2.C_2^4$ ($C_3:D_{12}$)

Order 6912: $C_2^8.C_3^3$ ($C_2^2:D_{12}$), $C_2^8.C_3^3$ ($D_6:D_4$)

Order 4608: $C_2^4\times C_2\times A_4^2$ ($S_3^2:C_2^2$), $C_2^3\times A_4^2.C_2^2$ ($D_6:D_6$), $C_2^2\times A_4^2.C_2^3$ ($D_6:D_6$), $C_2^2\times A_4^2.C_2^3$ ($C_6:D_{12}$)

Order 2304: $C_2^4\times A_4^2$ ($D_6\wr C_2$), $C_2^2\times A_4^2.C_2^2$ ($C_6^2:D_4$)

Order 1536: $C_2^9.C_3$ ($S_3^3:C_2$) x 2

Order 1152: $C_2\times S_4^2$ ($\POPlus(4,3)$)

Order 768: $C_2^8.C_3$ ($S_3^2:D_{12}$), $C_2^8.C_3$ ($D_6:\SOPlus(4,2)$)

Order 576: $A_4^2:C_2^2$ ($A_4^2:C_2^3$), $A_4^2:C_2^2$ ($S_4\wr C_2$), $S_4^2$ ($A_4^2:D_4$)

Order 512: $C_2^9$ ($S_3^2:S_3^2$)

Order 288: $C_2\times A_4^2$ ($C_2\wr S_3^2$), $C_2\times A_4^2$ ($S_4^2:C_2^2$), $\PSOPlus(4,3)$ ($C_2^6.S_3^2$), $\PSOPlus(4,3)$ ($C_2\wr S_3^2$)

Order 256: $C_2^8$ ($C_3^4.C_2^4.C_2$)

Order 144: $A_4^2$ ($C_2^7.S_3^2$), $A_4^2$ ($S_4^2:D_4$)

Order 96: $C_2^3:A_4$ ($S_4^2:D_6$) x 2

Order 48: $C_2^2:A_4$ ($(C_3\times S_4^2):D_4$), $C_2^2:A_4$ ($S_4^2:D_{12}$)

Order 32: $C_2^5$ ($C_3^2.A_4^2.C_2^3.C_2$), $C_2^5$ ($S_4^2:S_3^2$)

Order 16: $C_2^4$ ($(C_3\times S_4^2):D_{12}$), $C_2^4$ ($(S_3^2\times A_4^2).D_4$)

Order 2: $C_2$ ($S_4^2.\POPlus(4,3)$)

Order 1: $C_1$ ($A_4^2.S_4^2:D_4$)

Series

Derived series

$A_4^2.S_4^2:D_4$

$\rhd$

$A_4^2.A_4^2.C_2^2$

$\rhd$

$C_2^2\times A_4^2.C_2^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$A_4^2.S_4^2:D_4$

$\rhd$

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

$\rhd$

$A_4^2.A_4^2.C_2^3$

$\rhd$

$A_4^2.A_4^2.C_2^2$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8.C_3^3$

$\rhd$

$C_2^2\times A_4^2.C_2^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$A_4^2.S_4^2:D_4$

$\rhd$

$A_4^2.A_4^2.C_2^2$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

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

Character theory

Complex character table

The $410 \times 410$ character table is not available for this group.

Rational character table

The $400 \times 400$ rational character table is not available for this group.