Abstract group 331776.ce: $S_4^2.\POPlus(4,3)$

• Label: 331776.ce
• Order: \( 2^{12} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $16$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$S_4^2.\POPlus(4,3)$
Order:	\(331776\)\(\medspace = 2^{12} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^8.C_3^4.D_4^2$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \)
Composition factors:	$C_2$ x 12, $C_3$ x 4
Derived length:	$4$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	5775	6560	58224	92640	13824	127104	27648	331776
Conjugacy classes	1	28	12	55	65	4	53	2	220
Divisions	1	28	12	55	65	4	53	2	220
Autjugacy classes	1	24	8	41	41	2	32	1	150

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$32$
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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid b^{6}=c^{6}=d^{6}=e^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,3)(6,7)(9,10,12)(11,15)(14,16), (1,4,3,7)(2,6,5,8)(9,11,12,15)(10,14,13,16), (1,3,2,5)(4,6,7)(10,13)(11,16,14,15)\rangle$
Transitive group:	36T28193	36T28194	36T28195	36T28196	all 8
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:	$S_4^2$ . $\POPlus(4,3)$	$C_2^8$ . $(S_3^2:S_3^2)$	$C_2^4$ . $(S_4^2:S_3^2)$	$A_4^2$ . $(C_2\wr S_3^2)$	all 32

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 49 normal subgroups (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_4^2.\POPlus(4,3)$
Commutator:	$G' \simeq$ $A_4^2.A_4^2.C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_4^2.\POPlus(4,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8$	$G/\operatorname{Fit} \simeq$ $S_3^2:S_3^2$
Radical:	$R \simeq$ $S_4^2.\POPlus(4,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^8$	$G/\operatorname{soc} \simeq$ $S_3^2:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.D_4^2$
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 5184) are shown, as well as all normal subgroups of any index.

Order 331776: $S_4^2.\POPlus(4,3)$

Order 165888: $A_4^2.S_4\wr C_2$ x 2, $A_4^2.S_4\wr C_2$ x 2, $C_2^8.C_3.S_3^3$, $A_4^2\wr C_2.C_2^2$, $A_4^2.(C_2^5.S_3^2)$

Order 110592: $A_4^2.C_2\wr D_6$ x 2

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

Order 55296: $C_2^8.S_3^3$ x 3, $C_2^8.S_3^3$ x 2, $A_4^2.(C_2^5.D_6)$ 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, $C_2^6.C_3^3.C_2^4.C_2$, $C_2^6.C_3^3.C_2.C_2^4$

Order 41472: $A_4^2.A_4^2.C_2$ x 2, $A_4^2\wr C_2$ x 2, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^3.A_4.C_2$, $A_4.C_2^6.C_3^3.C_2$, $C_3.S_4^3$, $(S_3\times A_4).S_4^2$

Order 36864: $(C_2^3\times S_4^2).D_4$, $D_4^2.\POPlus(4,3)$

Order 27648: $C_2^6.C_3^3.C_2^3.C_2$ x 4, $C_2^8.C_3^2.D_6$ x 4, $C_2^8.(C_3\times S_3^2)$ x 4, $A_4^2.C_2^3:S_4$ x 4, $C_2^8.C_3^2.D_6$ x 3, $C_2^6.(A_4\times S_3^2)$ x 3, $A_4^2.C_2^3:S_4$ x 3, $C_2^6.C_3^3.C_2^4$ x 2, $C_2^8.C_3^2.D_6$ x 2, $C_2^6.(A_4\times S_3^2)$ x 2, $C_2^6.(S_3\times D_6:S_3)$ x 2, $C_2^6.C_3^3.D_4.C_2$ 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, $A_4^2.(C_2\times D_4\times A_4)$ x 2, $A_4^2.C_2^4.C_6.C_2$ x 2, $A_4^2:C_2^3.S_4$ x 2, $A_4^2:(C_2^4:C_{12})$ x 2, $S_4^3:C_2$ x 2, $C_2^4.A_4^2:D_6$ x 2, $C_2^6.(S_3\times C_3:D_{12})$, $C_2^6.C_3^3.C_2^3.C_2$, $A_4^2.(C_2\times D_4\times A_4)$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $(C_2^4\times A_4).D_6:D_6$, $C_2^6.(C_6\times S_3):D_6$, $C_2^6.C_3^3.D_4.C_2$, $A_4^2.A_4.C_2^4$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2.C_2^3$, $C_2^6.C_3^3.C_2^3.C_2$, $A_4.C_2^5.C_6^2.C_2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.(C_3:S_3\times D_{12})$, $C_2\times S_4^3$

Order 20736: $A_4^3:D_6$ x 3, $A_4^3:D_6$ x 2, $A_4^3.D_6$ x 2, $C_2^8.C_3^4$, $C_3\times A_4\times S_4^2$, $A_4^3:D_6$, $(C_3\times A_4).S_4^2$, $A_4^3.D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $S_3^2:\POPlus(4,3)$, $S_4^2:S_3^2$

Order 18432: $A_4^2.C_2\wr D_4$ x 4, $C_2^7.D_6^2$ x 3, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^5.S_4^2$ x 2, $(C_2^2\times S_4^2).D_4$ x 2, $A_4^2:C_2\wr D_4$ x 2, $(C_2^2\times S_4^2).D_4$ x 2, $(C_2^2\times S_4^2).D_4$ x 2, $A_4^2.C_2\wr D_4$ x 2, $C_2^3.C_2\wr S_3^2$ x 2, $(C_2^2\times S_4^2).D_4$ x 2, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $(C_2^5\times C_4).D_6^2$, $C_2^3.C_2\wr S_3^2$, $(C_2^3\times S_4^2).C_4$, $(C_2^3\times S_4^2).C_2^2$, $(C_4\times S_4^2).C_2^3$, $(C_2^2\times S_4^2).C_2^3$, $A_4^2:C_2^4.D_4$

Order 13824: $A_4^3.C_2^3$ x 14, $S_4^3$ x 8, $A_4^3.C_2^3$ x 6, $C_2^6.C_3^3.D_4$ x 5, $C_2^8.C_3^3.C_2$ x 5, $A_4.C_2^6.C_3.C_6$ x 4, $C_2^6.C_3^3.D_4$ x 4, $C_2^6.C_6^2.C_6$ x 4, $A_4^2.(D_4\times A_4)$ x 4, $(C_2^2\times A_4^2).S_4$ x 4, $C_2^8:(S_3\times C_3^2)$ x 4, $C_2^6.C_3^3.D_4$ x 3, $A_4^2.A_4.C_2^3$ x 3, $C_2^4:C_3.A_4^2.C_2$ x 3, $A_4.C_2^5.C_6^2$ x 3, $C_2^6.C_3^3.D_4$ x 3, $C_2^2:A_4^2.S_4$ x 3, $A_4^3.C_2^3$ x 3, $C_2^6.C_3^2.D_6.C_2$ x 2, $A_4^2.C_2^2\times C_2\times A_4$ x 2, $A_4.A_4^2.C_2^3$ x 2, $C_2^6.C_3^3.D_4$ x 2, $A_4^2.C_2^4.C_6$ x 2, $C_2^6.C_3^3.C_2^3$ x 2, $A_4^2.A_4.C_2^2.C_2$ x 2, $A_4^2.(D_4\times A_4)$ x 2, $A_4^2.(D_4\times A_4)$ x 2, $A_4.C_2^6.C_3.C_6$ x 2, $C_2^6.C_6^2.C_6$ x 2, $C_2^4:A_4^2.C_6$ x 2, $C_2^8.(S_3\times C_3^2)$ x 2, $C_{12}.(C_2\times S_4^2)$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^2.A_4^2:S_4$ x 2, $A_4^2:C_4\times S_4$ x 2, $A_4^2:\GL(2,\mathbb{Z}/4)$ x 2, $A_4^3:D_4$ x 2, $S_4^2:S_4$ x 2, $A_4^3:D_4$ x 2, $C_2^6.C_3^3.C_2^2.C_2$, $C_2^6.C_3^3.D_4$, $\PSOPlusPlus(4,3)\times C_2^4:C_3$, $C_2^6.C_3^2.D_6.C_2$, $A_4^2.A_4.C_2^2.C_2$, $C_2^6.C_3^3.C_4.C_2$, $C_2^6.C_3^3.C_2^2.C_2$, $C_2^6.C_3^2.D_6.C_2$, $A_4.C_2^5.C_6^2$, $C_2^7.C_3.C_6^2$, $A_4^2.(D_4\times A_4)$, $A_4.A_4^2.C_2^2.C_2$, $C_2^6.C_3^3.C_4.C_2$, $(D_4\times A_4^2).D_6$, $C_2^6.S_3^3$, $A_4^3.C_2^3$

Order 12288: $C_2^6.(D_4\times S_4)$ x 2

Order 10368: $A_4^3:C_6$ x 3, $A_4^3:S_3$ x 3, $A_4^3:S_3$ x 2, $A_4^3:S_3$ x 2, $A_4^3:C_6$ x 2, $C_3:S_4^2\times S_3$ x 2, $(S_3\times A_4)\wr C_2$ x 2, $A_4^2:(S_3\times D_6)$ x 2, $A_4^2:\SOPlus(4,2)$ x 2, $(C_3\times A_4^2):D_{12}$ x 2, $(C_3\times S_4)\wr C_2$ x 2, $A_4^3:C_6$, $S_3\times A_4^3$, $S_3^2\times \PSOPlus(4,3)$, $C_3:S_3\times S_4^2$, $\PSOPlus(4,3).S_3^2$, $C_6^2:C_2^3.S_3^2$

Order 9216: $C_2^4.S_4^2$ x 6, $A_4^2.C_2^4.C_2^2$ x 5, $C_2\times S_4^2:D_4$ x 5, $(C_2^3\times A_4^2):D_4$ x 5, $(C_2^3\times A_4^2):D_4$ x 5, $C_2^6.(S_3\times C_3:D_4)$ x 4, $A_4^2.D_4^2$ x 4, $C_2^6.C_6^2.C_2^2$ x 4, $A_4^2.D_4^2$ x 4, $A_4^2.C_2^3.C_2^3$ x 4, $A_4^2.D_4^2$ x 4, $A_4^2.C_2^3.C_2^3$ x 4, $A_4^2.D_4^2$ x 4, $A_4^2.D_4^2$ x 4, $C_2^6.D_6:D_6$ x 4, $C_2^6.D_6^2$ x 4, $C_2^4.S_4^2$ x 4, $C_2^5.(A_4\times S_4)$ x 4, $C_2^4.S_4^2$ x 4, $C_2^8.S_3^2$ x 4, $(C_2\times S_4^2).D_4$ x 4, $C_2^6.D_6^2$ x 4, $A_4^2:D_4^2$ x 4, $(D_4\times S_4^2):C_2$ x 4, $A_4^2.C_2\wr C_4$ x 4, $C_2^6.(C_4\times S_3^2)$ x 4, $A_4^2:C_2\wr C_4$ x 4, $A_4^2.C_2^5.C_2$ x 3, $A_4^2.D_4^2$ x 3, $C_2^2\wr S_3:D_{12}$ x 3, $C_2^7.(S_3\times D_6)$ x 3, $C_2^7:\SOPlus(4,2)$ x 3, $C_2^8:S_3^2$ x 3, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $C_2^6.(S_3\times D_{12})$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.C_2^5.C_2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4.C_2^5.C_6.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.D_4^2$ x 2, $C_2^6.(S_3\times D_{12})$ x 2, $(C_2\times A_4^2).C_2^5$ x 2, $A_4^2.D_4^2$ x 2, $C_2^8.S_3^2$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4.(C_2^4\times A_4).C_2^2$ x 2, $A_4^2.D_4^2$ x 2, $C_2^6.(C_4\times S_3^2)$ x 2, $C_2^4.(C_6\times A_4).C_2^3$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $C_2^6.C_4:S_3^2$ x 2, $C_2^8.S_3^2$ x 2, $(C_2^2:S_4)^2$ x 2, $C_2^6.C_6:S_4$ x 2, $C_2^5.\PSOPlus(4,3)$ x 2, $C_2^4.S_4^2$ x 2, $C_2^4.A_4^2:C_4$ x 2, $(C_2^2\times S_4^2):C_2^2$ x 2, $(C_2^3\times A_4^2).D_4$ x 2, $(C_2^3\times A_4^2).D_4$ x 2, $(C_2^2\times A_4^2).C_4^2$ x 2, $A_4^2.D_4:D_4$ x 2, $(D_4\times C_2^5).S_3^2$ x 2, $C_2^6.D_6^2$ x 2, $C_2^7:\SOPlus(4,2)$ x 2, $C_2^7.\SOPlus(4,2)$ x 2, $C_2\wr S_3^2:C_4$ x 2, $(C_2^2\times S_4^2):C_4$ x 2, $C_2^6.C_3^2.D_4.C_2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.D_4^2$, $A_4.(C_2^4\times A_4).C_2^2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.D_4^2$, $C_2^6.C_6^2.C_2^2$, $A_4^2.C_2^3.C_2^3$, $C_2^3\times A_4^2.C_2^3$, $C_4^2:\POPlus(4,3)$, $C_4^2:\POPlus(4,3)$, $A_4^2.D_4:D_4$, $(D_4\times C_2^5):S_3^2$, $(C_2\times S_4^2).C_2^3$, $(C_4\times S_4^2).C_2^2$, $A_4^2:C_4^2:C_2^2$, $A_4^2:C_2^4:C_4$, $C_4.C_2\wr S_3^2$

Order 6912: $A_4:S_4^2$ x 41, $A_4\times S_4^2$ x 19, $A_4.A_4^2.C_2^2$ x 10, $A_4^3:C_2^2$ x 9, $S_4\times C_3:\GL(2,\mathbb{Z}/4)$ x 8, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ x 6, $A_4^3:C_2^2$ x 6, $A_4:S_4^2$ x 6, $C_2^8.C_3^3$ x 4, $S_3\times A_4:\GL(2,\mathbb{Z}/4)$ x 4, $D_6:S_4^2$ x 4, $(A_4\times C_2^4):S_3^2$ x 4, $(C_2\times C_6):S_4^2$ x 4, $(C_2\times C_6):S_4^2$ x 4, $C_2^2\times C_3:S_4^2$ x 4, $A_4^3:C_2^2$ x 4, $A_4^3:C_2^2$ x 4, $A_4^2.A_4.C_4$ x 3, $A_4^2:S_3\times D_4$ x 3, $(C_2\times C_6):S_4^2$ x 3, $(C_2^2\times A_4^2):D_6$ x 3, $D_6\times S_4^2$ x 3, $C_2^2.A_4^3$ x 3, $A_4^3:C_2^2$ x 3, $A_4^3:C_2^2$ x 3, $A_4.A_4^2.C_4$ x 2, $C_2^6.C_3^3.C_2^2$ x 2, $C_2^6.C_3^3.C_4$ x 2, $A_4^2:S_3\times D_4$ x 2, $A_4\times \GL(2,\mathbb{Z}/4):C_6$ x 2, $A_4\times C_2^4:S_3^2$ x 2, $D_6.S_4^2$ x 2, $D_6.S_4^2$ x 2, $S_3\times A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $D_6.S_4^2$ x 2, $C_{12}:S_4^2$ x 2, $D_6:S_4^2$ x 2, $C_{12}:S_4^2$ x 2, $C_2^5.S_3^3$ x 2, $C_3\times S_4\times \GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times C_6):S_4^2$ x 2, $C_{12}:S_4^2$ x 2, $D_6:S_4^2$ x 2, $C_2^4.(A_4\times S_3^2)$ x 2, $A_4\times C_2^4:S_3^2$ x 2, $S_3\times A_4^2.C_2^3$ x 2, $(C_2\times C_6):S_4^2$ x 2, $D_6:S_4^2$ x 2, $C_3\times C_2^2:S_4^2$ x 2, $C_{12}:S_4^2$ x 2, $A_4^2:(S_3\times D_4)$ x 2, $C_2^2\times A_4^2.D_6$ x 2, $C_2^2.A_4^3$ x 2, $C_2^2\wr C_3:S_3^2$ x 2, $A_4^3:C_4$ x 2, $A_4^3:C_4$ x 2, $S_4^2:D_6$ x 2, $D_6^2:(C_2\times S_4)$ x 2, $C_2\times A_4^2\times C_2\times A_4$, $A_4^2\times C_2^4:C_3$, $C_2^6.C_3^3.C_4$, $S_3\times D_4\times A_4^2$, $A_4^2:C_6\times D_4$, $S_3\times A_4^2:D_4$, $C_4\times A_4^2:D_6$, $C_2^5.S_3^3$, $(C_2\times C_6):S_4^2$, $(A_4\times D_{12}):S_4$, $C_2^5.S_3^3$, $D_6:S_4^2$, $D_{12}\times \PSOPlus(4,3)$, $C_2\times C_6\times S_4^2$

Order 6144: $C_2^5.(D_4\times S_4)$ x 5, $C_2^8.D_{12}$ x 4, $C_2^6.(C_4\times S_4)$ x 4, $C_2^6.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:C_3.C_2^5.C_2^2$ x 2, $C_2^5.(D_4\times S_4)$ x 2, $C_2^5.(D_4\times S_4)$ x 2, $(D_4\times C_2^5):D_{12}$ x 2, $(C_2^2\times C_4^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^2.C_2\wr C_6$ x 2, $A_4.C_2^5.C_2^4$, $A_4.C_2^5.C_2^4$

Order 5184: $A_4^2:S_3^2$ x 3, $C_3^2:S_4^2$ x 3, $A_4^2:S_3^2$ x 3, $C_3\times A_4^2:D_6$ x 3, $C_3\times A_4^3$ x 2, $C_3^2:S_4^2$ x 2, $C_3^2:S_4^2$ x 2, $A_4^2:S_3^2$ x 2, $(A_4\times C_6^2):D_6$ x 2, $C_3^2:S_4^2$ x 2, $C_6^2:C_6\times S_4$ x 2, $(C_3^2\times A_4^2):C_4$ x 2, $(C_3\times A_4^2):C_{12}$ x 2, $A_4^2:S_3^2$ x 2, $A_4^2:S_3^2$, $S_3^2\times A_4^2$, $C_6^2:C_6\times S_4$, $C_3^2:S_4^2$, $A_4^2:C_6\times S_3$, $C_3^2\times S_4^2$

Order 4608: $C_2^4\times \PSOPlusPlus(4,3)$, $C_2^3\times A_4^2.C_2^2$

Order 4096: $C_2^6.D_4^2$

Order 2304: $C_2^4\times A_4^2$, $C_2^3\times C_2\times A_4^2$

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

Order 576: $S_4^2$

Order 288: $\PSOPlus(4,3)$ x 2

Order 256: $C_2^8$

Order 144: $A_4^2$ x 2

Order 81: $C_3^4$

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

Order 16: $C_2^4$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 64 (order at least 5184) are shown, as well as all normal subgroups of any index.

Order 331776: $S_4^2.\POPlus(4,3)$

Order 165888: $C_2^8.C_3.S_3^3$, $A_4^2\wr C_2.C_2^2$, $A_4^2.(C_2^5.S_3^2)$, $A_4^2.S_4\wr C_2$, $A_4^2.S_4\wr C_2$

Order 110592: $A_4^2.C_2\wr D_6$

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

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

Order 41472: $A_4^2.A_4^2.C_2$ x 2, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^3.A_4.C_2$, $A_4.C_2^6.C_3^3.C_2$, $C_3.S_4^3$, $(S_3\times A_4).S_4^2$, $A_4^2\wr C_2$

Order 36864: $(C_2^3\times S_4^2).D_4$, $D_4^2.\POPlus(4,3)$

Order 27648: $C_2^6.C_3^3.C_2^3.C_2$ x 3, $C_2^6.C_3^3.C_2^4$ x 2, $C_2^8.C_3^2.D_6$ 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, $C_2^8.(C_3\times S_3^2)$ x 2, $C_2^6.(A_4\times S_3^2)$ x 2, $A_4^2.C_2^3:S_4$ x 2, $A_4^2.C_2^3:S_4$ x 2, $C_2^6.(S_3\times C_3:D_{12})$, $C_2^6.C_3^3.C_2^3.C_2$, $A_4^2.(C_2\times D_4\times A_4)$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $(C_2^4\times A_4).D_6:D_6$, $C_2^8.C_3^2.D_6$, $C_2^6.(A_4\times S_3^2)$, $C_2^6.(S_3\times D_6:S_3)$, $C_2^6.(C_6\times S_3):D_6$, $C_2^6.C_3^3.D_4.C_2$, $C_2^8.C_3^2.D_6$, $C_2^6.C_3^3.D_4.C_2$, $A_4^2.A_4.C_2^4$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.C_3^3.C_2.C_2^3$, $A_4^2.(C_2\times D_4\times A_4)$, $A_4^2.C_2^4.C_6.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $A_4.C_2^5.C_6^2.C_2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^6.(C_3:S_3\times D_{12})$, $A_4^2:C_2^3.S_4$, $A_4^2:(C_2^4:C_{12})$, $S_4^3:C_2$, $C_2^4.A_4^2:D_6$, $C_2\times S_4^3$

Order 20736: $A_4^3:D_6$ x 2, $A_4^3.D_6$ x 2, $C_2^8.C_3^4$, $A_4^3:D_6$, $C_3\times A_4\times S_4^2$, $A_4^3:D_6$, $(C_3\times A_4).S_4^2$, $A_4^3.D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $S_3^2:\POPlus(4,3)$, $S_4^2:S_3^2$

Order 18432: $C_2^7.D_6^2$ x 3, $C_2^3.C_2\wr S_3^2$ x 2, $A_4^2.C_2\wr D_4$ x 2, $C_2^6.D_6^2.C_2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^4.C_2^3$, $A_4^2.C_2^4.C_2^3$, $C_2^5.S_4^2$, $(C_2^2\times S_4^2).D_4$, $(C_2^5\times C_4).D_6^2$, $C_2^3.C_2\wr S_3^2$, $A_4^2:C_2\wr D_4$, $(C_2^2\times S_4^2).D_4$, $(C_2^2\times S_4^2).D_4$, $(C_2^3\times S_4^2).C_4$, $(C_2^3\times S_4^2).C_2^2$, $(C_4\times S_4^2).C_2^3$, $(C_2^2\times S_4^2).C_2^3$, $A_4^2.C_2\wr D_4$, $A_4^2:C_2^4.D_4$, $(C_2^2\times S_4^2).D_4$

Order 13824: $A_4^3.C_2^3$ x 13, $S_4^3$ x 5, $C_2^6.C_3^3.D_4$ x 4, $A_4^3.C_2^3$ x 4, $C_2^6.C_3^3.D_4$ x 3, $C_2^6.C_3^3.D_4$ x 3, $C_2^8.C_3^3.C_2$ x 3, $A_4^2.(D_4\times A_4)$ x 3, $A_4^2.A_4.C_2^3$ x 3, $A_4.C_2^5.C_6^2$ x 3, $C_2^6.C_3^3.D_4$ x 3, $A_4.C_2^6.C_3.C_6$ x 2, $C_2^6.C_6^2.C_6$ x 2, $A_4.A_4^2.C_2^3$ x 2, $C_2^6.C_3^3.C_2^3$ x 2, $C_2^4:C_3.A_4^2.C_2$ x 2, $C_2^2:A_4^2.S_4$ x 2, $A_4^3.C_2^3$ x 2, $(C_2^2\times A_4^2).S_4$ x 2, $C_2^8:(S_3\times C_3^2)$ x 2, $C_2^6.C_3^3.C_2^2.C_2$, $C_2^6.C_3^2.D_6.C_2$, $A_4^2.C_2^2\times C_2\times A_4$, $C_2^6.C_3^3.D_4$, $C_2^6.C_3^3.D_4$, $\PSOPlusPlus(4,3)\times C_2^4:C_3$, $A_4^2.C_2^4.C_6$, $C_2^6.C_3^2.D_6.C_2$, $A_4^2.A_4.C_2^2.C_2$, $C_2^6.C_3^3.C_4.C_2$, $C_2^6.C_3^3.C_2^2.C_2$, $C_2^6.C_3^2.D_6.C_2$, $A_4.C_2^5.C_6^2$, $C_2^7.C_3.C_6^2$, $A_4^2.A_4.C_2^2.C_2$, $A_4^2.(D_4\times A_4)$, $A_4^2.(D_4\times A_4)$, $A_4.A_4^2.C_2^2.C_2$, $A_4^2.(D_4\times A_4)$, $A_4.C_2^6.C_3.C_6$, $C_2^6.C_6^2.C_6$, $C_2^6.C_3^3.C_4.C_2$, $C_2^4:A_4^2.C_6$, $C_2^8.(S_3\times C_3^2)$, $(D_4\times A_4^2).D_6$, $C_{12}.(C_2\times S_4^2)$, $C_2^6.S_3^3$, $C_2^6.S_3^3$, $C_2^6.S_3^3$, $A_4^3.C_2^3$, $C_2^2.A_4^2:S_4$, $A_4^2:C_4\times S_4$, $A_4^2:\GL(2,\mathbb{Z}/4)$, $A_4^3:D_4$, $S_4^2:S_4$, $A_4^3:D_4$

Order 12288: $C_2^6.(D_4\times S_4)$

Order 10368: $A_4^3:C_6$ x 2, $A_4^3:S_3$ x 2, $A_4^3:S_3$ x 2, $A_4^3:S_3$ x 2, $A_4^3:C_6$ x 2, $A_4^2:(S_3\times D_6)$ x 2, $A_4^3:C_6$, $S_3\times A_4^3$, $C_3:S_4^2\times S_3$, $S_3^2\times \PSOPlus(4,3)$, $C_3:S_3\times S_4^2$, $(S_3\times A_4)\wr C_2$, $\PSOPlus(4,3).S_3^2$, $A_4^2:\SOPlus(4,2)$, $C_6^2:C_2^3.S_3^2$, $(C_3\times A_4^2):D_{12}$, $(C_3\times S_4)\wr C_2$

Order 9216: $A_4^2.C_2^4.C_2^2$ x 3, $A_4^2.D_4^2$ x 3, $A_4^2.D_4^2$ x 3, $C_2^6.D_6^2$ x 3, $C_2^4.S_4^2$ x 3, $C_2\times S_4^2:D_4$ x 3, $C_2^6.D_6^2$ x 3, $C_2^6.(C_4\times S_3^2)$ x 3, $C_2^2\wr S_3:D_{12}$ x 3, $C_2^7.(S_3\times D_6)$ x 3, $(C_2^3\times A_4^2):D_4$ x 3, $(C_2^3\times A_4^2):D_4$ x 3, $C_2^8:S_3^2$ x 3, $C_2^6.(S_3\times C_3:D_4)$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $C_2^6.C_6^2.C_2^2$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.C_2^5.C_2$ x 2, $A_4^2.D_4^2$ x 2, $A_4^2.C_2^3.C_2^3$ x 2, $A_4^2.D_4^2$ x 2, $C_2^6.D_6:D_6$ x 2, $C_2^4.S_4^2$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^4.S_4^2$ x 2, $C_2^8.S_3^2$ x 2, $(C_2\times S_4^2).D_4$ x 2, $A_4^2:D_4^2$ x 2, $(D_4\times S_4^2):C_2$ x 2, $A_4^2.C_2\wr C_4$ x 2, $C_2^6.D_6^2$ x 2, $C_2^7:\SOPlus(4,2)$ x 2, $A_4^2:C_2\wr C_4$ x 2, $A_4^2.C_2^4.C_2^2$, $C_2^6.C_3^2.D_4.C_2$, $A_4^2.C_2^4.C_2^2$, $C_2^6.(S_3\times D_{12})$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2^3$, $A_4.C_2^5.C_6.C_2^2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^4.C_2^2$, $A_4^2.D_4^2$, $A_4^2.D_4^2$, $C_2^6.(S_3\times D_{12})$, $(C_2\times A_4^2).C_2^5$, $A_4^2.D_4^2$, $C_2^8.S_3^2$, $A_4.(C_2^4\times A_4).C_2^2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.D_4^2$, $A_4^2.D_4^2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.C_2^4.C_2^2$, $A_4^2.D_4^2$, $A_4.(C_2^4\times A_4).C_2^2$, $A_4^2.D_4^2$, $C_2^6.(C_4\times S_3^2)$, $C_2^4.(C_6\times A_4).C_2^3$, $C_2^6.C_6^2.C_2^2$, $A_4^2.C_2^3.C_2^3$, $A_4^2.D_4^2$, $A_4^2.C_2^4.C_2^2$, $C_2^3\times A_4^2.C_2^3$, $C_2^6.C_4:S_3^2$, $C_2^8.S_3^2$, $(C_2^2:S_4)^2$, $C_2^6.C_6:S_4$, $C_2^5.\PSOPlus(4,3)$, $C_2^4.S_4^2$, $C_2^4.A_4^2:C_4$, $C_4^2:\POPlus(4,3)$, $C_4^2:\POPlus(4,3)$, $A_4^2.D_4:D_4$, $(D_4\times C_2^5):S_3^2$, $(C_2\times S_4^2).C_2^3$, $(C_2^2\times S_4^2):C_2^2$, $(C_4\times S_4^2).C_2^2$, $(C_2^3\times A_4^2).D_4$, $(C_2^3\times A_4^2).D_4$, $A_4^2:C_4^2:C_2^2$, $A_4^2:C_2^4:C_4$, $(C_2^2\times A_4^2).C_4^2$, $C_4.C_2\wr S_3^2$, $A_4^2.D_4:D_4$, $(D_4\times C_2^5).S_3^2$, $C_2^7:\SOPlus(4,2)$, $C_2^7.\SOPlus(4,2)$, $C_2\wr S_3^2:C_4$, $(C_2^2\times S_4^2):C_4$

Order 6912: $A_4:S_4^2$ x 26, $A_4\times S_4^2$ x 10, $A_4^3:C_2^2$ x 8, $A_4.A_4^2.C_2^2$ x 8, $A_4^3:C_2^2$ x 6, $S_4\times C_3:\GL(2,\mathbb{Z}/4)$ x 4, $A_4:S_4^2$ x 4, $S_3\times A_4:\GL(2,\mathbb{Z}/4)$ x 3, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ x 3, $(C_2^2\times A_4^2):D_6$ x 3, $A_4^2.A_4.C_4$ x 2, $A_4.A_4^2.C_4$ x 2, $C_2^6.C_3^3.C_2^2$ x 2, $C_2^6.C_3^3.C_4$ x 2, $C_2^8.C_3^3$ x 2, $A_4^2:S_3\times D_4$ x 2, $D_6:S_4^2$ x 2, $(A_4\times C_2^4):S_3^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $A_4^2:(S_3\times D_4)$ x 2, $C_2^2\times A_4^2.D_6$ x 2, $C_2^2\times C_3:S_4^2$ x 2, $D_6\times S_4^2$ x 2, $C_2^2.A_4^3$ x 2, $A_4^3:C_2^2$ x 2, $A_4^3:C_2^2$ x 2, $A_4^3:C_2^2$ x 2, $A_4^3:C_2^2$ x 2, $C_2\times A_4^2\times C_2\times A_4$, $A_4^2\times C_2^4:C_3$, $C_2^6.C_3^3.C_4$, $A_4^2:S_3\times D_4$, $S_3\times D_4\times A_4^2$, $A_4\times \GL(2,\mathbb{Z}/4):C_6$, $A_4\times C_2^4:S_3^2$, $A_4^2:C_6\times D_4$, $D_6.S_4^2$, $D_6.S_4^2$, $S_3\times A_4\times \GL(2,\mathbb{Z}/4)$, $S_3\times A_4^2:D_4$, $D_6.S_4^2$, $C_4\times A_4^2:D_6$, $C_{12}:S_4^2$, $D_6:S_4^2$, $C_{12}:S_4^2$, $C_2^5.S_3^3$, $C_2^5.S_3^3$, $C_3\times S_4\times \GL(2,\mathbb{Z}/4)$, $(C_2\times C_6):S_4^2$, $C_{12}:S_4^2$, $D_6:S_4^2$, $C_2^4.(A_4\times S_3^2)$, $A_4\times C_2^4:S_3^2$, $S_3\times A_4^2.C_2^3$, $(C_2\times C_6):S_4^2$, $D_6:S_4^2$, $C_3\times C_2^2:S_4^2$, $(C_2\times C_6):S_4^2$, $(A_4\times D_{12}):S_4$, $C_{12}:S_4^2$, $C_2^5.S_3^3$, $D_6:S_4^2$, $D_{12}\times \PSOPlus(4,3)$, $C_2\times C_6\times S_4^2$, $C_2^2.A_4^3$, $C_2^2\wr C_3:S_3^2$, $A_4^3:C_4$, $A_4^3:C_4$, $S_4^2:D_6$, $D_6^2:(C_2\times S_4)$

Order 6144: $C_2^5.(D_4\times S_4)$ x 3, $C_2^8.D_{12}$ x 2, $C_2^6.(C_4\times S_4)$ x 2, $C_2^6.\GL(2,\mathbb{Z}/4)$ x 2, $A_4.C_2^5.C_2^4$, $A_4.C_2^5.C_2^4$, $C_2^4:C_3.C_2^5.C_2^2$, $C_2^5.(D_4\times S_4)$, $C_2^5.(D_4\times S_4)$, $(D_4\times C_2^5):D_{12}$, $(C_2^2\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_4^2.C_2\wr C_6$

Order 5184: $C_3\times A_4^3$ x 2, $A_4^2:S_3^2$ x 2, $(A_4\times C_6^2):D_6$ x 2, $C_3^2:S_4^2$ x 2, $A_4^2:S_3^2$ x 2, $A_4^2:S_3^2$ x 2, $C_3\times A_4^2:D_6$ x 2, $C_3^2:S_4^2$, $C_3^2:S_4^2$, $A_4^2:S_3^2$, $A_4^2:S_3^2$, $S_3^2\times A_4^2$, $C_6^2:C_6\times S_4$, $C_3^2:S_4^2$, $C_3^2:S_4^2$, $A_4^2:C_6\times S_3$, $C_6^2:C_6\times S_4$, $C_3^2\times S_4^2$, $(C_3^2\times A_4^2):C_4$, $(C_3\times A_4^2):C_{12}$

Order 4608: $C_2^4\times \PSOPlusPlus(4,3)$, $C_2^3\times A_4^2.C_2^2$

Order 4096: $C_2^6.D_4^2$

Order 2304: $C_2^4\times A_4^2$, $C_2^3\times C_2\times A_4^2$

Order 768: $C_2^8.C_3$

Order 576: $S_4^2$

Order 288: $\PSOPlus(4,3)$ x 2

Order 256: $C_2^8$

Order 144: $A_4^2$ x 2

Order 81: $C_3^4$

Order 48: $C_2^2:A_4$

Order 16: $C_2^4$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 331776: $S_4^2.\POPlus(4,3)$ ($C_1$)

Order 165888: $A_4^2.S_4\wr C_2$ ($C_2$) x 2, $A_4^2.S_4\wr C_2$ ($C_2$) x 2, $C_2^8.C_3.S_3^3$ ($C_2$), $A_4^2\wr C_2.C_2^2$ ($C_2$), $A_4^2.(C_2^5.S_3^2)$ ($C_2$)

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

Order 55296: $(C_2^2\times S_4^2).S_4$ ($S_3$) x 2

Order 41472: $A_4^2.A_4^2.C_2$ ($C_2^3$), $A_4^2.A_4^2.C_2$ ($D_4$), $C_2^8.C_3^4.C_2$ ($D_4$)

Order 27648: $A_4^2.C_2^4.C_6.C_2$ ($D_6$) x 2, $A_4^2:C_2^3.S_4$ ($D_6$) x 2, $A_4^2.C_2^3:S_4$ ($D_6$) x 2

Order 20736: $C_2^8.C_3^4$ ($C_2\times D_4$)

Order 13824: $A_4^2.C_2^4.C_6$ ($C_2\times D_6$) x 2

Order 9216: $C_2^3\times A_4^2.C_2^3$ ($S_3^2$)

Order 6912: $C_2^8.C_3^3$ ($S_3\times D_4$) x 2

Order 4608: $C_2^4\times \PSOPlusPlus(4,3)$ ($\SOPlus(4,2)$), $C_2^3\times A_4^2.C_2^2$ ($S_3\times D_6$)

Order 2304: $C_2^4\times A_4^2$ ($S_3^2:C_2^2$), $C_2^3\times C_2\times A_4^2$ ($D_6:D_6$)

Order 768: $C_2^8.C_3$ ($S_3^3:C_2$) x 2

Order 576: $S_4^2$ ($\POPlus(4,3)$)

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

Order 256: $C_2^8$ ($S_3^2:S_3^2$)

Order 144: $A_4^2$ ($C_2\wr S_3^2$), $A_4^2$ ($S_4^2:C_2^2$)

Order 48: $C_2^2:A_4$ ($S_4^2:D_6$) x 2

Order 16: $C_2^4$ ($S_3^2:\POPlus(4,3)$), $C_2^4$ ($S_4^2:S_3^2$)

Order 1: $C_1$ ($S_4^2.\POPlus(4,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 331776: $S_4^2.\POPlus(4,3)$ ($C_1$)

Order 165888: $C_2^8.C_3.S_3^3$ ($C_2$), $A_4^2\wr C_2.C_2^2$ ($C_2$), $A_4^2.(C_2^5.S_3^2)$ ($C_2$), $A_4^2.S_4\wr C_2$ ($C_2$), $A_4^2.S_4\wr C_2$ ($C_2$)

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

Order 55296: $(C_2^2\times S_4^2).S_4$ ($S_3$)

Order 41472: $A_4^2.A_4^2.C_2$ ($C_2^3$), $A_4^2.A_4^2.C_2$ ($D_4$), $C_2^8.C_3^4.C_2$ ($D_4$)

Order 27648: $A_4^2.C_2^4.C_6.C_2$ ($D_6$), $A_4^2:C_2^3.S_4$ ($D_6$), $A_4^2.C_2^3:S_4$ ($D_6$)

Order 20736: $C_2^8.C_3^4$ ($C_2\times D_4$)

Order 13824: $A_4^2.C_2^4.C_6$ ($C_2\times D_6$)

Order 9216: $C_2^3\times A_4^2.C_2^3$ ($S_3^2$)

Order 6912: $C_2^8.C_3^3$ ($S_3\times D_4$)

Order 4608: $C_2^4\times \PSOPlusPlus(4,3)$ ($\SOPlus(4,2)$), $C_2^3\times A_4^2.C_2^2$ ($S_3\times D_6$)

Order 2304: $C_2^4\times A_4^2$ ($S_3^2:C_2^2$), $C_2^3\times C_2\times A_4^2$ ($D_6:D_6$)

Order 768: $C_2^8.C_3$ ($S_3^3:C_2$)

Order 576: $S_4^2$ ($\POPlus(4,3)$)

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

Order 256: $C_2^8$ ($S_3^2:S_3^2$)

Order 144: $A_4^2$ ($C_2\wr S_3^2$), $A_4^2$ ($S_4^2:C_2^2$)

Order 48: $C_2^2:A_4$ ($S_4^2:D_6$)

Order 16: $C_2^4$ ($S_3^2:\POPlus(4,3)$), $C_2^4$ ($S_4^2:S_3^2$)

Order 1: $C_1$ ($S_4^2.\POPlus(4,3)$)

Series

Derived series

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

$\rhd$

$A_4^2.A_4^2.C_2$

$\rhd$

$C_2^3\times C_2\times A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_2^8.C_3.S_3^3$

$\rhd$

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

$\rhd$

$A_4^2.A_4^2.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8.C_3^3$

$\rhd$

$C_2^3\times C_2\times A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$A_4^2.A_4^2.C_2$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $220 \times 220$ rational character table is not available for this group.