Abstract group 41472.kr: $A_4^2:(C_6^2:D_4)$

• Label: 41472.kr
• Order: \( 2^{9} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $14$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$A_4^2:(C_6^2:D_4)$
Order:	\(41472\)\(\medspace = 2^{9} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^5:S_3^2$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \)
Composition factors:	$C_2$ x 9, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	1671	728	8568	18984	11520	41472
Conjugacy classes	1	23	11	26	61	22	144
Divisions	1	23	11	26	61	22	144
Autjugacy classes	1	14	5	17	25	10	72

Dimension	1	2	4	6	8	9	12	16	18	24	36	48
Irr. complex chars.	16	20	24	8	17	16	6	4	4	9	16	4	144
Irr. rational chars.	16	20	24	8	17	16	6	4	4	9	16	4	144

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=c^{2}=d^{12}=e^{2}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(1,7)(5,8,10), (2,6)(4,7)(5,8)(10,12)(11,14,13), (1,3)(2,4)(5,10)(6,7)(9,11,13,14) \!\cdots\! \rangle$
Direct product:	$\SOPlus(4,2)$ $\, \times\, $ $\POPlus(4,3)$
Semidirect product:	$A_4^2$ $\,\rtimes\,$ $(C_6^2:D_4)$	$(C_6^2:S_4)$ $\,\rtimes\,$ $(S_3\times D_4)$	$(C_6^2:S_4)$ $\,\rtimes\,$ $(S_3\times D_4)$	$(D_6^2:A_4)$ $\,\rtimes\,$ $(C_2\times D_6)$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2:C_2^3)$ . $D_6^2$	$(C_3:S_3)$ . $(A_4^2.C_2^4)$	$(A_4^2.C_6^2.C_2)$ . $C_2^2$	$(C_3^2.A_4^2.C_2)$ . $C_2^4$	all 7
Aut. group:	$\Aut(C_6^2:(C_2\times S_4))$	$\Aut(C_6^2:\GL(2,\mathbb{Z}/4))$	$\Aut(D_6^2:S_4)$	$\Aut(A_4^2:S_3^2)$	all 9

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 2156788 subgroups in 19124 conjugacy classes, 139 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $A_4^2:(C_6^2:D_4)$
Commutator:	$G' \simeq$ $C_3^2.A_4^2.C_2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_4^2:(C_6^2:D_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $D_4\times S_3^2$
Radical:	$R \simeq$ $A_4^2:(C_6^2:D_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{soc} \simeq$ $D_4\times S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

All subgroups of index up to 81 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 41472: $A_4^2:(C_6^2:D_4)$

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

Order 13824: $C_6^2.(C_2^3\times S_4).C_2$ x 2

Order 10368: $C_2^4:C_3.S_3^3$ x 4, $(C_2^2\times C_6^2).C_6^2.C_2$ x 4, $C_2^4:C_3.S_3^3$ x 4, $C_3^2.A_4^2.D_4$ x 4, $C_2^4.C_3^4.C_2^3$ x 4, $C_2^4.C_3^4.C_2^2.C_2$ x 2, $C_2^4.C_3^4.C_2^3$ x 2, $C_3^2.(C_2\times A_4^2).C_2^2$ x 2, $C_2^4:C_3.S_3^3$ x 2, $C_2^4.C_3^4.C_2^3$ x 2, $A_4^2.C_3:S_3.C_2^2$ x 2, $C_3^2.A_4^2.D_4$ x 2, $C_3^2.A_4^2.D_4$ x 2, $A_4^2.C_6^2.C_2$ x 2, $A_4^2.C_6^2.C_2$, $C_6^2.(C_3\times D_4\times A_4)$, $(C_6^2\times A_4).D_6.C_2$, $A_4^2.C_3:S_3.C_2^2$, $C_6^2.A_4:D_{12}$

Order 6912: $C_6^2.(D_4\times S_4)$ x 8, $C_6^2.(C_2^2\times S_4).C_2$ x 4, $C_6^2.(D_4\times S_4)$ x 4, $C_6^2.(D_4\times S_4)$ x 4, $C_2^4:C_3.D_6^2$ x 4, $C_2^4:C_3.C_3:S_3.C_2^3$ x 2, $C_2^4.S_3^3.C_2$ x 2, $C_6^2.(C_2\times C_4:S_4)$ x 2, $C_6^2.(C_2\times D_4\times A_4)$ x 2, $(C_3^2\times A_4).D_4^2$

Order 5184: $A_4^2.(C_6\times S_3)$ x 4, $A_4\wr C_2\times C_3\times S_3$ x 4, $C_2^4.C_3^4.C_2^2$ x 4, $(C_3\times A_4^2).D_6$ x 4, $C_2^4.C_3^4.C_2^2$ x 4, $(C_3\times A_4^2).D_6$ x 2, $C_2^4.C_3^4.C_2^2$ x 2, $A_4^2.(C_6\times S_3)$ x 2, $C_2^4.C_3^4.C_2^2$ x 2, $(C_3\times A_4^2).D_6$ x 2, $C_2^4.C_3^4.C_4$ x 2, $C_3^2.A_4^2.C_2^2$ x 2, $A_4^2.S_3^2$ x 2, $(C_3\times A_4^2).D_6$ x 2, $(C_3\times A_4^2).D_6$, $C_3^2\times A_4^2.C_2^2$, $(C_3\times A_4^2).D_6$, $C_6^2.C_6^2.C_2^2$, $C_3^2.A_4^2.C_4$, $A_4^2.C_3^2.C_4$

Order 4608: $A_4^2.C_2^4.C_2$, $C_6^2.C_2^4.C_2^3$

Order 3456: $C_2^4.S_3^3$ x 18, $C_2^4.S_3^3$ x 8, $C_6^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4.S_3^3$ x 8, $C_6^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4.S_3^3$ x 8, $C_6^2.(C_2^3\times A_4)$ x 4, $C_6^2.C_4:S_4$ x 4, $C_6^2.C_4:S_4$ x 4, $(C_2^4\times C_3:S_3).C_6.C_2$ x 4, $(C_3\times A_4^2).C_2^3$ x 4, $C_2^4:C_3.(C_2\times S_3^2)$ x 4, $C_2^4:C_3.(C_2\times S_3^2)$ x 4, $C_2^2:S_4:C_2\times C_3\times S_3$ x 4, $C_2^4.(S_3\times C_3:S_3).C_2$ x 4, $C_2^4.(S_3\times C_3:S_3).C_2$ x 4, $A_4^2.D_6.C_2$ x 4, $A_4^2.C_2^2\times S_3$ x 2, $C_6^2.(C_2\times C_4\times A_4)$ x 2, $(C_3^2\times A_4).C_2^2:D_4$ x 2, $C_2^4:C_3.C_6^2.C_2$ x 2, $C_2\times C_6^2.(S_3\times D_4)$ x 2, $C_3\times A_4^2.C_2^3$ x 2, $A_4.(D_4\times S_3^2)$ x 2, $C_6^2.C_2^4:S_3$ x 2, $(C_3\times A_4^2).C_2^3$ x 2, $C_2^4.(C_6\times C_3:S_3).C_2$ x 2, $A_4^2.D_6.C_2$ x 2, $C_6^2.(D_4\times A_4)$ x 2, $C_6^2.(D_4\times A_4)$ x 2, $C_6^2.C_2^3:D_6$ x 2, $(A_4\times S_3^2).C_2^3$, $(C_3^2\times A_4).C_4:D_4$, $(A_4\times S_3^2).C_2^3$, $C_6^2.(C_2\times C_6):D_4$, $C_6^2.D_6:D_4$, $A_4.(D_4\times C_3:S_3).C_2$, $C_6^2.(D_4\times C_3:C_4)$

Order 2592: $C_2^4.C_3^4.C_2$ x 4, $A_4^2\times C_3\times S_3$ x 2, $C_6^2.C_6^2.C_2$ x 2, $A_4^2.C_3^2.C_2$ x 2, $C_2^4.C_3^4.C_2$ x 2, $C_3^2\times A_4\wr C_2$ x 2, $(C_3^3\times A_4).C_2^3$ x 2, $C_6^2.C_3:D_{12}$, $C_3^2.A_4^2.C_2$, $C_3^4.C_2^4.C_2$, $C_3^2.(C_3\times D_4\times A_4)$, $C_3^2\times \PSOPlusPlus(4,3)$, $A_4^2.C_3:S_3$, $A_4.(S_3\times C_3:S_3).C_2$

Order 2304: $C_6^2.D_4^2$ x 4, $C_6^2.D_4^2$ x 4, $A_4^2.C_2^3.C_2$ x 4, $C_2^4.D_6^2$ x 4, $D_6^2.C_2^4$ x 3, $C_6^2.D_4^2$ x 3, $C_6^2.C_2^4.C_2^2$ x 2, $C_6^2.C_2^4.C_2^2$ x 2, $C_6^2.C_2^5.C_2$ x 2, $A_4^2.C_2^4$ x 2, $C_6^2.C_2^4.C_2^2$ x 2, $C_6^2.D_4^2$ x 2, $C_2^4.D_6:D_6$ x 2, $C_6^2.(C_2\times C_4\times D_4)$ x 2, $A_4^2.C_2^3.C_2$ x 2, $C_2^4.C_6^2.C_2^2$ x 2, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^3.C_2$, $D_6^2.C_2^4$, $C_6^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^4.C_2^2$

Order 1728: $A_4^2:D_6$ x 24, $A_4^2:D_6$ x 24, $D_6^2:D_6$ x 22, $A_4^2:D_6$ x 14, $A_4^2:D_6$ x 12, $C_3\times \POPlus(4,3)$ x 12, $C_6^2:(C_2\times S_4)$ x 8, $(C_6\times D_6):S_4$ x 8, $C_3\times C_2^4:S_3^2$ x 8, $D_6^2:D_6$ x 8, $(C_6\times D_6):S_4$ x 8, $C_3\times C_2^4:S_3^2$ x 8, $A_4:D_6^2$ x 6, $C_2\times D_6^2:C_6$ x 6, $D_6^2:A_4$ x 4, $C_6^2:(C_2\times S_4)$ x 4, $C_6^2:A_4:C_4$ x 4, $A_4^2:D_6$ x 4, $A_4\wr C_2\times C_6$ x 4, $(A_4\times S_3^2):C_4$ x 4, $(C_2^3\times C_6):C_6^2$ x 4, $(C_2\times C_6^2):S_4$ x 4, $D_6^2:A_4$ x 4, $D_6^2:A_4$ x 4, $(C_2^2\times C_6^2):D_6$ x 4, $(C_2^2\times C_6^2):D_6$ x 4, $C_2\times C_6^2:D_{12}$ x 3, $C_6^2:(C_4\times D_6)$ x 3, $A_4\times D_6^2$ x 2, $C_6^2:(C_4\times A_4)$ x 2, $D_6\times A_4^2$ x 2, $A_4^2:D_6$ x 2, $C_6\times \PSOPlus(4,3)$ x 2, $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$ x 2, $S_3\times C_3:\GL(2,\mathbb{Z}/4)$ x 2, $C_3\times S_3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.S_3^3$ x 2, $C_2^3.S_3^3$ x 2, $C_2^3.S_3^3$ x 2, $C_2^3.S_3^3$ x 2, $C_2^3.S_3^3$ x 2, $(C_2\times C_6^2).D_{12}$ x 2, $C_6^2.(C_2^2\times A_4)$ x 2, $(C_2\times C_6^2):S_4$ x 2, $C_3^2\times C_2^3:S_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $A_4\times C_6^2:C_4$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $(A_4\times C_6^2):C_4$, $D_6^2:D_6$, $(C_2\times C_6^2).D_{12}$, $A_4:C_6^2:C_4$, $C_3:S_3.\GL(2,\mathbb{Z}/4)$, $(C_3^2\times A_4):C_4^2$

Order 1536: $C_2^4:C_3.C_2^4.C_2$ x 2

Order 1296: $S_3^3:S_3$ x 4, $C_6^2:C_6^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $S_3^4$ x 2, $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$ x 2, $S_3^3:C_6$ x 2, $C_3^2\times A_4^2$, $C_6^2:S_3^2$, $C_3\times C_6^2:C_{12}$, $(A_4\times C_3^3):C_4$, $C_3^3:(S_3\times D_4)$, $(C_3\times S_3^2):D_6$, $C_3^2:C_4\times S_3^2$

Order 1152: $C_2^4.S_3^2.C_2$ x 16, $C_2^4:C_3.D_6.C_2$ x 12, $C_6^2.(C_4\times D_4)$ x 12, $C_2^4.S_3^2.C_2$ x 8, $C_2\times A_4\wr C_2.C_2$ x 8, $D_6^2.C_2^3$ x 8, $D_6^2.C_2^3$ x 7, $A_4^2.C_2^3$ x 7, $C_6^2.C_2^2\wr C_2$ x 6, $C_6^2.C_2^2:D_4$ x 6, $C_6^2.C_2^4.C_2$ x 6, $C_6^2.C_2^5$ x 6, $(C_2\times C_6^2).C_2^4$ x 6, $C_6^2.(C_4\times D_4)$ x 6, $C_6^2.C_2^4.C_2$ x 6, $(C_2\times C_6^2).C_2^4$ x 6, $D_6^2.C_2^3$ x 6, $(C_2\times C_6^2).C_2^4$ x 6, $(C_2\times C_6^2).C_2^4$ x 6, $C_6\times C_2^2:S_4:C_2$ x 4, $C_6^2.C_2^4.C_2$ x 4, $C_6^2.C_2^4.C_2$ x 4, $C_2^4:C_3.D_6.C_2$ x 4, $C_6^2.C_4:D_4$ x 4, $C_6^2.C_2^2\wr C_2$ x 4, $C_6^2.C_2^2:D_4$ x 4, $(C_2\times A_4^2).C_2^2$ x 4, $C_2^4.(C_2\times S_3^2)$ x 4, $C_2^2:S_4:C_2\times S_3$ x 4, $A_4^2.D_4$ x 4, $C_6^2.C_2^4.C_2$ x 4, $D_6^2.C_2^3$ x 4, $C_6^2.C_2^4.C_2$ x 4, $D_6^2.C_2^3$ x 4, $C_6^2.C_2^4.C_2$ x 4, $D_6^2.C_2^3$ x 4, $C_2\times A_4^2.C_2^2$ x 4, $(C_3\times C_6).C_2^5.C_2$ x 4, $(C_2\times C_6^2).C_2^4$ x 4, $(C_3\times C_2^4:C_3).C_2^3$ x 4, $C_2^4.S_3^2.C_2$ x 4, $C_2\times C_2^4.S_3^2$ x 4, $D_6^2.C_2^3$ x 4, $A_4^2.C_2^3$ x 4, $C_6^2.(C_2\times C_4^2)$ x 3, $C_6^2.C_2^4.C_2$ x 3, $C_6^2.(C_2\times C_4:C_4)$ x 3, $D_6^2.C_2^3$ x 3, $C_6^2.C_2^4.C_2$ x 3, $D_6^2.C_2^3$ x 3, $C_6^2.C_2^5$ x 2, $A_4^2.C_2^2.C_2$ x 2, $D_6^2.C_2^3$ x 2, $D_6^2.C_2^3$ x 2, $C_6^2.C_2^4.C_2$ x 2, $A_4^2.C_2^3$ x 2, $A_4^2.D_4$ x 2, $C_6^2.(C_2^2\times C_4).C_2$ x 2, $(C_6\times S_4).C_2^3$ x 2, $C_6^2.(C_2^2\times C_4).C_2$ x 2, $C_6^2.(C_4\times D_4)$ x 2, $(C_3\times C_6).C_2^5.C_2$ x 2, $A_4^2.C_2^2.C_2$ x 2, $A_4^2.D_4$ x 2, $C_2\wr C_2^2\times C_3\times S_3$ x 2, $D_6^2.C_2^3$ x 2, $C_6^2.C_2^4.C_2$ x 2, $C_6^2.(C_4\times D_4)$ x 2, $C_6^2.C_2^4.C_2$ x 2, $C_2^4.(C_3\times D_{12})$ x 2, $C_6^2.C_2^4.C_2$ x 2, $C_6^2.C_2^4.C_2$ x 2, $(C_2\times C_6^2).C_2^4$, $C_2^5.C_6^2$, $C_6^2.C_2^4.C_2$, $D_6^2.C_2^3$, $A_4^2.C_2^2.C_2$, $C_6^2.(C_2\times C_4:C_4)$, $C_6^2.C_2^4.C_2$, $(C_3\times C_6).C_2^5.C_2$, $A_4^2.D_4$, $A_4^2.C_2^2.C_2$

Order 864: $S_4\times S_3^2$ x 34, $S_3^2:S_4$ x 26, $A_4^2:S_3$ x 16, $A_4^2:C_6$ x 16, $D_6^2:C_6$ x 15, $C_6^2:D_{12}$ x 12, $C_3^2:C_4\times S_4$ x 12, $D_6^2:C_6$ x 10, $S_3\times A_4^2$ x 10, $A_4^2:S_3$ x 10, $A_4^2:C_6$ x 10, $C_6^2:S_4$ x 8, $S_3\times C_6:S_4$ x 8, $C_2\times A_4:S_3^2$ x 6, $C_6\times S_3\times S_4$ x 6, $(C_6\times D_6):A_4$ x 4, $C_6^2:S_4$ x 4, $C_6^2:S_4$ x 4, $C_2\times C_6^2:C_{12}$ x 4, $C_6^2:C_6:C_4$ x 4, $(C_6\times D_6):A_4$ x 4, $C_6:S_3\times S_4$ x 3, $S_3^3:C_2^2$ x 3, $C_6^2:(C_2\times A_4)$ x 2, $C_6\times A_4\times D_6$ x 2, $C_6\times A_4^2$ x 2, $C_6^2:S_4$ x 2, $D_6^2:S_3$ x 2, $D_6^2:S_3$ x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_6.(S_3\times S_4)$ x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $S_3\times C_6.S_4$ x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $S_3\times A_4:C_{12}$ x 2, $(C_3\times S_3^2):D_4$ x 2, $D_6:\SOPlus(4,2)$ x 2, $C_3^2\times C_2^3:A_4$ x 2, $(C_2^2\times C_6^2):C_6$ x 2, $C_6:D_6\times A_4$, $D_6^2:C_6$, $C_6^2:S_4$, $C_3^2\times \GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_6^2:(C_4\times S_3)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$, $(S_3\times D_6).D_6$, $D_6:\SOPlus(4,2)$, $C_3^3:(C_4\times D_4)$

Order 768: $C_2\wr D_6$ x 8, $C_2^6:D_6$ x 4, $C_2^5:S_4$ x 4, $(C_2^2\times D_4):S_4$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^3\times C_4):S_4$ x 2, $C_4\times C_2^3:S_4$ x 2, $D_4\times C_2^3:A_4$ x 2, $C_2\wr C_2^2\times D_6$ x 2, $D_4\times \GL(2,\mathbb{Z}/4)$

Order 648: $C_3:S_3^3$ x 7, $C_3:S_3^3$ x 6, $C_3\times S_3^3$ x 6, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $S_3\times C_3^2\times A_4$ x 2, $C_3^3:S_4$ x 2, $C_3:S_3^3$ x 2, $C_3^4:D_4$ x 2, $C_3^4:D_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:C_4\times S_3$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$

Order 576: $C_3^2:D_4^2$ x 89, $D_6^2.C_2^2$ x 54, $C_2^4:S_3^2$ x 40, $C_2^4:S_3^2$ x 40, $D_6^2:C_2^2$ x 36, $\POPlus(4,3)$ x 34, $A_4^2:C_2^2$ x 22, $D_6^2:C_2^2$ x 21, $S_3\times C_2^3:A_4$ x 20, $(C_2^2\times C_6):S_4$ x 20, $C_3\times C_2^3:S_4$ x 20, $D_6^2.C_2^2$ x 18, $D_6^2.C_2^2$ x 18, $S_3\times \GL(2,\mathbb{Z}/4)$ x 18, $D_6^2:C_4$ x 16, $C_4:D_6^2$ x 14, $D_6^2.C_2^2$ x 14, $D_6^2.C_2^2$ x 14, $A_4^2:C_2^2$ x 13, $D_6^2:C_2^2$ x 12, $D_6^2:C_2^2$ x 12, $D_6^2:C_2^2$ x 12, $D_6^2:C_4$ x 12, $(C_2\times D_6):D_{12}$ x 12, $D_6^2:C_4$ x 12, $D_6^2:C_4$ x 12, $D_6^2.C_2^2$ x 10, $D_6^2.C_2^2$ x 10, $Q_8:\SOPlus(4,2)$ x 9, $D_4:\SOPlus(4,2)$ x 9, $(C_6\times C_{12}):D_4$ x 9, $(C_6\times C_{12}):D_4$ x 9, $(C_2^2\times C_6):S_4$ x 8, $C_2^2:S_4\times C_6$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2:C_4$ x 8, $D_6^2.C_2^2$ x 8, $C_6^2:(C_2^2\times C_4)$ x 7, $D_6^2.C_2^2$ x 7, $D_6^2.C_2^2$ x 7, $C_2^2:D_6^2$ x 6, $D_6^2:C_2^2$ x 6, $D_6^2:C_2^2$ x 6, $C_4.D_6^2$ x 6, $C_6^2.(C_2^2\times C_4)$ x 6, $Q_8:\SOPlus(4,2)$ x 6, $C_6^2.C_2^4$ x 6, $C_2^4:S_3^2$ x 6, $C_2^4:S_3^2$ x 6, $D_6^2:C_2^2$ x 6, $D_6^2:C_2^2$ x 6, $C_3\times C_2^4:D_6$ x 6, $S_3\times C_2^3:C_{12}$ x 6, $C_3\times C_2^3:D_{12}$ x 6, $C_2^4:S_3^2$ x 6, $(C_2\times C_{12}):D_{12}$ x 6, $(C_2\times C_{12}):D_{12}$ x 6, $(C_6\times C_{12}):D_4$ x 6, $(C_6\times D_4).D_6$ x 6, $(C_6\times D_{12}):C_4$ x 6, $(C_2\times D_6):D_{12}$ x 6, $C_6^2.C_4^2$ x 6, $C_2^3.\SOPlus(4,2)$ x 6, $D_6^2.C_2^2$ x 5, $D_6^2.C_2^2$ x 5, $D_6^2.C_2^2$ x 5, $C_2^2:A_4\times D_6$ x 4, $C_2^4:C_6^2$ x 4, $C_6^2.(C_2^2\times C_4)$ x 4, $C_6^2.(C_2^2\times C_4)$ x 4, $C_6^2:C_4^2$ x 4, $C_2^4:(C_6\times S_3)$ x 4, $C_2^3.\SOPlus(4,2)$ x 4, $C_6^2.(C_2^2\times C_4)$ x 3, $C_6^2.(C_2^2\times C_4)$ x 3, $C_6^2.C_2^4$ x 3, $(C_2^2\times C_6^2):C_4$ x 3, $C_6^2.(C_2^2\times C_4)$ x 3, $(C_6\times C_{12}):D_4$ x 3, $(C_6\times D_4).D_6$ x 3, $(C_2\times C_{12}):D_{12}$ x 3, $C_2^2\times D_6^2$ x 2, $C_2^2\times A_4\times D_6$ x 2, $C_2^2\times A_4^2$ x 2, $C_6^2.C_2^4$ x 2, $C_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_6\times \GL(2,\mathbb{Z}/4)$ x 2, $C_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^2$ x 2, $A_4^2:C_4$ x 2, $(C_6\times C_{12}):D_4$ x 2, $C_2^2\times C_6^2:C_4$, $C_6^2.C_2^4$, $D_4\times C_3:S_4$, $Q_8\times \SOPlus(4,2)$, $C_4\times A_4^2$, $A_4^2:C_4$, $C_2^4:C_6^2$, $(C_6\times C_{12}):D_4$

Order 512: $C_2^6.C_2^3$

Order 432: $S_3^3:C_2$ x 20, $C_6^2:A_4$ x 2

Order 288: $C_6^2:D_4$ x 4, $A_4\wr C_2$ x 2, $D_4\times S_3^2$, $C_6^2:C_2^3$, $\PSOPlus(4,3)$

Order 144: $S_3^2:C_2^2$ x 44, $C_2^2\times C_6^2$, $A_4^2$

Order 96: $C_2^2:S_4$ x 2, $D_4\times D_6$ x 2

Order 81: $C_3^4$

Order 72: $\SOPlus(4,2)$ x 65, $S_3\times D_6$ x 12

Order 48: $S_3\times D_4$ x 20, $C_2^2:A_4$ x 2

Order 36: $S_3^2$ x 36, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$ x 4

Order 24: $C_2\times D_6$ x 20

Order 18: $C_3:S_3$

Order 16: $C_2\times D_4$ x 44, $C_2^4$

Order 12: $D_6$ x 84

Order 9: $C_3^2$

Order 8: $D_4$ x 65, $C_2^3$ x 16

Order 6: $S_3$ x 40

Order 4: $C_2^2$ x 44

Order 2: $C_2$ x 18

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 81 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 41472: $A_4^2:(C_6^2:D_4)$

Order 20736: $C_3^2.A_4^2.C_2^3.C_2$, $C_3^2.A_4^2.C_2^3.C_2$, $C_3^2.A_4^2.C_2^3.C_2$, $C_3^2.A_4^2.C_2^3.C_2$, $C_2^4.C_3^4.C_2^4$, $C_3^2.A_4^2.C_2^3.C_2$, $A_4^2.C_3:S_3.C_2^3$, $C_3^2.(C_2\times A_4^2).C_2^3$

Order 13824: $C_6^2.(C_2^3\times S_4).C_2$

Order 10368: $A_4^2.C_6^2.C_2$, $C_2^4:C_3.S_3^3$, $C_6^2.(C_3\times D_4\times A_4)$, $C_2^4.C_3^4.C_2^2.C_2$, $C_2^4.C_3^4.C_2^3$, $(C_2^2\times C_6^2).C_6^2.C_2$, $C_3^2.(C_2\times A_4^2).C_2^2$, $C_2^4:C_3.S_3^3$, $C_2^4:C_3.S_3^3$, $C_2^4.C_3^4.C_2^3$, $(C_6^2\times A_4).D_6.C_2$, $A_4^2.C_3:S_3.C_2^2$, $C_3^2.A_4^2.D_4$, $C_3^2.A_4^2.D_4$, $C_3^2.A_4^2.D_4$, $A_4^2.C_6^2.C_2$, $A_4^2.C_3:S_3.C_2^2$, $C_6^2.A_4:D_{12}$, $C_2^4.C_3^4.C_2^3$

Order 6912: $C_6^2.(D_4\times S_4)$ x 2, $C_6^2.(D_4\times S_4)$ x 2, $C_6^2.(D_4\times S_4)$ x 2, $C_2^4:C_3.C_3:S_3.C_2^3$, $C_2^4.S_3^3.C_2$, $C_6^2.(C_2\times C_4:S_4)$, $C_6^2.(C_2\times D_4\times A_4)$, $C_6^2.(C_2^2\times S_4).C_2$, $C_2^4:C_3.D_6^2$, $(C_3^2\times A_4).D_4^2$

Order 5184: $A_4^2.(C_6\times S_3)$, $(C_3\times A_4^2).D_6$, $C_2^4.C_3^4.C_2^2$, $(C_3\times A_4^2).D_6$, $C_3^2\times A_4^2.C_2^2$, $A_4\wr C_2\times C_3\times S_3$, $A_4^2.(C_6\times S_3)$, $C_2^4.C_3^4.C_2^2$, $C_2^4.C_3^4.C_2^2$, $(C_3\times A_4^2).D_6$, $C_2^4.C_3^4.C_4$, $(C_3\times A_4^2).D_6$, $(C_3\times A_4^2).D_6$, $C_3^2.A_4^2.C_2^2$, $A_4^2.S_3^2$, $C_2^4.C_3^4.C_2^2$, $C_6^2.C_6^2.C_2^2$, $C_3^2.A_4^2.C_4$, $(C_3\times A_4^2).D_6$, $A_4^2.C_3^2.C_4$

Order 4608: $A_4^2.C_2^4.C_2$, $C_6^2.C_2^4.C_2^3$

Order 3456: $C_2^4.S_3^3$ x 6, $C_6^2.C_4:S_4$ x 2, $C_2^4.S_3^3$ x 2, $C_6^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^3$ x 2, $C_6^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^3$ x 2, $C_2^4.(S_3\times C_3:S_3).C_2$ x 2, $C_2^4.(S_3\times C_3:S_3).C_2$ x 2, $A_4^2.C_2^2\times S_3$, $C_6^2.(C_2^3\times A_4)$, $C_6^2.(C_2\times C_4\times A_4)$, $(C_3^2\times A_4).C_2^2:D_4$, $C_2^4:C_3.C_6^2.C_2$, $C_6^2.C_4:S_4$, $C_2\times C_6^2.(S_3\times D_4)$, $(A_4\times S_3^2).C_2^3$, $C_3\times A_4^2.C_2^3$, $A_4.(D_4\times S_3^2)$, $(C_2^4\times C_3:S_3).C_6.C_2$, $C_6^2.C_2^4:S_3$, $(C_3\times A_4^2).C_2^3$, $(C_3^2\times A_4).C_4:D_4$, $(C_3\times A_4^2).C_2^3$, $(A_4\times S_3^2).C_2^3$, $C_2^4:C_3.(C_2\times S_3^2)$, $C_2^4:C_3.(C_2\times S_3^2)$, $C_2^4.(C_6\times C_3:S_3).C_2$, $C_6^2.(C_2\times C_6):D_4$, $C_2^2:S_4:C_2\times C_3\times S_3$, $C_6^2.D_6:D_4$, $A_4^2.D_6.C_2$, $C_6^2.(D_4\times A_4)$, $A_4.(D_4\times C_3:S_3).C_2$, $C_6^2.(D_4\times C_3:C_4)$, $C_6^2.(D_4\times A_4)$, $C_6^2.C_2^3:D_6$, $A_4^2.D_6.C_2$

Order 2592: $C_2^4.C_3^4.C_2$, $C_6^2.C_3:D_{12}$, $A_4^2\times C_3\times S_3$, $C_3^2.A_4^2.C_2$, $C_6^2.C_6^2.C_2$, $A_4^2.C_3^2.C_2$, $C_3^4.C_2^4.C_2$, $C_2^4.C_3^4.C_2$, $C_3^2.(C_3\times D_4\times A_4)$, $C_3^2\times A_4\wr C_2$, $C_3^2\times \PSOPlusPlus(4,3)$, $(C_3^3\times A_4).C_2^3$, $A_4^2.C_3:S_3$, $A_4.(S_3\times C_3:S_3).C_2$

Order 2304: $D_6^2.C_2^4$ x 2, $C_6^2.D_4^2$ x 2, $C_6^2.D_4^2$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2$, $C_6^2.D_4^2$, $C_6^2.C_2^5.C_2$, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^4$, $C_6^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2$, $D_6^2.C_2^4$, $C_6^2.D_4^2$, $C_6^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2$, $C_2^4.D_6:D_6$, $C_6^2.C_2^4.C_2^2$, $C_6^2.(C_2\times C_4\times D_4)$, $C_6^2.C_2^4.C_2^2$, $A_4^2.C_2^3.C_2$, $C_6^2.C_2^4.C_2^2$, $C_2^4.D_6^2$, $C_2^4.C_6^2.C_2^2$

Order 1728: $D_6^2:D_6$ x 10, $A_4^2:D_6$ x 6, $A_4^2:D_6$ x 6, $A_4^2:D_6$ x 6, $C_2\times D_6^2:C_6$ x 4, $A_4^2:D_6$ x 4, $C_3\times \POPlus(4,3)$ x 4, $D_6^2:D_6$ x 3, $C_6^2:(C_2\times S_4)$ x 2, $(C_6\times D_6):S_4$ x 2, $C_6^2:(C_2\times S_4)$ x 2, $C_3\times C_2^4:S_3^2$ x 2, $C_6^2:A_4:C_4$ x 2, $A_4:D_6^2$ x 2, $C_2\times C_6^2:D_{12}$ x 2, $C_6^2:(C_4\times D_6)$ x 2, $(A_4\times S_3^2):C_4$ x 2, $(C_6\times D_6):S_4$ x 2, $(C_2^2\times C_6^2):D_6$ x 2, $C_3\times C_2^4:S_3^2$ x 2, $D_6^2:A_4$, $A_4\times D_6^2$, $C_6^2:(C_4\times A_4)$, $A_4\times C_6^2:C_4$, $D_6\times A_4^2$, $A_4^2:D_6$, $C_6\times \PSOPlus(4,3)$, $A_4^2:D_6$, $A_4\wr C_2\times C_6$, $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$, $S_3\times C_3:\GL(2,\mathbb{Z}/4)$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $C_3\times S_3\times \GL(2,\mathbb{Z}/4)$, $(A_4\times C_6^2):C_4$, $D_6^2:D_6$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $(C_2\times C_6^2).D_{12}$, $(C_2\times C_6^2).D_{12}$, $A_4:C_6^2:C_4$, $C_3:S_3.\GL(2,\mathbb{Z}/4)$, $(C_3^2\times A_4):C_4^2$, $C_6^2.(C_2^2\times A_4)$, $(C_2^3\times C_6):C_6^2$, $(C_2\times C_6^2):S_4$, $(C_2\times C_6^2):S_4$, $C_3^2\times C_2^3:S_4$, $D_6^2:A_4$, $D_6^2:A_4$, $(C_2^2\times C_6^2):D_6$, $(C_2^2\times C_6^2):C_{12}$

Order 1536: $C_2^4:C_3.C_2^4.C_2$

Order 1296: $C_3^2\times A_4^2$, $C_6^2:C_6^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_3\times C_6^2:C_{12}$, $(A_4\times C_3^3):C_4$, $S_3^4$, $C_3^3:(S_3\times D_4)$, $S_3\times C_3^2:D_{12}$, $S_3^2:S_3^2$, $(C_3\times S_3^2):D_6$, $S_3^3:S_3$, $S_3^3:C_6$, $C_3^2:C_4\times S_3^2$

Order 1152: $D_6^2.C_2^3$ x 5, $C_2^4.S_3^2.C_2$ x 4, $C_2^4:C_3.D_6.C_2$ x 3, $A_4^2.C_2^3$ x 3, $C_6^2.(C_4\times D_4)$ x 3, $C_6^2.(C_4\times D_4)$ x 3, $D_6^2.C_2^3$ x 3, $C_6^2.C_2^2\wr C_2$ x 2, $C_6^2.C_2^2:D_4$ x 2, $D_6^2.C_2^3$ x 2, $C_2^4.S_3^2.C_2$ x 2, $C_6^2.C_2^4.C_2$ x 2, $C_6^2.C_2^5$ x 2, $C_6^2.(C_2\times C_4^2)$ x 2, $D_6^2.C_2^3$ x 2, $(C_2\times C_6^2).C_2^4$ x 2, $D_6^2.C_2^3$ x 2, $C_6^2.C_2^4.C_2$ x 2, $D_6^2.C_2^3$ x 2, $(C_3\times C_6).C_2^5.C_2$ x 2, $C_2\times A_4^2.C_2^2$ x 2, $C_2\times A_4\wr C_2.C_2$ x 2, $C_6^2.(C_2\times C_4:C_4)$ x 2, $D_6^2.C_2^3$ x 2, $D_6^2.C_2^3$ x 2, $(C_3\times C_6).C_2^5.C_2$ x 2, $C_6^2.C_2^4.C_2$ x 2, $(C_2\times C_6^2).C_2^4$ x 2, $D_6^2.C_2^3$ x 2, $C_6^2.C_2^4.C_2$ x 2, $D_6^2.C_2^3$ x 2, $D_6^2.C_2^3$ x 2, $(C_2\times C_6^2).C_2^4$ x 2, $(C_2\times C_6^2).C_2^4$ x 2, $C_6^2.C_2^5$, $(C_2\times C_6^2).C_2^4$, $A_4^2.C_2^2.C_2$, $C_6\times C_2^2:S_4:C_2$, $D_6^2.C_2^3$, $C_6^2.C_2^4.C_2$, $C_6^2.C_2^4.C_2$, $A_4^2.C_2^3$, $A_4^2.D_4$, $C_6^2.C_2^4.C_2$, $C_6^2.(C_2^2\times C_4).C_2$, $C_2^4:C_3.D_6.C_2$, $C_6^2.C_4:D_4$, $C_6^2.C_2^2\wr C_2$, $C_6^2.C_2^2:D_4$, $(C_2\times A_4^2).C_2^2$, $C_2^5.C_6^2$, $C_2^4.(C_2\times S_3^2)$, $C_2^2:S_4:C_2\times S_3$, $(C_6\times S_4).C_2^3$, $A_4^2.D_4$, $C_6^2.C_2^4.C_2$, $D_6^2.C_2^3$, $C_6^2.C_2^4.C_2$, $C_6^2.(C_2^2\times C_4).C_2$, $A_4^2.C_2^2.C_2$, $C_6^2.C_2^4.C_2$, $C_6^2.(C_4\times D_4)$, $C_6^2.(C_2\times C_4:C_4)$, $C_6^2.C_2^4.C_2$, $A_4^2.C_2^2.C_2$, $A_4^2.D_4$, $C_2\wr C_2^2\times C_3\times S_3$, $C_6^2.C_2^4.C_2$, $C_6^2.C_2^4.C_2$, $C_6^2.(C_4\times D_4)$, $(C_3\times C_6).C_2^5.C_2$, $C_6^2.C_2^4.C_2$, $(C_2\times C_6^2).C_2^4$, $(C_3\times C_2^4:C_3).C_2^3$, $C_2^4.S_3^2.C_2$, $C_2\times C_2^4.S_3^2$, $A_4^2.D_4$, $C_2^4.(C_3\times D_{12})$, $C_6^2.C_2^4.C_2$, $A_4^2.C_2^3$, $C_6^2.C_2^4.C_2$, $A_4^2.C_2^2.C_2$

Order 864: $S_4\times S_3^2$ x 10, $D_6^2:C_6$ x 8, $S_3^2:S_4$ x 8, $C_3^2:C_4\times S_4$ x 7, $C_6^2:D_{12}$ x 6, $D_6^2:C_6$ x 4, $S_3\times A_4^2$ x 4, $A_4^2:S_3$ x 4, $A_4^2:C_6$ x 4, $A_4^2:S_3$ x 4, $A_4^2:C_6$ x 4, $S_3\times C_6:S_4$ x 3, $C_2\times C_6^2:C_{12}$ x 3, $C_6^2:C_6:C_4$ x 3, $C_6^2:S_4$ x 2, $C_6^2:S_4$ x 2, $C_6^2:S_4$ x 2, $C_2\times A_4:S_3^2$ x 2, $C_6:S_3\times S_4$ x 2, $C_6\times S_3\times S_4$ x 2, $S_3^3:C_2^2$ x 2, $C_6^2:(C_2\times A_4)$, $(C_6\times D_6):A_4$, $C_6:D_6\times A_4$, $C_6\times A_4\times D_6$, $D_6^2:C_6$, $C_6\times A_4^2$, $C_6^2:S_4$, $C_6^2:S_4$, $C_3^2\times \GL(2,\mathbb{Z}/4)$, $D_6^2:S_3$, $C_6^2:D_{12}$, $D_6^2:S_3$, $C_6^2:(C_4\times S_3)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $S_3\times C_6.S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_{12}$, $(S_3\times D_6).D_6$, $(C_3\times S_3^2):D_4$, $D_6:\SOPlus(4,2)$, $D_6:\SOPlus(4,2)$, $C_3^3:(C_4\times D_4)$, $C_3^2\times C_2^3:A_4$, $(C_2^2\times C_6^2):C_6$, $(C_6\times D_6):A_4$

Order 768: $C_2^6:D_6$ x 2, $(C_2^2\times D_4):S_4$ x 2, $C_2\wr D_6$ x 2, $C_2^5:S_4$, $D_4\times \GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4):S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^3:S_4$, $D_4\times C_2^3:A_4$, $C_2\wr C_2^2\times D_6$

Order 648: $C_3:S_3^3$ x 3, $C_3:S_3^3$ x 2, $C_3\times S_3^3$ x 2, $C_3\times C_6^2:C_6$, $S_3\times C_3^2\times A_4$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3:S_3^3$, $C_3^3:D_{12}$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3\wr D_4$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^3:C_4\times S_3$, $S_3\times C_3^2:C_{12}$

Order 576: $C_3^2:D_4^2$ x 34, $D_6^2.C_2^2$ x 14, $D_6^2:C_2^2$ x 13, $\POPlus(4,3)$ x 12, $C_2^4:S_3^2$ x 10, $D_6^2.C_2^2$ x 10, $C_2^4:S_3^2$ x 10, $D_6^2:C_2^2$ x 9, $S_3\times \GL(2,\mathbb{Z}/4)$ x 9, $D_6^2:C_2^2$ x 8, $A_4^2:C_2^2$ x 6, $A_4^2:C_2^2$ x 6, $D_6^2:C_4$ x 6, $C_4:D_6^2$ x 5, $C_6^2:(C_2^2\times C_4)$ x 5, $D_4:\SOPlus(4,2)$ x 5, $D_6^2.C_2^2$ x 5, $(C_6\times C_{12}):D_4$ x 5, $D_6^2.C_2^2$ x 5, $D_6^2.C_2^2$ x 5, $D_6^2.C_2^2$ x 5, $D_6^2.C_2^2$ x 5, $S_3\times C_2^3:A_4$ x 5, $(C_2^2\times C_6):S_4$ x 5, $C_3\times C_2^3:S_4$ x 5, $D_6^2:C_2^2$ x 4, $D_6^2:C_2^2$ x 4, $C_6^2.(C_2^2\times C_4)$ x 4, $Q_8:\SOPlus(4,2)$ x 4, $(C_6\times C_{12}):D_4$ x 4, $D_6^2.C_2^2$ x 4, $D_6^2.C_2^2$ x 4, $D_6^2.C_2^2$ x 4, $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_4$ x 3, $C_6^2.(C_2^2\times C_4)$ x 3, $C_6^2.(C_2^2\times C_4)$ x 3, $C_6^2:C_4^2$ x 3, $D_6^2.C_2^2$ x 3, $C_2^3.\SOPlus(4,2)$ x 3, $D_6^2.C_2^2$ x 3, $D_6^2:C_4$ x 3, $D_6^2.C_2^2$ x 3, $(C_2\times D_6):D_{12}$ x 3, $D_6^2:C_4$ x 3, $D_6^2:C_4$ x 3, $C_6^2.C_4^2$ x 3, $(C_2^2\times C_6):S_4$ x 2, $C_2^2:S_4\times C_6$ x 2, $C_2^2:D_6^2$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_2^2$ x 2, $C_4.D_6^2$ x 2, $C_6^2.(C_2^2\times C_4)$ x 2, $C_6^2.(C_2^2\times C_4)$ x 2, $Q_8:\SOPlus(4,2)$ x 2, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_2^4:S_3^2$ x 2, $C_2^4:S_3^2$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_6^2.(C_2^2\times C_4)$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_2^2$ x 2, $(C_6\times C_{12}):D_4$ x 2, $(C_6\times D_4).D_6$ x 2, $(C_2\times C_{12}):D_{12}$ x 2, $C_3\times C_2^4:D_6$ x 2, $S_3\times C_2^3:C_{12}$ x 2, $C_3\times C_2^3:D_{12}$ x 2, $C_2^4:S_3^2$ x 2, $(C_2\times C_{12}):D_{12}$ x 2, $(C_2\times C_{12}):D_{12}$ x 2, $(C_6\times C_{12}):D_4$ x 2, $(C_6\times D_4).D_6$ x 2, $(C_6\times D_{12}):C_4$ x 2, $(C_2\times D_6):D_{12}$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $C_2^2\times D_6^2$, $C_2^2\times C_6^2:C_4$, $C_2^2:A_4\times D_6$, $C_2^2\times A_4\times D_6$, $C_2^2\times A_4^2$, $C_6^2.C_2^4$, $C_6^2.C_2^4$, $C_6:\GL(2,\mathbb{Z}/4)$, $D_4\times C_3:S_4$, $C_6\times \GL(2,\mathbb{Z}/4)$, $Q_8\times \SOPlus(4,2)$, $C_4\times A_4^2$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $A_4^2:C_4$, $A_4^2:C_4$, $C_2^4:C_6^2$, $C_2^4:(C_6\times S_3)$, $C_2^4:C_6^2$, $(C_6\times C_{12}):D_4$, $(C_6\times C_{12}):D_4$

Order 512: $C_2^6.C_2^3$

Order 432: $S_3^3:C_2$ x 9, $C_6^2:A_4$

Order 288: $C_6^2:D_4$ x 3, $D_4\times S_3^2$, $C_6^2:C_2^3$, $\PSOPlus(4,3)$, $A_4\wr C_2$

Order 144: $S_3^2:C_2^2$ x 21, $C_2^2\times C_6^2$, $A_4^2$

Order 96: $C_2^2:S_4$, $D_4\times D_6$

Order 81: $C_3^4$

Order 72: $\SOPlus(4,2)$ x 30, $S_3\times D_6$ x 4

Order 48: $S_3\times D_4$ x 9, $C_2^2:A_4$

Order 36: $S_3^2$ x 13, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$ x 3

Order 24: $C_2\times D_6$ x 5

Order 18: $C_3:S_3$

Order 16: $C_2\times D_4$ x 21, $C_2^4$

Order 12: $D_6$ x 26

Order 9: $C_3^2$

Order 8: $D_4$ x 30, $C_2^3$ x 6

Order 6: $S_3$ x 15

Order 4: $C_2^2$ x 17

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 41472: $A_4^2:(C_6^2:D_4)$ ($C_1$)

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

Order 10368: $C_2^4:C_3.S_3^3$ ($C_2^2$) x 4, $(C_2^2\times C_6^2).C_6^2.C_2$ ($C_2^2$) x 4, $C_3^2.A_4^2.D_4$ ($C_2^2$) x 4, $C_2^4.C_3^4.C_2^3$ ($C_2^2$) x 4, $C_2^4.C_3^4.C_2^2.C_2$ ($C_2^2$) x 2, $C_3^2.(C_2\times A_4^2).C_2^2$ ($C_2^2$) x 2, $C_2^4:C_3.S_3^3$ ($C_2^2$) x 2, $C_2^4.C_3^4.C_2^3$ ($C_2^2$) x 2, $A_4^2.C_3:S_3.C_2^2$ ($C_2^2$) x 2, $C_3^2.A_4^2.D_4$ ($C_2^2$) x 2, $C_3^2.A_4^2.D_4$ ($C_2^2$) x 2, $A_4^2.C_6^2.C_2$ ($C_2^2$), $C_6^2.(C_3\times D_4\times A_4)$ ($C_2^2$), $(C_6^2\times A_4).D_6.C_2$ ($C_2^2$), $A_4^2.C_3:S_3.C_2^2$ ($C_2^2$), $C_6^2.A_4:D_{12}$ ($C_2^2$)

Order 6912: $C_6^2.(D_4\times S_4)$ ($S_3$) x 2

Order 5184: $C_2^4.C_3^4.C_2^2$ ($C_2^3$) x 4, $C_2^4.C_3^4.C_2^2$ ($C_2^3$) x 2, $C_2^4.C_3^4.C_4$ ($C_2^3$) x 2, $C_3^2.A_4^2.C_2^2$ ($C_2^3$) x 2, $A_4^2.S_3^2$ ($C_2^3$) x 2, $(C_3\times A_4^2).D_6$ ($D_4$) x 2, $(C_3\times A_4^2).D_6$ ($D_4$), $C_3^2\times A_4^2.C_2^2$ ($D_4$), $(C_3\times A_4^2).D_6$ ($C_2^3$), $C_3^2.A_4^2.C_4$ ($C_2^3$), $A_4^2.C_3^2.C_4$ ($C_2^3$)

Order 3456: $C_6^2.\GL(2,\mathbb{Z}/4)$ ($D_6$) x 4, $C_2^4.S_3^3$ ($D_6$) x 4, $C_6^2.C_4:S_4$ ($D_6$) x 2, $C_2^4.(S_3\times C_3:S_3).C_2$ ($D_6$) x 2, $C_6^2.(D_4\times A_4)$ ($D_6$) x 2

Order 2592: $C_2^4.C_3^4.C_2$ ($C_2\times D_4$) x 2, $C_3^2\times A_4\wr C_2$ ($C_2\times D_4$) x 2, $C_3^2.A_4^2.C_2$ ($C_2^4$), $C_3^2\times \PSOPlusPlus(4,3)$ ($C_2\times D_4$), $A_4^2.C_3:S_3$ ($C_2\times D_4$)

Order 1728: $D_6^2:A_4$ ($C_2\times D_6$) x 4, $C_6^2:(C_2\times S_4)$ ($C_2\times D_6$) x 4, $C_6^2:(C_2\times S_4)$ ($C_2\times D_6$) x 2, $C_6^2:(C_4\times A_4)$ ($C_2\times D_6$) x 2, $C_6^2:A_4:C_4$ ($C_2\times D_6$) x 2

Order 1296: $C_3^2\times A_4^2$ ($C_2^2\times D_4$)

Order 1152: $D_6^2.C_2^3$ ($S_3^2$)

Order 864: $C_6^2:(C_2\times A_4)$ ($C_2^2\times D_6$) x 2, $C_6^2:S_4$ ($S_3\times D_4$) x 2, $C_6^2:S_4$ ($S_3\times D_4$) x 2

Order 576: $C_2^2\times D_6^2$ ($S_3\times D_6$) x 2, $C_2^2\times C_6^2:C_4$ ($S_3\times D_6$), $\POPlus(4,3)$ ($\SOPlus(4,2)$)

Order 432: $C_6^2:A_4$ ($D_4\times D_6$) x 2

Order 288: $A_4\wr C_2$ ($S_3^2:C_2^2$) x 2, $C_6^2:C_2^3$ ($D_6^2$), $\PSOPlus(4,3)$ ($S_3^2:C_2^2$)

Order 144: $C_2^2\times C_6^2$ ($D_4\times S_3^2$), $A_4^2$ ($C_6^2:D_4$)

Order 96: $C_2^2:S_4$ ($S_3^3:C_2$) x 2

Order 72: $\SOPlus(4,2)$ ($\POPlus(4,3)$)

Order 48: $C_2^2:A_4$ ($S_3^3:C_2^2$) x 2

Order 36: $S_3^2$ ($A_4^2.C_2^3$) x 2, $C_3^2:C_4$ ($A_4^2.C_2^3$)

Order 18: $C_3:S_3$ ($A_4^2.C_2^4$)

Order 16: $C_2^4$ ($C_3^4.C_2^3.C_2^2$)

Order 9: $C_3^2$ ($A_4^2.C_2^4.C_2$)

Order 1: $C_1$ ($A_4^2:(C_6^2:D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 41472: $A_4^2:(C_6^2:D_4)$ ($C_1$)

Order 20736: $C_3^2.A_4^2.C_2^3.C_2$ ($C_2$), $C_3^2.A_4^2.C_2^3.C_2$ ($C_2$), $C_3^2.A_4^2.C_2^3.C_2$ ($C_2$), $C_3^2.A_4^2.C_2^3.C_2$ ($C_2$), $C_2^4.C_3^4.C_2^4$ ($C_2$), $C_3^2.A_4^2.C_2^3.C_2$ ($C_2$), $A_4^2.C_3:S_3.C_2^3$ ($C_2$), $C_3^2.(C_2\times A_4^2).C_2^3$ ($C_2$)

Order 10368: $A_4^2.C_6^2.C_2$ ($C_2^2$), $C_2^4:C_3.S_3^3$ ($C_2^2$), $C_6^2.(C_3\times D_4\times A_4)$ ($C_2^2$), $C_2^4.C_3^4.C_2^2.C_2$ ($C_2^2$), $(C_2^2\times C_6^2).C_6^2.C_2$ ($C_2^2$), $C_3^2.(C_2\times A_4^2).C_2^2$ ($C_2^2$), $C_2^4:C_3.S_3^3$ ($C_2^2$), $C_2^4.C_3^4.C_2^3$ ($C_2^2$), $(C_6^2\times A_4).D_6.C_2$ ($C_2^2$), $A_4^2.C_3:S_3.C_2^2$ ($C_2^2$), $C_3^2.A_4^2.D_4$ ($C_2^2$), $C_3^2.A_4^2.D_4$ ($C_2^2$), $C_3^2.A_4^2.D_4$ ($C_2^2$), $A_4^2.C_3:S_3.C_2^2$ ($C_2^2$), $C_6^2.A_4:D_{12}$ ($C_2^2$), $C_2^4.C_3^4.C_2^3$ ($C_2^2$)

Order 6912: $C_6^2.(D_4\times S_4)$ ($S_3$)

Order 5184: $(C_3\times A_4^2).D_6$ ($D_4$), $C_3^2\times A_4^2.C_2^2$ ($D_4$), $C_2^4.C_3^4.C_2^2$ ($C_2^3$), $C_2^4.C_3^4.C_4$ ($C_2^3$), $(C_3\times A_4^2).D_6$ ($C_2^3$), $C_3^2.A_4^2.C_2^2$ ($C_2^3$), $A_4^2.S_3^2$ ($C_2^3$), $C_2^4.C_3^4.C_2^2$ ($C_2^3$), $C_3^2.A_4^2.C_4$ ($C_2^3$), $(C_3\times A_4^2).D_6$ ($D_4$), $A_4^2.C_3^2.C_4$ ($C_2^3$)

Order 3456: $C_6^2.C_4:S_4$ ($D_6$), $C_6^2.\GL(2,\mathbb{Z}/4)$ ($D_6$), $C_2^4.S_3^3$ ($D_6$), $C_2^4.(S_3\times C_3:S_3).C_2$ ($D_6$), $C_6^2.(D_4\times A_4)$ ($D_6$)

Order 2592: $C_3^2.A_4^2.C_2$ ($C_2^4$), $C_2^4.C_3^4.C_2$ ($C_2\times D_4$), $C_3^2\times A_4\wr C_2$ ($C_2\times D_4$), $C_3^2\times \PSOPlusPlus(4,3)$ ($C_2\times D_4$), $A_4^2.C_3:S_3$ ($C_2\times D_4$)

Order 1728: $D_6^2:A_4$ ($C_2\times D_6$), $C_6^2:(C_2\times S_4)$ ($C_2\times D_6$), $C_6^2:(C_2\times S_4)$ ($C_2\times D_6$), $C_6^2:(C_4\times A_4)$ ($C_2\times D_6$), $C_6^2:A_4:C_4$ ($C_2\times D_6$)

Order 1296: $C_3^2\times A_4^2$ ($C_2^2\times D_4$)

Order 1152: $D_6^2.C_2^3$ ($S_3^2$)

Order 864: $C_6^2:(C_2\times A_4)$ ($C_2^2\times D_6$), $C_6^2:S_4$ ($S_3\times D_4$), $C_6^2:S_4$ ($S_3\times D_4$)

Order 576: $C_2^2\times D_6^2$ ($S_3\times D_6$), $C_2^2\times C_6^2:C_4$ ($S_3\times D_6$), $\POPlus(4,3)$ ($\SOPlus(4,2)$)

Order 432: $C_6^2:A_4$ ($D_4\times D_6$)

Order 288: $C_6^2:C_2^3$ ($D_6^2$), $\PSOPlus(4,3)$ ($S_3^2:C_2^2$), $A_4\wr C_2$ ($S_3^2:C_2^2$)

Order 144: $C_2^2\times C_6^2$ ($D_4\times S_3^2$), $A_4^2$ ($C_6^2:D_4$)

Order 96: $C_2^2:S_4$ ($S_3^3:C_2$)

Order 72: $\SOPlus(4,2)$ ($\POPlus(4,3)$)

Order 48: $C_2^2:A_4$ ($S_3^3:C_2^2$)

Order 36: $C_3^2:C_4$ ($A_4^2.C_2^3$), $S_3^2$ ($A_4^2.C_2^3$)

Order 18: $C_3:S_3$ ($A_4^2.C_2^4$)

Order 16: $C_2^4$ ($C_3^4.C_2^3.C_2^2$)

Order 9: $C_3^2$ ($A_4^2.C_2^4.C_2$)

Order 1: $C_1$ ($A_4^2:(C_6^2:D_4)$)

Series

Derived series

$A_4^2:(C_6^2:D_4)$

$\rhd$

$C_3^2.A_4^2.C_2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

Chief series

$A_4^2:(C_6^2:D_4)$

$\rhd$

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

$\rhd$

$C_2^4.C_3^4.C_2^3$

$\rhd$

$C_3^2.A_4^2.C_2^2$

$\rhd$

$C_3^2.A_4^2.C_2$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$A_4^2:(C_6^2:D_4)$

$\rhd$

$C_3^2.A_4^2.C_2$

$\rhd$

$C_3^2\times A_4^2$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 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

See the $144 \times 144$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.