Abstract group 2903040.j: $S_3\times S_4\times A_8$

• Label: 2903040.j
• Order: \( 2^{10} \cdot 3^{4} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{4} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$S_3\times S_4\times A_8$
Order:	\(2903040\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$S_3\times S_4\times S_8$, of order \(5806080\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $A_8$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	28	30	42	60	84
Elements	1	12639	33290	249504	1344	665574	5760	52416	531984	224640	107520	32256	149760	138240	274176	241920	112896	69120	2903040
Conjugacy classes	1	17	11	22	1	55	2	5	29	10	11	2	6	4	19	6	7	2	210
Divisions	1	17	11	22	1	55	1	5	29	5	7	2	3	2	11	3	4	1	180
Autjugacy classes	1	17	11	22	1	55	1	5	29	5	7	2	3	2	11	3	4	1	180

Dimension	1	2	3	4	6	7	14	20	21	28	35	40	42	45	56	60	63	64	70	80	84	90	105	112	120	126	128	135	140	168	180	192	210	224	252	256	270	280	336	360	384	420	540
Irr. complex chars.	4	4	4	1	2	4	8	4	16	9	4	4	18	8	9	4	12	4	8	1	9	8	4	5	2	6	4	8	5	6	2	4	6	1	0	1	4	1	2	0	2	2	0	210
Irr. rational chars.	4	4	4	1	2	4	8	4	8	9	4	4	14	0	9	4	4	4	8	1	11	4	4	5	2	6	4	0	5	7	4	4	6	1	2	1	4	1	2	1	2	2	2	180

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	42	42	42
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(5,7), (4,7)(5,6), (1,2), (4,5,7), (2,3)(4,6)(8,15,11,10,13), (1,3,2), (1,3)(4,7)(8,10)(9,15)(11,12)(13,14), (4,5)(6,7)\rangle$
Direct product:	$S_3$ $\, \times\, $ $S_4$ $\, \times\, $ $A_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(S_4\times A_8)$ . $S_3$	$(S_3\times A_8)$ . $S_4$	$A_8$ . $(S_3\times S_4)$	$(S_3\times S_4)$ . $A_8$	all 24
Aut. group:	$\Aut(C_2^4:C_6^2)$

Elements of the group are displayed as permutations of degree 15.

Homology

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

Subgroups

There are 26 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3\times S_4\times A_8$
Commutator:	$G' \simeq$ $C_3\times A_4\times A_8$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3\times S_4\times A_8$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6$	$G/\operatorname{Fit} \simeq$ $D_6\times A_8$
Radical:	$R \simeq$ $S_3\times S_4$	$G/R \simeq$ $A_8$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6\times A_8$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 630 (order at least 4608) are shown, as well as all normal subgroups of any index.

Order 2903040: $S_3\times S_4\times A_8$

Order 1451520: $C_3:S_4\times A_8$, $A_8\times S_3\times A_4$, $C_3\times S_4\times A_8$

Order 967680: $S_3\times A_8\times D_4$, $(C_2\times S_4).A_8$

Order 725760: $C_3\times A_4\times A_8$, $S_3^2\times A_8$

Order 483840: $S_4\times A_8$ x 3, $C_2^2\times S_3\times A_8$ x 2, $A_8\times C_3:D_4$ x 2, $C_2\times A_4\times A_8$, $C_4\times S_3\times A_8$, $C_3\times A_8\times D_4$, $A_8\times D_{12}$

Order 362880: $C_3\times S_3\times A_8$ x 2, $S_3\times S_4\times A_7$, $C_3:S_3\times A_8$

Order 322560: $C_2\times A_8\times D_4$

Order 241920: $D_6\times A_8$ x 6, $C_2\times C_6\times A_8$ x 2, $A_4\times A_8$ x 2, $C_3:C_4\times A_8$, $C_{12}\times A_8$

Order 193536: $C_2^5.(S_3^2\times \GL(3,2))$ x 2

Order 181440: $C_3:S_4\times A_7$, $C_3\times S_4\times A_7$, $S_3\times A_4\times A_7$, $C_3^2\times A_8$

Order 161280: $A_8\times D_4$ x 4, $C_2^3\times A_8$ x 2, $C_2\times C_4\times A_8$

Order 120960: $S_3\times A_8$ x 6, $C_6\times A_8$ x 3, $C_2\times S_4\times A_7$, $S_3\times D_4\times A_7$

Order 103680: $S_3\times S_4\times S_6$

Order 96768: $S_3\times A_4\times C_2^3:\GL(3,2)$ x 2, $(C_2^2\times C_6):(S_4\times \GL(3,2))$ x 2, $C_3\times S_4\times C_2^3:\GL(3,2)$ x 2

Order 90720: $C_3\times A_4\times A_7$, $S_3^2\times A_7$

Order 82944: $C_2^6.S_3^4$

Order 80640: $C_2^2\times A_8$ x 7, $C_4\times A_8$ x 2

Order 64512: $C_2^5.(D_6\times \GL(3,2))$ x 2, $(C_2^4\times S_4).\GL(3,2)$ x 2

Order 60480: $S_4\times A_7$ x 3, $C_3\times A_8$ x 3, $C_3:D_4\times A_7$ x 2, $C_2\times D_6\times A_7$ x 2, $C_3\times D_4\times A_7$, $D_{12}\times A_7$, $C_4\times S_3\times A_7$, $C_2\times A_4\times A_7$

Order 51840: $C_3\times S_4\times S_6$, $S_4\times A_5:S_3^2$, $C_3:(S_4\times S_6)$, $C_3:S_4\times S_6$, $S_3\times S_4\times A_6$, $S_3\times A_4:S_6$, $S_4\times C_3:S_6$, $S_3\times A_4\times S_6$

Order 48384: $S_3^2\times C_2^3:\GL(3,2)$ x 2, $C_2^5:(C_3^2\times \GL(3,2))$ x 2

Order 45360: $C_3\times S_3\times A_7$ x 2, $C_3:S_3\times A_7$

Order 41472: $S_4\times A_4^2:D_6$ x 2, $A_4^2.(D_6\times S_4)$ x 2, $A_4^2.(D_6\times S_4)$ x 2, $S_3\times A_4^3:C_2^2$ x 2, $A_4^3.(C_2^2\times C_6)$, $S_4\times A_4^2:D_6$, $A_4^2.(D_6\times S_4)$, $S_3\times A_4^3:C_2^2$, $A_4^3:C_2^2\times S_3$, $A_4^3.(C_2\times D_6)$, $(S_3\times A_4).S_4^2$

Order 40320: $C_2\times A_8$ x 5, $C_2\times D_4\times A_7$

Order 34560: $C_2\times S_4\times S_6$, $S_3\times D_4\times S_6$

Order 32256: $S_4\times C_2^3:\GL(3,2)$ x 6, $(D_6\times C_2^4).\GL(3,2)$ x 4, $C_2^5:(S_3\times \GL(3,2))$ x 4, $A_4\times C_2^4:\GL(3,2)$ x 2, $D_{12}\times C_2^3:\GL(3,2)$ x 2, $C_2^4.(D_6\times \GL(3,2))$ x 2, $C_3\times D_4\times C_2^3.\PSL(2,7)$ x 2

Order 30240: $D_6\times A_7$ x 6, $C_2\times C_6\times A_7$ x 2, $A_4\times A_7$ x 2, $C_{12}\times A_7$, $C_3:C_4\times A_7$

Order 27648: $(C_2^2\times D_6).S_4^2$ x 2, $(C_{12}\times A_4^2).C_2^4$, $(C_2^2\times D_6).S_4^2$, $A_4^3.C_2^4$

Order 25920: $S_3^2\times S_6$, $S_4\times C_3^2:S_5$, $C_3:S_4\times A_6$, $C_3^2:(S_4\times S_5)$, $C_3^2:(S_4\times S_5)$, $C_3^2:S_5\times S_4$, $A_4\times C_3:S_6$, $(C_3\times A_4):S_6$, $C_3\times S_4\times A_6$, $C_3\times A_4\times S_6$, $S_3\times A_4\times A_6$, $C_3\times A_4:S_6$, $S_3\times \GL(2,4):S_4$, $A_4\times A_5:S_3^2$, $S_3\times S_4\times \GL(2,4)$

Order 24192: $C_6^2:C_2^2:\GL(3,2)$ x 4, $S_3\times S_4\times \GL(3,2)$ x 3, $F_8:C_6^2:D_6$ x 2, $C_6^2:C_2^2:\GL(3,2)$ x 2

Order 22680: $C_3^2\times A_7$

Order 21504: $(D_4\times C_2^4).\GL(3,2)$ x 2

Order 20736: $A_4^2:C_6\times S_4$ 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, $A_4^3:D_6$ x 2, $A_4^3:D_6$ x 2, $A_4^3:D_6$ x 2, $A_4^3:D_6$ x 2, $A_4^3:D_6$ x 2, $C_3\times A_4^3:C_2^2$ x 2, $A_4^3:D_6$ x 2, $A_4^3:D_6$, $(C_3\times A_4).S_4^2$, $A_4^3.D_6$, $A_4^3.D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $C_3\times A_4^3:C_2^2$, $C_3\times A_4^3:C_2^2$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^2:D_6^2$

Order 20160: $D_4\times A_7$ x 4, $C_2^3\times A_7$ x 2, $C_2\times C_4\times A_7$, $A_8$

Order 17280: $S_4\times S_6$ x 5, $C_3:(D_4\times S_6)$ x 2, $C_3:D_4\times S_6$ x 2, $S_3\times C_2^2:S_6$ x 2, $C_2\times D_6\times S_6$ x 2, $S_3\times S_4\times S_5$ x 2, $C_2\times A_4:S_6$, $C_2\times S_4\times A_6$, $C_2\times A_4\times S_6$, $D_{12}:S_6$, $S_3\times C_4:S_6$, $C_3\times D_4\times S_6$, $S_3\times D_4\times A_6$, $D_4\times C_3:S_6$, $D_{12}\times S_6$, $C_4\times S_3\times S_6$, $C_6:S_5\times S_4$, $(C_3\times S_3\times D_4).S_5$

Order 16128: $S_3\times C_2^4:\GL(3,2)$ x 12, $C_6\times C_2^4:\GL(3,2)$ x 4, $A_4\times C_2^3:\GL(3,2)$ x 4, $C_2^3:\GL(3,2)\times C_{12}$ x 2, $C_2^4.(S_3\times \GL(3,2))$ x 2

Order 15120: $S_3\times A_7$ x 6, $C_6\times A_7$ x 3

Order 13824: $A_4^3.C_2^3$ x 11, $C_2^6:S_3^3$ x 4, $C_2^6:S_3^3$ x 4, $C_2^6.S_3^3$ x 4, $C_2^6.S_3^3$ x 4, $C_2^6.S_3^3$ x 4, $C_2^6.S_3^3$ x 4, $D_4\times A_4^2:D_6$ x 2, $A_4^2:(D_4\times D_6)$ x 2, $C_2^5:(A_4\times S_3^2)$ x 2, $D_4\times A_4^2:D_6$ x 2, $C_2^6.S_3^3$ x 2, $C_2^3.A_4^2:D_6$ x 2, $C_2^6:(C_6\times S_3^2)$ x 2, $C_2^6:S_3^3$ x 2, $D_{12}:\POPlus(4,3)$ x 2, $C_2^6:S_3^3$ x 2, $(C_2^2\times C_6).S_4^2$ x 2, $(C_2^2\times C_6).S_4^2$ x 2, $(C_2^2\times C_6).S_4^2$ x 2, $(C_2^2\times C_6).S_4^2$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6:S_3^3$ x 2, $A_4^3.C_2^3$ x 2, $A_4^3.C_2^3$ x 2, $(D_4\times A_4^2).D_6$, $D_4\times A_4^2:D_6$, $(C_{12}\times A_4^2):C_2^3$, $C_3\times C_2^3:S_4^2$, $Q_8:A_4^2:D_6$, $Q_8:A_4^2:D_6$, $(C_2^2\times C_6):S_4^2$, $(C_2^2\times C_6):S_4^2$, $C_2^5.(A_4\times S_3^2)$, $C_4\times C_2^4:S_3^3$, $A_4^2:(D_4\times D_6)$, $A_4^3.C_2^3$, $D_{12}\times \POPlus(4,3)$, $D_{12}:\POPlus(4,3)$, $C_2^6.S_3^3$, $(C_2^2\times C_6):S_4^2$, $A_4^3:C_2^3$, $A_4^3:C_2^3$

Order 12960: $A_6:S_3^2$ x 2, $C_3\times S_3\times S_6$ x 2, $C_3:S_4\times \GL(2,4)$, $C_3\times \GL(2,4):S_4$, $A_4\times C_3^2:S_5$, $S_3\times A_4\times \GL(2,4)$, $A_4\times C_3^2:S_5$, $(C_3\times \GL(2,4)):S_4$, $C_3\times S_4\times \GL(2,4)$, $C_3\times A_4\times A_6$, $C_3:S_3\times S_6$, $A_5:S_3^3$, $A_6:S_3^2$, $S_3^2\times A_6$

Order 12096: $C_3:S_4\times \GL(3,2)$ x 3, $C_3\times S_4\times \GL(3,2)$ x 3, $S_3\times A_4\times \GL(3,2)$ x 3, $F_8:C_6^2:C_6$ x 2, $F_8:C_3^2:S_4$ x 2, $F_8:C_3^2\times S_4$ x 2, $C_3^2\times C_2^3:\GL(3,2)$ x 2

Order 11520: $C_2\times D_4\times S_6$

Order 10752: $D_4\times C_2^3:\GL(3,2)$ x 8, $C_2^6.\GL(3,2)$ x 4, $C_4\times C_2^4:\GL(3,2)$ x 2

Order 10368: $S_3\times A_4^2:D_6$ x 4, $A_4^3:C_6$ x 2, $A_4^3:C_6$ x 2, $A_4^3:S_3$ x 2, $A_4^3:S_3$ x 2, $A_4^2:(S_3\times D_6)$ x 2, $A_4\wr C_2\times S_3^2$ x 2, $C_3\times C_2^4:S_3^3$ x 2, $S_3\times A_4^2:D_6$ x 2, $A_4^3:C_6$, $A_4^3:S_3$, $A_4^3:S_3$, $A_4^3:C_6$, $A_4^3:S_3$, $S_3\times A_4^3$, $A_4^3:C_6$, $(C_2^3\times C_6):S_3^3$, $C_3:S_4^2\times S_3$, $S_3^2\times \PSOPlus(4,3)$, $S_3^3:C_2\times S_4$, $A_4^2:(S_3\times D_6)$

Order 10080: $C_2^2\times A_7$ x 7, $C_4\times A_7$ x 2

Order 9216: $C_2^6.D_6^2$ x 2, $C_2^4.S_4^2$ x 2, $C_2^5.(D_6\times S_4)$, $C_2^4.S_4^2$, $C_2^6.D_6^2$, $C_2^6.D_6^2$

Order 8640: $D_6\times S_6$ x 12, $S_4\times C_3:S_5$ x 8, $S_4\times A_6$ x 3, $A_4:S_6$ x 3, $A_4\times S_6$ x 3, $S_3\times A_4:S_5$ x 3, $C_6^2:(C_2\times S_5)$ x 2, $C_6^2:(C_2\times S_5)$ x 2, $(C_6\times S_6):C_2$ x 2, $D_6:S_6$ x 2, $S_3\times A_4\times S_5$ x 2, $(C_2\times C_6):S_6$ x 2, $C_3\times C_2^2:S_6$ x 2, $D_6:S_6$ x 2, $C_3:D_4\times A_6$ x 2, $C_2\times D_6\times A_6$ x 2, $C_2\times C_6:S_6$ x 2, $C_2\times C_6\times S_6$ x 2, $S_3\times S_4\times A_5$ x 2, $C_3:S_4\times S_5$ x 2, $C_3:(S_4\times S_5)$ x 2, $A_5:D_6^2$ x 2, $S_3\times \GL(2,4):D_4$ x 2, $C_3\times S_4\times S_5$ x 2, $C_2\times A_4\times A_6$, $C_4\times S_3\times A_6$, $S_3\times D_4\times \GL(2,4)$, $D_4\times C_3^2:S_5$, $D_4\times C_3^2:S_5$, $C_3:(C_4\times S_6)$, $D_{12}\times A_6$, $C_{12}:(S_3\times S_5)$, $D_{12}\times C_3:S_5$, $C_3:C_4\times S_6$, $D_6.S_6$, $C_{12}:S_6$, $D_5.S_3^2\times S_4$, $C_{12}\times S_6$, $C_{12}:S_6$, $C_{12}:S_6$, $A_4\times C_6:S_5$, $C_2\times \GL(2,4):S_4$, $C_3\times D_4\times A_6$, $C_6\times S_4\times A_5$, $S_3\times C_{12}:S_5$, $S_3\times C_{12}:S_5$

Order 8064: $S_3\times C_2^3:\GL(3,2)$ x 12, $C_6\times C_2^3:\GL(3,2)$ x 6, $S_3\times D_4\times \GL(3,2)$ x 3, $C_2\times S_4\times \GL(3,2)$ x 3, $S_3\times S_4\times F_8$ x 2, $(D_{12}\times F_8):C_6$ x 2, $F_8:C_6\times S_4$ x 2

Order 7560: $C_3\times A_7$ x 3

Order 6912: $C_2^5:S_3^3$ x 24, $A_4^3:C_2^2$ x 16, $A_4^3:C_2^2$ x 14, $A_4:S_4^2$ x 8, $A_4^3:C_2^2$ x 7, $A_4^3:C_2^2$ x 7, $(A_4\times C_2^4):S_3^2$ x 5, $A_4^2:(S_3\times D_4)$ x 4, $(C_2^2\times A_4^2):D_6$ x 4, $C_2^2\times A_4^2:D_6$ x 4, $A_4\times C_2^4:S_3^2$ x 4, $(A_4\times C_2^4):S_3^2$ x 4, $C_2^5.S_3^3$ x 4, $C_2^5.S_3^3$ x 4, $A_4^2:(S_3\times D_4)$ x 4, $(C_2^2\times A_4^2):D_6$ x 4, $A_4^2:C_2^2\times D_6$ 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, $C_2^5.S_3^3$ x 4, $C_2^5.S_3^3$ x 4, $C_2^5.S_3^3$ x 4, $C_2^5.S_3^3$ x 4, $(C_2\times C_6).S_4^2$ x 4, $(C_2\times C_6).S_4^2$ x 4, $C_3\times C_2^2:S_4^2$ x 4, $C_3\times C_2^2.S_4^2$ x 4, $(C_2\times C_6):S_4^2$ x 4, $(C_2\times C_6).S_4^2$ x 4, $A_4^2:(S_3\times D_4)$ x 4, $C_3\times C_2^6:S_3^2$ x 4, $D_6\times S_4^2$ x 4, $C_2^5.S_3^3$ x 4, $C_2^5.S_3^3$ x 4, $A_4^2:(S_3\times D_4)$ x 4, $S_3\times A_4^2:D_4$ x 4, $S_3\times A_4:\GL(2,\mathbb{Z}/4)$ x 3, $C_4\times A_4^2:D_6$ x 2, $S_3\times A_4^2:D_4$ x 2, $(C_4\times A_4^2):D_6$ x 2, $C_2^6:(C_3\times S_3^2)$ x 2, $C_2^2:A_4^2:D_6$ x 2, $C_{12}:\POPlus(4,3)$ x 2, $A_4^2:C_6\times D_4$ x 2, $S_3\times C_2^3:A_4^2$ x 2, $C_{12}:\POPlus(4,3)$ x 2, $C_{12}:\POPlus(4,3)$ x 2, $C_{12}:\POPlus(4,3)$ x 2, $A_4^2:S_3\times D_4$ x 2, $C_2^6:(C_3\times S_3^2)$ x 2, $(A_4\times C_2^4):C_6^2$ x 2, $C_4\times A_4^2:D_6$ x 2, $C_{12}:\POPlus(4,3)$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $C_2^6:(C_3\times S_3^2)$ x 2, $C_2^4.(A_4\times S_3^2)$ x 2, $C_2^3:S_4\times S_3^2$ x 2, $A_4\wr C_2\times D_{12}$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5:(C_3^2\times S_4)$ x 2, $C_2^3:C_6^2:S_4$ x 2, $C_2^3:C_6^2:S_4$ x 2, $C_2^3:C_6^2:S_4$ x 2, $(C_2^2\times A_4^2):D_6$ x 2, $A_4^2:(S_3\times D_4)$ x 2, $C_2^2\times A_4^2.D_6$ x 2, $C_6\times A_4^2:C_2^3$ x 2, $C_2^2\times A_4^2:D_6$ x 2, $(C_2^5\times C_6):S_3^2$ x 2, $(C_2^5\times C_6):S_3^2$ x 2, $C_3\times C_2^6:S_3^2$ x 2, $(C_2^5\times C_6):S_3^2$ x 2, $C_2^4.(A_4\times S_3^2)$ x 2, $C_2^6:(C_3\times S_3^2)$ x 2, $C_2^2\wr C_3:S_3^2$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $(C_2^2\times A_4^2):D_6$ x 2, $C_3\times C_2\wr S_3^2$ x 2, $A_4^3:C_2^2$ x 2, $A_4^3:C_2^2$ x 2, $S_3\times D_4\times A_4^2$, $A_4^2:C_6\times D_4$, $A_4^2:S_3\times D_4$, $C_{12}:\POPlus(4,3)$, $C_{12}\times \POPlus(4,3)$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:S_3$, $Q_8:A_4^2:S_3$, $C_2^5:(C_3^2\times S_4)$, $D_6.S_4^2$, $D_6.S_4^2$, $C_{12}:\POPlus(4,3)$, $C_4\times A_4^2:D_6$, $Q_8:A_4^2:C_6$, $S_3\times A_4\times \GL(2,\mathbb{Z}/4)$, $S_3\times A_4^2:D_4$, $D_6.S_4^2$, $C_4\times A_4^2:D_6$, $S_4\times C_3:\GL(2,\mathbb{Z}/4)$, $D_6:S_4^2$, $D_6:S_4^2$, $C_3\times S_4\times \GL(2,\mathbb{Z}/4)$, $(C_2\times C_6):S_4^2$, $D_6:S_4^2$, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$, $S_3\times Q_8:A_4^2$, $Q_8:A_4^2:S_3$, $C_2^5.S_3^3$, $C_2^5.S_3^3$, $S_3\times A_4^2.C_2^3$, $A_4^3:C_2^2$, $A_4^3:C_2^2$, $D_6:S_4^2$, $(A_4\times D_{12}):S_4$, $D_{12}\times \PSOPlus(4,3)$, $A_4.A_4^2.C_2^2$, $C_2^3:S_4\times S_3^2$, $C_2^5.S_3^3$

Order 6480: $\GL(2,4):S_3^2$ x 2, $\GL(2,4):S_3^2$ x 2, $C_3^2:S_6$ x 2, $C_3\times S_3\times A_6$ x 2, $C_6^2:\GL(2,4)$, $C_3:S_3\times A_6$, $C_3^2:S_6$, $C_3^2\times S_6$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $S_3^2\times \GL(2,4)$

Order 6048: $S_3^2\times \GL(3,2)$ x 3, $C_3\times A_4\times \GL(3,2)$ x 3, $A_4\times F_8:C_3^2$ x 2, $F_8:C_3\times S_3^2$ x 2

Order 5760: $D_4\times S_6$ x 8, $C_2^3\times S_6$ x 2, $C_2^3:S_6$ x 2, $S_3\times D_4\times S_5$ x 2, $C_2\times S_4\times S_5$ x 2, $C_6:S_5\times D_4$, $C_2\times D_4\times A_6$, $C_2\times C_4:S_6$, $C_2\times C_4\times S_6$

Order 5376: $C_2^5:\GL(3,2)$ x 14, $C_4\times C_2^3:\GL(3,2)$ x 4

Order 5184: $A_4^2:S_3^2$ x 4, $A_4^2:C_6\times S_3$ x 4, $A_4^2:S_3^2$ x 4, $C_3\times A_4^2:D_6$ x 4, $A_4^2:S_3^2$ x 3, $(C_3\times A_4^2):D_6$ x 2, $(C_3\times A_4^2):D_6$ x 2, $A_4^2:C_6\times S_3$ x 2, $S_4\times S_3^3$ x 2, $D_6^2:S_3^2$ x 2, $D_6^2:S_3^2$ x 2, $S_3^3:S_4$ 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, $C_3\times A_4^3$, $(C_3\times A_4^2):D_6$, $C_3^2:S_4^2$, $C_3^2:S_4^2$, $A_4^2:S_3^2$, $A_4^2:S_3^2$, $(A_4\times C_6^2):D_6$, $S_3^2\times A_4^2$, $C_3^2:S_4^2$, $C_3^2:S_4^2$, $C_6^2:C_6\times S_4$, $S_3^2:C_6\times S_4$, $C_3^3:(D_4\times S_4)$, $S_4\times C_3^2:D_{12}$, $A_4^2:C_6^2$, $C_3^3:(D_4\times S_4)$, $S_3\times C_6^2:D_{12}$, $C_6^2:(C_4\times S_3^2)$, $D_6^2:C_6\times S_3$

Order 5040: $C_2\times A_7$ x 5

Order 4608: $C_2^3:S_4^2$ x 20, $C_2^5.D_6^2$ x 18, $C_2^5.D_6^2$ x 8, $C_2^5.D_6^2$ x 8, $C_2^3:S_4^2$ x 6, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^5:D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^7:S_3^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^3:S_4^2$ x 4, $C_2^3.S_4^2$ x 4, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5:(S_3\times S_4)$ x 2, $(C_2^2\times D_6).\GL(2,\mathbb{Z}/4)$ x 2, $D_6\times C_2^4:S_4$ x 2, $S_3\times C_2^5:S_4$ x 2, $A_4^2:C_2^2\times D_4$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2\wr C_6.D_6$ x 2, $C_2^5:D_6^2$ x 2, $C_2\wr C_6:D_6$ x 2, $C_2^4.(D_6\times S_4)$ x 2, $C_2^5:(C_6\times S_4)$ x 2, $(A_4\times C_2^4):S_4$ x 2, $C_2^4.(A_4\times S_4)$ x 2, $A_4\times C_2^4:S_4$ x 2, $C_2^7:S_3^2$ x 2, $C_2^7:S_3^2$ x 2, $(C_6\times A_4).C_2^6$, $C_2^5.D_6^2$, $C_{12}:C_2^4:S_4$, $C_2^5:(C_6\times S_4)$, $(C_6\times \GL(2,\mathbb{Z}/4)):D_4$, $C_2^3:S_4\times D_{12}$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $S_3\times C_2^5:S_4$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_6:\GL(2,\mathbb{Z}/4):D_4$, $C_2^5.(A_4\times D_6)$, $C_2^5.D_6^2$, $C_2.C_2^6.C_3.D_6$, $A_4\times C_2^4:S_4$, $C_2^3:S_4^2$, $C_2^4:A_4\times S_4$, $C_2^5.D_6^2$, $C_2^6:(C_6\times D_6)$, $C_2^5.D_6^2$, $(C_2^3\times D_{12}):S_4$, $A_4^2:C_2^2\times D_4$

Order 1024: $C_2^4:D_4^2$

Order 144: $S_3\times S_4$

Order 81: $C_3^4$

Order 72: $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 36: $C_3\times A_4$

Order 24: $C_2\times D_6$, $S_4$

Order 12: $C_2\times C_6$, $A_4$

Order 7: $C_7$

Order 6: $S_3$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 630 (order at least 4608) are shown, as well as all normal subgroups of any index.

Order 2903040: $S_3\times S_4\times A_8$

Order 1451520: $C_3:S_4\times A_8$, $A_8\times S_3\times A_4$, $C_3\times S_4\times A_8$

Order 967680: $S_3\times A_8\times D_4$, $(C_2\times S_4).A_8$

Order 725760: $C_3\times A_4\times A_8$, $S_3^2\times A_8$

Order 483840: $S_4\times A_8$ x 3, $C_2^2\times S_3\times A_8$ x 2, $A_8\times C_3:D_4$ x 2, $C_2\times A_4\times A_8$, $C_4\times S_3\times A_8$, $C_3\times A_8\times D_4$, $A_8\times D_{12}$

Order 362880: $C_3\times S_3\times A_8$ x 2, $S_3\times S_4\times A_7$, $C_3:S_3\times A_8$

Order 322560: $C_2\times A_8\times D_4$

Order 241920: $D_6\times A_8$ x 6, $C_2\times C_6\times A_8$ x 2, $A_4\times A_8$ x 2, $C_3:C_4\times A_8$, $C_{12}\times A_8$

Order 193536: $C_2^5.(S_3^2\times \GL(3,2))$

Order 181440: $C_3:S_4\times A_7$, $C_3\times S_4\times A_7$, $S_3\times A_4\times A_7$, $C_3^2\times A_8$

Order 161280: $A_8\times D_4$ x 4, $C_2^3\times A_8$ x 2, $C_2\times C_4\times A_8$

Order 120960: $S_3\times A_8$ x 6, $C_6\times A_8$ x 3, $C_2\times S_4\times A_7$, $S_3\times D_4\times A_7$

Order 103680: $S_3\times S_4\times S_6$

Order 96768: $S_3\times A_4\times C_2^3:\GL(3,2)$, $(C_2^2\times C_6):(S_4\times \GL(3,2))$, $C_3\times S_4\times C_2^3:\GL(3,2)$

Order 90720: $C_3\times A_4\times A_7$, $S_3^2\times A_7$

Order 82944: $C_2^6.S_3^4$

Order 80640: $C_2^2\times A_8$ x 7, $C_4\times A_8$ x 2

Order 64512: $C_2^5.(D_6\times \GL(3,2))$, $(C_2^4\times S_4).\GL(3,2)$

Order 60480: $S_4\times A_7$ x 3, $C_3\times A_8$ x 3, $C_3:D_4\times A_7$ x 2, $C_2\times D_6\times A_7$ x 2, $C_3\times D_4\times A_7$, $D_{12}\times A_7$, $C_4\times S_3\times A_7$, $C_2\times A_4\times A_7$

Order 51840: $C_3\times S_4\times S_6$, $S_4\times A_5:S_3^2$, $C_3:(S_4\times S_6)$, $C_3:S_4\times S_6$, $S_3\times S_4\times A_6$, $S_3\times A_4:S_6$, $S_4\times C_3:S_6$, $S_3\times A_4\times S_6$

Order 48384: $S_3^2\times C_2^3:\GL(3,2)$, $C_2^5:(C_3^2\times \GL(3,2))$

Order 45360: $C_3\times S_3\times A_7$ x 2, $C_3:S_3\times A_7$

Order 41472: $S_4\times A_4^2:D_6$, $A_4^3.(C_2^2\times C_6)$, $S_4\times A_4^2:D_6$, $A_4^2.(D_6\times S_4)$, $A_4^2.(D_6\times S_4)$, $A_4^2.(D_6\times S_4)$, $S_3\times A_4^3:C_2^2$, $S_3\times A_4^3:C_2^2$, $A_4^3:C_2^2\times S_3$, $A_4^3.(C_2\times D_6)$, $(S_3\times A_4).S_4^2$

Order 40320: $C_2\times A_8$ x 5, $C_2\times D_4\times A_7$

Order 34560: $C_2\times S_4\times S_6$, $S_3\times D_4\times S_6$

Order 32256: $S_4\times C_2^3:\GL(3,2)$ x 3, $(D_6\times C_2^4).\GL(3,2)$ x 2, $C_2^5:(S_3\times \GL(3,2))$ x 2, $A_4\times C_2^4:\GL(3,2)$, $D_{12}\times C_2^3:\GL(3,2)$, $C_2^4.(D_6\times \GL(3,2))$, $C_3\times D_4\times C_2^3.\PSL(2,7)$

Order 30240: $D_6\times A_7$ x 6, $C_2\times C_6\times A_7$ x 2, $A_4\times A_7$ x 2, $C_{12}\times A_7$, $C_3:C_4\times A_7$

Order 27648: $(C_{12}\times A_4^2).C_2^4$, $(C_2^2\times D_6).S_4^2$, $(C_2^2\times D_6).S_4^2$, $A_4^3.C_2^4$

Order 25920: $S_3^2\times S_6$, $S_4\times C_3^2:S_5$, $C_3:S_4\times A_6$, $C_3^2:(S_4\times S_5)$, $C_3^2:(S_4\times S_5)$, $C_3^2:S_5\times S_4$, $A_4\times C_3:S_6$, $(C_3\times A_4):S_6$, $C_3\times S_4\times A_6$, $C_3\times A_4\times S_6$, $S_3\times A_4\times A_6$, $C_3\times A_4:S_6$, $S_3\times \GL(2,4):S_4$, $A_4\times A_5:S_3^2$, $S_3\times S_4\times \GL(2,4)$

Order 24192: $C_6^2:C_2^2:\GL(3,2)$ x 2, $S_3\times S_4\times \GL(3,2)$ x 2, $F_8:C_6^2:D_6$, $C_6^2:C_2^2:\GL(3,2)$

Order 22680: $C_3^2\times A_7$

Order 21504: $(D_4\times C_2^4).\GL(3,2)$

Order 20736: $A_4^2:C_6\times S_4$, $A_4^3:D_6$, $A_4^3:D_6$, $(C_3\times A_4).S_4^2$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3.D_6$, $A_4^3:D_6$, $A_4^3.D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^3:D_6$, $C_3\times A_4^3:C_2^2$, $A_4^3:D_6$, $C_3\times A_4^3:C_2^2$, $C_3\times A_4^3:C_2^2$, $A_4^3:D_6$, $A_4^3:D_6$, $A_4^2:D_6^2$

Order 20160: $D_4\times A_7$ x 4, $C_2^3\times A_7$ x 2, $C_2\times C_4\times A_7$, $A_8$

Order 17280: $S_4\times S_6$ x 5, $C_3:(D_4\times S_6)$ x 2, $C_3:D_4\times S_6$ x 2, $S_3\times C_2^2:S_6$ x 2, $C_2\times D_6\times S_6$ x 2, $S_3\times S_4\times S_5$ x 2, $C_2\times A_4:S_6$, $C_2\times S_4\times A_6$, $C_2\times A_4\times S_6$, $D_{12}:S_6$, $S_3\times C_4:S_6$, $C_3\times D_4\times S_6$, $S_3\times D_4\times A_6$, $D_4\times C_3:S_6$, $D_{12}\times S_6$, $C_4\times S_3\times S_6$, $C_6:S_5\times S_4$, $(C_3\times S_3\times D_4).S_5$

Order 16128: $S_3\times C_2^4:\GL(3,2)$ x 6, $C_6\times C_2^4:\GL(3,2)$ x 2, $A_4\times C_2^3:\GL(3,2)$ x 2, $C_2^3:\GL(3,2)\times C_{12}$, $C_2^4.(S_3\times \GL(3,2))$

Order 15120: $S_3\times A_7$ x 6, $C_6\times A_7$ x 3

Order 13824: $A_4^3.C_2^3$ x 8, $C_2^6.S_3^3$ x 2, $C_2^6:S_3^3$ x 2, $C_2^6:S_3^3$ x 2, $C_2^6:S_3^3$ x 2, $C_2^6:S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6:S_3^3$ x 2, $D_4\times A_4^2:D_6$, $(D_4\times A_4^2).D_6$, $D_4\times A_4^2:D_6$, $(C_{12}\times A_4^2):C_2^3$, $A_4^2:(D_4\times D_6)$, $C_2^5:(A_4\times S_3^2)$, $D_4\times A_4^2:D_6$, $C_3\times C_2^3:S_4^2$, $Q_8:A_4^2:D_6$, $Q_8:A_4^2:D_6$, $C_2^3.A_4^2:D_6$, $C_2^6:(C_6\times S_3^2)$, $(C_2^2\times C_6):S_4^2$, $(C_2^2\times C_6):S_4^2$, $C_2^5.(A_4\times S_3^2)$, $C_4\times C_2^4:S_3^3$, $A_4^2:(D_4\times D_6)$, $D_{12}:\POPlus(4,3)$, $A_4^3.C_2^3$, $D_{12}\times \POPlus(4,3)$, $D_{12}:\POPlus(4,3)$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$, $C_2^6.S_3^3$, $(C_2^2\times C_6):S_4^2$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$, $C_2^6.S_3^3$, $A_4^3.C_2^3$, $A_4^3:C_2^3$, $A_4^3.C_2^3$, $A_4^3:C_2^3$

Order 12960: $A_6:S_3^2$ x 2, $C_3\times S_3\times S_6$ x 2, $C_3:S_4\times \GL(2,4)$, $C_3\times \GL(2,4):S_4$, $A_4\times C_3^2:S_5$, $S_3\times A_4\times \GL(2,4)$, $A_4\times C_3^2:S_5$, $(C_3\times \GL(2,4)):S_4$, $C_3\times S_4\times \GL(2,4)$, $C_3\times A_4\times A_6$, $C_3:S_3\times S_6$, $A_5:S_3^3$, $A_6:S_3^2$, $S_3^2\times A_6$

Order 12096: $C_3:S_4\times \GL(3,2)$ x 2, $C_3\times S_4\times \GL(3,2)$ x 2, $S_3\times A_4\times \GL(3,2)$ x 2, $F_8:C_6^2:C_6$, $F_8:C_3^2:S_4$, $F_8:C_3^2\times S_4$, $C_3^2\times C_2^3:\GL(3,2)$

Order 11520: $C_2\times D_4\times S_6$

Order 10752: $D_4\times C_2^3:\GL(3,2)$ x 4, $C_2^6.\GL(3,2)$ x 2, $C_4\times C_2^4:\GL(3,2)$

Order 10368: $S_3\times A_4^2:D_6$ x 2, $C_3\times C_2^4:S_3^3$ x 2, $S_3\times A_4^2:D_6$ x 2, $A_4^3:C_6$, $A_4^3:C_6$, $A_4^3:C_6$, $A_4^3:S_3$, $A_4^3:S_3$, $A_4^3:S_3$, $A_4^3:S_3$, $A_4^3:C_6$, $A_4^3:S_3$, $S_3\times A_4^3$, $A_4^3:C_6$, $(C_2^3\times C_6):S_3^3$, $A_4^2:(S_3\times D_6)$, $A_4\wr C_2\times S_3^2$, $C_3:S_4^2\times S_3$, $S_3^2\times \PSOPlus(4,3)$, $S_3^3:C_2\times S_4$, $A_4^2:(S_3\times D_6)$

Order 10080: $C_2^2\times A_7$ x 7, $C_4\times A_7$ x 2

Order 9216: $C_2^5.(D_6\times S_4)$, $C_2^6.D_6^2$, $C_2^4.S_4^2$, $C_2^6.D_6^2$, $C_2^6.D_6^2$, $C_2^4.S_4^2$

Order 8640: $D_6\times S_6$ x 12, $S_4\times C_3:S_5$ x 8, $S_4\times A_6$ x 3, $A_4:S_6$ x 3, $A_4\times S_6$ x 3, $S_3\times A_4:S_5$ x 3, $C_6^2:(C_2\times S_5)$ x 2, $C_6^2:(C_2\times S_5)$ x 2, $(C_6\times S_6):C_2$ x 2, $D_6:S_6$ x 2, $S_3\times A_4\times S_5$ x 2, $(C_2\times C_6):S_6$ x 2, $C_3\times C_2^2:S_6$ x 2, $D_6:S_6$ x 2, $C_3:D_4\times A_6$ x 2, $C_2\times D_6\times A_6$ x 2, $C_2\times C_6:S_6$ x 2, $C_2\times C_6\times S_6$ x 2, $S_3\times S_4\times A_5$ x 2, $C_3:S_4\times S_5$ x 2, $C_3:(S_4\times S_5)$ x 2, $A_5:D_6^2$ x 2, $S_3\times \GL(2,4):D_4$ x 2, $C_3\times S_4\times S_5$ x 2, $C_2\times A_4\times A_6$, $C_4\times S_3\times A_6$, $S_3\times D_4\times \GL(2,4)$, $D_4\times C_3^2:S_5$, $D_4\times C_3^2:S_5$, $C_3:(C_4\times S_6)$, $D_{12}\times A_6$, $C_{12}:(S_3\times S_5)$, $D_{12}\times C_3:S_5$, $C_3:C_4\times S_6$, $D_6.S_6$, $C_{12}:S_6$, $D_5.S_3^2\times S_4$, $C_{12}\times S_6$, $C_{12}:S_6$, $C_{12}:S_6$, $A_4\times C_6:S_5$, $C_2\times \GL(2,4):S_4$, $C_3\times D_4\times A_6$, $C_6\times S_4\times A_5$, $S_3\times C_{12}:S_5$, $S_3\times C_{12}:S_5$

Order 8064: $S_3\times C_2^3:\GL(3,2)$ x 6, $C_6\times C_2^3:\GL(3,2)$ x 3, $S_3\times D_4\times \GL(3,2)$ x 2, $C_2\times S_4\times \GL(3,2)$ x 2, $S_3\times S_4\times F_8$, $(D_{12}\times F_8):C_6$, $F_8:C_6\times S_4$

Order 7560: $C_3\times A_7$ x 3

Order 6912: $C_2^5:S_3^3$ x 18, $A_4^3:C_2^2$ x 8, $A_4:S_4^2$ x 7, $A_4^3:C_2^2$ x 7, $A_4^3:C_2^2$ x 5, $A_4^3:C_2^2$ x 5, $S_3\times A_4:\GL(2,\mathbb{Z}/4)$ x 3, $(A_4\times C_2^4):S_3^2$ x 3, $D_6\times S_4^2$ x 3, $A_4^2:(S_3\times D_4)$ x 2, $(C_2^2\times A_4^2):D_6$ x 2, $C_2^2\times A_4^2:D_6$ x 2, $A_4\times C_2^4:S_3^2$ x 2, $(A_4\times C_2^4):S_3^2$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $A_4^2:(S_3\times D_4)$ x 2, $(C_2^2\times A_4^2):D_6$ x 2, $C_2^5.S_3^3$ x 2, $A_4^2:C_2^2\times D_6$ 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, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $(C_2\times C_6).S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $C_3\times C_2^2:S_4^2$ x 2, $C_3\times C_2^2.S_4^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $(C_2^2\times A_4^2):D_6$ x 2, $A_4^2:(S_3\times D_4)$ x 2, $C_2^2\times A_4^2.D_6$ x 2, $A_4^2:(S_3\times D_4)$ x 2, $C_3\times C_2^6:S_3^2$ x 2, $C_6\times A_4^2:C_2^3$ x 2, $C_2^2\times A_4^2:D_6$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $C_2^5.S_3^3$ x 2, $A_4^2:(S_3\times D_4)$ x 2, $(C_2^2\times A_4^2):D_6$ x 2, $S_3\times A_4^2:D_4$ x 2, $C_3\times C_2\wr S_3^2$ x 2, $C_4\times A_4^2:D_6$, $S_3\times A_4^2:D_4$, $(C_4\times A_4^2):D_6$, $C_2^6:(C_3\times S_3^2)$, $C_2^2:A_4^2:D_6$, $S_3\times D_4\times A_4^2$, $A_4^2:C_6\times D_4$, $A_4^2:S_3\times D_4$, $C_{12}:\POPlus(4,3)$, $C_{12}:\POPlus(4,3)$, $C_{12}\times \POPlus(4,3)$, $Q_8:A_4^2:C_6$, $A_4^2:C_6\times D_4$, $Q_8:A_4^2:S_3$, $Q_8:A_4^2:S_3$, $C_2^5:(C_3^2\times S_4)$, $D_6.S_4^2$, $D_6.S_4^2$, $S_3\times C_2^3:A_4^2$, $C_{12}:\POPlus(4,3)$, $C_{12}:\POPlus(4,3)$, $C_{12}:\POPlus(4,3)$, $C_{12}:\POPlus(4,3)$, $C_4\times A_4^2:D_6$, $A_4^2:S_3\times D_4$, $Q_8:A_4^2:C_6$, $S_3\times A_4\times \GL(2,\mathbb{Z}/4)$, $S_3\times A_4^2:D_4$, $D_6.S_4^2$, $C_4\times A_4^2:D_6$, $S_4\times C_3:\GL(2,\mathbb{Z}/4)$, $D_6:S_4^2$, $D_6:S_4^2$, $C_3\times S_4\times \GL(2,\mathbb{Z}/4)$, $C_2^6:(C_3\times S_3^2)$, $(A_4\times C_2^4):C_6^2$, $(C_2\times C_6):S_4^2$, $D_6:S_4^2$, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$, $S_3\times Q_8:A_4^2$, $Q_8:A_4^2:S_3$, $C_4\times A_4^2:D_6$, $C_{12}:\POPlus(4,3)$, $C_2^5.S_3^3$, $C_2^5.S_3^3$, $C_2^5.S_3^3$, $C_2^5.S_3^3$, $C_2^6:(C_3\times S_3^2)$, $C_2^4.(A_4\times S_3^2)$, $S_3\times A_4^2.C_2^3$, $C_2^3:S_4\times S_3^2$, $A_4^3:C_2^2$, $A_4^3:C_2^2$, $A_4\wr C_2\times D_{12}$, $C_2^5.S_3^3$, $C_2^5.S_3^3$, $D_6:S_4^2$, $C_2^5:(C_3^2\times S_4)$, $C_2^3:C_6^2:S_4$, $C_2^3:C_6^2:S_4$, $C_2^3:C_6^2:S_4$, $(A_4\times D_{12}):S_4$, $D_{12}\times \PSOPlus(4,3)$, $A_4.A_4^2.C_2^2$, $C_2^3:S_4\times S_3^2$, $(C_2^5\times C_6):S_3^2$, $(C_2^5\times C_6):S_3^2$, $C_3\times C_2^6:S_3^2$, $(C_2^5\times C_6):S_3^2$, $C_2^4.(A_4\times S_3^2)$, $C_2^6:(C_3\times S_3^2)$, $C_2^2\wr C_3:S_3^2$, $C_2^5.S_3^3$, $A_4^3:C_2^2$, $A_4^3:C_2^2$

Order 6480: $\GL(2,4):S_3^2$ x 2, $\GL(2,4):S_3^2$ x 2, $C_3^2:S_6$ x 2, $C_3\times S_3\times A_6$ x 2, $C_6^2:\GL(2,4)$, $C_3:S_3\times A_6$, $C_3^2:S_6$, $C_3^2\times S_6$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $S_3^2\times \GL(2,4)$

Order 6048: $S_3^2\times \GL(3,2)$ x 2, $C_3\times A_4\times \GL(3,2)$ x 2, $A_4\times F_8:C_3^2$, $F_8:C_3\times S_3^2$

Order 5760: $D_4\times S_6$ x 8, $C_2^3\times S_6$ x 2, $C_2^3:S_6$ x 2, $S_3\times D_4\times S_5$ x 2, $C_2\times S_4\times S_5$ x 2, $C_6:S_5\times D_4$, $C_2\times D_4\times A_6$, $C_2\times C_4:S_6$, $C_2\times C_4\times S_6$

Order 5376: $C_2^5:\GL(3,2)$ x 7, $C_4\times C_2^3:\GL(3,2)$ x 2

Order 5184: $A_4^2:S_3^2$ x 3, $A_4^2:S_3^2$ x 2, $A_4^2:C_6\times S_3$ x 2, $A_4^2:S_3^2$ x 2, $A_4^2:C_6\times S_3$ x 2, $C_3\times A_4^2:D_6$ x 2, $S_4\times S_3^3$ x 2, $D_6^2:S_3^2$ x 2, $D_6^2:S_3^2$ x 2, $S_3^3:S_4$ x 2, $C_3\times A_4^2:D_6$ x 2, $C_3\times A_4^3$, $(C_3\times A_4^2):D_6$, $(C_3\times A_4^2):D_6$, $(C_3\times A_4^2):D_6$, $C_3^2:S_4^2$, $C_3^2:S_4^2$, $A_4^2:S_3^2$, $A_4^2:S_3^2$, $(A_4\times C_6^2):D_6$, $S_3^2\times A_4^2$, $C_3^2:S_4^2$, $C_3^2:S_4^2$, $C_6^2:C_6\times S_4$, $S_3^2:C_6\times S_4$, $C_3^3:(D_4\times S_4)$, $S_4\times C_3^2:D_{12}$, $A_4^2:C_6^2$, $C_3^3:(D_4\times S_4)$, $S_3\times C_6^2:D_{12}$, $C_6^2:(C_4\times S_3^2)$, $D_6^2:C_6\times S_3$, $A_4^2:S_3^2$, $A_4^2:S_3^2$

Order 5040: $C_2\times A_7$ x 5

Order 4608: $C_2^5.D_6^2$ x 13, $C_2^3:S_4^2$ x 10, $C_2^3:S_4^2$ x 6, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5:(S_3\times S_4)$ x 2, $(C_2^2\times D_6).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_6^2$ x 2, $D_6\times C_2^4:S_4$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5:D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5:D_6^2$ x 2, $C_2^7:S_3^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^4.(D_6\times S_4)$ x 2, $C_2^3:S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $C_2^7:S_3^2$ x 2, $(C_6\times A_4).C_2^6$, $C_2^5.D_6^2$, $C_{12}:C_2^4:S_4$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5:(C_6\times S_4)$, $(C_6\times \GL(2,\mathbb{Z}/4)):D_4$, $C_2^3:S_4\times D_{12}$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $S_3\times C_2^5:S_4$, $C_2^5.D_6^2$, $S_3\times C_2^5:S_4$, $C_2^5.D_6^2$, $C_6:\GL(2,\mathbb{Z}/4):D_4$, $C_2^5.(A_4\times D_6)$, $A_4^2:C_2^2\times D_4$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2\wr C_6.D_6$, $C_2.C_2^6.C_3.D_6$, $A_4\times C_2^4:S_4$, $C_2^3:S_4^2$, $C_2^4:A_4\times S_4$, $C_2^5.D_6^2$, $C_2^6:(C_6\times D_6)$, $C_2\wr C_6:D_6$, $C_2^5.D_6^2$, $C_2^5:(C_6\times S_4)$, $(C_2^3\times D_{12}):S_4$, $A_4^2:C_2^2\times D_4$, $(A_4\times C_2^4):S_4$, $C_2^4.(A_4\times S_4)$, $A_4\times C_2^4:S_4$, $C_2^7:S_3^2$

Order 1024: $C_2^4:D_4^2$

Order 144: $S_3\times S_4$

Order 81: $C_3^4$

Order 72: $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 36: $C_3\times A_4$

Order 24: $C_2\times D_6$, $S_4$

Order 12: $C_2\times C_6$, $A_4$

Order 7: $C_7$

Order 6: $S_3$

Order 5: $C_5$

Order 4: $C_2^2$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2903040: $S_3\times S_4\times A_8$ ($C_1$)

Order 1451520: $C_3:S_4\times A_8$ ($C_2$), $A_8\times S_3\times A_4$ ($C_2$), $C_3\times S_4\times A_8$ ($C_2$)

Order 725760: $C_3\times A_4\times A_8$ ($C_2^2$)

Order 483840: $C_2^2\times S_3\times A_8$ ($S_3$), $S_4\times A_8$ ($S_3$)

Order 241920: $C_2\times C_6\times A_8$ ($D_6$), $A_4\times A_8$ ($D_6$)

Order 120960: $S_3\times A_8$ ($S_4$)

Order 80640: $C_2^2\times A_8$ ($S_3^2$)

Order 60480: $C_3\times A_8$ ($C_2\times S_4$)

Order 20160: $A_8$ ($S_3\times S_4$)

Order 144: $S_3\times S_4$ ($A_8$)

Order 72: $S_3\times A_4$ ($C_2\times A_8$), $C_3:S_4$ ($C_2\times A_8$), $C_3\times S_4$ ($C_2\times A_8$)

Order 36: $C_3\times A_4$ ($C_2^2\times A_8$)

Order 24: $C_2\times D_6$ ($S_3\times A_8$), $S_4$ ($S_3\times A_8$)

Order 12: $C_2\times C_6$ ($D_6\times A_8$), $A_4$ ($D_6\times A_8$)

Order 6: $S_3$ ($S_4\times A_8$)

Order 4: $C_2^2$ ($S_3^2\times A_8$)

Order 3: $C_3$ ($(C_2\times S_4).A_8$)

Order 1: $C_1$ ($S_3\times S_4\times A_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2903040: $S_3\times S_4\times A_8$ ($C_1$)

Order 1451520: $C_3:S_4\times A_8$ ($C_2$), $A_8\times S_3\times A_4$ ($C_2$), $C_3\times S_4\times A_8$ ($C_2$)

Order 725760: $C_3\times A_4\times A_8$ ($C_2^2$)

Order 483840: $C_2^2\times S_3\times A_8$ ($S_3$), $S_4\times A_8$ ($S_3$)

Order 241920: $C_2\times C_6\times A_8$ ($D_6$), $A_4\times A_8$ ($D_6$)

Order 120960: $S_3\times A_8$ ($S_4$)

Order 80640: $C_2^2\times A_8$ ($S_3^2$)

Order 60480: $C_3\times A_8$ ($C_2\times S_4$)

Order 20160: $A_8$ ($S_3\times S_4$)

Order 144: $S_3\times S_4$ ($A_8$)

Order 72: $S_3\times A_4$ ($C_2\times A_8$), $C_3:S_4$ ($C_2\times A_8$), $C_3\times S_4$ ($C_2\times A_8$)

Order 36: $C_3\times A_4$ ($C_2^2\times A_8$)

Order 24: $C_2\times D_6$ ($S_3\times A_8$), $S_4$ ($S_3\times A_8$)

Order 12: $C_2\times C_6$ ($D_6\times A_8$), $A_4$ ($D_6\times A_8$)

Order 6: $S_3$ ($S_4\times A_8$)

Order 4: $C_2^2$ ($S_3^2\times A_8$)

Order 3: $C_3$ ($(C_2\times S_4).A_8$)

Order 1: $C_1$ ($S_3\times S_4\times A_8$)

Series

Derived series

$S_3\times S_4\times A_8$

$\rhd$

$C_3\times A_4\times A_8$

$\rhd$

$C_2^2\times A_8$

$\rhd$

$A_8$

Chief series

$S_3\times S_4\times A_8$

$\rhd$

$S_3\times S_4$

$\rhd$

$S_3\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_3\times S_4\times A_8$

$\rhd$

$C_3\times A_4\times A_8$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $210 \times 210$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $180 \times 180$ rational character table (warning: may be slow to load).