Abstract group 1088640.d: $S_6\times {}^2G(2,3)$

• Label: 1088640.d
• Order: \( 2^{7} \cdot 3^{5} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{5} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$S_6\times {}^2G(2,3)$
Order:	\(1088640\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \cdot 7 \)
Exponent:	\(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$\SL(2,8).A_6.C_6.C_2$, of order \(2177280\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$, $C_3$, $A_6$, $\SL(2,8)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	9	10	12	14	15	18	21	28	30	35	36	42	45
Elements	1	4863	18224	11520	144	290544	216	40824	9072	131040	16200	32256	158760	17280	38880	72576	31104	90720	51840	72576	1088640
Conjugacy classes	1	7	11	4	1	37	1	9	1	10	3	3	15	2	2	2	1	6	2	3	121
Divisions	1	7	8	4	1	24	1	6	1	6	3	2	10	2	2	1	1	4	2	2	88
Autjugacy classes	1	5	7	4	1	22	1	6	1	10	2	3	9	1	2	2	1	6	1	3	88

Dimension	1	2	5	7	8	9	10	14	16	18	20	21	27	32	35	40	63	70	72	80	105	112	126	128	135	140	144	160	189	210	224	243	256	270	336	432
Irr. complex chars.	6	0	12	6	6	6	6	0	3	0	0	2	2	0	12	12	6	6	6	6	4	3	0	3	4	0	0	0	2	2	0	2	0	2	1	1	121
Irr. rational chars.	2	2	4	2	2	2	6	2	3	2	2	2	2	1	4	4	2	6	2	6	4	1	2	1	4	2	2	2	2	2	1	2	1	2	1	1	88

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$180624$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	35	35	35
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(8,13)(9,12)(10,11)(14,15), (7,14)(8,10)(9,13)(11,12), (7,8,9,10,11,12,13), (1,2,3,4,5,6), (1,4)(5,6), (8,9,11)(10,13,12), (1,2,6,4)(3,5)\rangle$
Direct product:	$S_6$ $\, \times\, $ ${}^2G(2,3)$
Semidirect product:	$(S_6\times \SL(2,8))$ $\,\rtimes\,$ $C_3$	$\SL(2,8)$ $\,\rtimes\,$ $(C_3\times S_6)$	$(A_6\times \SL(2,8))$ $\,\rtimes\,$ $C_6$	$A_6$ $\,\rtimes\,$ $(\SL(2,8):C_6)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 7056720 subgroups in 2752 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_6\times {}^2G(2,3)$
Commutator:	$G' \simeq$ $A_6\times \SL(2,8)$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_6\times {}^2G(2,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $S_6\times {}^2G(2,3)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $S_6\times {}^2G(2,3)$
Socle:	$\operatorname{soc} \simeq$ $A_6\times \SL(2,8)$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9:C_3^3$
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: 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

Order 1088640: $S_6\times {}^2G(2,3)$

Order 544320: $A_6\times {}^2G(2,3)$

Order 362880: $S_6\times \SL(2,8)$

Order 181440: $S_5\times {}^2G(2,3)$ x 2, $A_6\times \SL(2,8)$

Order 120960: $F_8:C_3\times S_6$

Order 108864: ${}^2G(2,3)\times \SOPlus(4,2)$

Order 90720: $A_5\times {}^2G(2,3)$ x 2

Order 72576: $S_4\times \SL(2,8):C_6$ x 2

Order 60480: $S_5\times \SL(2,8)$ x 2, $F_8:C_3\times A_6$

Order 54432: $S_3^2\times {}^2G(2,3)$ x 2, $(C_3^2\times \SL(2,8)):C_{12}$

Order 40320: $F_8\times S_6$

Order 38880: $C_9:C_6\times S_6$

Order 36288: $S_4\times {}^2G(2,3)$ x 4, $A_4\times \SL(2,8):C_6$ x 2, $\SL(2,8)\times \SOPlus(4,2)$

Order 30240: $A_5\times \SL(2,8)$ x 2, $F_5\times {}^2G(2,3)$, $F_7\times S_6$

Order 27216: $C_3\times S_3\times {}^2G(2,3)$ x 2, $(C_3^2\times \SL(2,8)):C_6$

Order 24192: $C_2\times S_4\times \SL(2,8)$ x 2, $D_4\times \SL(2,8):C_6$

Order 20160: $F_8:C_3\times S_5$ x 2, $F_8\times A_6$

Order 19440: $C_9:C_3\times S_6$, $C_9:(C_3\times S_6)$, $C_9:C_6\times A_6$

Order 18144: $S_3^2\times \SL(2,8)$ x 2, $D_6\times {}^2G(2,3)$ x 2, $A_4\times {}^2G(2,3)$ x 2, $C_3^2:C_4\times \SL(2,8)$

Order 17280: $C_2\times A_4\times S_6$

Order 15120: $D_5\times {}^2G(2,3)$, $A_6:F_7$, $F_7\times A_6$, $C_7:C_3\times S_6$

Order 13608: $C_3^2\times {}^2G(2,3)$

Order 12960: $D_9\times S_6$, $C_3\times S_3\times S_6$

Order 12096: $S_4\times \SL(2,8)$ x 4, $D_4\times {}^2G(2,3)$ x 4, $C_2^3:{}^2G(2,3)$ x 2, $C_2^3\times {}^2G(2,3)$ x 2, $C_2\times A_4\times \SL(2,8)$ x 2, $C_2\times \SL(2,8):C_{12}$, $(S_3^2\times F_8):C_6$

Order 10080: $F_8:\GL(2,4)$ x 2, $F_5\times \SL(2,8)$, $D_7\times S_6$

Order 9720: $C_9:C_3\times A_6$

Order 9072: $S_3\times {}^2G(2,3)$ x 6, $C_3\times S_3\times \SL(2,8)$ x 2, $C_6\times {}^2G(2,3)$ x 2, $C_3:S_3\times \SL(2,8)$

Order 8640: $A_4\times S_6$ x 2, $C_2\times A_4\times A_6$

Order 8064: $F_8:C_6\times S_4$ x 2, $C_2\times D_4\times \SL(2,8)$

Order 7560: $\SL(2,8):C_{15}$, $C_7:C_3\times A_6$

Order 6720: $F_8\times S_5$ x 2

Order 6480: $C_9\times S_6$ x 2, $C_9:C_6\times S_5$ x 2, $C_9:S_6$, $D_9\times A_6$, $C_3^2\times S_6$, $C_3^2:S_6$, $C_3\times S_3\times A_6$

Order 6048: $C_2^2\times {}^2G(2,3)$ x 5, $\SL(2,8):A_4$ x 2, $A_4\times \SL(2,8)$ x 2, $\SL(2,8):C_{12}$ x 2, $F_8:C_3\times S_3^2$ x 2, $D_6\times \SL(2,8)$ x 2, $(C_3^2\times F_8):C_{12}$

Order 5760: $C_2^3\times S_6$

Order 5040: $F_7\times S_5$ x 2, $C_7:S_6$, $C_7\times S_6$, $D_5\times \SL(2,8)$, $D_7\times A_6$

Order 4536: $C_3\times {}^2G(2,3)$ x 4, $C_3^2\times \SL(2,8)$

Order 4320: $C_6\times S_6$, $A_4\times A_6$, $S_3\times S_6$

Order 4032: $D_4\times \SL(2,8)$ x 4, $F_8:C_3\times S_4$ x 4, $C_2^3\times \SL(2,8)$ x 2, $A_4\times F_8:C_6$ x 2, $F_8\times \SOPlus(4,2)$, $C_2\times C_4\times \SL(2,8)$

Order 3888: $(D_9\times S_3^2):C_6$

Order 3360: $F_8\times A_5$ x 2, $F_5\times F_8:C_3$

Order 3240: $C_9\times A_6$ x 2, $C_9:C_3\times S_5$ x 2, $D_9:\GL(2,4)$ x 2, $C_9:(C_3\times S_5)$ x 2, $C_3^2\times A_6$

Order 3024: $\SL(2,8):C_6$ x 5, $S_3\times \SL(2,8)$ x 4, $S_3\times F_8:C_3^2$ x 2, $C_6\times \SL(2,8)$ x 2, $F_7\times \SOPlus(4,2)$, $(C_3^2\times F_8):C_6$

Order 2880: $C_2\times A_4\times S_5$ x 2, $C_2^2\times S_6$ x 2, $C_2^3\times A_6$

Order 2688: $C_2\times S_4\times F_8$ x 2, $D_4\times F_8:C_6$

Order 2592: $C_{18}:C_6\times S_4$ x 2

Order 2520: $F_7\times A_5$ x 2, $A_5:F_7$ x 2, $C_7:C_3\times S_5$ x 2, $C_5\times \SL(2,8)$, $C_7\times A_6$

Order 2160: $C_3\times S_6$ x 3, $D_9\times S_5$ x 2, $\GL(2,4):D_6$ x 2, $C_6\times A_6$, $C_3:S_6$, $S_3\times A_6$

Order 2016: $C_2^2\times \SL(2,8)$ x 5, $S_3^2\times F_8$ x 2, $C_2\times S_4\times F_7$ x 2, $F_8:C_6\times S_3$ x 2, $C_4\times \SL(2,8)$ x 2, $A_4\times F_8:C_3$ x 2, $C_3^2:C_4\times F_8$

Order 1944: $C_3^2.S_3^3$ x 2, $(C_9\times S_3^2):C_6$ x 2, $C_3^4.D_{12}$, $(C_9\times S_3^2):C_6$, $(C_3^2\times D_9):C_{12}$

Order 1728: $C_2\times D_6^2:C_6$

Order 1680: $D_7\times S_5$ x 2, $D_5\times F_8:C_3$

Order 1620: $C_9:\GL(2,4)$ x 2

Order 1512: ${}^2G(2,3)$ x 3, $S_3^2\times F_7$ x 2, $S_3^2:F_7$ x 2, $C_3\times \SL(2,8)$ x 2, $F_8:C_3^3$, $C_3^2:C_{28}:C_6$, $(C_7\times S_3^2):C_6$, $C_3^2:C_4\times F_7$

Order 1440: $A_4\times S_5$ x 4, $C_2^3:\GL(2,4)$ x 2, $C_2\times S_6$ x 2, $C_2^2\times A_6$

Order 1344: $S_4\times F_8$ x 4, $D_4\times F_8:C_3$ x 4, $C_2^5:C_{42}$ x 2, $C_2\times F_8:A_4$ x 2, $C_2^2\times F_8:C_6$ x 2, $C_2\times F_8:C_{12}$

Order 1296: $C_6^2.S_3^2$ x 8, $A_4\times C_{18}:C_6$ x 2, $C_9:(C_6\times S_4)$ x 2, $C_9:C_6\times S_4$ x 2, $S_3^3:C_6$, $S_3^2:D_{18}$

Order 1260: $C_7:\GL(2,4)$ x 2

Order 1152: $A_4^2:C_2^3$ x 2

Order 1120: $F_5\times F_8$

Order 1080: $C_9\times S_5$ x 4, $S_3\times \GL(2,4)$ x 2, $C_3^2\times S_5$ x 2, $C_3^2:S_5$ x 2, $C_3\times A_6$ x 2, $D_9\times A_5$ x 2, $C_9:S_5$ x 2, $D_{45}:C_{12}$

Order 1008: $S_4\times F_7$ x 8, $F_8:C_3\times S_3$ x 6, $C_2\times \SL(2,8)$ x 3, $C_3\times S_3\times F_8$ x 2, $C_2\times A_4\times F_7$ x 2, $C_2\times A_4:F_7$ x 2, $C_7:C_6\times S_4$ x 2, $C_3\times F_8:C_6$ x 2, $C_3:S_3\times F_8$, $S_3^2:D_{14}$

Order 972: $C_3^3.S_3^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.C_6^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.S_3^2$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$

Order 960: $C_2^3\times S_5$ x 2

Order 896: $C_2^6:C_{14}$

Order 864: $D_6^2:C_6$ x 4, $D_6^2:C_6$ x 2, $C_6\times S_3\times S_4$ x 2, $S_4\times D_{18}$ x 2, $C_2\times C_6^2:C_{12}$, $C_2\times D_{36}:C_6$

Order 840: $D_7\times A_5$ x 2, $C_7:S_5$ x 2, $C_7\times S_5$ x 2, $F_8:C_{15}$, $F_5\times F_7$

Order 756: $C_{21}:(C_6\times S_3)$ x 2, $C_{21}:C_6\times S_3$ x 2, $C_{21}:C_6\times S_3$ x 2, $C_{21}:C_6^2$ x 2, $C_3:S_3\times F_7$, $(C_3\times C_{21}):C_{12}$, $(C_3\times C_{21}):C_{12}$

Order 720: $C_6\times S_5$ x 2, $A_4\times A_5$ x 2, $S_3\times S_5$ x 2, $S_6$ x 2, $C_2\times A_6$

Order 672: $C_2\times F_8:C_6$ x 5, $S_4\times D_{14}$ x 2, $D_6\times F_8$ x 2, $A_4\times F_8$ x 2, $F_8:A_4$ x 2, $F_8:C_{12}$ x 2, $C_2\times D_4\times F_7$

Order 648: $A_4\times C_9:C_6$ x 4, $C_9:(C_3\times S_4)$ x 4, $C_9:C_3\times S_4$ x 4, $C_3\times S_3^3$ x 2, $C_3\wr D_4$ x 2, $C_9:C_6\times A_4$ x 2, $C_{18}:C_6\times S_3$ x 2, $D_9\times S_3^2$ x 2, $S_3^2:D_9$ x 2, $S_3^2:C_{18}$ x 2, $C_3^3:D_{12}$, $C_3^4:D_4$, $S_3\times C_3^2:C_{12}$, $C_3^2:D_{36}$, $C_3^2:C_4\times D_9$

Order 576: $A_4^2:C_2^2$ x 12, $C_2^2\times A_4^2$ x 2, $D_6^2:C_2^2$

Order 560: $D_5\times F_8$

Order 540: $C_9\times A_5$ x 4, $C_3\times \GL(2,4)$ x 2, $D_{45}:C_6$, $C_{45}:C_{12}$, $C_{45}:C_{12}$

Order 504: $A_4\times F_7$ x 4, $A_4:F_7$ x 4, $C_7:C_3\times S_4$ x 4, $F_8:C_3^2$ x 4, $D_7\times S_3^2$ x 2, $S_3^2:D_7$ x 2, $C_7:C_6\times A_4$ x 2, $D_6\times F_7$ x 2, $C_3^2\times F_8$, $C_3^2:D_{28}$, $S_3^2:C_{14}$, $C_3^2:C_4\times D_7$, $\SL(2,8)$

Order 486: $C_3^4.S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^4.S_3$, $C_3^4.C_6$, $D_9:C_3^3$

Order 480: $C_2^2\times S_5$ x 4, $C_2^3\times A_5$ x 2, $C_2\times A_4\times F_5$

Order 448: $D_4\times F_8$ x 4, $C_2^3\times F_8$ x 2, $C_2^4:C_{28}$

Order 432: $C_6^2:D_6$ x 8, $D_9\times S_4$ x 8, $D_{36}:C_6$ x 8, $A_4\times S_3^2$ x 6, $C_6\times S_3\times A_4$ x 4, $C_{18}\times S_4$ x 4, $C_6^2:D_6$ x 2, $A_4:C_6^2$ x 2, $C_6^2:C_{12}$ x 2, $C_6^2.D_6$ x 2, $A_4\times D_{18}$ x 2, $D_{18}:A_4$ x 2, $C_{18}:S_4$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6:S_3\times A_4$, $C_6\times \SOPlus(4,2)$, $S_3^3:C_2$, $C_{12}.C_6^2$, $D_{36}:C_6$, $D_{18}:C_{12}$

Order 420: $C_7\times A_5$ x 2, $D_5\times F_7$, $C_{35}:C_{12}$, $C_{35}:C_{12}$

Order 384: $C_2^4\times S_4$ x 2, $C_2^6:C_6$

Order 378: $C_3^2:F_7$ x 2, $C_3\times C_{21}:C_6$ x 2, $C_3^2:F_7$, $(C_3\times C_{21}):C_6$, $C_3^2\times F_7$

Order 360: $C_3\times S_5$ x 6, $C_6\times A_5$ x 2, $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $D_9\times F_5$, $D_{15}:C_{12}$, $A_6$

Order 336: $D_7\times S_4$ x 8, $D_4\times F_7$ x 8, $F_8:C_6$ x 5, $S_3\times F_8$ x 4, $D_{14}:A_4$ x 2, $A_4\times D_{14}$ x 2, $C_2^3\times F_7$ x 2, $C_{14}:S_4$ x 2, $C_{14}\times S_4$ x 2, $C_6\times F_8$ x 2, $C_2^3:F_7$ x 2, $C_2\times C_{28}:C_6$, $D_{28}:C_6$, $D_{14}:C_{12}$

Order 324: $C_3^2.S_3^2$ x 12, $C_3^2\times S_3^2$ x 4, $C_9\times S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3\wr C_2^2$ x 2, $C_3^3.D_6$ x 2, $C_3^2.C_6^2$ x 2, $C_9:C_6^2$ x 2, $C_6^2.C_3^2$ x 2, $C_9:S_3^2$ x 2, $C_9:S_3^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^2:C_{36}$ x 2, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^2:D_{18}$, $(C_3^2\times C_9):C_4$

Order 288: $A_4\times S_4$ x 8, $C_2\times A_4^2$ x 6, $C_6^2:D_4$ x 4, $C_2\times D_6^2$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6\times S_4$ x 2, $D_6\times S_4$ x 2, $C_6^2:C_2^3$, $D_4\times D_{18}$, $C_2\times C_6^2:C_4$

Order 280: $C_5\times F_8$, $D_7\times F_5$

Order 270: $C_{45}:C_6$, $C_{45}:C_6$, $C_9:C_{30}$

Order 252: $S_3\times F_7$ x 10, $C_{21}:D_6$ x 2, $S_3\times D_{21}$ x 2, $C_7\times S_3^2$ x 2, $C_{21}:D_6$ x 2, $C_{42}:C_6$ x 2, $C_{42}:C_6$ x 2, $C_6\times F_7$ x 2, $A_4\times C_7:C_3$ x 2, $C_{21}:D_6$, $(C_3\times C_{21}):C_4$, $C_3^2:C_{28}$

Order 243: $C_9:C_3^3$

Order 240: $C_2\times S_5$ x 4, $A_4\times F_5$ x 2, $C_2^2\times A_5$ x 2, $A_4\times D_{10}$

Order 224: $C_2^2\times F_8$ x 5, $C_4\times F_8$ x 2, $D_4\times D_{14}$

Order 216: $D_{18}:C_6$ x 11, $C_9\times S_4$ x 8, $D_{18}:C_6$ x 8, $C_6^2:C_6$ x 8, $S_3^2:C_6$ x 5, $A_4\times D_9$ x 4, $D_9:A_4$ x 4, $C_9:S_4$ x 4, $C_3^2.S_4$ x 4, $C_{36}:C_6$ x 4, $C_{36}:C_6$ x 4, $C_{36}:C_6$ x 4, $C_6\times S_3^2$ x 4, $C_3^2:S_4$ x 4, $C_3^2\times S_4$ x 4, $C_{18}:A_4$ x 4, $A_4\times C_{18}$ x 4, $C_6^2:C_6$ x 3, $C_6^2:C_6$ x 2, $S_3^3$ x 2, $S_3^2:S_3$ x 2, $C_6.C_6^2$ x 2, $C_6^2.C_6$ x 2, $S_3\times D_{18}$ x 2, $C_{18}:C_{12}$, $C_{18}:C_{12}$, $C_2\times C_3^2:C_{12}$, $C_3^2:D_{12}$, $C_3^2:C_4\times S_3$

Order 210: $C_{35}:C_6$, $C_{35}:C_6$, $C_5\times F_7$

Order 192: $C_2^3\times S_4$ x 12, $C_2^5:C_6$ x 12, $A_4\times C_2^4$ x 4, $C_2^2\wr C_3$ x 2, $C_2^4:C_{12}$

Order 189: $C_7:C_3^3$

Order 180: $\GL(2,4)$ x 4, $C_9\times F_5$ x 2, $D_5\times D_9$, $C_{45}:C_4$, $C_{15}:D_6$, $C_{15}:C_{12}$, $C_3^2\times F_5$

Order 168: $C_2^2\times F_7$ x 11, $D_{14}:C_6$ x 8, $C_{28}:C_6$ x 4, $C_4\times F_7$ x 4, $D_7:A_4$ x 4, $A_4\times D_7$ x 4, $C_7:S_4$ x 4, $C_7\times S_4$ x 4, $C_{28}:C_6$ x 4, $F_8:C_3$ x 3, $C_{14}:A_4$ x 2, $A_4\times C_{14}$ x 2, $C_2\times C_{14}:C_6$ x 2, $S_3\times D_{14}$ x 2, $C_3\times F_8$ x 2, $C_{14}:C_{12}$, $C_{14}:C_{12}$

Order 162: $C_3^3.C_6$ x 10, $C_3^3.S_3$ x 8, $D_9:C_3^2$ x 8, $C_3^2:C_{18}$ x 4, $C_3^2\wr C_2$ x 3, $S_3\times C_3^3$ x 3, $C_{18}:C_3^2$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:D_9$ x 2, $C_3^3:C_6$, $C_3^2:D_9$, $C_3^2\times D_9$

Order 160: $C_2^3\times F_5$

Order 144: $A_4\times D_6$ x 18, $C_6\times S_4$ x 16, $D_4\times D_9$ x 8, $S_3\times S_4$ x 8, $C_{12}:D_6$ x 8, $D_6^2$ x 6, $S_3^2:C_2^2$ x 6, $C_6^2:C_2^2$ x 4, $C_2^2:C_6^2$ x 4, $D_4\times C_{18}$ x 2, $C_{18}:D_4$ x 2, $C_6^2:C_4$ x 2, $C_6:S_4$ x 2, $A_4^2$ x 2, $C_6^2:C_2^2$ x 2, $C_2^2\times D_{18}$ x 2, $C_2^2:D_{18}$ x 2, $C_2\times D_{36}$, $C_4\times D_{18}$, $C_6^2:C_2^2$, $C_4:C_6^2$, $C_6\times D_{12}$, $C_{12}\times D_6$

Order 140: $D_5\times D_7$, $C_{35}:C_4$, $C_7\times F_5$

Order 135: $C_9:C_{15}$

Order 128: $D_4\times C_2^4$

Order 126: $C_{21}:C_6$ x 6, $C_{21}:C_6$ x 6, $C_3\times F_7$ x 6, $C_3\times D_{21}$ x 2, $S_3\times C_{21}$ x 2, $C_{21}:C_6$ x 2, $C_3:D_{21}$, $C_{21}:S_3$, $C_3^2\times D_7$

Order 120: $S_5$ x 4, $D_5\times A_4$ x 2, $C_2\times A_5$ x 2, $C_{10}\times A_4$, $C_6\times F_5$, $S_3\times F_5$

Order 112: $D_4\times D_7$ x 8, $C_2\times F_8$ x 3, $C_2^2\times D_{14}$ x 2, $C_{14}:D_4$ x 2, $D_4\times C_{14}$, $C_2\times D_{28}$, $C_4\times D_{14}$

Order 108: $C_3\times S_3^2$ x 22, $C_{18}:C_6$ x 16, $S_3\times D_9$ x 10, $C_3^2\times D_6$ x 6, $C_{18}:C_6$ x 5, $S_3\times C_{18}$ x 4, $C_9:A_4$ x 4, $C_9\times A_4$ x 4, $C_3^2:D_6$ x 3, $C_3^2\times A_4$ x 3, $C_3:S_3^2$ x 3, $C_3^2:C_{12}$ x 3, $C_9:C_{12}$ x 2, $C_3:S_3^2$ x 2, $C_3:D_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3^2.A_4$ x 2, $C_9:C_{12}$ x 2, $C_3^3:C_4$

Order 105: $C_7:C_{15}$

Order 96: $C_2^2\times S_4$ x 20, $C_2^3\times A_4$ x 19, $D_4\times A_4$ x 16, $C_2^3:A_4$ x 6, $C_2^3:C_{12}$ x 6, $C_2^3\times D_6$ x 2, $C_{12}:C_2^3$, $D_4\times D_6$

Order 90: $C_9\times D_5$ x 2, $C_3\times D_{15}$, $S_3\times C_{15}$, $C_3^2\times D_5$, $D_{45}$, $C_5\times D_9$

Order 84: $C_2\times F_7$ x 14, $S_3\times D_7$ x 8, $C_{14}:C_6$ x 5, $C_7:C_{12}$ x 2, $D_{42}$ x 2, $S_3\times C_{14}$ x 2, $C_3\times D_{14}$ x 2, $C_7:A_4$ x 2, $C_7\times A_4$ x 2, $C_7:C_{12}$ x 2

Order 81: $C_9:C_3^2$ x 8, $C_3^2\times C_9$ x 2, $C_3^4$

Order 80: $C_2^2\times F_5$ x 2, $C_2^2\times D_{10}$

Order 72: $C_6\times A_4$ x 20, $S_3\times A_4$ x 20, $C_6\times D_6$ x 17, $C_3\times S_4$ x 16, $C_2\times D_{18}$ x 11, $S_3\times D_6$ x 10, $C_9:D_4$ x 8, $C_6\wr C_2$ x 8, $D_4\times C_9$ x 8, $\SOPlus(4,2)$ x 6, $D_{36}$ x 4, $C_4\times D_9$ x 4, $C_3:S_4$ x 4, $D_4\times C_3^2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_2^2\times C_{18}$ x 4, $C_2^2:C_{18}$ x 4, $C_2^2:D_9$ x 4, $C_2\times C_6^2$ x 3, $C_2\times C_{36}$ x 2, $C_6:D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_{18}:C_4$, $C_6\times C_{12}$, $C_6:C_{12}$

Order 70: $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 64: $D_4\times C_2^3$ x 12, $C_2^6$ x 2, $C_2^4\times C_4$

Order 63: $C_{21}:C_3$ x 4, $C_3\times C_{21}$

Order 60: $C_3\times F_5$ x 3, $A_5$ x 2, $C_5\times A_4$, $S_3\times D_5$, $C_{15}:C_4$, $C_3\times D_{10}$

Order 56: $C_2\times D_{14}$ x 11, $C_7:D_4$ x 8, $C_7\times D_4$ x 4, $D_{28}$ x 4, $C_4\times D_7$ x 4, $C_2^2\times C_{14}$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$, $F_8$

Order 54: $S_3\times C_3^2$ x 23, $C_9:C_6$ x 14, $C_3^2:C_6$ x 13, $S_3\times C_9$ x 12, $C_9:C_6$ x 9, $C_3:D_9$ x 6, $C_3\times D_9$ x 6, $C_3\times C_{18}$ x 4, $C_3^2\times C_6$ x 3, $C_3^2:S_3$

Order 48: $C_2^2\times A_4$ x 29, $C_2\times S_4$ x 22, $C_2^2\times D_6$ x 14, $C_6\times D_4$ x 13, $S_3\times D_4$ x 8, $C_2^3\times C_6$ x 4, $C_4\times A_4$ x 4, $C_2^2:A_4$ x 2, $C_6:D_4$ x 2, $C_2^2\times C_{12}$, $C_2\times D_{12}$, $C_4\times D_6$

Order 45: $C_{45}$ x 2, $C_3\times C_{15}$

Order 42: $F_7$ x 8, $C_7:C_6$ x 5, $D_{21}$ x 4, $C_3\times D_7$ x 4, $S_3\times C_7$ x 4, $C_{42}$ x 2

Order 40: $C_2\times D_{10}$ x 2, $C_2\times F_5$ x 2, $C_2^2\times C_{10}$

Order 36: $C_6\times S_3$ x 38, $S_3^2$ x 18, $D_{18}$ x 14, $C_3\times A_4$ x 12, $C_2\times C_{18}$ x 10, $C_6^2$ x 8, $C_2^2:C_9$ x 4, $C_{36}$ x 4, $C_6:S_3$ x 4, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_9:C_4$ x 2

Order 35: $C_{35}$

Order 32: $C_2^2\times D_4$ x 38, $C_2^5$ x 13, $C_2^3\times C_4$ x 6

Order 30: $C_3\times D_5$ x 3, $C_{30}$, $D_{15}$, $C_5\times S_3$

Order 28: $D_{14}$ x 12, $C_2\times C_{14}$ x 5, $C_{28}$ x 2, $C_7:C_4$ x 2

Order 27: $C_9:C_3$ x 11, $C_3^3$ x 8, $C_3\times C_9$ x 8

Order 24: $C_2\times A_4$ x 33, $C_2\times D_6$ x 31, $C_2^2\times C_6$ x 21, $C_3\times D_4$ x 20, $S_4$ x 12, $C_3:D_4$ x 8, $C_2\times C_{12}$ x 7, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_6:C_4$

Order 21: $C_7:C_3$ x 3, $C_{21}$ x 2

Order 20: $D_{10}$ x 2, $F_5$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 42, $C_3\times C_6$ x 18, $C_{18}$ x 10, $C_3:S_3$ x 8, $D_9$ x 8

Order 16: $C_2\times D_4$ x 54, $C_2^4$ x 33, $C_2^2\times C_4$ x 10

Order 15: $C_{15}$ x 2

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 12: $D_6$ x 34, $C_2\times C_6$ x 30, $A_4$ x 11, $C_{12}$ x 6, $C_3:C_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 16, $C_9$ x 6

Order 8: $C_2^3$ x 47, $D_4$ x 24, $C_2\times C_4$ x 10

Order 7: $C_7$

Order 6: $C_6$ x 24, $S_3$ x 16

Order 5: $C_5$

Order 4: $C_2^2$ x 27, $C_4$ x 4

Order 3: $C_3$ x 8

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1088640: $S_6\times {}^2G(2,3)$

Order 544320: $A_6\times {}^2G(2,3)$

Order 362880: $S_6\times \SL(2,8)$

Order 181440: $A_6\times \SL(2,8)$, $S_5\times {}^2G(2,3)$

Order 120960: $F_8:C_3\times S_6$

Order 108864: ${}^2G(2,3)\times \SOPlus(4,2)$

Order 90720: $A_5\times {}^2G(2,3)$

Order 72576: $S_4\times \SL(2,8):C_6$

Order 60480: $F_8:C_3\times A_6$, $S_5\times \SL(2,8)$

Order 54432: $(C_3^2\times \SL(2,8)):C_{12}$, $S_3^2\times {}^2G(2,3)$

Order 40320: $F_8\times S_6$

Order 38880: $C_9:C_6\times S_6$

Order 36288: $S_4\times {}^2G(2,3)$ x 2, $\SL(2,8)\times \SOPlus(4,2)$, $A_4\times \SL(2,8):C_6$

Order 30240: $F_5\times {}^2G(2,3)$, $A_5\times \SL(2,8)$, $F_7\times S_6$

Order 27216: $(C_3^2\times \SL(2,8)):C_6$, $C_3\times S_3\times {}^2G(2,3)$

Order 24192: $D_4\times \SL(2,8):C_6$, $C_2\times S_4\times \SL(2,8)$

Order 20160: $F_8\times A_6$, $F_8:C_3\times S_5$

Order 19440: $C_9:C_3\times S_6$, $C_9:(C_3\times S_6)$, $C_9:C_6\times A_6$

Order 18144: $C_3^2:C_4\times \SL(2,8)$, $S_3^2\times \SL(2,8)$, $D_6\times {}^2G(2,3)$, $A_4\times {}^2G(2,3)$

Order 17280: $C_2\times A_4\times S_6$

Order 15120: $D_5\times {}^2G(2,3)$, $A_6:F_7$, $F_7\times A_6$, $C_7:C_3\times S_6$

Order 13608: $C_3^2\times {}^2G(2,3)$

Order 12960: $D_9\times S_6$, $C_3\times S_3\times S_6$

Order 12096: $D_4\times {}^2G(2,3)$ x 3, $S_4\times \SL(2,8)$ x 2, $C_2\times \SL(2,8):C_{12}$, $C_2^3:{}^2G(2,3)$, $C_2^3\times {}^2G(2,3)$, $C_2\times A_4\times \SL(2,8)$, $(S_3^2\times F_8):C_6$

Order 10080: $F_5\times \SL(2,8)$, $D_7\times S_6$, $F_8:\GL(2,4)$

Order 9720: $C_9:C_3\times A_6$

Order 9072: $S_3\times {}^2G(2,3)$ x 3, $C_3\times S_3\times \SL(2,8)$, $C_3:S_3\times \SL(2,8)$, $C_6\times {}^2G(2,3)$

Order 8640: $A_4\times S_6$ x 2, $C_2\times A_4\times A_6$

Order 8064: $C_2\times D_4\times \SL(2,8)$, $F_8:C_6\times S_4$

Order 7560: $\SL(2,8):C_{15}$, $C_7:C_3\times A_6$

Order 6720: $F_8\times S_5$

Order 6480: $C_9\times S_6$ x 2, $C_9:S_6$, $D_9\times A_6$, $C_3^2\times S_6$, $C_9:C_6\times S_5$, $C_3^2:S_6$, $C_3\times S_3\times A_6$

Order 6048: $C_2^2\times {}^2G(2,3)$ x 3, $\SL(2,8):C_{12}$ x 2, $\SL(2,8):A_4$, $A_4\times \SL(2,8)$, $(C_3^2\times F_8):C_{12}$, $F_8:C_3\times S_3^2$, $D_6\times \SL(2,8)$

Order 5760: $C_2^3\times S_6$

Order 5040: $C_7:S_6$, $C_7\times S_6$, $F_7\times S_5$, $D_5\times \SL(2,8)$, $D_7\times A_6$

Order 4536: $C_3\times {}^2G(2,3)$ x 2, $C_3^2\times \SL(2,8)$

Order 4320: $C_6\times S_6$, $A_4\times A_6$, $S_3\times S_6$

Order 4032: $D_4\times \SL(2,8)$ x 3, $F_8:C_3\times S_4$ x 2, $F_8\times \SOPlus(4,2)$, $C_2^3\times \SL(2,8)$, $C_2\times C_4\times \SL(2,8)$, $A_4\times F_8:C_6$

Order 3888: $(D_9\times S_3^2):C_6$

Order 3360: $F_5\times F_8:C_3$, $F_8\times A_5$

Order 3240: $C_9\times A_6$ x 2, $C_3^2\times A_6$, $C_9:C_3\times S_5$, $D_9:\GL(2,4)$, $C_9:(C_3\times S_5)$

Order 3024: $\SL(2,8):C_6$ x 3, $S_3\times \SL(2,8)$ x 2, $F_7\times \SOPlus(4,2)$, $S_3\times F_8:C_3^2$, $(C_3^2\times F_8):C_6$, $C_6\times \SL(2,8)$

Order 2880: $C_2^2\times S_6$ x 2, $C_2^3\times A_6$, $C_2\times A_4\times S_5$

Order 2688: $D_4\times F_8:C_6$, $C_2\times S_4\times F_8$

Order 2592: $C_{18}:C_6\times S_4$

Order 2520: $C_5\times \SL(2,8)$, $C_7\times A_6$, $F_7\times A_5$, $A_5:F_7$, $C_7:C_3\times S_5$

Order 2160: $C_3\times S_6$ x 3, $D_9\times S_5$, $C_6\times A_6$, $\GL(2,4):D_6$, $C_3:S_6$, $S_3\times A_6$

Order 2016: $C_2^2\times \SL(2,8)$ x 3, $C_4\times \SL(2,8)$ x 2, $S_3^2\times F_8$, $C_3^2:C_4\times F_8$, $C_2\times S_4\times F_7$, $F_8:C_6\times S_3$, $A_4\times F_8:C_3$

Order 1944: $C_3^2.S_3^3$, $C_3^4.D_{12}$, $(C_9\times S_3^2):C_6$, $(C_9\times S_3^2):C_6$, $(C_3^2\times D_9):C_{12}$

Order 1728: $C_2\times D_6^2:C_6$

Order 1680: $D_5\times F_8:C_3$, $D_7\times S_5$

Order 1620: $C_9:\GL(2,4)$

Order 1512: ${}^2G(2,3)$ x 2, $F_8:C_3^3$, $S_3^2\times F_7$, $C_3^2:C_{28}:C_6$, $S_3^2:F_7$, $(C_7\times S_3^2):C_6$, $C_3^2:C_4\times F_7$, $C_3\times \SL(2,8)$

Order 1440: $A_4\times S_5$ x 2, $C_2\times S_6$ x 2, $C_2^3:\GL(2,4)$, $C_2^2\times A_6$

Order 1344: $D_4\times F_8:C_3$ x 3, $S_4\times F_8$ x 2, $C_2^5:C_{42}$, $C_2\times F_8:A_4$, $C_2^2\times F_8:C_6$, $C_2\times F_8:C_{12}$

Order 1296: $C_6^2.S_3^2$ x 4, $S_3^3:C_6$, $A_4\times C_{18}:C_6$, $C_9:(C_6\times S_4)$, $C_9:C_6\times S_4$, $S_3^2:D_{18}$

Order 1260: $C_7:\GL(2,4)$

Order 1152: $A_4^2:C_2^3$

Order 1120: $F_5\times F_8$

Order 1080: $C_3\times A_6$ x 2, $C_9\times S_5$ x 2, $S_3\times \GL(2,4)$, $C_3^2\times S_5$, $C_3^2:S_5$, $D_{45}:C_{12}$, $D_9\times A_5$, $C_9:S_5$

Order 1008: $S_4\times F_7$ x 4, $F_8:C_3\times S_3$ x 3, $C_2\times \SL(2,8)$ x 2, $C_3:S_3\times F_8$, $C_3\times S_3\times F_8$, $S_3^2:D_{14}$, $C_2\times A_4\times F_7$, $C_2\times A_4:F_7$, $C_7:C_6\times S_4$, $C_3\times F_8:C_6$

Order 972: $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.C_6^2$, $C_3^3.S_3^2$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$

Order 960: $C_2^3\times S_5$

Order 896: $C_2^6:C_{14}$

Order 864: $D_6^2:C_6$ x 3, $D_6^2:C_6$, $C_6\times S_3\times S_4$, $C_2\times C_6^2:C_{12}$, $S_4\times D_{18}$, $C_2\times D_{36}:C_6$

Order 840: $F_8:C_{15}$, $F_5\times F_7$, $D_7\times A_5$, $C_7:S_5$, $C_7\times S_5$

Order 756: $C_{21}:(C_6\times S_3)$, $C_{21}:C_6\times S_3$, $C_{21}:C_6\times S_3$, $C_3:S_3\times F_7$, $C_{21}:C_6^2$, $(C_3\times C_{21}):C_{12}$, $(C_3\times C_{21}):C_{12}$

Order 720: $S_6$ x 2, $C_6\times S_5$, $A_4\times A_5$, $S_3\times S_5$, $C_2\times A_6$

Order 672: $C_2\times F_8:C_6$ x 3, $F_8:C_{12}$ x 2, $S_4\times D_{14}$, $D_6\times F_8$, $A_4\times F_8$, $F_8:A_4$, $C_2\times D_4\times F_7$

Order 648: $A_4\times C_9:C_6$ x 2, $C_9:(C_3\times S_4)$ x 2, $C_9:C_3\times S_4$ x 2, $S_3^2:C_{18}$ x 2, $C_3\times S_3^3$, $C_3^3:D_{12}$, $C_3\wr D_4$, $C_3^4:D_4$, $S_3\times C_3^2:C_{12}$, $C_9:C_6\times A_4$, $C_{18}:C_6\times S_3$, $D_9\times S_3^2$, $C_3^2:D_{36}$, $S_3^2:D_9$, $C_3^2:C_4\times D_9$

Order 576: $A_4^2:C_2^2$ x 6, $D_6^2:C_2^2$, $C_2^2\times A_4^2$

Order 560: $D_5\times F_8$

Order 540: $C_9\times A_5$ x 2, $C_3\times \GL(2,4)$, $D_{45}:C_6$, $C_{45}:C_{12}$, $C_{45}:C_{12}$

Order 504: $A_4\times F_7$ x 2, $A_4:F_7$ x 2, $C_7:C_3\times S_4$ x 2, $F_8:C_3^2$ x 2, $C_3^2\times F_8$, $D_7\times S_3^2$, $C_3^2:D_{28}$, $S_3^2:D_7$, $S_3^2:C_{14}$, $C_3^2:C_4\times D_7$, $C_7:C_6\times A_4$, $D_6\times F_7$, $\SL(2,8)$

Order 486: $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4.S_3$, $C_3^4.C_6$, $D_9:C_3^3$

Order 480: $C_2^2\times S_5$ x 2, $C_2\times A_4\times F_5$, $C_2^3\times A_5$

Order 448: $D_4\times F_8$ x 3, $C_2^3\times F_8$, $C_2^4:C_{28}$

Order 432: $D_{36}:C_6$ x 6, $C_6^2:D_6$ x 4, $D_9\times S_4$ x 4, $A_4\times S_3^2$ x 3, $C_6\times S_3\times A_4$ x 2, $C_6^2:C_{12}$ x 2, $C_{18}\times S_4$ x 2, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_6\times \SOPlus(4,2)$, $S_3^3:C_2$, $C_6^2.D_6$, $A_4\times D_{18}$, $D_{18}:A_4$, $C_{18}:S_4$, $C_6^2.D_6$, $C_{12}.C_6^2$, $C_6^2.D_6$, $D_{36}:C_6$, $D_{18}:C_{12}$

Order 420: $D_5\times F_7$, $C_{35}:C_{12}$, $C_{35}:C_{12}$, $C_7\times A_5$

Order 384: $C_2^4\times S_4$, $C_2^6:C_6$

Order 378: $C_3^2:F_7$, $(C_3\times C_{21}):C_6$, $C_3^2:F_7$, $C_3\times C_{21}:C_6$, $C_3^2\times F_7$

Order 360: $C_3\times S_5$ x 3, $D_9\times F_5$, $D_{15}:C_{12}$, $C_6\times A_5$, $S_3\times A_5$, $C_3:S_5$, $A_6$

Order 336: $D_4\times F_7$ x 6, $D_7\times S_4$ x 4, $F_8:C_6$ x 3, $S_3\times F_8$ x 2, $D_{14}:A_4$, $A_4\times D_{14}$, $C_2^3\times F_7$, $C_{14}:S_4$, $C_{14}\times S_4$, $C_6\times F_8$, $C_2\times C_{28}:C_6$, $C_2^3:F_7$, $D_{28}:C_6$, $D_{14}:C_{12}$

Order 324: $C_3^2.S_3^2$ x 6, $C_3^2:S_3^2$ x 2, $C_3^2\times S_3^2$ x 2, $C_9\times S_3^2$ x 2, $C_3^2:C_{36}$ x 2, $C_3\wr C_2^2$, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^3.D_6$, $C_3^2.C_6^2$, $C_9:C_6^2$, $C_6^2.C_3^2$, $C_9:S_3^2$, $C_9:S_3^2$, $C_3^2:D_{18}$, $C_3^2:D_{18}$, $(C_3^2\times C_9):C_4$

Order 288: $A_4\times S_4$ x 4, $C_6^2:D_4$ x 3, $C_2\times A_4^2$ x 3, $C_6^2:C_2^3$, $D_4\times D_{18}$, $C_2\times D_6^2$, $C_2\times C_6^2:C_4$, $C_2\times A_4\times D_6$, $C_2\times C_6\times S_4$, $D_6\times S_4$

Order 280: $C_5\times F_8$, $D_7\times F_5$

Order 270: $C_{45}:C_6$, $C_{45}:C_6$, $C_9:C_{30}$

Order 252: $S_3\times F_7$ x 5, $C_{21}:D_6$, $S_3\times D_{21}$, $C_7\times S_3^2$, $C_{21}:D_6$, $C_{21}:D_6$, $(C_3\times C_{21}):C_4$, $C_3^2:C_{28}$, $C_{42}:C_6$, $C_{42}:C_6$, $C_6\times F_7$, $A_4\times C_7:C_3$

Order 243: $C_9:C_3^3$

Order 240: $A_4\times F_5$ x 2, $C_2\times S_5$ x 2, $A_4\times D_{10}$, $C_2^2\times A_5$

Order 224: $C_2^2\times F_8$ x 3, $C_4\times F_8$ x 2, $D_4\times D_{14}$

Order 216: $D_{18}:C_6$ x 6, $C_9\times S_4$ x 4, $D_{18}:C_6$ x 4, $C_{36}:C_6$ x 4, $C_6^2:C_6$ x 4, $S_3^2:C_6$ x 4, $C_{36}:C_6$ x 3, $C_{36}:C_6$ x 3, $A_4\times D_9$ x 2, $D_9:A_4$ x 2, $C_9:S_4$ x 2, $C_3^2.S_4$ x 2, $C_6^2:C_6$ x 2, $C_6\times S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_{18}:A_4$ x 2, $A_4\times C_{18}$ x 2, $C_{18}:C_{12}$, $C_{18}:C_{12}$, $C_2\times C_3^2:C_{12}$, $S_3^3$, $C_3^2:D_{12}$, $S_3^2:S_3$, $C_3^2:C_4\times S_3$, $C_6.C_6^2$, $C_6^2.C_6$, $S_3\times D_{18}$

Order 210: $C_{35}:C_6$, $C_{35}:C_6$, $C_5\times F_7$

Order 192: $C_2^5:C_6$ x 9, $C_2^3\times S_4$ x 6, $A_4\times C_2^4$ x 2, $C_2^2\wr C_3$, $C_2^4:C_{12}$

Order 189: $C_7:C_3^3$

Order 180: $C_9\times F_5$ x 2, $\GL(2,4)$ x 2, $D_5\times D_9$, $C_{45}:C_4$, $C_{15}:D_6$, $C_{15}:C_{12}$, $C_3^2\times F_5$

Order 168: $C_2^2\times F_7$ x 6, $C_4\times F_7$ x 4, $D_{14}:C_6$ x 4, $C_{28}:C_6$ x 3, $C_{28}:C_6$ x 3, $D_7:A_4$ x 2, $A_4\times D_7$ x 2, $C_7:S_4$ x 2, $C_7\times S_4$ x 2, $F_8:C_3$ x 2, $C_{14}:A_4$, $A_4\times C_{14}$, $C_2\times C_{14}:C_6$, $S_3\times D_{14}$, $C_3\times F_8$, $C_{14}:C_{12}$, $C_{14}:C_{12}$

Order 162: $C_3^3.C_6$ x 5, $C_3^3.S_3$ x 4, $D_9:C_3^2$ x 4, $C_3^2\wr C_2$ x 2, $S_3\times C_3^3$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:C_{18}$ x 2, $C_3^3:C_6$, $C_{18}:C_3^2$, $C_3^2:D_9$, $C_3^2:D_9$, $C_3^2\times D_9$

Order 160: $C_2^3\times F_5$

Order 144: $A_4\times D_6$ x 9, $C_6\times S_4$ x 8, $D_4\times D_9$ x 6, $C_{12}:D_6$ x 6, $S_3^2:C_2^2$ x 4, $S_3\times S_4$ x 4, $D_6^2$ x 3, $D_4\times C_{18}$ x 2, $C_6^2:C_2^2$ x 2, $C_2^2:C_6^2$ x 2, $C_6^2:C_4$ x 2, $C_{18}:D_4$, $C_2\times D_{36}$, $C_4\times D_{18}$, $C_6^2:C_2^2$, $C_6:S_4$, $A_4^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_{12}\times D_6$, $C_2^2\times D_{18}$, $C_2^2:D_{18}$

Order 140: $D_5\times D_7$, $C_{35}:C_4$, $C_7\times F_5$

Order 135: $C_9:C_{15}$

Order 128: $D_4\times C_2^4$

Order 126: $C_{21}:C_6$ x 3, $C_{21}:C_6$ x 3, $C_3\times F_7$ x 3, $C_3:D_{21}$, $C_{21}:S_3$, $C_3\times D_{21}$, $S_3\times C_{21}$, $C_3^2\times D_7$, $C_{21}:C_6$

Order 120: $D_5\times A_4$ x 2, $S_5$ x 2, $C_{10}\times A_4$, $C_6\times F_5$, $S_3\times F_5$, $C_2\times A_5$

Order 112: $D_4\times D_7$ x 6, $C_2\times F_8$ x 2, $C_2^2\times D_{14}$, $D_4\times C_{14}$, $C_{14}:D_4$, $C_2\times D_{28}$, $C_4\times D_{14}$

Order 108: $C_3\times S_3^2$ x 11, $C_{18}:C_6$ x 9, $S_3\times D_9$ x 5, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_{18}:C_6$ x 3, $C_9:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3:S_3^2$ x 2, $S_3\times C_{18}$ x 2, $C_9:A_4$ x 2, $C_9\times A_4$ x 2, $C_9:C_{12}$ x 2, $C_3:S_3^2$, $C_3^3:C_4$, $C_3:D_{18}$, $C_3\times D_{18}$, $C_3^2.A_4$

Order 105: $C_7:C_{15}$

Order 96: $C_2^3\times A_4$ x 10, $C_2^2\times S_4$ x 10, $D_4\times A_4$ x 10, $C_2^3:C_{12}$ x 6, $C_2^3:A_4$ x 3, $C_2^3\times D_6$, $C_{12}:C_2^3$, $D_4\times D_6$

Order 90: $C_9\times D_5$ x 2, $C_3\times D_{15}$, $S_3\times C_{15}$, $C_3^2\times D_5$, $D_{45}$, $C_5\times D_9$

Order 84: $C_2\times F_7$ x 8, $S_3\times D_7$ x 4, $C_{14}:C_6$ x 3, $C_7:C_{12}$ x 2, $C_7:C_{12}$ x 2, $D_{42}$, $S_3\times C_{14}$, $C_3\times D_{14}$, $C_7:A_4$, $C_7\times A_4$

Order 81: $C_9:C_3^2$ x 4, $C_3^2\times C_9$ x 2, $C_3^4$

Order 80: $C_2^2\times F_5$ x 2, $C_2^2\times D_{10}$

Order 72: $C_6\times A_4$ x 10, $S_3\times A_4$ x 10, $C_6\times D_6$ x 9, $C_3\times S_4$ x 8, $C_2\times D_{18}$ x 6, $D_4\times C_9$ x 6, $S_3\times D_6$ x 5, $C_9:D_4$ x 4, $C_4\times D_9$ x 4, $\SOPlus(4,2)$ x 4, $C_6\wr C_2$ x 4, $S_3\times C_{12}$ x 4, $D_{36}$ x 3, $D_4\times C_3^2$ x 3, $C_3\times D_{12}$ x 3, $C_2\times C_{36}$ x 2, $C_2\times C_6^2$ x 2, $C_6:D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_3:S_4$ x 2, $C_2^2\times C_{18}$ x 2, $C_2^2:C_{18}$ x 2, $C_2^2:D_9$ x 2, $C_{18}:C_4$, $C_6\times C_{12}$, $C_6:C_{12}$

Order 70: $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 64: $D_4\times C_2^3$ x 9, $C_2^6$, $C_2^4\times C_4$

Order 63: $C_{21}:C_3$ x 2, $C_3\times C_{21}$

Order 60: $C_3\times F_5$ x 3, $C_5\times A_4$, $S_3\times D_5$, $C_{15}:C_4$, $A_5$, $C_3\times D_{10}$

Order 56: $C_2\times D_{14}$ x 6, $C_7:D_4$ x 4, $C_4\times D_7$ x 4, $C_7\times D_4$ x 3, $D_{28}$ x 3, $C_2\times C_{28}$, $C_{14}:C_4$, $C_2^2\times C_{14}$, $F_8$

Order 54: $S_3\times C_3^2$ x 12, $C_9:C_6$ x 8, $C_3^2:C_6$ x 8, $S_3\times C_9$ x 6, $C_9:C_6$ x 5, $C_3:D_9$ x 3, $C_3\times D_9$ x 3, $C_3\times C_{18}$ x 2, $C_3^2\times C_6$ x 2, $C_3^2:S_3$

Order 48: $C_2^2\times A_4$ x 16, $C_2\times S_4$ x 11, $C_6\times D_4$ x 10, $C_2^2\times D_6$ x 7, $S_3\times D_4$ x 6, $C_4\times A_4$ x 4, $C_2^3\times C_6$ x 2, $C_2^2:A_4$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 45: $C_{45}$ x 2, $C_3\times C_{15}$

Order 42: $F_7$ x 5, $C_7:C_6$ x 3, $D_{21}$ x 2, $C_3\times D_7$ x 2, $S_3\times C_7$ x 2, $C_{42}$

Order 40: $C_2\times D_{10}$ x 2, $C_2\times F_5$ x 2, $C_2^2\times C_{10}$

Order 36: $C_6\times S_3$ x 20, $S_3^2$ x 9, $D_{18}$ x 8, $C_2\times C_{18}$ x 6, $C_3\times A_4$ x 6, $C_6^2$ x 5, $C_{36}$ x 4, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_2^2:C_9$ x 2, $C_9:C_4$ x 2

Order 35: $C_{35}$

Order 32: $C_2^2\times D_4$ x 24, $C_2^5$ x 7, $C_2^3\times C_4$ x 6

Order 30: $C_3\times D_5$ x 3, $C_{30}$, $D_{15}$, $C_5\times S_3$

Order 28: $D_{14}$ x 7, $C_2\times C_{14}$ x 3, $C_{28}$ x 2, $C_7:C_4$ x 2

Order 27: $C_9:C_3$ x 6, $C_3^3$ x 5, $C_3\times C_9$ x 4

Order 24: $C_2\times A_4$ x 18, $C_2\times D_6$ x 16, $C_3\times D_4$ x 13, $C_2^2\times C_6$ x 11, $C_2\times C_{12}$ x 7, $S_4$ x 6, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $D_{12}$ x 3, $C_6:C_4$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 20: $D_{10}$ x 2, $F_5$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 22, $C_3\times C_6$ x 10, $C_{18}$ x 6, $C_3:S_3$ x 5, $D_9$ x 5

Order 16: $C_2\times D_4$ x 32, $C_2^4$ x 18, $C_2^2\times C_4$ x 10

Order 15: $C_{15}$ x 2

Order 14: $D_7$ x 3, $C_{14}$ x 2

Order 12: $D_6$ x 18, $C_2\times C_6$ x 17, $A_4$ x 6, $C_{12}$ x 6, $C_3:C_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 9, $C_9$ x 4

Order 8: $C_2^3$ x 26, $D_4$ x 14, $C_2\times C_4$ x 10

Order 7: $C_7$

Order 6: $C_6$ x 14, $S_3$ x 9

Order 5: $C_5$

Order 4: $C_2^2$ x 16, $C_4$ x 4

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1088640: $S_6\times {}^2G(2,3)$ ($C_1$)

Order 544320: $A_6\times {}^2G(2,3)$ ($C_2$)

Order 362880: $S_6\times \SL(2,8)$ ($C_3$)

Order 181440: $A_6\times \SL(2,8)$ ($C_6$)

Order 1512: ${}^2G(2,3)$ ($S_6$)

Order 720: $S_6$ (${}^2G(2,3)$)

Order 504: $\SL(2,8)$ ($C_3\times S_6$)

Order 360: $A_6$ ($\SL(2,8):C_6$)

Order 1: $C_1$ ($S_6\times {}^2G(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1088640: $S_6\times {}^2G(2,3)$ ($C_1$)

Order 544320: $A_6\times {}^2G(2,3)$ ($C_2$)

Order 362880: $S_6\times \SL(2,8)$ ($C_3$)

Order 181440: $A_6\times \SL(2,8)$ ($C_6$)

Order 1512: ${}^2G(2,3)$ ($S_6$)

Order 720: $S_6$ (${}^2G(2,3)$)

Order 504: $\SL(2,8)$ ($C_3\times S_6$)

Order 360: $A_6$ ($\SL(2,8):C_6$)

Order 1: $C_1$ ($S_6\times {}^2G(2,3)$)

Series

Derived series

$S_6\times {}^2G(2,3)$

$\rhd$

$S_6\times {}^2G(2,3)$

$\rhd$

$A_6\times \SL(2,8)$

$\rhd$

$A_6\times \SL(2,8)$

Chief series

$S_6\times {}^2G(2,3)$

$\rhd$

$S_6\times {}^2G(2,3)$

$\rhd$

$A_6\times {}^2G(2,3)$

$\rhd$

$A_6\times {}^2G(2,3)$

$\rhd$

$A_6\times \SL(2,8)$

$\rhd$

$A_6\times \SL(2,8)$

$\rhd$

$\SL(2,8)$

$\rhd$

$\SL(2,8)$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$S_6\times {}^2G(2,3)$

$\rhd$

$S_6\times {}^2G(2,3)$

$\rhd$

$A_6\times \SL(2,8)$

$\rhd$

$A_6\times \SL(2,8)$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 1 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 $121 \times 121$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $88 \times 88$ rational character table.