Abstract group 17280.bh: $D_6^2:S_5$

• Label: 17280.bh
• Order: \( 2^{7} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$D_6^2:S_5$
Order:	\(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$D_6^2.C_2^3.S_5$, of order \(138240\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	1343	188	2240	24	5812	1512	4240	192	1728	17280
Conjugacy classes	1	27	7	12	1	64	15	24	3	17	171
Divisions	1	27	7	12	1	64	11	24	3	12	162
Autjugacy classes	1	16	5	9	1	35	6	15	2	6	96

Dimension	1	2	4	5	6	8	10	12	16	20	24	32	40	48
Irr. complex chars.	16	20	24	16	16	21	20	16	8	8	4	1	1	0	171
Irr. rational chars.	16	20	24	16	8	21	20	12	8	8	6	1	1	1	162

Minimal presentations

Permutation degree:	$15$
Transitive degree:	not computed
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	32
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(10,14)(11,12)(13,15), (1,2)(3,4), (1,2)(12,13,14), (1,4)(2,3)(5,6)(10,13,15,12,11,14) \!\cdots\! \rangle$
Direct product:	$S_3$ ${}^2$ $\, \times\, $ $(C_2^2:S_5)$
Semidirect product:	$D_6^2$ $\,\rtimes\,$ $S_5$	$(D_6\times S_5)$ $\,\rtimes\,$ $D_6$	$(D_6.S_5)$ $\,\rtimes\,$ $D_6$	$(D_6:S_5)$ $\,\rtimes\,$ $D_6$	all 79
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6$ . $(D_6\times S_5)$	$(C_2\times A_5)$ . $D_6^2$	$C_2$ . $(A_5:D_6^2)$	$(S_3\times D_6)$ . $(C_2\times S_5)$	all 14

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

Homology

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

Subgroups

There are 716430 subgroups in 11621 conjugacy classes, 168 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_5:D_6^2$
Commutator:	$G' \simeq$ $C_6\times \GL(2,4)$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_5:D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2\times S_5$
Radical:	$R \simeq$ $D_6^2$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_6\times \GL(2,4)$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

All subgroups of index up to 90 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 17280: $D_6^2:S_5$

Order 8640: $S_3\times D_6:S_5$ x 2, $S_3\times D_6:S_5$ x 2, $S_3\times \GL(2,4):D_4$ x 2, $S_3\times \GL(2,4):D_4$ x 2, $A_5\times D_6^2$, $(S_3\times D_6):S_5$, $(C_6\times S_5):D_6$, $(S_3\times D_6):S_5$, $C_6^2:(C_2\times S_5)$, $A_5:C_4\times S_3^2$, $A_5:D_6^2$

Order 5760: $S_3\times C_2^3:S_5$ x 2

Order 4320: $S_3\times D_6\times A_5$ x 8, $S_3^2\times S_5$ x 4, $S_3\times \GL(2,4):C_4$ x 2, $(C_6\times S_3):S_5$ x 2, $\GL(2,4):D_{12}$ x 2, $\GL(2,4):D_{12}$ x 2, $C_6^2:S_5$ x 2, $\GL(2,4):D_{12}$ x 2, $C_3\times D_6:S_5$ x 2, $C_6\times D_6\times A_5$ x 2, $S_3\times C_6:S_5$ x 2, $S_3\times A_5:C_{12}$ x 2, $C_6\times S_3\times S_5$ x 2, $C_6:D_6\times A_5$, $\GL(2,4):(C_4\times S_3)$, $(C_6\times S_5):S_3$, $\GL(2,4):D_{12}$, $C_6^2:S_5$, $\GL(2,4):(C_4\times S_3)$, $C_6^2:S_5$, $C_2\times A_5:S_3^2$, $C_6:S_3\times S_5$

Order 3456: $S_3^2\times \GL(2,\mathbb{Z}/4)$

Order 2880: $S_3\times C_2^2:S_5$ x 17, $\GL(2,4):C_2^4$ x 2, $A_5:C_4\times D_6$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times D_6:S_5$ x 2, $C_2\times \GL(2,4):D_4$ x 2, $C_2^2:S_5\times C_6$ x 2, $\GL(2,4):C_2^4$ x 2, $D_6^2:F_5$

Order 2160: $S_3^2\times A_5$ x 10, $D_6\times \GL(2,4)$ x 8, $\GL(2,4):D_6$ x 4, $A_5:S_3^2$ x 4, $\GL(2,4):D_6$ x 4, $\GL(2,4):C_{12}$ x 2, $\GL(2,4):D_6$ x 2, $\GL(2,4):D_6$ x 2, $A_5:S_3^2$ x 2, $\GL(2,4):C_{12}$, $(C_3\times \GL(2,4)):C_4$, $C_6^2\times A_5$, $A_5:C_6^2$, $\GL(2,4):D_6$

Order 1920: $C_2^4:S_5$

Order 1728: $S_3\times C_3:\GL(2,\mathbb{Z}/4)$ 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, $A_4\times D_6^2$, $A_4:D_6^2$, $D_4\times S_3^3$, $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $C_2^3.S_3^3$

Order 1440: $\GL(2,4):C_2^3$ x 21, $D_6\times S_5$ x 17, $\GL(2,4):D_4$ x 9, $\GL(2,4):D_4$ x 9, $D_6:S_5$ x 9, $D_6:S_5$ x 9, $D_6.S_5$ x 9, $D_{30}:C_4\times S_3$ x 4, $D_5.D_6^2$ x 2, $C_2^3\times \GL(2,4)$ x 2, $\GL(2,4):C_2^3$ x 2, $\GL(2,4):C_2^3$ x 2, $(C_6\times D_{30}):C_4$ x 2, $S_3\times D_{10}:C_{12}$ x 2, $(S_3\times D_{30}):C_4$ x 2, $C_2\times \GL(2,4):C_4$ x 2, $A_5:C_4\times C_6$ x 2, $D_5\times D_6^2$, $C_6^2:(C_2\times F_5)$, $(C_6\times F_5):D_6$

Order 1152: $D_6\times \GL(2,\mathbb{Z}/4)$ x 2, $D_6^2:D_4$

Order 1080: $S_3\times \GL(2,4)$ x 6, $\GL(2,4):S_3$ x 3, $C_6\times \GL(2,4)$ x 2, $C_3^2:S_5$ x 2, $C_3^2\times S_5$, $C_3^2:S_5$

Order 960: $C_2^3:S_5$ x 12, $D_6\times D_{10}:C_4$ x 2, $C_2^4\times A_5$, $C_2^3\times S_5$, $C_2^3.S_5$

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

Order 720: $D_6\times A_5$ x 40, $S_3\times S_5$ x 18, $D_{10}\times S_3^2$ x 16, $C_2^2\times \GL(2,4)$ x 11, $C_6:S_5$ x 9, $C_6\times S_5$ x 9, $F_5\times S_3^2$ x 8, $\GL(2,4):C_4$ x 5, $A_5:C_{12}$ x 5, $D_{10}.S_3^2$ x 4, $D_{30}:C_{12}$ x 4, $D_{10}.S_3^2$ x 4, $D_{30}:C_{12}$ x 4, $D_6\times D_{30}$ x 2, $C_6^2:D_{10}$ x 2, $D_{10}.S_3^2$ x 2, $C_6:S_3\times F_5$ x 2, $C_6^2:F_5$ x 2, $D_{10}.S_3^2$ x 2, $(C_6\times F_5):S_3$ x 2, $C_5:D_6^2$, $C_5\times D_6^2$, $C_6^2:D_{10}$, $C_6^2:F_5$, $C_6^2:F_5$

Order 576: $S_3\times \GL(2,\mathbb{Z}/4)$ x 19, $D_6^2:C_2^2$ x 6, $D_6^2:C_2^2$ x 6, $C_4:D_6^2$ x 5, $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_4$ x 3, $C_2^2\times A_4\times D_6$ x 2, $C_2^2:D_6^2$ 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, $C_6^2.C_2^4$ x 2, $C_2^4:S_3^2$ x 2, $C_2^2\times D_6^2$, $C_6^2.C_2^4$, $C_2^4:S_3^2$

Order 540: $C_3\times \GL(2,4)$

Order 480: $D_{10}:C_4\times S_3$ x 17, $C_2^2:S_5$ x 16, $C_2^3\times A_5$ x 11, $C_2^2\times S_5$ x 10, $C_2^2.S_5$ x 6, $C_2\times D_6\times F_5$ x 4, $D_{10}.D_{12}$ x 4, $C_{15}:C_2^5$ x 2, $(C_6\times D_{10}):C_4$ x 2, $C_2\times D_{10}:C_{12}$ x 2

Order 432: $C_2\times S_3^3$ x 31, $C_6^2:D_6$ x 12, $D_6:S_3^2$ x 12, $A_4\times S_3^2$ x 10, $C_6\times S_3\times A_4$ x 8, $C_3:D_6^2$ x 6, $C_3\times D_6^2$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 6, $C_{12}:S_3^2$ x 6, $D_6:S_3^2$ x 6, $D_6:S_3^2$ x 6, $D_6:S_3^2$ x 6, $D_6:S_3^2$ x 6, $D_6:S_3^2$ x 6, $C_6:S_3\times A_4$ x 4, $A_4:S_3^2$ x 4, $C_6^2:D_6$ x 4, $C_6^2:D_6$ x 3, $C_{12}:C_6^2$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}:S_3^2$ x 3, $C_{12}\times S_3^2$ x 3, $C_2.S_3^3$ x 3, $C_2.S_3^3$ x 3, $C_3:D_6^2$ x 2, $C_6^2:D_6$ x 2, $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_3\times C_6.S_4$ x 2, $A_4\times C_6^2$, $C_6^2:D_6$, $A_4:C_6^2$, $C_6^2:D_6$, $C_{12}:S_3^2$, $(C_3^2\times A_4):C_4$, $C_3\times A_4:C_{12}$, $C_2.S_3^3$

Order 384: $C_2^5:D_6$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$

Order 360: $D_5\times S_3^2$ x 36, $S_3\times D_{30}$ x 16, $C_{30}:D_6$ x 16, $S_3\times A_5$ x 15, $C_6\times A_5$ x 12, $C_{30}:D_6$ x 8, $C_{10}\times S_3^2$ x 8, $C_{30}:D_6$ x 8, $D_5.S_3^2$ x 8, $D_{15}:C_{12}$ x 8, $C_3:S_5$ x 5, $C_3\times S_5$ x 5, $C_{30}:C_{12}$ x 4, $D_5.S_3^2$ x 4, $C_3:S_3\times F_5$ x 4, $C_6\times D_{30}$ x 2, $D_6\times C_{30}$ x 2, $C_2\times C_3^2:F_5$ x 2, $C_{30}:C_{12}$ x 2, $C_6:D_{30}$, $C_{30}:D_6$, $D_5\times C_6^2$

Order 320: $C_2^4:F_5$

Order 288: $D_4\times S_3^2$ x 59, $C_2\times D_6^2$ x 25, $C_2\times A_4\times D_6$ x 23, $D_6^2:C_2$ x 20, $D_6\times S_4$ x 19, $C_3:\GL(2,\mathbb{Z}/4)$ x 12, $D_6:D_{12}$ x 12, $D_6.D_{12}$ x 12, $C_3\times \GL(2,\mathbb{Z}/4)$ x 11, $D_6:S_4$ x 11, $D_6:S_4$ x 11, $D_6.S_4$ x 11, $C_6^2:C_2^3$ x 10, $D_6^2:C_2$ x 10, $D_6\times D_{12}$ x 10, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_6^2.C_2^3$ x 6, $D_6^2:C_2$ x 6, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_2^3.S_3^2$ x 6, $D_6:D_{12}$ x 6, $D_6^2.C_2$ x 6, $D_{12}:D_6$ x 5, $C_2.D_6^2$ x 5, $C_6^2:C_2^3$ x 5, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $C_2^3.S_3^2$ x 3, $C_2\times C_6.S_4$ x 2, $C_2\times A_4:C_{12}$ x 2, $C_6^2:D_4$ x 2, $C_6^2:C_2^3$ x 2, $C_2^3:C_6^2$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:C_2^3$

Order 240: $D_6\times D_{10}$ x 41, $D_6\times F_5$ x 34, $C_2^2\times A_5$ x 23, $D_{30}:C_4$ x 18, $C_2\times S_5$ x 16, $D_{10}.D_6$ x 9, $D_{10}:C_{12}$ x 9, $A_5:C_4$ x 4, $D_{10}.D_6$ x 4, $D_{10}:C_{12}$ x 4, $C_2^2\times D_{30}$ x 2, $C_{15}:C_2^4$ x 2, $C_{15}:C_2^4$ x 2

Order 216: $S_3^3$ x 64, $C_6:S_3^2$ x 45, $C_6\times S_3^2$ x 45, $C_6:S_3^2$ x 15, $C_3^2:D_{12}$ x 12, $C_6^2:C_6$ x 6, $S_3\times C_6^2$ x 6, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 6, $C_3^2:D_{12}$ x 6, $C_3^2:D_{12}$ x 6, $C_3^2:D_{12}$ x 6, $C_3^3:D_4$ x 6, $C_3^3:D_4$ x 6, $C_6.S_3^2$ x 6, $C_6^2:C_6$ x 3, $C_3^2:D_{12}$ x 3, $C_3:S_3\times C_{12}$ x 3, $C_3^2\times D_{12}$ x 3, $C_6.C_6^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6^2:S_3$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:S_3$ x 2, $C_3^2:S_4$, $C_3^2\times S_4$, $D_4\times C_3^3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$

Order 192: $C_2^4:D_6$ x 33, $C_2\times \GL(2,\mathbb{Z}/4)$ x 14, $C_{12}:C_2^4$ x 7, $C_2^4:D_6$ x 6, $C_2^3:D_{12}$ x 6, $C_2^4.D_6$ x 6, $D_6\times C_2^4$ x 2, $C_2^5:C_6$ x 2, $C_2^4:D_6$ x 2, $A_4\times C_2^4$, $C_2^3\times S_4$, $C_2^3.S_4$

Order 180: $\GL(2,4)$ x 2

Order 144: $D_6^2$

Order 128: $C_2^4:D_4$

Order 120: $S_5$ x 4, $C_2\times A_5$

Order 96: $D_4\times D_6$ x 34

Order 72: $S_3\times D_6$ x 124, $C_6\times D_6$ x 2, $C_6:D_6$

Order 60: $A_5$

Order 48: $S_3\times D_4$ x 153

Order 36: $S_3^2$ x 100, $C_6\times S_3$ x 4, $C_6:S_3$ x 2, $C_6^2$

Order 32: $C_2^2\times D_4$ x 16

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 226

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 2, $C_3\times C_6$

Order 16: $C_2\times D_4$ x 96

Order 12: $D_6$ x 272, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 104, $D_4$ x 64

Order 6: $S_3$ x 55, $C_6$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 117

Order 3: $C_3$ x 2

Order 2: $C_2$ x 25

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 90 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 17280: $D_6^2:S_5$

Order 8640: $A_5\times D_6^2$, $(S_3\times D_6):S_5$, $(C_6\times S_5):D_6$, $(S_3\times D_6):S_5$, $C_6^2:(C_2\times S_5)$, $A_5:C_4\times S_3^2$, $S_3\times D_6:S_5$, $S_3\times D_6:S_5$, $S_3\times \GL(2,4):D_4$, $S_3\times \GL(2,4):D_4$, $A_5:D_6^2$

Order 5760: $S_3\times C_2^3:S_5$

Order 4320: $S_3\times D_6\times A_5$ x 4, $C_6:D_6\times A_5$, $S_3\times \GL(2,4):C_4$, $(C_6\times S_3):S_5$, $\GL(2,4):D_{12}$, $\GL(2,4):(C_4\times S_3)$, $(C_6\times S_5):S_3$, $\GL(2,4):D_{12}$, $C_6^2:S_5$, $\GL(2,4):D_{12}$, $C_3\times D_6:S_5$, $C_6\times D_6\times A_5$, $\GL(2,4):D_{12}$, $C_6^2:S_5$, $\GL(2,4):(C_4\times S_3)$, $C_6^2:S_5$, $S_3\times C_6:S_5$, $S_3\times A_5:C_{12}$, $C_2\times A_5:S_3^2$, $C_6:S_3\times S_5$, $C_6\times S_3\times S_5$, $S_3^2\times S_5$

Order 3456: $S_3^2\times \GL(2,\mathbb{Z}/4)$

Order 2880: $S_3\times C_2^2:S_5$ x 9, $\GL(2,4):C_2^4$, $A_5:C_4\times D_6$, $C_2\times D_6:S_5$, $C_2\times D_6:S_5$, $C_2\times \GL(2,4):D_4$, $C_2^2:S_5\times C_6$, $D_6^2:F_5$, $\GL(2,4):C_2^4$

Order 2160: $\GL(2,4):D_6$ x 3, $D_6\times \GL(2,4)$ x 3, $S_3^2\times A_5$ x 3, $\GL(2,4):C_{12}$, $\GL(2,4):C_{12}$, $(C_3\times \GL(2,4)):C_4$, $C_6^2\times A_5$, $A_5:C_6^2$, $\GL(2,4):D_6$, $\GL(2,4):D_6$, $\GL(2,4):D_6$, $A_5:S_3^2$, $A_5:S_3^2$, $\GL(2,4):D_6$

Order 1920: $C_2^4:S_5$

Order 1728: $A_4\times D_6^2$, $A_4:D_6^2$, $D_4\times S_3^3$, $C_3:(S_3\times \GL(2,\mathbb{Z}/4))$, $S_3\times C_3:\GL(2,\mathbb{Z}/4)$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $C_3\times S_3\times \GL(2,\mathbb{Z}/4)$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $C_2^3.S_3^3$, $C_2^3.S_3^3$

Order 1440: $\GL(2,4):C_2^3$ x 8, $D_6\times S_5$ x 6, $\GL(2,4):D_4$ x 5, $\GL(2,4):D_4$ x 5, $D_6:S_5$ x 5, $D_6:S_5$ x 5, $D_6.S_5$ x 5, $D_5.D_6^2$ x 2, $(S_3\times D_{30}):C_4$ x 2, $D_{30}:C_4\times S_3$ x 2, $D_5\times D_6^2$, $C_2^3\times \GL(2,4)$, $\GL(2,4):C_2^3$, $\GL(2,4):C_2^3$, $C_6^2:(C_2\times F_5)$, $(C_6\times D_{30}):C_4$, $(C_6\times F_5):D_6$, $S_3\times D_{10}:C_{12}$, $C_2\times \GL(2,4):C_4$, $A_5:C_4\times C_6$

Order 1152: $D_6\times \GL(2,\mathbb{Z}/4)$, $D_6^2:D_4$

Order 1080: $C_6\times \GL(2,4)$ x 2, $S_3\times \GL(2,4)$ x 2, $\GL(2,4):S_3$ x 2, $C_3^2\times S_5$, $C_3^2:S_5$, $C_3^2:S_5$

Order 960: $C_2^3:S_5$ x 8, $C_2^4\times A_5$, $C_2^3\times S_5$, $D_6\times D_{10}:C_4$, $C_2^3.S_5$

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

Order 720: $D_6\times A_5$ x 13, $D_{10}\times S_3^2$ x 8, $C_2^2\times \GL(2,4)$ x 5, $S_3\times S_5$ x 5, $C_6:S_5$ x 4, $C_6\times S_5$ x 4, $\GL(2,4):C_4$ x 3, $A_5:C_{12}$ x 3, $D_{10}.S_3^2$ x 2, $D_{10}.S_3^2$ x 2, $C_6:S_3\times F_5$ x 2, $D_{30}:C_{12}$ x 2, $F_5\times S_3^2$ x 2, $D_{10}.S_3^2$ x 2, $(C_6\times F_5):S_3$ x 2, $D_{30}:C_{12}$ x 2, $C_5:D_6^2$, $D_6\times D_{30}$, $C_5\times D_6^2$, $C_6^2:D_{10}$, $C_6^2:D_{10}$, $C_6^2:F_5$, $C_6^2:F_5$, $C_6^2:F_5$, $D_{10}.S_3^2$

Order 576: $S_3\times \GL(2,\mathbb{Z}/4)$ x 10, $C_4:D_6^2$ 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, $C_2^2\times D_6^2$, $C_2^2\times A_4\times D_6$, $C_2^2:D_6^2$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_6\times \GL(2,\mathbb{Z}/4)$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_6^2.C_2^4$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_2^4:S_3^2$

Order 540: $C_3\times \GL(2,4)$

Order 480: $C_2^2:S_5$ x 10, $D_{10}:C_4\times S_3$ x 9, $C_2^3\times A_5$ x 6, $C_2^2\times S_5$ x 5, $C_2^2.S_5$ x 4, $C_2\times D_6\times F_5$ x 2, $D_{10}.D_{12}$ x 2, $C_{15}:C_2^5$, $(C_6\times D_{10}):C_4$, $C_2\times D_{10}:C_{12}$

Order 432: $C_2\times S_3^3$ x 15, $C_6^2:D_6$ x 6, $D_6:S_3^2$ x 6, $C_3:D_6^2$ x 4, $C_3\times D_6^2$ x 4, $C_6^2:D_6$ x 4, $C_6^2:D_6$ x 4, $C_6^2:D_6$ x 4, $C_6^2:D_6$ x 4, $C_6^2:D_6$ x 4, $D_6:S_3^2$ x 4, $D_6:S_3^2$ x 4, $D_6:S_3^2$ x 4, $C_6:S_3\times A_4$ x 3, $C_6\times S_3\times A_4$ x 3, $A_4\times S_3^2$ x 3, $C_{12}:S_3^2$ x 3, $D_6:S_3^2$ x 3, $D_6:S_3^2$ x 3, $C_3:D_6^2$ x 2, $C_6^2:D_6$ x 2, $C_{12}:C_6^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}\times S_3^2$ x 2, $C_2.S_3^3$ x 2, $C_2.S_3^3$ x 2, $A_4\times C_6^2$, $C_6^2:D_6$, $C_6^2:D_6$, $A_4:C_6^2$, $A_4:S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_{12}:S_3^2$, $(C_3^2\times A_4):C_4$, $C_3\times C_6.S_4$, $C_3\times A_4:C_{12}$, $C_2.S_3^3$

Order 384: $C_2^2\times \GL(2,\mathbb{Z}/4)$, $C_2^5:D_6$

Order 360: $D_5\times S_3^2$ x 9, $C_{30}:D_6$ x 6, $C_{30}:D_6$ x 6, $C_6\times A_5$ x 6, $S_3\times A_5$ x 6, $S_3\times D_{30}$ x 5, $C_{30}:D_6$ x 4, $C_{10}\times S_3^2$ x 4, $C_3:S_5$ x 3, $C_3\times S_5$ x 3, $C_2\times C_3^2:F_5$ x 2, $C_{30}:C_{12}$ x 2, $C_{30}:C_{12}$ x 2, $D_5.S_3^2$ x 2, $D_5.S_3^2$ x 2, $C_3:S_3\times F_5$ x 2, $D_{15}:C_{12}$ x 2, $C_6:D_{30}$, $C_{30}:D_6$, $C_6\times D_{30}$, $D_6\times C_{30}$, $D_5\times C_6^2$

Order 320: $C_2^4:F_5$

Order 288: $D_4\times S_3^2$ x 27, $C_2\times D_6^2$ x 14, $D_6^2:C_2$ x 10, $C_2\times A_4\times D_6$ x 9, $D_6^2:C_2$ x 8, $C_3:\GL(2,\mathbb{Z}/4)$ x 7, $D_6\times S_4$ x 7, $C_3\times \GL(2,\mathbb{Z}/4)$ x 6, $D_6:S_4$ x 6, $D_6:S_4$ x 6, $D_6.S_4$ x 6, $D_6:D_{12}$ x 6, $D_6^2.C_2$ x 6, $D_6.D_{12}$ x 6, $C_6^2:C_2^3$ x 5, $D_6\times D_{12}$ x 5, $D_{12}:D_6$ x 4, $C_2.D_6^2$ x 4, $C_6^2:C_2^3$ x 4, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $D_6^2:C_2$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_2^3.S_3^2$ x 3, $C_2^3.S_3^2$ x 3, $D_6:D_{12}$ x 3, $C_2\times C_6.S_4$, $C_2\times A_4:C_{12}$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:C_2^3$, $C_6^2:C_2^3$, $C_2^3:C_6^2$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 240: $D_6\times D_{10}$ x 15, $D_6\times F_5$ x 12, $D_{30}:C_4$ x 10, $C_2^2\times A_5$ x 10, $C_2\times S_5$ x 7, $D_{10}.D_6$ x 5, $D_{10}:C_{12}$ x 5, $A_5:C_4$ x 3, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$

Order 216: $C_6:S_3^2$ x 20, $C_6\times S_3^2$ x 18, $S_3^3$ x 16, $C_6:S_3^2$ x 7, $C_3^2:D_{12}$ x 6, $C_6^2:C_6$ x 4, $S_3\times C_6^2$ x 4, $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 4, $C_3^2:D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_3^3:D_4$ x 4, $C_3^2:D_{12}$ x 3, $C_3^3:D_4$ x 3, $C_6.S_3^2$ x 3, $C_6^2:S_3$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6^2:S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_3:S_3\times C_{12}$ x 2, $C_3^2\times D_{12}$ x 2, $C_6.C_6^2$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2\times S_4$, $D_4\times C_3^3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$

Order 192: $C_2^4:D_6$ x 17, $C_2\times \GL(2,\mathbb{Z}/4)$ x 9, $C_{12}:C_2^4$ x 4, $C_2^4:D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $D_6\times C_2^4$, $A_4\times C_2^4$, $C_2^3\times S_4$, $C_2^3.S_4$, $C_2^5:C_6$, $C_2^4:D_6$

Order 180: $\GL(2,4)$

Order 144: $D_6^2$

Order 128: $C_2^4:D_4$

Order 120: $S_5$ x 3, $C_2\times A_5$

Order 96: $D_4\times D_6$ x 10

Order 72: $S_3\times D_6$ x 19, $C_6:D_6$, $C_6\times D_6$

Order 60: $A_5$

Order 48: $S_3\times D_4$ x 54

Order 36: $S_3^2$ x 24, $C_6:S_3$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 32: $C_2^2\times D_4$ x 7

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 22

Order 18: $C_3\times C_6$, $C_3:S_3$, $C_3\times S_3$

Order 16: $C_2\times D_4$ x 38

Order 12: $D_6$ x 62, $C_2\times C_6$

Order 9: $C_3^2$

Order 8: $D_4$ x 30, $C_2^3$ x 14

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

Order 5: $C_5$

Order 4: $C_2^2$ x 32

Order 3: $C_3$

Order 2: $C_2$ x 14

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 17280: $D_6^2:S_5$ ($C_1$)

Order 8640: $S_3\times D_6:S_5$ ($C_2$) x 2, $S_3\times D_6:S_5$ ($C_2$) x 2, $S_3\times \GL(2,4):D_4$ ($C_2$) x 2, $S_3\times \GL(2,4):D_4$ ($C_2$) x 2, $A_5\times D_6^2$ ($C_2$), $(S_3\times D_6):S_5$ ($C_2$), $(C_6\times S_5):D_6$ ($C_2$), $(S_3\times D_6):S_5$ ($C_2$), $C_6^2:(C_2\times S_5)$ ($C_2$), $A_5:C_4\times S_3^2$ ($C_2$), $A_5:D_6^2$ ($C_2$)

Order 4320: $S_3\times D_6\times A_5$ ($C_2^2$) x 4, $S_3\times \GL(2,4):C_4$ ($C_2^2$) x 2, $(C_6\times S_3):S_5$ ($C_2^2$) x 2, $\GL(2,4):D_{12}$ ($C_2^2$) x 2, $\GL(2,4):D_{12}$ ($C_2^2$) x 2, $C_6^2:S_5$ ($C_2^2$) x 2, $\GL(2,4):D_{12}$ ($C_2^2$) x 2, $C_3\times D_6:S_5$ ($C_2^2$) x 2, $C_6\times D_6\times A_5$ ($C_2^2$) x 2, $S_3\times C_6:S_5$ ($C_2^2$) x 2, $S_3\times A_5:C_{12}$ ($C_2^2$) x 2, $C_6\times S_3\times S_5$ ($C_2^2$) x 2, $C_6:D_6\times A_5$ ($C_2^2$), $\GL(2,4):(C_4\times S_3)$ ($C_2^2$), $(C_6\times S_5):S_3$ ($C_2^2$), $\GL(2,4):D_{12}$ ($C_2^2$), $C_6^2:S_5$ ($C_2^2$), $\GL(2,4):(C_4\times S_3)$ ($C_2^2$), $C_6^2:S_5$ ($C_2^2$), $C_2\times A_5:S_3^2$ ($C_2^2$), $C_6:S_3\times S_5$ ($C_2^2$)

Order 2880: $S_3\times C_2^2:S_5$ ($S_3$) x 2

Order 2160: $D_6\times \GL(2,4)$ ($C_2^3$) x 4, $S_3^2\times A_5$ ($D_4$) x 4, $\GL(2,4):C_{12}$ ($C_2^3$) x 2, $\GL(2,4):D_6$ ($C_2^3$) x 2, $\GL(2,4):D_6$ ($C_2^3$) x 2, $\GL(2,4):C_{12}$ ($C_2^3$), $(C_3\times \GL(2,4)):C_4$ ($C_2^3$), $C_6^2\times A_5$ ($C_2^3$), $A_5:C_6^2$ ($C_2^3$), $\GL(2,4):D_6$ ($C_2^3$)

Order 1440: $\GL(2,4):C_2^3$ ($D_6$) x 2, $D_6\times S_5$ ($D_6$) x 2, $\GL(2,4):D_4$ ($D_6$) x 2, $\GL(2,4):D_4$ ($D_6$) x 2, $D_6:S_5$ ($D_6$) x 2, $D_6:S_5$ ($D_6$) x 2, $D_6.S_5$ ($D_6$) x 2

Order 1080: $S_3\times \GL(2,4)$ ($C_2\times D_4$) x 4, $\GL(2,4):S_3$ ($C_2\times D_4$) x 2, $C_6\times \GL(2,4)$ ($C_2^4$)

Order 720: $D_6\times A_5$ ($C_2\times D_6$) x 4, $C_2^2\times \GL(2,4)$ ($C_2\times D_6$) x 2, $C_6:S_5$ ($C_2\times D_6$) x 2, $C_6\times S_5$ ($C_2\times D_6$) x 2, $\GL(2,4):C_4$ ($C_2\times D_6$) x 2, $A_5:C_{12}$ ($C_2\times D_6$) x 2

Order 540: $C_3\times \GL(2,4)$ ($C_2^2\times D_4$)

Order 480: $C_2^2:S_5$ ($S_3^2$)

Order 360: $S_3\times A_5$ ($S_3\times D_4$) x 4, $C_6\times A_5$ ($C_2^2\times D_6$) x 2

Order 240: $A_5:C_4$ ($S_3\times D_6$), $C_2^2\times A_5$ ($S_3\times D_6$), $C_2\times S_5$ ($S_3\times D_6$)

Order 180: $\GL(2,4)$ ($D_4\times D_6$) x 2

Order 144: $D_6^2$ ($S_5$)

Order 120: $C_2\times A_5$ ($D_6^2$)

Order 72: $S_3\times D_6$ ($C_2\times S_5$) x 4, $C_6\times D_6$ ($C_2\times S_5$) x 2, $C_6:D_6$ ($C_2\times S_5$)

Order 60: $A_5$ ($D_4\times S_3^2$)

Order 36: $C_6\times S_3$ ($C_2^2\times S_5$) x 4, $S_3^2$ ($C_2^2:S_5$) x 4, $C_6:S_3$ ($C_2^2\times S_5$) x 2, $C_6^2$ ($C_2^2\times S_5$)

Order 24: $C_2\times D_6$ ($S_3\times S_5$) x 2

Order 18: $C_3\times S_3$ ($C_2^3:S_5$) x 4, $C_3:S_3$ ($C_2^3:S_5$) x 2, $C_3\times C_6$ ($C_2^3\times S_5$)

Order 12: $D_6$ ($D_6\times S_5$) x 4, $C_2\times C_6$ ($D_6\times S_5$) x 2

Order 9: $C_3^2$ ($C_2^4:S_5$)

Order 6: $S_3$ ($S_3\times C_2^2:S_5$) x 4, $C_6$ ($\GL(2,4):C_2^4$) x 2

Order 4: $C_2^2$ ($S_3^2\times S_5$)

Order 3: $C_3$ ($S_3\times C_2^3:S_5$) x 2

Order 2: $C_2$ ($A_5:D_6^2$)

Order 1: $C_1$ ($D_6^2:S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 17280: $D_6^2:S_5$ ($C_1$)

Order 8640: $A_5\times D_6^2$ ($C_2$), $(S_3\times D_6):S_5$ ($C_2$), $(C_6\times S_5):D_6$ ($C_2$), $(S_3\times D_6):S_5$ ($C_2$), $C_6^2:(C_2\times S_5)$ ($C_2$), $A_5:C_4\times S_3^2$ ($C_2$), $S_3\times D_6:S_5$ ($C_2$), $S_3\times D_6:S_5$ ($C_2$), $S_3\times \GL(2,4):D_4$ ($C_2$), $S_3\times \GL(2,4):D_4$ ($C_2$), $A_5:D_6^2$ ($C_2$)

Order 4320: $S_3\times D_6\times A_5$ ($C_2^2$) x 3, $C_6:D_6\times A_5$ ($C_2^2$), $S_3\times \GL(2,4):C_4$ ($C_2^2$), $(C_6\times S_3):S_5$ ($C_2^2$), $\GL(2,4):D_{12}$ ($C_2^2$), $\GL(2,4):(C_4\times S_3)$ ($C_2^2$), $(C_6\times S_5):S_3$ ($C_2^2$), $\GL(2,4):D_{12}$ ($C_2^2$), $C_6^2:S_5$ ($C_2^2$), $\GL(2,4):D_{12}$ ($C_2^2$), $C_3\times D_6:S_5$ ($C_2^2$), $C_6\times D_6\times A_5$ ($C_2^2$), $\GL(2,4):D_{12}$ ($C_2^2$), $C_6^2:S_5$ ($C_2^2$), $\GL(2,4):(C_4\times S_3)$ ($C_2^2$), $C_6^2:S_5$ ($C_2^2$), $S_3\times C_6:S_5$ ($C_2^2$), $S_3\times A_5:C_{12}$ ($C_2^2$), $C_2\times A_5:S_3^2$ ($C_2^2$), $C_6:S_3\times S_5$ ($C_2^2$), $C_6\times S_3\times S_5$ ($C_2^2$)

Order 2880: $S_3\times C_2^2:S_5$ ($S_3$)

Order 2160: $\GL(2,4):D_6$ ($C_2^3$) x 2, $D_6\times \GL(2,4)$ ($C_2^3$) x 2, $\GL(2,4):C_{12}$ ($C_2^3$), $\GL(2,4):C_{12}$ ($C_2^3$), $(C_3\times \GL(2,4)):C_4$ ($C_2^3$), $C_6^2\times A_5$ ($C_2^3$), $A_5:C_6^2$ ($C_2^3$), $\GL(2,4):D_6$ ($C_2^3$), $\GL(2,4):D_6$ ($C_2^3$), $S_3^2\times A_5$ ($D_4$)

Order 1440: $\GL(2,4):C_2^3$ ($D_6$), $D_6\times S_5$ ($D_6$), $\GL(2,4):D_4$ ($D_6$), $\GL(2,4):D_4$ ($D_6$), $D_6:S_5$ ($D_6$), $D_6:S_5$ ($D_6$), $D_6.S_5$ ($D_6$)

Order 1080: $C_6\times \GL(2,4)$ ($C_2^4$), $S_3\times \GL(2,4)$ ($C_2\times D_4$), $\GL(2,4):S_3$ ($C_2\times D_4$)

Order 720: $D_6\times A_5$ ($C_2\times D_6$) x 2, $C_2^2\times \GL(2,4)$ ($C_2\times D_6$), $C_6:S_5$ ($C_2\times D_6$), $C_6\times S_5$ ($C_2\times D_6$), $\GL(2,4):C_4$ ($C_2\times D_6$), $A_5:C_{12}$ ($C_2\times D_6$)

Order 540: $C_3\times \GL(2,4)$ ($C_2^2\times D_4$)

Order 480: $C_2^2:S_5$ ($S_3^2$)

Order 360: $C_6\times A_5$ ($C_2^2\times D_6$), $S_3\times A_5$ ($S_3\times D_4$)

Order 240: $A_5:C_4$ ($S_3\times D_6$), $C_2^2\times A_5$ ($S_3\times D_6$), $C_2\times S_5$ ($S_3\times D_6$)

Order 180: $\GL(2,4)$ ($D_4\times D_6$)

Order 144: $D_6^2$ ($S_5$)

Order 120: $C_2\times A_5$ ($D_6^2$)

Order 72: $S_3\times D_6$ ($C_2\times S_5$) x 3, $C_6:D_6$ ($C_2\times S_5$), $C_6\times D_6$ ($C_2\times S_5$)

Order 60: $A_5$ ($D_4\times S_3^2$)

Order 36: $C_6:S_3$ ($C_2^2\times S_5$) x 2, $C_6\times S_3$ ($C_2^2\times S_5$) x 2, $C_6^2$ ($C_2^2\times S_5$), $S_3^2$ ($C_2^2:S_5$)

Order 24: $C_2\times D_6$ ($S_3\times S_5$)

Order 18: $C_3\times C_6$ ($C_2^3\times S_5$), $C_3:S_3$ ($C_2^3:S_5$), $C_3\times S_3$ ($C_2^3:S_5$)

Order 12: $D_6$ ($D_6\times S_5$) x 2, $C_2\times C_6$ ($D_6\times S_5$)

Order 9: $C_3^2$ ($C_2^4:S_5$)

Order 6: $C_6$ ($\GL(2,4):C_2^4$), $S_3$ ($S_3\times C_2^2:S_5$)

Order 4: $C_2^2$ ($S_3^2\times S_5$)

Order 3: $C_3$ ($S_3\times C_2^3:S_5$)

Order 2: $C_2$ ($A_5:D_6^2$)

Order 1: $C_1$ ($D_6^2:S_5$)

Series

Derived series

$D_6^2:S_5$

$\rhd$

$D_6^2:S_5$

$\rhd$

$C_6\times \GL(2,4)$

$\rhd$

$C_6\times \GL(2,4)$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$D_6^2:S_5$

$\rhd$

$D_6^2:S_5$

$\rhd$

$A_5\times D_6^2$

$\rhd$

$A_5\times D_6^2$

$\rhd$

$D_6^2$

$\rhd$

$D_6^2$

$\rhd$

$C_6:D_6$

$\rhd$

$C_6:D_6$

$\rhd$

$C_6:S_3$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$D_6^2:S_5$

$\rhd$

$D_6^2:S_5$

$\rhd$

$C_6\times \GL(2,4)$

$\rhd$

$C_6\times \GL(2,4)$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$C_3\times \GL(2,4)$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 4 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 $171 \times 171$ 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 $162 \times 162$ rational character table (warning: may be slow to load).