Abstract group 23040.e: $(C_2\times S_4):\GL(2,5)$

• Label: 23040.e
• Order: \( 2^{9} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $32$
• Trans deg.: $288$
• Rank: $2$

Group information

Description:	$(C_2\times S_4):\GL(2,5)$
Order:	\(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^4\times S_5\times C_2\times S_4$, of order \(92160\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_3$, $A_5$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	639	188	5248	24	1764	1280	936	4832	192	2112	4480	576	768	23040
Conjugacy classes	1	14	3	37	1	17	8	11	32	1	20	28	3	4	180
Divisions	1	14	3	23	1	17	4	9	16	1	8	6	3	2	108
Autjugacy classes	1	13	3	18	1	16	4	8	14	1	7	6	3	2	97

Dimension	1	2	3	4	5	6	8	10	12	15	16	18	20	24	30	32	36	48	72
Irr. complex chars.	8	10	8	24	8	18	24	10	38	8	0	16	0	4	2	0	2	0	0	180
Irr. rational chars.	4	6	4	7	4	8	8	6	14	4	9	4	3	9	4	2	6	5	1	108

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$288$
Rank:	$2$
Inequivalent generating pairs:	$684$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $32$ $\langle(1,4)(2,12)(3,17)(5,14)(6,11)(7,23)(9,10)(13,16)(15,18)(20,22), (25,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 8 & 11 \\ 17 & 11 \end{array}\right), \left(\begin{array}{rr} 7 & 6 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 0 & 3 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2\times \GL(2,5))$ $\,\rtimes\,$ $S_4$	$(C_2\times S_4)$ $\,\rtimes\,$ $\GL(2,5)$	$(C_2^3\times \GL(2,5))$ $\,\rtimes\,$ $S_3$	$A_4$ $\,\rtimes\,$ $(C_2^2:\GL(2,5))$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^4.D_6)$ . $S_5$	$C_2^4$ . $(D_6.S_5)$	$C_2$ . $(S_4\times \GL(2,5))$	$(C_2\times C_4)$ . $(S_4\times S_5)$	all 33

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 305232 subgroups in 5046 conjugacy classes, 52 normal (50 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $S_4\times S_5$
Commutator:	$G' \simeq$ $C_2^4.\GL(2,4)$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_4\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_4$	$G/\operatorname{Fit} \simeq$ $S_3\times S_5$
Radical:	$R \simeq$ $C_2^4.D_6$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $D_6.S_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 23040: $(C_2\times S_4):\GL(2,5)$

Order 11520: $S_4\times \SL(2,5):C_2^2$, $C_2\times A_4:\GL(2,5)$, $C_2\times A_4\times \GL(2,5)$

Order 7680: $(C_2\times D_4):\GL(2,5)$

Order 5760: $S_4\times \SL(2,5):C_2$ x 2, $D_6:\GL(2,5)$, $A_4\times \GL(2,5)$, $A_4:\GL(2,5)$, $C_2\times S_4\times \SL(2,5)$, $C_2\times \SL(2,5):S_4$, $A_4\times \SL(2,5):C_2^2$

Order 4608: $(C_2\times C_4).S_4^2$

Order 3840: $C_2^3:\GL(2,5)$ x 3, $(C_2\times C_4):\GL(2,5)$ x 2, $C_2^3\times \GL(2,5)$, $D_4\times \SL(2,5):C_2^2$, $(C_{10}\times S_4):C_4^2$

Order 2880: $A_4\times \SL(2,5):C_2$ x 2, $S_4\times \SL(2,5)$, $\SL(2,5):S_4$, $C_2^4.\GL(2,4)$, $C_2\times \SL(2,5):D_6$, $C_6\times \GL(2,5)$, $C_6:\GL(2,5)$

Order 2304: $(C_2^3\times \SL(2,3)):C_{12}$, $C_2^3:C_{12}.S_4$, $S_4\times \SL(2,3):C_2^2$, $(C_2^3\times C_{24}):D_6$

Order 1920: $C_2^2\times \GL(2,5)$ x 6, $C_2^2:\GL(2,5)$ x 6, $D_4\times \SL(2,5):C_2$ x 4, $\SL(2,5):C_2^4$ x 2, $C_2\times \SL(2,5):D_4$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $D_{10}.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times \SL(2,5):D_4$, $C_2\times D_4\times \SL(2,5)$, $C_4\times \SL(2,5):C_2^2$, $C_4\times D_{10}\times S_4$, $A_4\times C_{10}:C_4^2$, $C_4\times D_{10}.S_4$

Order 1536: $(C_2^2\times C_4).\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^2):D_6$

Order 1440: $\SL(2,5):D_6$ x 2, $C_2\times \SL(2,5):C_6$, $\SL(2,5):D_6$, $D_6\times \SL(2,5)$, $A_4\times \SL(2,5)$, $C_3:\GL(2,5)$, $C_3\times \GL(2,5)$

Order 1280: $(C_2\times D_{20}):C_4^2$

Order 1152: $S_4\times \SL(2,3):C_2$ x 2, $A_4\times \SL(2,3):C_2^2$, $C_2^3.D_6^2$, $C_2\times S_4\times \SL(2,3)$, $C_2\times \SL(2,3):S_4$, $C_8:D_6\times A_4$, $(C_6\times A_4):\OD_{16}$, $A_4:\Unitary(2,3)$, $A_4\times \Unitary(2,3)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$

Order 960: $C_2\times \GL(2,5)$ x 13, $\SL(2,5):C_2^3$ x 13, $D_5.\GL(2,\mathbb{Z}/4)$ x 8, $\SL(2,5):D_4$ x 4, $C_4\times D_5\times S_4$ x 4, $D_4\times \SL(2,5)$ x 2, $C_2\times C_4\times \SL(2,5)$ x 2, $\SL(2,5):D_4$ x 2, $C_4^2.A_5$ x 2, $C_2^2\times A_4\times F_5$ x 2, $C_2^2\times A_4:F_5$ x 2, $C_4\times A_4\times F_5$ x 2, $C_4\times A_4:F_5$ x 2, $C_2^3\times \SL(2,5)$ x 2, $D_{30}:C_4^2$, $C_2\times D_{10}\times S_4$, $C_4\times A_4\times D_{10}$, $C_2^4.D_{30}$, $C_2\times C_{20}\times S_4$, $A_4:C_4\times D_{10}$, $C_{10}:(C_4\times S_4)$, $C_{10}:C_4\times S_4$

Order 768: $C_2\times C_4.\GL(2,\mathbb{Z}/4)$ x 3, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times C_8):D_6$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^3:\Unitary(2,3)$, $C_2^3\times \Unitary(2,3)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^5.D_{12}$, $(C_2\times S_4):\OD_{16}$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4\times C_2.\GL(2,\mathbb{Z}/4)$, $D_4\times \SL(2,3):C_2^2$, $A_4\times C_4^2:C_2^2$

Order 720: $\SL(2,5):C_6$ x 2, $C_6\times \SL(2,5)$, $\SL(2,5):S_3$, $S_3\times \SL(2,5)$

Order 640: $C_2\times D_{10}:C_4^2$ x 3, $(C_2\times C_{20}):C_4^2$ x 2, $(C_2\times D_{10}):C_4^2$ x 2, $(C_2^3\times F_5):C_4$ x 2, $C_{10}.C_4^3$ x 2, $(D_4\times D_{10}):C_4$ x 2, $D_{10}.C_2^5$, $C_4\times D_4\times D_{10}$

Order 576: $C_4\times S_3\times S_4$ x 4, $C_4.A_4^2$ x 2, $A_4\times C_{24}:C_2$ x 2, $C_{12}.(C_2\times S_4)$ x 2, $C_2^2:D_6^2$, $C_2\times \SL(2,3):D_6$, $C_4\times A_4\times D_6$, $C_2\times C_{12}:S_4$, $C_2\times C_{12}\times S_4$, $C_2^2.A_4^2$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $S_4\times \SL(2,3)$, $\SL(2,3):S_4$, $C_2\times A_4\times C_{24}$, $C_6:C_8\times A_4$, $C_2\times C_{12}.S_4$, $C_2\times C_{12}.S_4$, $C_6\times \Unitary(2,3)$, $C_2\times A_4:C_{24}$, $D_6^2.C_4$

Order 512: $C_2^3.D_4^2$

Order 480: $\SL(2,5):C_2^2$ x 16, $D_{10}\times S_4$ x 8, $C_2\times A_4\times F_5$ x 8, $D_{10}.S_4$ x 8, $C_2^2\times \SL(2,5)$ x 6, $\GL(2,5)$ x 4, $C_4\times D_5\times A_4$ x 4, $D_{10}.S_4$ x 2, $C_5:(C_4\times S_4)$ x 2, $C_5:C_4\times S_4$ x 2, $D_{10}.D_{12}$ x 2, $C_4\times \SL(2,5)$ x 2, $C_{20}:S_4$ x 2, $C_{20}\times S_4$ x 2, $D_{10}.D_{12}$ x 2, $C_2\times A_4\times D_{10}$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$, $C_2^3:C_{60}$, $C_{60}:C_2^3$, $C_{30}:C_4^2$, $C_{30}:C_4^2$, $C_{10}:C_4\times A_4$, $C_2^3.D_{30}$, $C_2\times A_4:C_{20}$

Order 384: $C_4.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^2\times \Unitary(2,3)$ x 6, $\GL(2,\mathbb{Z}/4):C_2^2$ x 4, $D_4\times \SL(2,3):C_2$ x 4, $C_4.\GL(2,\mathbb{Z}/4)$ x 3, $C_2\times D_6:\OD_{16}$ x 3, $A_4\times C_4\wr C_2$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $\OD_{16}:S_4$ x 2, $C_4^2:S_4$ x 2, $(C_2\times C_{24}):D_4$ x 2, $\SL(2,3):C_2^4$ x 2, $C_2\times \SL(2,3):D_4$ x 2, $C_2^2:\Unitary(2,3)$ x 2, $(C_2\times C_8):D_{12}$ x 2, $(C_2\times C_{12}):\OD_{16}$ x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$, $(C_2\times D_4):C_{24}$, $(C_2\times C_{24}):D_4$, $(C_6\times D_4):C_8$, $(D_4\times D_6):C_4$, $C_6.(C_8\times D_4)$, $C_2^4.(C_2\times A_4)$, $D_4:C_2^2\times A_4$, $C_2\times Q_8:S_4$, $C_2\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $\GL(2,\mathbb{Z}/4):C_2^2$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^5.D_6$, $C_{24}:C_2^4$, $C_4\times D_4\times D_6$, $C_2^5.C_{12}$, $C_2\times \SL(2,3):D_4$, $C_2\times D_4\times \SL(2,3)$, $C_4\times \SL(2,3):C_2^2$, $C_2\times A_4\times C_4^2$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^5.D_6$, $C_2\times A_4:\OD_{16}$

Order 360: $C_3\times \SL(2,5)$

Order 320: $(C_2\times D_{20}):C_4$ x 16, $D_{10}.C_4^2$ x 14, $D_{10}:C_4^2$ x 12, $D_{10}:C_4^2$ x 10, $C_4^2:D_{10}$ x 8, $C_2^4:F_5$ x 6, $C_2^4\times F_5$ x 2, $C_{20}:C_2^4$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4.D_{10}$ x 2, $C_{20}:C_2^4$, $C_2\times D_4\times C_{20}$, $C_2^4.D_{10}$, $C_2\times D_{20}:C_4$, $C_4:C_4\times D_{10}$, $C_4^2:D_{10}$, $C_4^2\times D_{10}$

Order 288: $D_6\times S_4$ x 6, $C_4\times S_3\times A_4$ x 4, $\SL(2,3):D_6$ x 2, $C_{12}:S_4$ x 2, $C_{12}\times S_4$ x 2, $D_6.S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $Q_8.C_6^2$, $C_2^3.C_6^2$, $C_2.D_6^2$, $C_6:C_4\times A_4$, $D_6\times \SL(2,3)$, $\SL(2,3):D_6$, $C_2\times C_6.S_4$, $C_2\times A_4:C_{12}$, $A_4\times \SL(2,3)$, $C_{24}:D_6$, $A_4\times C_{24}$, $C_2\times C_{12}.D_6$, $A_4\times C_3:C_8$, $C_{12}.S_4$, $C_{12}.S_4$, $\SL(2,3):C_{12}$, $A_4:C_{24}$, $C_{12}.D_{12}$, $D_6:C_{24}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 256: $C_2^5.D_4$ x 3, $(C_2\times C_4^2):D_4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times D_4).C_4^2$ x 2, $C_2^4.C_2^4$, $D_4^2:C_2^2$, $C_4^2:C_2^4$, $C_2^5.D_4$, $C_2^5.D_4$, $(C_2\times D_4):\OD_{16}$

Order 240: $D_{30}:C_4$ x 8, $D_5\times S_4$ x 8, $A_4\times D_{10}$ x 6, $\SL(2,5):C_2$ x 5, $C_2\times \SL(2,5)$ x 4, $C_{10}:S_4$ x 4, $C_{10}\times S_4$ x 4, $A_4\times F_5$ x 4, $A_4:F_5$ x 4, $C_{12}:D_{10}$ x 4, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2, $A_4\times C_{20}$ x 2, $C_{60}:C_4$ x 2, $C_{12}\times F_5$ x 2, $A_4\times C_5:C_4$ x 2, $C_2^3:C_{30}$, $D_6\times D_{10}$, $C_4\times D_{30}$, $D_6\times C_{20}$, $C_{12}\times D_{10}$, $D_{30}:C_4$, $D_6.D_{10}$, $D_{10}.D_6$, $C_{10}.S_4$, $A_4:C_{20}$

Order 192: $C_2\times \Unitary(2,3)$ x 15, $\SL(2,3):C_2^3$ x 13, $D_6:\OD_{16}$ x 12, $C_{24}:C_2^3$ x 10, $C_2^4.D_6$ x 9, $C_4^2:D_6$ x 8, $C_2\times D_6:C_8$ x 6, $\GL(2,\mathbb{Z}/4):C_2$ x 6, $C_2.\GL(2,\mathbb{Z}/4)$ x 4, $A_4\times D_4:C_2$ x 4, $D_4:S_4$ x 4, $D_4\times S_4$ x 4, $(C_2\times C_{24}):C_4$ x 4, $\SL(2,3):D_4$ x 4, $C_2^3:C_{24}$ x 3, $C_2\times C_{12}.D_4$ x 3, $C_4^2.A_4$ x 2, $C_2\times C_4\times \SL(2,3)$ x 2, $A_4\times C_4^2$ x 2, $A_4:C_4^2$ x 2, $A_4:\OD_{16}$ x 2, $C_2\times A_4:C_8$ x 2, $(C_2\times C_{12}):C_8$ x 2, $C_{12}:C_2^4$ x 2, $Q_8.(C_2\times A_4)$ x 2, $C_2^3\times \SL(2,3)$ x 2, $C_2^4:C_{12}$ x 2, $C_2^3.S_4$ x 2, $Q_8:S_4$ x 2, $Q_8\times S_4$ x 2, $(C_2\times C_4):C_{24}$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $A_4\times \OD_{16}$ x 2, $C_2^3:C_{24}$ x 2, $\SL(2,3):D_4$ x 2, $D_4\times \SL(2,3)$ x 2, $C_4\wr C_2\times C_6$, $(C_6\times D_4):C_4$, $\OD_{16}:D_6$, $D_6:\OD_{16}$, $D_6:C_4^2$, $C_4^2:D_6$, $C_2^3\times S_4$, $D_{12}:C_2^3$, $C_{12}:C_2^4$, $C_2^3:\SL(2,3)$, $C_2\times Q_8\times A_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^3\times C_{24}$, $C_2\times D_4\times C_{12}$, $C_2^4.D_6$, $C_{12}.C_2^4$, $C_2\times D_{12}:C_4$, $C_4:C_4\times D_6$, $C_4^2:D_6$, $D_6\times C_4^2$

Order 160: $C_2^3:F_5$ x 48, $C_{10}.C_4^2$ x 36, $C_{10}:C_4^2$ x 35, $C_{20}:C_2^3$ x 29, $C_2^3\times F_5$ x 24, $D_4\times D_{10}$ x 16, $C_{20}:D_4$ x 8, $C_2^3.D_{10}$ x 8, $D_{10}.D_4$ x 8, $C_4\times D_{20}$ x 4, $D_5\times C_4^2$ x 4, $D_4\times C_{20}$ x 4, $C_2^3.D_{10}$ x 4, $C_2^2.D_{20}$ x 4, $D_{20}:C_4$ x 4, $D_{10}.D_4$ x 4, $C_{10}:C_4^2$ x 3, $C_2^3\times D_{10}$ x 2, $C_2^3\times C_{20}$ x 2, $C_2^3:D_{10}$ x 2, $C_{10}.C_2^4$ x 2, $C_2^3:C_{20}$ x 2, $C_2^3.D_{10}$ x 2, $C_2^3.D_{10}$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_2\times C_4:C_{20}$, $C_2\times C_4\times C_{20}$, $C_2^2.D_{20}$

Order 144: $S_3\times S_4$ x 8, $A_4\times D_6$ x 4, $C_6:S_4$ x 4, $C_6\times S_4$ x 4, $C_4\times S_3^2$ x 4, $C_6:C_{24}$ x 2, $C_{24}:C_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}\times D_6$ x 2, $\SL(2,3):C_6$ x 2, $C_{12}\times A_4$ x 2, $D_6.D_6$ x 2, $A_4\times C_3:C_4$ x 2, $C_{12}.D_6$, $C_2^2:C_6^2$, $D_6^2$, $C_{12}:D_6$, $C_6\times \SL(2,3)$, $C_2^2.S_3^2$, $S_3\times \SL(2,3)$, $\SL(2,3):S_3$, $C_6.S_4$, $A_4:C_{12}$, $C_6\times C_{24}$

Order 128: $C_4^2:C_2^3$ x 16, $C_2^4.D_4$ x 12, $D_4^2:C_2$ x 8, $C_4^2:D_4$ x 4, $\OD_{16}:D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^3:\OD_{16}$ x 3, $C_2^3:C_4^2$ x 3, $C_2^3:\OD_{16}$ x 2, $C_2^3.\OD_{16}$ x 2, $C_2^3.C_4^2$ x 2, $D_4:C_2^4$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $C_2.C_4^3$ x 2, $C_2^3\times \OD_{16}$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $C_2^3\times C_4^2$, $C_2^3:C_4^2$, $C_2^4.C_2^3$, $C_2^3.C_4^2$, $C_2^4.D_4$

Order 120: $S_3\times D_{10}$ x 8, $C_{30}:C_4$ x 8, $C_6\times F_5$ x 8, $D_5\times A_4$ x 4, $D_5\times C_{12}$ x 4, $C_{10}\times A_4$ x 3, $C_6.D_{10}$ x 2, $C_6.D_{10}$ x 2, $C_5:S_4$ x 2, $C_5\times S_4$ x 2, $C_4\times D_{15}$ x 2, $S_3\times C_{20}$ x 2, $D_{15}:C_4$ x 2, $\SL(2,5)$, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 96: $C_8:D_6$ x 35, $C_{12}:C_2^3$ x 30, $\SL(2,3):C_2^2$ x 18, $D_6:C_8$ x 15, $D_4\times D_6$ x 13, $C_2^3:C_{12}$ x 11, $C_4\times S_4$ x 10, $C_2^2.S_4$ x 9, $C_2^3.D_6$ x 8, $D_6.D_4$ x 8, $\Unitary(2,3)$ x 8, $C_2^2\times S_4$ x 8, $C_{12}:D_4$ x 8, $C_2^2:C_{24}$ x 6, $C_{12}.D_4$ x 6, $C_2^2\times \SL(2,3)$ x 6, $C_2^2\times C_{24}$ x 6, $C_{12}.C_2^3$ x 6, $C_2^2.D_{12}$ x 5, $D_6.D_4$ x 4, $C_4\times D_{12}$ x 4, $S_3\times C_4^2$ x 4, $D_4:D_6$ x 4, $\GL(2,\mathbb{Z}/4)$ x 4, $D_4\times C_{12}$ x 4, $C_2^3.D_6$ x 4, $C_6:C_4^2$ x 4, $D_{12}:C_4$ x 4, $D_4\times A_4$ x 3, $C_4:S_4$ x 3, $A_4:Q_8$ x 3, $C_8\times A_4$ x 2, $C_4\times \SL(2,3)$ x 2, $A_4:C_8$ x 2, $D_4:C_{12}$ x 2, $C_{12}.D_4$ x 2, $D_{12}:C_4$ x 2, $C_2^3\times D_6$ x 2, $C_2^3\times C_{12}$ x 2, $C_2^3:D_6$ x 2, $C_6.C_2^4$ x 2, $Q_8\times A_4$ x 2, $C_2^3:C_{12}$ x 2, $C_2\times C_4\times C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3.D_6$ x 2, $D_{12}:C_4$ x 2, $C_6.C_4^2$, $C_2^3\times A_4$, $C_6.C_2^4$, $C_{12}:C_2^3$, $Q_8:D_6$, $Q_8\times D_6$, $D_4:D_6$, $D_{12}:C_2^2$, $C_2^2\times D_{12}$, $Q_8:A_4$, $C_6\times \OD_{16}$, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$, $C_6:\OD_{16}$

Order 80: $C_2^2\times F_5$ x 94, $C_4\times D_{10}$ x 79, $D_{10}:C_4$ x 48, $D_4\times D_5$ x 32, $C_2^2\times D_{10}$ x 31, $C_4\times F_5$ x 20, $C_{10}:D_4$ x 16, $C_2^2\times C_{20}$ x 15, $C_2^2.D_{10}$ x 15, $D_4\times C_{10}$ x 8, $C_2\times D_{20}$ x 8, $D_{10}:C_4$ x 8, $C_{20}:C_4$ x 6, $C_2^2:C_{20}$ x 4, $C_{10}.D_4$ x 4, $C_{10}.D_4$ x 4, $C_2^3\times C_{10}$ x 2, $C_4:C_{20}$ x 2, $C_4\times C_{20}$ x 2, $C_{20}:C_4$ x 2

Order 72: $S_3\times D_6$ x 6, $S_3\times C_{12}$ x 6, $C_6.D_6$ x 4, $C_6\times A_4$ x 3, $C_6\times D_6$ x 2, $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3\times S_4$ x 2, $C_{12}:S_3$ x 2, $C_6:C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $C_6:D_6$, $C_6\times C_{12}$, $C_6.D_6$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_4^2:C_2^2$ x 63, $D_4:C_2^3$ x 27, $C_4^2:C_2^2$ x 20, $C_4^2:C_2^2$ x 19, $D_4:D_4$ x 16, $C_2^2:\OD_{16}$ x 10, $C_2^2:C_4^2$ x 10, $C_2^2\times C_4^2$ x 10, $C_2^2\times \OD_{16}$ x 9, $C_2^3.D_4$ x 8, $Q_8:D_4$ x 8, $D_4:D_4$ x 8, $D_4^2$ x 8, $C_4^2.C_2^2$ x 8, $C_4^2:C_4$ x 8, $C_2^2.C_4^2$ x 7, $C_2^3:C_8$ x 6, $C_2^3:D_4$ x 6, $C_2^4:C_4$ x 5, $Q_8:D_4$ x 4, $D_4\times Q_8$ x 4, $C_2^3:Q_8$ x 4, $C_4^2.C_2^2$ x 4, $C_2^2.C_4^2$ x 4, $D_4\times C_2^3$ x 3, $C_2^4\times C_4$ x 3, $Q_8\times C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_2^3:D_4$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3\times C_8$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3.D_4$

Order 60: $S_3\times D_5$ x 8, $C_3\times D_{10}$ x 6, $C_{15}:C_4$ x 4, $C_3\times F_5$ x 4, $D_{30}$ x 4, $S_3\times C_{10}$ x 4, $C_{60}$ x 2, $C_5:C_{12}$ x 2, $C_5\times A_4$, $C_{15}:C_4$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_4\times D_6$ x 84, $S_3\times D_4$ x 36, $C_2^2\times D_6$ x 24, $C_{24}:C_2$ x 20, $C_2^2\times C_{12}$ x 17, $C_6.C_2^3$ x 17, $C_6:C_8$ x 15, $C_2\times S_4$ x 15, $C_6:D_4$ x 15, $C_2\times C_{24}$ x 15, $D_6:C_4$ x 12, $C_6\times D_4$ x 9, $\SL(2,3):C_2$ x 9, $C_4\times A_4$ x 8, $C_{12}:C_4$ x 8, $C_2\times D_{12}$ x 7, $C_2^2\times A_4$ x 6, $D_{12}:C_2$ x 6, $C_2\times \SL(2,3)$ x 6, $A_4:C_4$ x 5, $D_4:C_6$ x 4, $D_4:S_3$ x 4, $C_2^2:C_{12}$ x 4, $C_4\times C_{12}$ x 4, $C_6.D_4$ x 4, $C_6.D_4$ x 4, $C_2^3\times C_6$ x 2, $D_{12}:C_2$ x 2, $S_3\times Q_8$ x 2, $C_3\times \OD_{16}$ x 2, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_3:\OD_{16}$ x 2, $C_6\times Q_8$, $C_6:Q_8$

Order 40: $C_2\times F_5$ x 92, $C_2\times D_{10}$ x 91, $C_4\times D_5$ x 44, $C_2\times C_{20}$ x 23, $C_{10}:C_4$ x 23, $C_5:D_4$ x 16, $C_2^2\times C_{10}$ x 15, $D_{20}$ x 8, $C_5\times D_4$ x 8

Order 36: $C_6\times S_3$ x 8, $S_3^2$ x 8, $C_6:S_3$ x 4, $C_3:C_{12}$ x 3, $C_3\times C_{12}$ x 2, $C_3^2:C_4$, $C_6^2$, $C_3\times A_4$

Order 32: $D_4:C_2^2$ x 94, $C_4\times D_4$ x 48, $C_2^3\times C_4$ x 45, $C_4\wr C_2$ x 44, $C_2^3:C_4$ x 43, $C_2\times C_4^2$ x 37, $C_2^2\times D_4$ x 36, $C_4:D_4$ x 36, $C_2\times \OD_{16}$ x 29, $C_2^2:Q_8$ x 20, $C_2.C_4^2$ x 18, $C_2^2.D_4$ x 16, $C_2^2\wr C_2$ x 16, $C_2^2\times Q_8$ x 13, $C_2^2:C_8$ x 12, $C_2^2.D_4$ x 11, $C_4^2:C_2$ x 8, $C_2^2\times C_8$ x 6, $C_4^2:C_2$ x 6, $C_4:Q_8$ x 4, $C_4:D_4$ x 4, $C_4\times Q_8$ x 4, $C_2^5$ x 2

Order 30: $C_3\times D_5$ x 4, $C_{30}$ x 3, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 24: $C_2\times D_6$ x 75, $C_4\times S_3$ x 54, $C_2\times C_{12}$ x 34, $C_6:C_4$ x 32, $C_3:D_4$ x 20, $C_2^2\times C_6$ x 16, $D_{12}$ x 11, $C_3\times D_4$ x 11, $C_2\times A_4$ x 8, $C_{24}$ x 6, $C_3:C_8$ x 6, $S_4$ x 5, $C_3:Q_8$ x 3, $\SL(2,3)$ x 3, $C_3\times Q_8$ x 2

Order 20: $D_{10}$ x 64, $C_2\times C_{10}$ x 22, $F_5$ x 16, $C_{20}$ x 8, $C_5:C_4$ x 8

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3, $C_3:S_3$ x 2

Order 16: $C_2^2\times C_4$ x 151, $C_2\times D_4$ x 98, $D_4:C_2$ x 82, $C_2^2:C_4$ x 52, $C_4^2$ x 25, $C_2\times Q_8$ x 25, $C_2^4$ x 24, $C_4:C_4$ x 18, $\OD_{16}$ x 17, $C_2\times C_8$ x 13

Order 15: $C_{15}$

Order 12: $D_6$ x 63, $C_2\times C_6$ x 28, $C_{12}$ x 16, $C_3:C_4$ x 13, $A_4$ x 2

Order 10: $D_5$ x 10, $C_{10}$ x 9

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 129, $C_2^3$ x 71, $D_4$ x 51, $Q_8$ x 13, $C_8$ x 4

Order 6: $C_6$ x 17, $S_3$ x 11

Order 5: $C_5$

Order 4: $C_2^2$ x 59, $C_4$ x 23

Order 3: $C_3$ x 3

Order 2: $C_2$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 23040: $(C_2\times S_4):\GL(2,5)$

Order 11520: $S_4\times \SL(2,5):C_2^2$, $C_2\times A_4:\GL(2,5)$, $C_2\times A_4\times \GL(2,5)$

Order 7680: $(C_2\times D_4):\GL(2,5)$

Order 5760: $S_4\times \SL(2,5):C_2$ x 2, $D_6:\GL(2,5)$, $A_4\times \GL(2,5)$, $A_4:\GL(2,5)$, $C_2\times S_4\times \SL(2,5)$, $C_2\times \SL(2,5):S_4$, $A_4\times \SL(2,5):C_2^2$

Order 4608: $(C_2\times C_4).S_4^2$

Order 3840: $C_2^3:\GL(2,5)$ x 3, $(C_2\times C_4):\GL(2,5)$ x 2, $C_2^3\times \GL(2,5)$, $D_4\times \SL(2,5):C_2^2$, $(C_{10}\times S_4):C_4^2$

Order 2880: $A_4\times \SL(2,5):C_2$ x 2, $S_4\times \SL(2,5)$, $\SL(2,5):S_4$, $C_2^4.\GL(2,4)$, $C_2\times \SL(2,5):D_6$, $C_6\times \GL(2,5)$, $C_6:\GL(2,5)$

Order 2304: $(C_2^3\times \SL(2,3)):C_{12}$, $C_2^3:C_{12}.S_4$, $S_4\times \SL(2,3):C_2^2$, $(C_2^3\times C_{24}):D_6$

Order 1920: $C_2^2\times \GL(2,5)$ x 6, $C_2^2:\GL(2,5)$ x 5, $D_4\times \SL(2,5):C_2$ x 4, $\SL(2,5):C_2^4$ x 2, $C_2\times \SL(2,5):D_4$ x 2, $C_2\times \SL(2,5):D_4$, $C_2\times D_4\times \SL(2,5)$, $C_4\times \SL(2,5):C_2^2$, $C_4\times D_{10}\times S_4$, $A_4\times C_{10}:C_4^2$, $C_4\times D_{10}.S_4$, $D_{10}.\GL(2,\mathbb{Z}/4)$, $D_{10}.\GL(2,\mathbb{Z}/4)$

Order 1536: $(C_2^2\times C_4).\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^2):D_6$

Order 1440: $\SL(2,5):D_6$ x 2, $C_2\times \SL(2,5):C_6$, $\SL(2,5):D_6$, $D_6\times \SL(2,5)$, $A_4\times \SL(2,5)$, $C_3:\GL(2,5)$, $C_3\times \GL(2,5)$

Order 1280: $(C_2\times D_{20}):C_4^2$

Order 1152: $S_4\times \SL(2,3):C_2$ x 2, $A_4\times \SL(2,3):C_2^2$, $C_2^3.D_6^2$, $C_2\times S_4\times \SL(2,3)$, $C_2\times \SL(2,3):S_4$, $C_8:D_6\times A_4$, $(C_6\times A_4):\OD_{16}$, $A_4:\Unitary(2,3)$, $A_4\times \Unitary(2,3)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$

Order 960: $\SL(2,5):C_2^3$ x 13, $C_2\times \GL(2,5)$ x 12, $\SL(2,5):D_4$ x 4, $C_4\times D_5\times S_4$ x 4, $D_4\times \SL(2,5)$ x 2, $C_2\times C_4\times \SL(2,5)$ x 2, $\SL(2,5):D_4$ x 2, $C_4^2.A_5$ x 2, $C_4\times A_4\times F_5$ x 2, $C_4\times A_4:F_5$ x 2, $C_2^3\times \SL(2,5)$ x 2, $D_{30}:C_4^2$, $C_2\times D_{10}\times S_4$, $C_2^2\times A_4\times F_5$, $C_2^2\times A_4:F_5$, $C_4\times A_4\times D_{10}$, $C_2^4.D_{30}$, $C_2\times C_{20}\times S_4$, $A_4:C_4\times D_{10}$, $C_{10}:(C_4\times S_4)$, $C_{10}:C_4\times S_4$, $D_5.\GL(2,\mathbb{Z}/4)$

Order 768: $C_2\times C_4.\GL(2,\mathbb{Z}/4)$ x 3, $(C_2\times C_4).\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times C_8):D_6$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^3:\Unitary(2,3)$, $C_2^3\times \Unitary(2,3)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^5.D_{12}$, $(C_2\times S_4):\OD_{16}$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4\times C_2.\GL(2,\mathbb{Z}/4)$, $D_4\times \SL(2,3):C_2^2$, $A_4\times C_4^2:C_2^2$

Order 720: $\SL(2,5):C_6$ x 2, $C_6\times \SL(2,5)$, $\SL(2,5):S_3$, $S_3\times \SL(2,5)$

Order 640: $C_2\times D_{10}:C_4^2$ x 3, $C_{10}.C_4^3$ x 2, $(C_2\times C_{20}):C_4^2$, $(C_2\times D_{10}):C_4^2$, $(C_2^3\times F_5):C_4$, $D_{10}.C_2^5$, $C_4\times D_4\times D_{10}$, $(D_4\times D_{10}):C_4$

Order 576: $C_4\times S_3\times S_4$ x 4, $C_4.A_4^2$ x 2, $A_4\times C_{24}:C_2$ x 2, $C_{12}.(C_2\times S_4)$ x 2, $C_2^2:D_6^2$, $C_2\times \SL(2,3):D_6$, $C_4\times A_4\times D_6$, $C_2\times C_{12}:S_4$, $C_2\times C_{12}\times S_4$, $C_2^2.A_4^2$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $S_4\times \SL(2,3)$, $\SL(2,3):S_4$, $C_2\times A_4\times C_{24}$, $C_6:C_8\times A_4$, $C_2\times C_{12}.S_4$, $C_2\times C_{12}.S_4$, $C_6\times \Unitary(2,3)$, $C_2\times A_4:C_{24}$, $D_6^2.C_4$

Order 512: $C_2^3.D_4^2$

Order 480: $\SL(2,5):C_2^2$ x 16, $C_2^2\times \SL(2,5)$ x 6, $\GL(2,5)$ x 4, $D_{10}\times S_4$ x 4, $C_4\times D_5\times A_4$ x 4, $C_2\times A_4\times F_5$ x 3, $D_{10}.S_4$ x 3, $D_{10}.S_4$ x 2, $C_5:(C_4\times S_4)$ x 2, $C_5:C_4\times S_4$ x 2, $C_4\times \SL(2,5)$ x 2, $C_{20}:S_4$ x 2, $C_{20}\times S_4$ x 2, $D_{10}.D_{12}$, $C_2\times A_4\times D_{10}$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$, $C_2^3:C_{60}$, $C_{60}:C_2^3$, $C_{30}:C_4^2$, $C_{30}:C_4^2$, $C_{10}:C_4\times A_4$, $C_2^3.D_{30}$, $C_2\times A_4:C_{20}$, $D_{10}.D_{12}$

Order 384: $C_2^2\times \Unitary(2,3)$ x 6, $C_4.\GL(2,\mathbb{Z}/4)$ x 5, $\GL(2,\mathbb{Z}/4):C_2^2$ x 4, $D_4\times \SL(2,3):C_2$ x 4, $C_4.\GL(2,\mathbb{Z}/4)$ x 3, $C_2\times D_6:\OD_{16}$ x 3, $A_4\times C_4\wr C_2$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times C_{24}):D_4$ x 2, $\SL(2,3):C_2^4$ x 2, $C_2\times \SL(2,3):D_4$ x 2, $(C_2\times C_8):D_{12}$ x 2, $(C_2\times C_{12}):\OD_{16}$ x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$, $\OD_{16}:S_4$, $C_4^2:S_4$, $(C_2\times D_4):C_{24}$, $(C_2\times C_{24}):D_4$, $(C_6\times D_4):C_8$, $(D_4\times D_6):C_4$, $C_6.(C_8\times D_4)$, $C_2^4.(C_2\times A_4)$, $D_4:C_2^2\times A_4$, $C_2\times Q_8:S_4$, $C_2\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $\GL(2,\mathbb{Z}/4):C_2^2$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^5.D_6$, $C_{24}:C_2^4$, $C_4\times D_4\times D_6$, $C_2^5.C_{12}$, $C_2\times \SL(2,3):D_4$, $C_2\times D_4\times \SL(2,3)$, $C_4\times \SL(2,3):C_2^2$, $C_2\times A_4\times C_4^2$, $C_2^2:\Unitary(2,3)$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^5.D_6$, $C_2\times A_4:\OD_{16}$

Order 360: $C_3\times \SL(2,5)$

Order 320: $D_{10}:C_4^2$ x 9, $D_{10}:C_4^2$ x 9, $C_4^2:D_{10}$ x 8, $D_{10}.C_4^2$ x 7, $C_2^4:F_5$ x 3, $C_{20}:C_2^4$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4.D_{10}$ x 2, $C_2^4.D_{10}$ x 2, $(C_2\times D_{20}):C_4$ x 2, $C_2^4\times F_5$, $C_{20}:C_2^4$, $C_2\times D_4\times C_{20}$, $C_2^4.D_{10}$, $C_2\times D_{20}:C_4$, $C_4:C_4\times D_{10}$, $C_4^2:D_{10}$, $C_4^2\times D_{10}$

Order 288: $C_4\times S_3\times A_4$ x 4, $D_6\times S_4$ x 4, $\SL(2,3):D_6$ x 2, $C_{12}:S_4$ x 2, $C_{12}\times S_4$ x 2, $D_6.S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $Q_8.C_6^2$, $C_2^3.C_6^2$, $C_2.D_6^2$, $C_6:C_4\times A_4$, $D_6\times \SL(2,3)$, $\SL(2,3):D_6$, $C_2\times C_6.S_4$, $C_2\times A_4:C_{12}$, $A_4\times \SL(2,3)$, $C_{24}:D_6$, $A_4\times C_{24}$, $C_2\times C_{12}.D_6$, $A_4\times C_3:C_8$, $C_{12}.S_4$, $C_{12}.S_4$, $\SL(2,3):C_{12}$, $A_4:C_{24}$, $C_{12}.D_{12}$, $D_6:C_{24}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 256: $C_2^5.D_4$ x 3, $(C_2\times C_4^2):D_4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times D_4).C_4^2$ x 2, $C_2^4.C_2^4$, $D_4^2:C_2^2$, $C_4^2:C_2^4$, $C_2^5.D_4$, $C_2^5.D_4$, $(C_2\times D_4):\OD_{16}$

Order 240: $\SL(2,5):C_2$ x 5, $C_2\times \SL(2,5)$ x 4, $A_4\times D_{10}$ x 4, $C_{12}:D_{10}$ x 4, $C_{10}:S_4$ x 3, $C_{10}\times S_4$ x 3, $D_5\times S_4$ x 2, $A_4\times C_{20}$ x 2, $C_{60}:C_4$ x 2, $C_{12}\times F_5$ x 2, $A_4\times C_5:C_4$ x 2, $D_{30}:C_4$, $C_2^3:C_{30}$, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $A_4\times F_5$, $A_4:F_5$, $C_4\times D_{30}$, $D_6\times C_{20}$, $C_{12}\times D_{10}$, $D_{30}:C_4$, $D_6.D_{10}$, $D_{10}.D_6$, $C_{10}.S_4$, $A_4:C_{20}$

Order 192: $C_2\times \Unitary(2,3)$ x 14, $\SL(2,3):C_2^3$ x 13, $D_6:\OD_{16}$ x 10, $C_{24}:C_2^3$ x 10, $C_2^4.D_6$ x 9, $C_4^2:D_6$ x 8, $C_2\times D_6:C_8$ x 6, $A_4\times D_4:C_2$ x 4, $\GL(2,\mathbb{Z}/4):C_2$ x 4, $(C_2\times C_{24}):C_4$ x 4, $\SL(2,3):D_4$ x 4, $C_2^3:C_{24}$ x 3, $C_2\times C_{12}.D_4$ x 3, $D_4:S_4$ x 3, $D_4\times S_4$ x 3, $C_4^2.A_4$ x 2, $C_2\times C_4\times \SL(2,3)$ x 2, $A_4\times C_4^2$ x 2, $A_4:C_4^2$ x 2, $A_4:\OD_{16}$ x 2, $C_2\times A_4:C_8$ x 2, $(C_2\times C_{12}):C_8$ x 2, $C_{12}:C_2^4$ x 2, $Q_8.(C_2\times A_4)$ x 2, $C_2^3\times \SL(2,3)$ x 2, $C_2^4:C_{12}$ x 2, $C_2^3.S_4$ x 2, $Q_8:S_4$ x 2, $Q_8\times S_4$ x 2, $(C_2\times C_4):C_{24}$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $A_4\times \OD_{16}$ x 2, $C_2^3:C_{24}$ x 2, $\SL(2,3):D_4$ x 2, $D_4\times \SL(2,3)$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$, $C_4\wr C_2\times C_6$, $(C_6\times D_4):C_4$, $\OD_{16}:D_6$, $D_6:\OD_{16}$, $D_6:C_4^2$, $C_4^2:D_6$, $C_2^3\times S_4$, $D_{12}:C_2^3$, $C_{12}:C_2^4$, $C_2^3:\SL(2,3)$, $C_2\times Q_8\times A_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^3\times C_{24}$, $C_2\times D_4\times C_{12}$, $C_2^4.D_6$, $C_{12}.C_2^4$, $C_2\times D_{12}:C_4$, $C_4:C_4\times D_6$, $C_4^2:D_6$, $D_6\times C_4^2$

Order 160: $C_{20}:C_2^3$ x 29, $C_{10}:C_4^2$ x 27, $C_2^3:F_5$ x 13, $C_2^3\times F_5$ x 10, $C_{10}.C_4^2$ x 9, $D_4\times D_{10}$ x 8, $C_{20}:D_4$ x 8, $C_2^3.D_{10}$ x 8, $D_{10}.D_4$ x 8, $C_4\times D_{20}$ x 4, $D_5\times C_4^2$ x 4, $D_4\times C_{20}$ x 4, $C_2^3.D_{10}$ x 4, $C_2^2.D_{20}$ x 4, $D_{20}:C_4$ x 4, $D_{10}.D_4$ x 4, $C_{10}:C_4^2$ x 3, $C_2^3\times D_{10}$ x 2, $C_2^3\times C_{20}$ x 2, $C_2^3:D_{10}$ x 2, $C_{10}.C_2^4$ x 2, $C_2^3:C_{20}$ x 2, $C_2^3.D_{10}$ x 2, $C_2^3.D_{10}$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_2\times C_4:C_{20}$, $C_2\times C_4\times C_{20}$, $C_2^2.D_{20}$

Order 144: $A_4\times D_6$ x 4, $C_4\times S_3^2$ x 4, $C_6:S_4$ x 3, $C_6\times S_4$ x 3, $C_6:C_{24}$ x 2, $C_{24}:C_6$ x 2, $C_{12}.D_6$ x 2, $S_3\times S_4$ x 2, $C_{12}\times D_6$ x 2, $\SL(2,3):C_6$ x 2, $C_{12}\times A_4$ x 2, $D_6.D_6$ x 2, $A_4\times C_3:C_4$ x 2, $C_{12}.D_6$, $C_2^2:C_6^2$, $D_6^2$, $C_{12}:D_6$, $C_6\times \SL(2,3)$, $C_2^2.S_3^2$, $S_3\times \SL(2,3)$, $\SL(2,3):S_3$, $C_6.S_4$, $A_4:C_{12}$, $C_6\times C_{24}$

Order 128: $C_4^2:C_2^3$ x 16, $C_2^4.D_4$ x 10, $D_4^2:C_2$ x 8, $C_2^4.D_4$ x 4, $C_2^3:\OD_{16}$ x 3, $C_2^3:C_4^2$ x 3, $C_4^2:D_4$ x 2, $\OD_{16}:D_4$ x 2, $C_2^3:\OD_{16}$ x 2, $C_2^3.\OD_{16}$ x 2, $C_2^3.C_4^2$ x 2, $D_4:C_2^4$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $C_2.C_4^3$ x 2, $C_2^3\times \OD_{16}$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $C_2^3\times C_4^2$, $C_2^3:C_4^2$, $C_2^4.C_2^3$, $C_2^3.C_4^2$, $C_2^4.D_4$

Order 120: $S_3\times D_{10}$ x 4, $D_5\times C_{12}$ x 4, $C_{10}\times A_4$ x 3, $C_{30}:C_4$ x 3, $C_6\times F_5$ x 3, $C_6.D_{10}$ x 2, $C_6.D_{10}$ x 2, $D_5\times A_4$ x 2, $C_4\times D_{15}$ x 2, $S_3\times C_{20}$ x 2, $D_{15}:C_4$ x 2, $\SL(2,5)$, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_5:S_4$, $C_5\times S_4$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 96: $C_8:D_6$ x 31, $C_{12}:C_2^3$ x 30, $\SL(2,3):C_2^2$ x 18, $D_6:C_8$ x 13, $C_2^3:C_{12}$ x 10, $C_4\times S_4$ x 10, $D_4\times D_6$ x 9, $C_2^3.D_6$ x 8, $D_6.D_4$ x 8, $C_2^2.S_4$ x 8, $C_{12}:D_4$ x 8, $C_2^2\times S_4$ x 7, $\Unitary(2,3)$ x 6, $C_2^2\times \SL(2,3)$ x 6, $C_2^2\times C_{24}$ x 6, $C_{12}.C_2^3$ x 6, $C_2^2:C_{24}$ x 5, $C_{12}.D_4$ x 5, $C_2^2.D_{12}$ x 5, $D_6.D_4$ x 4, $C_4\times D_{12}$ x 4, $S_3\times C_4^2$ x 4, $D_4:D_6$ x 4, $D_4\times C_{12}$ x 4, $C_2^3.D_6$ x 4, $C_6:C_4^2$ x 4, $D_{12}:C_4$ x 4, $D_4\times A_4$ x 3, $C_8\times A_4$ x 2, $C_4\times \SL(2,3)$ x 2, $A_4:C_8$ x 2, $D_4:C_{12}$ x 2, $C_{12}.D_4$ x 2, $C_2^3\times D_6$ x 2, $C_2^3\times C_{12}$ x 2, $C_2^3:D_6$ x 2, $C_6.C_2^4$ x 2, $Q_8\times A_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_4:S_4$ x 2, $A_4:Q_8$ x 2, $C_2^3:C_{12}$ x 2, $C_2\times C_4\times C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3.D_6$ x 2, $C_6.C_4^2$, $D_{12}:C_4$, $C_2^3\times A_4$, $C_6.C_2^4$, $C_{12}:C_2^3$, $Q_8:D_6$, $Q_8\times D_6$, $D_4:D_6$, $D_{12}:C_2^2$, $C_2^2\times D_{12}$, $Q_8:A_4$, $C_6\times \OD_{16}$, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$, $C_6:\OD_{16}$, $D_{12}:C_4$

Order 80: $C_4\times D_{10}$ x 73, $C_2^2\times F_5$ x 33, $C_2^2\times D_{10}$ x 21, $C_4\times F_5$ x 16, $C_2^2\times C_{20}$ x 15, $C_2^2.D_{10}$ x 15, $C_{10}:D_4$ x 10, $D_4\times D_5$ x 8, $D_{10}:C_4$ x 8, $D_{10}:C_4$ x 7, $D_4\times C_{10}$ x 6, $C_{20}:C_4$ x 6, $C_2\times D_{20}$ x 4, $C_2^2:C_{20}$ x 4, $C_{10}.D_4$ x 4, $C_{10}.D_4$ x 4, $C_2^3\times C_{10}$ x 2, $C_4:C_{20}$ x 2, $C_4\times C_{20}$ x 2, $C_{20}:C_4$ x 2

Order 72: $S_3\times C_{12}$ x 6, $S_3\times D_6$ x 4, $C_6.D_6$ x 4, $C_6\times A_4$ x 3, $C_6\times D_6$ x 2, $S_3\times A_4$ x 2, $C_{12}:S_3$ x 2, $C_6:C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $C_6:D_6$, $C_3:S_4$, $C_3\times S_4$, $C_6\times C_{12}$, $C_6.D_6$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_4^2:C_2^2$ x 53, $D_4:C_2^3$ x 27, $C_4^2:C_2^2$ x 19, $C_4^2:C_2^2$ x 14, $D_4:D_4$ x 12, $C_2^2\times C_4^2$ x 10, $C_2^2:\OD_{16}$ x 9, $C_2^2:C_4^2$ x 9, $C_2^2\times \OD_{16}$ x 9, $Q_8:D_4$ x 8, $C_2^2.C_4^2$ x 7, $C_2^3:C_8$ x 6, $D_4:D_4$ x 6, $D_4^2$ x 6, $C_4^2.C_2^2$ x 6, $C_2^3:D_4$ x 6, $C_2^4:C_4$ x 5, $Q_8:D_4$ x 4, $D_4\times Q_8$ x 4, $C_2^3:Q_8$ x 4, $C_4^2:C_4$ x 4, $C_4^2.C_2^2$ x 4, $C_2^2.C_4^2$ x 4, $D_4\times C_2^3$ x 3, $C_2^4\times C_4$ x 3, $C_2^3.D_4$ x 2, $Q_8\times C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_2^3:D_4$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3\times C_8$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3.D_4$

Order 60: $C_3\times D_{10}$ x 4, $D_{30}$ x 3, $S_3\times C_{10}$ x 3, $S_3\times D_5$ x 2, $C_{60}$ x 2, $C_5:C_{12}$ x 2, $C_5\times A_4$, $C_{15}:C_4$, $C_3\times F_5$, $C_{15}:C_4$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 48: $C_4\times D_6$ x 82, $C_2^2\times D_6$ x 22, $C_{24}:C_2$ x 18, $C_2^2\times C_{12}$ x 17, $C_6.C_2^3$ x 17, $C_6:C_8$ x 14, $C_2\times C_{24}$ x 14, $C_2\times S_4$ x 11, $C_6:D_4$ x 11, $S_3\times D_4$ x 11, $\SL(2,3):C_2$ x 9, $D_6:C_4$ x 9, $C_{12}:C_4$ x 8, $C_6\times D_4$ x 7, $C_4\times A_4$ x 7, $C_2^2\times A_4$ x 6, $C_2\times \SL(2,3)$ x 6, $C_2\times D_{12}$ x 5, $D_4:C_6$ x 4, $D_{12}:C_2$ x 4, $A_4:C_4$ x 4, $C_2^2:C_{12}$ x 4, $C_4\times C_{12}$ x 4, $C_6.D_4$ x 4, $C_6.D_4$ x 4, $D_4:S_3$ x 3, $C_2^3\times C_6$ x 2, $D_{12}:C_2$ x 2, $S_3\times Q_8$ x 2, $C_3\times \OD_{16}$ x 2, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_3:\OD_{16}$ x 2, $C_6\times Q_8$, $C_6:Q_8$

Order 40: $C_2\times D_{10}$ x 50, $C_4\times D_5$ x 36, $C_2\times F_5$ x 28, $C_2\times C_{20}$ x 22, $C_{10}:C_4$ x 22, $C_2^2\times C_{10}$ x 14, $C_5:D_4$ x 6, $C_5\times D_4$ x 4, $D_{20}$ x 2

Order 36: $C_6\times S_3$ x 7, $C_3:C_{12}$ x 3, $C_6:S_3$ x 3, $C_3\times C_{12}$ x 2, $S_3^2$ x 2, $C_3^2:C_4$, $C_6^2$, $C_3\times A_4$

Order 32: $D_4:C_2^2$ x 87, $C_2^3\times C_4$ x 43, $C_4\times D_4$ x 42, $C_2\times C_4^2$ x 35, $C_2^2\times D_4$ x 33, $C_2^3:C_4$ x 32, $C_4\wr C_2$ x 28, $C_2\times \OD_{16}$ x 27, $C_4:D_4$ x 22, $C_2^2:Q_8$ x 14, $C_2^2\times Q_8$ x 13, $C_2^2.D_4$ x 11, $C_2^2:C_8$ x 10, $C_2.C_4^2$ x 9, $C_2^2.D_4$ x 8, $C_2^2\wr C_2$ x 8, $C_2^2\times C_8$ x 6, $C_4^2:C_2$ x 6, $C_4\times Q_8$ x 4, $C_4^2:C_2$ x 4, $C_4:Q_8$ x 3, $C_4:D_4$ x 3, $C_2^5$ x 2

Order 30: $C_{30}$ x 3, $C_3\times D_5$ x 2, $D_{15}$, $C_5\times S_3$

Order 24: $C_2\times D_6$ x 57, $C_4\times S_3$ x 46, $C_2\times C_{12}$ x 32, $C_6:C_4$ x 30, $C_2^2\times C_6$ x 15, $C_3:D_4$ x 8, $C_2\times A_4$ x 8, $C_3\times D_4$ x 7, $C_{24}$ x 6, $C_3:C_8$ x 6, $D_{12}$ x 4, $\SL(2,3)$ x 3, $S_4$ x 3, $C_3:Q_8$ x 2, $C_3\times Q_8$ x 2

Order 20: $D_{10}$ x 33, $C_2\times C_{10}$ x 19, $C_{20}$ x 7, $C_5:C_4$ x 7, $F_5$ x 4

Order 18: $C_3\times C_6$ x 3, $C_3\times S_3$ x 3, $C_3:S_3$

Order 16: $C_2^2\times C_4$ x 136, $C_2\times D_4$ x 75, $D_4:C_2$ x 67, $C_2^2:C_4$ x 29, $C_4^2$ x 24, $C_2^4$ x 23, $C_2\times Q_8$ x 23, $\OD_{16}$ x 16, $C_4:C_4$ x 14, $C_2\times C_8$ x 12

Order 15: $C_{15}$

Order 12: $D_6$ x 44, $C_2\times C_6$ x 25, $C_{12}$ x 14, $C_3:C_4$ x 11, $A_4$ x 2

Order 10: $C_{10}$ x 8, $D_5$ x 5

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 108, $C_2^3$ x 64, $D_4$ x 35, $Q_8$ x 11, $C_8$ x 4

Order 6: $C_6$ x 16, $S_3$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 52, $C_4$ x 18

Order 3: $C_3$ x 3

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 23040: $(C_2\times S_4):\GL(2,5)$ ($C_1$)

Order 11520: $S_4\times \SL(2,5):C_2^2$ ($C_2$), $C_2\times A_4:\GL(2,5)$ ($C_2$), $C_2\times A_4\times \GL(2,5)$ ($C_2$)

Order 5760: $C_2\times S_4\times \SL(2,5)$ ($C_4$), $C_2\times \SL(2,5):S_4$ ($C_4$), $A_4\times \SL(2,5):C_2^2$ ($C_2^2$)

Order 3840: $C_2^3\times \GL(2,5)$ ($S_3$)

Order 2880: $A_4\times \SL(2,5):C_2$ ($D_4$) x 2, $C_2^4.\GL(2,4)$ ($C_2\times C_4$)

Order 1920: $\SL(2,5):C_2^4$ ($D_6$)

Order 1440: $A_4\times \SL(2,5)$ ($C_2^2:C_4$)

Order 960: $C_2\times \GL(2,5)$ ($S_4$), $C_2^3\times \SL(2,5)$ ($C_4\times S_3$), $\SL(2,5):C_2^3$ ($C_3:D_4$), $\SL(2,5):C_2^3$ ($D_{12}$)

Order 480: $C_2^2\times \SL(2,5)$ ($D_6:C_4$), $\SL(2,5):C_2^2$ ($C_2\times S_4$)

Order 240: $C_2\times \SL(2,5)$ ($C_4\times S_4$), $\SL(2,5):C_2$ ($\GL(2,\mathbb{Z}/4)$), $\SL(2,5):C_2$ ($C_4:S_4$)

Order 192: $C_2^4.D_6$ ($S_5$)

Order 120: $\SL(2,5)$ ($C_2.\GL(2,\mathbb{Z}/4)$)

Order 96: $C_2^2\times S_4$ ($A_5:C_4$), $C_2^3:C_{12}$ ($C_2\times S_5$), $C_2^2.S_4$ ($A_5:C_4$)

Order 48: $C_2\times S_4$ ($\GL(2,5)$) x 2, $C_4\times A_4$ ($C_2^2:S_5$) x 2, $C_2^2\times A_4$ ($C_2^2.S_5$)

Order 32: $C_2^3\times C_4$ ($S_3\times S_5$)

Order 24: $C_2\times A_4$ ($C_2^3.S_5$), $C_2\times A_4$ ($\GL(2,5):C_2$), $C_2\times A_4$ ($C_2\times \GL(2,5)$)

Order 16: $C_2^4$ ($D_6.S_5$), $C_2^2\times C_4$ ($D_6:S_5$), $C_2^2\times C_4$ ($D_6:S_5$)

Order 12: $A_4$ ($C_2^2:\GL(2,5)$)

Order 8: $C_2^3$ ($D_6.A_5:C_4$), $C_2^3$ ($(C_4\times S_3).S_5$), $C_2^3$ ($S_3\times \GL(2,5)$), $C_2\times C_4$ ($S_4\times S_5$)

Order 4: $C_2^2$ ($D_6:\GL(2,5)$), $C_2^2$ ($A_5:C_4\times S_4$), $C_4$ ($(C_2\times S_4):S_5$), $C_4$ ($A_5:\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($(C_2^2\times S_4).S_5$), $C_2$ ($(C_4\times S_4).S_5$), $C_2$ ($S_4\times \GL(2,5)$)

Order 1: $C_1$ ($(C_2\times S_4):\GL(2,5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 23040: $(C_2\times S_4):\GL(2,5)$ ($C_1$)

Order 11520: $S_4\times \SL(2,5):C_2^2$ ($C_2$), $C_2\times A_4:\GL(2,5)$ ($C_2$), $C_2\times A_4\times \GL(2,5)$ ($C_2$)

Order 5760: $C_2\times S_4\times \SL(2,5)$ ($C_4$), $C_2\times \SL(2,5):S_4$ ($C_4$), $A_4\times \SL(2,5):C_2^2$ ($C_2^2$)

Order 3840: $C_2^3\times \GL(2,5)$ ($S_3$)

Order 2880: $A_4\times \SL(2,5):C_2$ ($D_4$) x 2, $C_2^4.\GL(2,4)$ ($C_2\times C_4$)

Order 1920: $\SL(2,5):C_2^4$ ($D_6$)

Order 1440: $A_4\times \SL(2,5)$ ($C_2^2:C_4$)

Order 960: $C_2\times \GL(2,5)$ ($S_4$), $C_2^3\times \SL(2,5)$ ($C_4\times S_3$), $\SL(2,5):C_2^3$ ($C_3:D_4$), $\SL(2,5):C_2^3$ ($D_{12}$)

Order 480: $C_2^2\times \SL(2,5)$ ($D_6:C_4$), $\SL(2,5):C_2^2$ ($C_2\times S_4$)

Order 240: $C_2\times \SL(2,5)$ ($C_4\times S_4$), $\SL(2,5):C_2$ ($\GL(2,\mathbb{Z}/4)$), $\SL(2,5):C_2$ ($C_4:S_4$)

Order 192: $C_2^4.D_6$ ($S_5$)

Order 120: $\SL(2,5)$ ($C_2.\GL(2,\mathbb{Z}/4)$)

Order 96: $C_2^2\times S_4$ ($A_5:C_4$), $C_2^3:C_{12}$ ($C_2\times S_5$), $C_2^2.S_4$ ($A_5:C_4$)

Order 48: $C_4\times A_4$ ($C_2^2:S_5$) x 2, $C_2^2\times A_4$ ($C_2^2.S_5$), $C_2\times S_4$ ($\GL(2,5)$)

Order 32: $C_2^3\times C_4$ ($S_3\times S_5$)

Order 24: $C_2\times A_4$ ($C_2^3.S_5$), $C_2\times A_4$ ($\GL(2,5):C_2$), $C_2\times A_4$ ($C_2\times \GL(2,5)$)

Order 16: $C_2^4$ ($D_6.S_5$), $C_2^2\times C_4$ ($D_6:S_5$), $C_2^2\times C_4$ ($D_6:S_5$)

Order 12: $A_4$ ($C_2^2:\GL(2,5)$)

Order 8: $C_2^3$ ($D_6.A_5:C_4$), $C_2^3$ ($(C_4\times S_3).S_5$), $C_2^3$ ($S_3\times \GL(2,5)$), $C_2\times C_4$ ($S_4\times S_5$)

Order 4: $C_2^2$ ($D_6:\GL(2,5)$), $C_2^2$ ($A_5:C_4\times S_4$), $C_4$ ($(C_2\times S_4):S_5$), $C_4$ ($A_5:\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($(C_2^2\times S_4).S_5$), $C_2$ ($(C_4\times S_4).S_5$), $C_2$ ($S_4\times \GL(2,5)$)

Order 1: $C_1$ ($(C_2\times S_4):\GL(2,5)$)

Series

Derived series

$(C_2\times S_4):\GL(2,5)$

$\rhd$

$C_2^4.\GL(2,4)$

$\rhd$

$C_2^2\times \SL(2,5)$

$\rhd$

$\SL(2,5)$

Chief series

$(C_2\times S_4):\GL(2,5)$

$\rhd$

$S_4\times \SL(2,5):C_2^2$

$\rhd$

$C_2^4.D_6$

$\rhd$

$C_2^2\times S_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(C_2\times S_4):\GL(2,5)$

$\rhd$

$C_2^4.\GL(2,4)$

$\rhd$

$A_4\times \SL(2,5)$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

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 $180 \times 180$ 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 $108 \times 108$ rational character table.