Abstract group 2304.ou: $S_3\times \SD_{16}\times S_4$

• Label: 2304.ou
• Order: \( 2^{8} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $4$

Group information

Description:	$S_3\times \SD_{16}\times S_4$
Order:	\(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$(C_6\times A_4).C_2^6.C_2$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Composition factors:	$C_2$ x 8, $C_3$ x 2
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	239	26	528	382	256	552	320	2304
Conjugacy classes	1	17	3	22	15	16	17	14	105
Divisions	1	17	3	22	15	8	17	7	90
Autjugacy classes	1	13	3	20	13	8	16	7	81

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	16	28	16	16	20	3	6	0	0	105
Irr. rational chars.	16	20	16	12	12	5	6	1	2	90

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$4$
Inequivalent generating quadruples:	$858574080$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=c^{2}=d^{6}=e^{2}=f^{8}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(5,7), (4,7)(5,6), (1,3)(4,5,6,7)(8,9,12,13,15,10,14,11), (1,2), (4,5,7) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $\SD_{16}$ $\, \times\, $ $S_4$
Semidirect product:	$(Q_8\times S_4)$ $\,\rtimes\,$ $D_6$	$(Q_8:S_4)$ $\,\rtimes\,$ $D_6$	$Q_8$ $\,\rtimes\,$ $(D_6\times S_4)$	$(C_8\times S_4)$ $\,\rtimes\,$ $D_6$	all 95
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_6\times S_4)$ . $D_4$	$(D_6.S_4)$ . $D_4$	$(D_4\times S_4)$ . $D_6$	$D_6$ . $(D_4\times S_4)$	all 57

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 38232 subgroups in 3362 conjugacy classes, 169 normal (153 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\GL(2,\mathbb{Z}/4):D_6$
Commutator:	$G' \simeq$ $C_{12}\times A_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^2:D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{24}:C_2^3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $S_3\times \SD_{16}\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $D_4\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 2304: $S_3\times \SD_{16}\times S_4$

Order 1152: $C_2^3.D_6^2$, $\GL(2,\mathbb{Z}/4):D_6$, $S_3\times A_4\times \SD_{16}$, $C_{24}:(C_2\times S_4)$, $C_3\times \SD_{16}\times S_4$, $Q_8:(S_3\times S_4)$, $Q_8:S_3\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_3:(\SD_{16}\times S_4)$, $C_3:\SD_{16}\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:(C_2\times S_4)$, $C_{24}:C_2\times S_4$, $S_3\times C_8:S_4$, $S_3\times C_8\times S_4$

Order 768: $C_2\times \SD_{16}\times S_4$, $D_6.D_4^2$

Order 576: $C_4\times S_3\times S_4$ x 2, $S_3\times A_4:Q_8$ x 2, $C_3:(Q_8\times S_4)$ x 2, $C_3:Q_8\times S_4$ x 2, $C_2^2:D_6^2$, $S_3\times Q_8\times A_4$, $S_3\times D_4\times A_4$, $Q_8\times C_3:S_4$, $D_4\times C_3:S_4$, $C_3\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_6$, $\GL(2,\mathbb{Z}/4):S_3$, $C_2^4:S_3^2$, $S_3\times \GL(2,\mathbb{Z}/4)$, $D_{12}:S_4$, $S_4\times D_{12}$, $S_3\times C_4:S_4$, $C_3\times A_4\times \SD_{16}$, $\SD_{16}\times S_3^2$, $A_4\times Q_8:S_3$, $A_4\times C_3:\SD_{16}$, $A_4\times C_{24}:C_2$, $S_3\times C_8\times A_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}:S_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}\times S_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_8$, $C_{12}.(C_2\times S_4)$, $C_{12}.(C_2\times S_4)$, $C_3:(C_8\times S_4)$, $C_3:C_8\times S_4$

Order 384: $\SD_{16}\times S_4$ x 9, $C_{24}:C_2^4$ x 2, $C_6.D_4^2$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_8:D_4$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_2^2:\SD_{16}$ x 2, $C_6.D_4^2$ x 2, $S_3\times Q_8:D_4$ x 2, $C_2\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $S_3\times D_4\times Q_8$, $S_3\times D_4^2$, $C_2\times A_4\times \SD_{16}$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{12}$, $C_2\times C_8\times S_4$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $D_{12}:\SD_{16}$, $S_3\times C_8:D_4$, $D_{12}:\SD_{16}$, $S_3\times C_4:\SD_{16}$, $D_{12}:\SD_{16}$, $S_3\times Q_8:D_4$, $\SD_{16}\times D_{12}$, $C_4\times S_3\times \SD_{16}$, $C_6.D_4^2$, $S_3\times C_8\times D_4$

Order 288: $D_6\times S_4$ x 8, $C_2^3.S_3^2$ x 3, $C_4\times S_3\times A_4$ x 2, $A_4\times C_3:Q_8$ x 2, $C_{12}:S_4$ x 2, $C_{12}.S_4$ x 2, $C_{12}\times S_4$ x 2, $C_{12}.S_4$ x 2, $D_6.S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_{24}:D_6$ x 2, $Q_8:S_3^2$ x 2, $D_{12}.D_6$ x 2, $C_{24}:D_6$ x 2, $C_2^3.C_6^2$, $C_2^3.C_6^2$, $Q_8\times S_3^2$, $D_4\times S_3^2$, $A_4\times C_3:D_4$, $A_4\times D_{12}$, $C_3:\GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $D_6:S_4$, $D_6:S_4$, $C_2^3.S_3^2$, $C_{24}:D_6$, $A_4\times C_{24}$, $Q_8:S_3^2$, $D_4.S_3^2$, $C_{24}:D_6$, $C_8\times S_3^2$, $A_4\times C_3:C_8$, $C_{12}.S_4$, $A_4:C_{24}$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 256: $C_4.D_4^2$

Order 192: $D_6\times \SD_{16}$ x 26, $Q_8\times S_4$ x 7, $D_4\times S_4$ x 7, $Q_8:S_4$ x 5, $A_4:\SD_{16}$ x 5, $C_8:S_4$ x 5, $C_8\times S_4$ x 5, $A_4\times \SD_{16}$ x 5, $C_{24}:D_4$ x 4, $D_6:\SD_{16}$ x 4, $D_6:\SD_{16}$ x 4, $Q_8:D_{12}$ x 4, $D_6:\SD_{16}$ x 4, $C_3:(D_4\times Q_8)$ x 4, $S_3\times C_2^2:Q_8$ x 4, $D_6.Q_{16}$ x 3, $D_6.D_8$ x 3, $C_{12}:C_2^4$ x 3, $C_2^4:D_6$ x 3, $D_{12}:D_4$ x 3, $S_3\times C_4:D_4$ x 3, $C_4^2:D_6$ x 3, $C_{24}:D_4$ x 2, $C_3\times C_2^2:\SD_{16}$ x 2, $C_3\times Q_8:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_{24}:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_{24}:D_4$ x 2, $C_{24}:D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $D_{12}.D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_8:D_{12}$ x 2, $D_{12}.D_4$ x 2, $D_6:\SD_{16}$ x 2, $D_6:\SD_{16}$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_2^3.D_{12}$ x 2, $D_{12}.D_4$ x 2, $C_3:(C_8\times D_4)$ x 2, $D_6.\OD_{16}$ x 2, $C_{12}.C_2^4$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_{24}:C_2^3$ x 2, $C_3:D_4\times Q_8$ x 2, $C_{12}.C_2^4$ x 2, $C_{12}.C_2^4$ x 2, $C_{24}:C_2^3$ x 2, $C_{24}:C_2^3$ x 2, $D_{12}:Q_8$ x 2, $C_4^2.D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $D_4\times D_{12}$ x 2, $C_4^2.D_6$ x 2, $C_2\times A_4:C_8$, $C_{24}:D_4$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}\times \SD_{16}$, $D_4\times C_{24}$, $C_3:C_4\times \SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $Q_8:D_{12}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_4^2.D_6$, $D_6.\SD_{16}$, $C_8:(C_4\times S_3)$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $D_{12}:C_8$, $C_4^2.D_6$, $Q_8:(C_4\times S_3)$, $C_3:(C_4\times \SD_{16})$, $C_8:D_{12}$, $C_{12}:\SD_{16}$, $C_8\times D_{12}$, $D_6.C_4^2$, $C_2^3\times S_4$, $C_2\times Q_8\times A_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $C_3\times D_4\times Q_8$, $C_3\times D_4^2$, $C_3:D_4^2$, $C_4^2:D_6$, $Q_8\times D_{12}$, $C_4^2.D_6$, $C_2^3:C_{24}$

Order 144: $S_3\times S_4$ x 10, $A_4\times D_6$ x 4, $C_6:S_4$ x 4, $C_6\times S_4$ x 4, $C_{12}.D_6$ x 4, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$ x 2, $C_{12}\times A_4$ x 2, $D_6:D_6$ x 2, $S_3\times D_{12}$ x 2, $C_4\times S_3^2$ x 2, $C_{12}.D_6$ x 2, $A_4\times C_3:C_4$ x 2, $C_6.S_4$ x 2, $A_4:C_{12}$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_{12}.D_6$, $C_2^2:C_6^2$, $D_6^2$, $C_{12}.D_6$, $C_{12}:D_6$, $D_6:D_6$, $C_{12}:D_6$, $C_3^2\times \SD_{16}$

Order 128: $D_4\times \SD_{16}$ x 16, $C_2^3\times \SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$

Order 96: $S_3\times \SD_{16}$ x 56, $D_4\times D_6$ x 35, $Q_8\times D_6$ x 18, $C_6\times \SD_{16}$ x 14, $Q_8:D_6$ x 14, $C_6:\SD_{16}$ x 14, $C_8:D_6$ x 14, $C_8\times D_6$ x 13, $C_2^2\times S_4$ x 11, $C_4\times S_4$ x 10, $A_4:Q_8$ x 9, $D_6.D_4$ x 8, $D_6.D_4$ x 7, $C_{12}:Q_8$ x 6, $C_2^2:D_{12}$ x 6, $C_2^3.D_6$ x 6, $D_6:D_4$ x 6, $D_6:Q_8$ x 6, $C_2^3.D_6$ x 5, $C_{12}:C_2^3$ x 5, $\GL(2,\mathbb{Z}/4)$ x 5, $C_{12}:D_4$ x 5, $D_6:D_4$ x 4, $D_6:C_8$ x 4, $C_2^3:D_6$ x 4, $C_6.C_2^4$ x 4, $Q_8\times A_4$ x 4, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $C_{12}.D_4$ x 4, $D_6:Q_8$ x 4, $C_{12}.D_4$ x 4, $C_{12}:D_4$ x 4, $C_2^3.D_6$ x 4, $C_{12}.D_4$ x 4, $D_6:Q_8$ x 4, $C_4\times D_{12}$ x 3, $C_8\times A_4$ x 3, $A_4:C_8$ x 3, $Q_8:C_{12}$ x 3, $D_4:C_{12}$ x 3, $C_6.Q_{16}$ x 3, $C_6.D_8$ x 3, $D_{12}:C_4$ x 3, $C_6.Q_{16}$ x 3, $C_{12}:C_2^3$ x 3, $C_2^2\times D_{12}$ x 3, $C_4.D_{12}$ x 3, $C_{12}:D_4$ x 3, $D_4\times C_{12}$ x 3, $D_{12}:C_4$ x 3, $C_{12}:D_4$ x 3, $C_2^3.D_6$ x 3, $C_{12}:D_4$ x 3, $C_4:D_{12}$ x 3, $D_{12}:C_4$ x 3, $C_3:(C_4\times Q_8)$ x 2, $S_3\times C_4^2$ x 2, $C_{12}:Q_8$ x 2, $C_{12}:Q_8$ x 2, $C_2^2:C_{24}$ x 2, $C_{12}.D_4$ x 2, $C_2^3\times D_6$ x 2, $C_6.C_2^4$ x 2, $C_{12}.D_4$ x 2, $C_{24}:C_4$ x 2, $C_2^3:C_{12}$ x 2, $C_2^2.S_4$ x 2, $C_2^2\times C_{24}$ x 2, $C_{12}:Q_8$ x 2, $C_2^4:C_6$ x 2, $C_2^4:S_3$ x 2, $C_{12}.Q_8$ x 2, $C_{12}.C_2^3$ x 2, $C_{12}:C_8$, $C_4:D_{12}$, $C_8:C_{12}$, $C_4:C_{24}$, $C_4\times C_{24}$, $C_{24}:C_4$, $C_2^3\times A_4$, $C_{12}:D_4$, $Q_8\times C_{12}$, $C_3:C_4\times Q_8$, $C_{12}:C_8$

Order 72: $S_3\times D_6$ x 8, $S_3\times C_{12}$ x 4, $C_3^2:Q_8$ x 4, $C_6.D_6$ x 4, $S_3\times A_4$ x 3, $C_3:S_4$ x 3, $C_3\times S_4$ x 3, $C_3^2:Q_8$ x 3, $C_6\times D_6$ x 2, $C_6\times A_4$ x 2, $C_{12}:S_3$ x 2, $C_3^2:Q_8$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $C_3:D_{12}$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $C_6:D_6$, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6^2:C_2$, $C_3:D_{12}$, $D_6:S_3$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_2^2\times \SD_{16}$ x 33, $C_8:D_4$ x 16, $C_2^2:\SD_{16}$ x 16, $Q_8:D_4$ x 16, $D_4\times Q_8$ x 12, $D_4^2$ x 12, $C_8:D_4$ x 8, $C_4:\SD_{16}$ x 8, $Q_8:D_4$ x 8, $C_4\times \SD_{16}$ x 8, $C_8\times D_4$ x 8, $C_2^3:Q_8$ x 4, $C_2^2.Q_{16}$ x 3, $C_2^2.D_8$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3:C_8$ x 2, $Q_8\times C_2^3$ x 2, $C_2^3\times C_8$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3:D_4$ x 2, $C_2\times C_4\times C_8$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_2^3.D_4$, $C_4^2.C_4$

Order 48: $S_3\times D_4$ x 74, $S_3\times Q_8$ x 36, $C_4\times D_6$ x 34, $C_6:D_4$ x 30, $C_2^2\times D_6$ x 26, $C_2\times S_4$ x 22, $C_6:Q_8$ x 21, $C_{24}:C_2$ x 18, $C_6\times D_4$ x 18, $S_3\times C_8$ x 18, $C_2\times D_{12}$ x 18, $C_3\times \SD_{16}$ x 18, $Q_8:S_3$ x 18, $C_3:\SD_{16}$ x 18, $D_6:C_4$ x 14, $C_6.D_4$ x 14, $C_6\times Q_8$ x 9, $C_4:C_{12}$ x 8, $C_{12}:C_4$ x 8, $C_6:C_8$ x 7, $C_2\times C_{24}$ x 7, $C_2^2:C_{12}$ x 7, $C_6.D_4$ x 7, $C_2^2\times A_4$ x 6, $C_4\times A_4$ x 6, $A_4:C_4$ x 6, $C_{12}:C_4$ x 6, $C_2^2\times C_{12}$ x 5, $C_6.C_2^3$ x 5, $C_2^3\times C_6$ x 2, $C_4\times C_{12}$ x 2

Order 36: $S_3^2$ x 10, $C_6\times S_3$ x 8, $C_3:C_{12}$ x 4, $C_6:S_3$ x 4, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_6^2$, $C_3\times A_4$

Order 32: $C_2\times \SD_{16}$ x 99, $C_2^2\times D_4$ x 44, $C_2^2:Q_8$ x 28, $C_2^2\times Q_8$ x 23, $C_4\times D_4$ x 22, $C_4:D_4$ x 20, $C_2^2\times C_8$ x 16, $C_2^2\wr C_2$ x 16, $C_4:Q_8$ x 14, $D_4:C_4$ x 12, $Q_8:C_4$ x 12, $C_2^2:C_8$ x 8, $C_2^2.D_4$ x 8, $C_2^3:C_4$ x 7, $C_4:D_4$ x 6, $C_4\times Q_8$ x 6, $C_2^3\times C_4$ x 5, $C_4\times C_8$ x 4, $C_8:C_4$ x 4, $C_4:C_8$ x 4, $C_2^5$ x 2, $C_2\times C_4^2$ x 2

Order 24: $C_2\times D_6$ x 87, $C_4\times S_3$ x 46, $C_3:D_4$ x 38, $C_3:Q_8$ x 28, $D_{12}$ x 23, $C_3\times D_4$ x 23, $C_2\times C_{12}$ x 18, $C_6:C_4$ x 18, $C_2^2\times C_6$ x 13, $C_3\times Q_8$ x 12, $S_4$ x 9, $C_{24}$ x 7, $C_2\times A_4$ x 7, $C_3:C_8$ x 7

Order 18: $C_3\times S_3$ x 6, $C_3:S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 130, $C_2\times Q_8$ x 56, $\SD_{16}$ x 52, $C_2^2\times C_4$ x 42, $C_4:C_4$ x 30, $C_2^2:C_4$ x 28, $C_2^4$ x 27, $C_2\times C_8$ x 25, $C_4^2$ x 8

Order 12: $D_6$ x 78, $C_2\times C_6$ x 26, $C_{12}$ x 17, $C_3:C_4$ x 17, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 88, $D_4$ x 71, $C_2\times C_4$ x 68, $Q_8$ x 27, $C_8$ x 8

Order 6: $S_3$ x 18, $C_6$ x 15

Order 4: $C_2^2$ x 76, $C_4$ x 22

Order 3: $C_3$ x 3

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2304: $S_3\times \SD_{16}\times S_4$

Order 1152: $C_2^3.D_6^2$, $\GL(2,\mathbb{Z}/4):D_6$, $S_3\times A_4\times \SD_{16}$, $C_{24}:(C_2\times S_4)$, $C_3\times \SD_{16}\times S_4$, $Q_8:(S_3\times S_4)$, $Q_8:S_3\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_3:(\SD_{16}\times S_4)$, $C_3:\SD_{16}\times S_4$, $D_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:(C_2\times S_4)$, $C_{24}:C_2\times S_4$, $S_3\times C_8:S_4$, $S_3\times C_8\times S_4$

Order 768: $C_2\times \SD_{16}\times S_4$, $D_6.D_4^2$

Order 576: $C_4\times S_3\times S_4$ x 2, $S_3\times A_4:Q_8$ x 2, $C_3:(Q_8\times S_4)$ x 2, $C_3:Q_8\times S_4$ x 2, $C_2^2:D_6^2$, $S_3\times Q_8\times A_4$, $S_3\times D_4\times A_4$, $Q_8\times C_3:S_4$, $D_4\times C_3:S_4$, $C_3\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_6$, $\GL(2,\mathbb{Z}/4):S_3$, $C_2^4:S_3^2$, $S_3\times \GL(2,\mathbb{Z}/4)$, $D_{12}:S_4$, $S_4\times D_{12}$, $S_3\times C_4:S_4$, $C_3\times A_4\times \SD_{16}$, $\SD_{16}\times S_3^2$, $A_4\times Q_8:S_3$, $A_4\times C_3:\SD_{16}$, $A_4\times C_{24}:C_2$, $S_3\times C_8\times A_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}:S_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{24}:S_4$, $C_{24}\times S_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $S_3\times A_4:C_8$, $C_{12}.(C_2\times S_4)$, $C_{12}.(C_2\times S_4)$, $C_3:(C_8\times S_4)$, $C_3:C_8\times S_4$

Order 384: $\SD_{16}\times S_4$ x 9, $C_{24}:C_2^4$ x 2, $C_6.D_4^2$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_8:D_4$ x 2, $C_6.D_4^2$ x 2, $S_3\times C_2^2:\SD_{16}$ x 2, $C_6.D_4^2$ x 2, $S_3\times Q_8:D_4$ x 2, $C_2\times Q_8\times S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $S_3\times D_4\times Q_8$, $S_3\times D_4^2$, $C_2\times A_4\times \SD_{16}$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{12}$, $C_2\times C_8\times S_4$, $C_6.D_4^2$, $C_6.D_4^2$, $C_6.D_4^2$, $D_{12}:\SD_{16}$, $S_3\times C_8:D_4$, $D_{12}:\SD_{16}$, $S_3\times C_4:\SD_{16}$, $D_{12}:\SD_{16}$, $S_3\times Q_8:D_4$, $\SD_{16}\times D_{12}$, $C_4\times S_3\times \SD_{16}$, $C_6.D_4^2$, $S_3\times C_8\times D_4$

Order 288: $D_6\times S_4$ x 5, $C_2^3.S_3^2$ x 3, $C_4\times S_3\times A_4$ x 2, $A_4\times C_3:Q_8$ x 2, $C_{12}:S_4$ x 2, $C_{12}.S_4$ x 2, $C_{12}\times S_4$ x 2, $C_{12}.S_4$ x 2, $D_6.S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_{24}:D_6$ x 2, $Q_8:S_3^2$ x 2, $D_{12}.D_6$ x 2, $C_{24}:D_6$ x 2, $C_2^3.C_6^2$, $C_2^3.C_6^2$, $Q_8\times S_3^2$, $D_4\times S_3^2$, $A_4\times C_3:D_4$, $A_4\times D_{12}$, $C_3:\GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $D_6:S_4$, $D_6:S_4$, $C_2^3.S_3^2$, $C_{24}:D_6$, $A_4\times C_{24}$, $Q_8:S_3^2$, $D_4.S_3^2$, $C_{24}:D_6$, $C_8\times S_3^2$, $A_4\times C_3:C_8$, $C_{12}.S_4$, $A_4:C_{24}$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 256: $C_4.D_4^2$

Order 192: $D_6\times \SD_{16}$ x 26, $Q_8\times S_4$ x 7, $D_4\times S_4$ x 7, $Q_8:S_4$ x 5, $A_4:\SD_{16}$ x 5, $C_8:S_4$ x 5, $C_8\times S_4$ x 5, $A_4\times \SD_{16}$ x 5, $C_{24}:D_4$ x 4, $D_6:\SD_{16}$ x 4, $D_6:\SD_{16}$ x 4, $Q_8:D_{12}$ x 4, $D_6:\SD_{16}$ x 4, $C_3:(D_4\times Q_8)$ x 4, $S_3\times C_2^2:Q_8$ x 4, $D_6.Q_{16}$ x 3, $D_6.D_8$ x 3, $C_{12}:C_2^4$ x 3, $C_2^4:D_6$ x 3, $D_{12}:D_4$ x 3, $S_3\times C_4:D_4$ x 3, $C_4^2:D_6$ x 3, $C_{24}:D_4$ x 2, $C_3\times C_2^2:\SD_{16}$ x 2, $C_3\times Q_8:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_{24}:D_4$ x 2, $(C_3\times Q_8):D_4$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_{24}:D_4$ x 2, $C_{24}:D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $D_{12}.D_4$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $(C_2\times C_6):\SD_{16}$ x 2, $C_8:D_{12}$ x 2, $D_{12}.D_4$ x 2, $D_6:\SD_{16}$ x 2, $D_6:\SD_{16}$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_2^3.D_{12}$ x 2, $D_{12}.D_4$ x 2, $C_3:(C_8\times D_4)$ x 2, $D_6.\OD_{16}$ x 2, $C_{12}.C_2^4$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_{24}:C_2^3$ x 2, $C_3:D_4\times Q_8$ x 2, $C_{12}.C_2^4$ x 2, $C_{12}.C_2^4$ x 2, $C_{24}:C_2^3$ x 2, $C_{24}:C_2^3$ x 2, $D_{12}:Q_8$ x 2, $C_4^2.D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $D_4\times D_{12}$ x 2, $C_4^2.D_6$ x 2, $C_2\times A_4:C_8$, $C_{24}:D_4$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}\times \SD_{16}$, $D_4\times C_{24}$, $C_3:C_4\times \SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $Q_8:D_{12}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $C_4^2.D_6$, $D_6.\SD_{16}$, $C_8:(C_4\times S_3)$, $C_{12}:\SD_{16}$, $C_{12}:\SD_{16}$, $D_{12}:C_8$, $C_4^2.D_6$, $Q_8:(C_4\times S_3)$, $C_3:(C_4\times \SD_{16})$, $C_8:D_{12}$, $C_{12}:\SD_{16}$, $C_8\times D_{12}$, $D_6.C_4^2$, $C_2^3\times S_4$, $C_2\times Q_8\times A_4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $C_3\times D_4\times Q_8$, $C_3\times D_4^2$, $C_3:D_4^2$, $C_4^2:D_6$, $Q_8\times D_{12}$, $C_4^2.D_6$, $C_2^3:C_{24}$

Order 144: $S_3\times S_4$ x 4, $C_{12}.D_6$ x 4, $A_4\times D_6$ x 3, $C_6:S_4$ x 3, $C_6\times S_4$ x 3, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$ x 2, $C_{12}\times A_4$ x 2, $D_6:D_6$ x 2, $S_3\times D_{12}$ x 2, $C_4\times S_3^2$ x 2, $C_{12}.D_6$ x 2, $A_4\times C_3:C_4$ x 2, $C_6.S_4$ x 2, $A_4:C_{12}$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:S_3$, $C_{24}:S_3$, $C_{12}.D_6$, $C_2^2:C_6^2$, $D_6^2$, $C_{12}.D_6$, $C_{12}:D_6$, $D_6:D_6$, $C_{12}:D_6$, $C_3^2\times \SD_{16}$

Order 128: $D_4\times \SD_{16}$ x 16, $C_2^3\times \SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$

Order 96: $S_3\times \SD_{16}$ x 56, $D_4\times D_6$ x 29, $Q_8\times D_6$ x 18, $C_6\times \SD_{16}$ x 14, $Q_8:D_6$ x 14, $C_6:\SD_{16}$ x 14, $C_8:D_6$ x 14, $C_8\times D_6$ x 13, $C_4\times S_4$ x 10, $A_4:Q_8$ x 9, $D_6.D_4$ x 8, $C_2^2\times S_4$ x 8, $D_6.D_4$ x 7, $C_{12}:Q_8$ x 6, $C_2^2:D_{12}$ x 6, $C_2^3.D_6$ x 6, $D_6:D_4$ x 6, $D_6:Q_8$ x 6, $C_2^3.D_6$ x 5, $C_{12}:C_2^3$ x 5, $\GL(2,\mathbb{Z}/4)$ x 5, $C_{12}:D_4$ x 5, $D_6:D_4$ x 4, $D_6:C_8$ x 4, $C_2^3:D_6$ x 4, $C_6.C_2^4$ x 4, $Q_8\times A_4$ x 4, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $C_{12}.D_4$ x 4, $D_6:Q_8$ x 4, $C_{12}.D_4$ x 4, $C_{12}:D_4$ x 4, $C_2^3.D_6$ x 4, $C_{12}.D_4$ x 4, $D_6:Q_8$ x 4, $C_4\times D_{12}$ x 3, $C_8\times A_4$ x 3, $A_4:C_8$ x 3, $Q_8:C_{12}$ x 3, $D_4:C_{12}$ x 3, $C_6.Q_{16}$ x 3, $C_6.D_8$ x 3, $D_{12}:C_4$ x 3, $C_6.Q_{16}$ x 3, $C_{12}:C_2^3$ x 3, $C_2^2\times D_{12}$ x 3, $C_4.D_{12}$ x 3, $C_{12}:D_4$ x 3, $D_4\times C_{12}$ x 3, $D_{12}:C_4$ x 3, $C_{12}:D_4$ x 3, $C_2^3.D_6$ x 3, $C_{12}:D_4$ x 3, $C_4:D_{12}$ x 3, $D_{12}:C_4$ x 3, $C_3:(C_4\times Q_8)$ x 2, $S_3\times C_4^2$ x 2, $C_{12}:Q_8$ x 2, $C_{12}:Q_8$ x 2, $C_2^2:C_{24}$ x 2, $C_{12}.D_4$ x 2, $C_2^3\times D_6$ x 2, $C_6.C_2^4$ x 2, $C_{12}.D_4$ x 2, $C_{24}:C_4$ x 2, $C_2^3:C_{12}$ x 2, $C_2^2.S_4$ x 2, $C_2^2\times C_{24}$ x 2, $C_{12}:Q_8$ x 2, $C_2^4:C_6$ x 2, $C_2^4:S_3$ x 2, $C_{12}.Q_8$ x 2, $C_{12}.C_2^3$ x 2, $C_{12}:C_8$, $C_4:D_{12}$, $C_8:C_{12}$, $C_4:C_{24}$, $C_4\times C_{24}$, $C_{24}:C_4$, $C_2^3\times A_4$, $C_{12}:D_4$, $Q_8\times C_{12}$, $C_3:C_4\times Q_8$, $C_{12}:C_8$

Order 72: $S_3\times D_6$ x 5, $S_3\times C_{12}$ x 4, $C_3^2:Q_8$ x 4, $C_6.D_6$ x 4, $C_3^2:Q_8$ x 3, $C_6\times D_6$ x 2, $C_6\times A_4$ 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, $C_3^2:Q_8$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $C_3:D_{12}$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $C_6:D_6$, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6^2:C_2$, $C_3:D_{12}$, $D_6:S_3$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_2^2\times \SD_{16}$ x 33, $C_8:D_4$ x 16, $C_2^2:\SD_{16}$ x 16, $Q_8:D_4$ x 16, $D_4\times Q_8$ x 12, $D_4^2$ x 12, $C_8:D_4$ x 8, $C_4:\SD_{16}$ x 8, $Q_8:D_4$ x 8, $C_4\times \SD_{16}$ x 8, $C_8\times D_4$ x 8, $C_2^3:Q_8$ x 4, $C_2^2.Q_{16}$ x 3, $C_2^2.D_8$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3:C_8$ x 2, $Q_8\times C_2^3$ x 2, $C_2^3\times C_8$ x 2, $C_4^2.C_2^2$ x 2, $C_2^3:D_4$ x 2, $C_2\times C_4\times C_8$, $C_4^2:C_2^2$, $C_4^2.C_2^2$, $C_2^3.D_4$, $C_4^2.C_4$

Order 48: $S_3\times D_4$ x 49, $S_3\times Q_8$ x 36, $C_4\times D_6$ x 31, $C_6:D_4$ x 25, $C_6:Q_8$ x 21, $C_2^2\times D_6$ x 20, $C_{24}:C_2$ x 18, $S_3\times C_8$ x 18, $C_3\times \SD_{16}$ x 18, $Q_8:S_3$ x 18, $C_3:\SD_{16}$ x 18, $C_6\times D_4$ x 16, $C_2\times D_{12}$ x 15, $D_6:C_4$ x 14, $C_6.D_4$ x 14, $C_2\times S_4$ x 13, $C_6\times Q_8$ x 9, $C_4:C_{12}$ x 8, $C_{12}:C_4$ x 8, $C_6:C_8$ x 7, $C_2\times C_{24}$ x 7, $C_2^2:C_{12}$ x 7, $C_6.D_4$ x 7, $C_4\times A_4$ x 6, $A_4:C_4$ x 6, $C_{12}:C_4$ x 6, $C_2^2\times A_4$ x 5, $C_2^2\times C_{12}$ x 5, $C_6.C_2^3$ x 5, $C_2^3\times C_6$ x 2, $C_4\times C_{12}$ x 2

Order 36: $C_6\times S_3$ x 6, $C_3:C_{12}$ x 4, $S_3^2$ x 4, $C_6:S_3$ x 3, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_6^2$, $C_3\times A_4$

Order 32: $C_2\times \SD_{16}$ x 99, $C_2^2\times D_4$ x 38, $C_2^2:Q_8$ x 28, $C_2^2\times Q_8$ x 23, $C_4\times D_4$ x 22, $C_4:D_4$ x 20, $C_2^2\times C_8$ x 16, $C_2^2\wr C_2$ x 16, $C_4:Q_8$ x 14, $D_4:C_4$ x 12, $Q_8:C_4$ x 12, $C_2^2:C_8$ x 8, $C_2^2.D_4$ x 8, $C_2^3:C_4$ x 7, $C_4:D_4$ x 6, $C_4\times Q_8$ x 6, $C_2^3\times C_4$ x 5, $C_4\times C_8$ x 4, $C_8:C_4$ x 4, $C_4:C_8$ x 4, $C_2^5$ x 2, $C_2\times C_4^2$ x 2

Order 24: $C_2\times D_6$ x 55, $C_4\times S_3$ x 38, $C_3:D_4$ x 28, $C_3:Q_8$ x 28, $C_3\times D_4$ x 19, $C_2\times C_{12}$ x 17, $C_6:C_4$ x 17, $D_{12}$ x 17, $C_2^2\times C_6$ x 12, $C_3\times Q_8$ x 12, $C_{24}$ x 7, $C_3:C_8$ x 7, $C_2\times A_4$ x 6, $S_4$ x 6

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

Order 16: $C_2\times D_4$ x 95, $C_2\times Q_8$ x 56, $\SD_{16}$ x 52, $C_2^2\times C_4$ x 39, $C_4:C_4$ x 30, $C_2^2:C_4$ x 28, $C_2\times C_8$ x 25, $C_2^4$ x 21, $C_4^2$ x 8

Order 12: $D_6$ x 46, $C_2\times C_6$ x 22, $C_{12}$ x 16, $C_3:C_4$ x 16, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 58, $C_2\times C_4$ x 58, $D_4$ x 51, $Q_8$ x 27, $C_8$ x 8

Order 6: $C_6$ x 13, $S_3$ x 12

Order 4: $C_2^2$ x 50, $C_4$ x 20

Order 3: $C_3$ x 3

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2304: $S_3\times \SD_{16}\times S_4$ ($C_1$)

Order 1152: $C_2^3.D_6^2$ ($C_2$), $\GL(2,\mathbb{Z}/4):D_6$ ($C_2$), $S_3\times A_4\times \SD_{16}$ ($C_2$), $C_{24}:(C_2\times S_4)$ ($C_2$), $C_3\times \SD_{16}\times S_4$ ($C_2$), $Q_8:(S_3\times S_4)$ ($C_2$), $Q_8:S_3\times S_4$ ($C_2$), $D_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_3:(\SD_{16}\times S_4)$ ($C_2$), $C_3:\SD_{16}\times S_4$ ($C_2$), $D_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_{24}:(C_2\times S_4)$ ($C_2$), $C_{24}:C_2\times S_4$ ($C_2$), $S_3\times C_8:S_4$ ($C_2$), $S_3\times C_8\times S_4$ ($C_2$)

Order 576: $S_3\times Q_8\times A_4$ ($C_2^2$), $S_3\times D_4\times A_4$ ($C_2^2$), $Q_8\times C_3:S_4$ ($C_2^2$), $D_4\times C_3:S_4$ ($C_2^2$), $C_3\times Q_8\times S_4$ ($C_2^2$), $\GL(2,\mathbb{Z}/4):C_6$ ($C_2^2$), $D_{12}:S_4$ ($C_2^2$), $S_4\times D_{12}$ ($C_2^2$), $S_3\times C_4:S_4$ ($C_2^2$), $C_4\times S_3\times S_4$ ($C_2^2$), $S_3\times A_4:Q_8$ ($C_2^2$), $C_3:(Q_8\times S_4)$ ($C_2^2$), $C_3:Q_8\times S_4$ ($C_2^2$), $C_3\times A_4\times \SD_{16}$ ($C_2^2$), $A_4\times Q_8:S_3$ ($C_2^2$), $A_4\times C_3:\SD_{16}$ ($C_2^2$), $A_4\times C_{24}:C_2$ ($C_2^2$), $S_3\times C_8\times A_4$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_{24}:S_4$ ($C_2^2$), $C_{24}:S_4$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_{24}:S_4$ ($C_2^2$), $C_{24}\times S_4$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $S_3\times A_4:C_8$ ($C_2^2$), $C_{12}.(C_2\times S_4)$ ($C_2^2$), $C_{12}.(C_2\times S_4)$ ($C_2^2$), $C_3:(C_8\times S_4)$ ($C_2^2$), $C_3:C_8\times S_4$ ($C_2^2$)

Order 384: $C_{24}:C_2^4$ ($S_3$), $\SD_{16}\times S_4$ ($S_3$)

Order 288: $C_2^3.C_6^2$ ($C_2^3$), $C_2^3.C_6^2$ ($C_2^3$), $A_4\times D_{12}$ ($C_2^3$), $C_4\times S_3\times A_4$ ($C_2^3$), $A_4\times C_3:Q_8$ ($C_2^3$), $C_{12}:S_4$ ($C_2^3$), $C_{12}:S_4$ ($C_2^3$), $C_{12}.S_4$ ($C_2^3$), $C_{12}:S_4$ ($C_2^3$), $C_{12}\times S_4$ ($C_2^3$), $C_{12}.S_4$ ($C_2^3$), $D_6.S_4$ ($D_4$), $C_2^3.S_3^2$ ($D_4$), $C_2^3.S_3^2$ ($D_4$), $A_4\times C_{24}$ ($C_2^3$), $A_4\times C_3:C_8$ ($C_2^3$), $C_{12}.S_4$ ($C_2^3$), $A_4:C_{24}$ ($C_2^3$), $D_6\times S_4$ ($D_4$)

Order 192: $Q_8:S_4$ ($D_6$), $A_4:\SD_{16}$ ($D_6$), $C_8:S_4$ ($D_6$), $C_8\times S_4$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $C_{12}:C_2^4$ ($D_6$), $Q_8\times S_4$ ($D_6$), $D_4\times S_4$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $A_4\times \SD_{16}$ ($D_6$)

Order 144: $S_3\times S_4$ ($\SD_{16}$) x 4, $A_4\times D_6$ ($C_2\times D_4$), $C_6:S_4$ ($C_2\times D_4$), $C_6\times S_4$ ($C_2\times D_4$), $C_{12}\times A_4$ ($C_2^4$), $A_4\times C_3:C_4$ ($C_2\times D_4$), $C_6.S_4$ ($C_2\times D_4$), $A_4:C_{12}$ ($C_2\times D_4$)

Order 96: $C_8\times A_4$ ($C_2\times D_6$), $A_4:C_8$ ($C_2\times D_6$), $C_6.C_2^4$ ($C_2\times D_6$), $C_{12}:C_2^3$ ($C_2\times D_6$), $C_2^2\times D_{12}$ ($C_2\times D_6$), $C_{12}:C_2^3$ ($C_2\times D_6$), $C_6.C_2^4$ ($C_2\times D_6$), $Q_8\times A_4$ ($C_2\times D_6$), $D_4\times A_4$ ($C_2\times D_6$), $C_4:S_4$ ($C_2\times D_6$), $C_4\times S_4$ ($C_2\times D_6$), $A_4:Q_8$ ($C_2\times D_6$), $C_2^2\times C_{24}$ ($C_2\times D_6$), $C_{12}.C_2^3$ ($C_2\times D_6$), $S_3\times \SD_{16}$ ($S_4$)

Order 72: $S_3\times A_4$ ($C_2\times \SD_{16}$) x 2, $C_3:S_4$ ($C_2\times \SD_{16}$) x 2, $C_3\times S_4$ ($C_2\times \SD_{16}$) x 2, $C_6\times A_4$ ($C_2^2\times D_4$)

Order 64: $C_2^2\times \SD_{16}$ ($S_3^2$)

Order 48: $C_{24}:C_2$ ($C_2\times S_4$), $C_2^2\times D_6$ ($S_3\times D_4$), $C_2\times S_4$ ($S_3\times D_4$), $C_2^2\times C_{12}$ ($C_2^2\times D_6$), $C_6.C_2^3$ ($S_3\times D_4$), $S_3\times Q_8$ ($C_2\times S_4$), $S_3\times C_8$ ($C_2\times S_4$), $S_3\times D_4$ ($C_2\times S_4$), $C_4\times A_4$ ($C_2^2\times D_6$), $A_4:C_4$ ($S_3\times D_4$), $C_3\times \SD_{16}$ ($C_2\times S_4$), $Q_8:S_3$ ($C_2\times S_4$), $C_3:\SD_{16}$ ($C_2\times S_4$)

Order 36: $C_3\times A_4$ ($C_2^2\times \SD_{16}$)

Order 32: $C_2^2\times Q_8$ ($S_3\times D_6$), $C_2^2\times D_4$ ($S_3\times D_6$), $C_2^2\times C_8$ ($S_3\times D_6$)

Order 24: $C_2\times D_6$ ($S_3\times \SD_{16}$) x 2, $S_4$ ($S_3\times \SD_{16}$) x 2, $D_{12}$ ($C_2^2\times S_4$), $C_4\times S_3$ ($C_2^2\times S_4$), $C_3:Q_8$ ($C_2^2\times S_4$), $C_{24}$ ($C_2^2\times S_4$), $C_2^2\times C_6$ ($D_4\times D_6$), $C_2\times A_4$ ($D_4\times D_6$), $C_3\times Q_8$ ($C_2^2\times S_4$), $C_3\times D_4$ ($C_2^2\times S_4$), $C_3:C_8$ ($C_2^2\times S_4$)

Order 16: $\SD_{16}$ ($S_3\times S_4$), $C_2^2\times C_4$ ($D_6^2$)

Order 12: $C_2\times C_6$ ($D_6\times \SD_{16}$), $D_6$ ($D_4\times S_4$), $A_4$ ($D_6\times \SD_{16}$), $C_{12}$ ($C_2^3\times S_4$), $C_3:C_4$ ($D_4\times S_4$)

Order 8: $C_2^3$ ($D_4\times S_3^2$), $Q_8$ ($D_6\times S_4$), $D_4$ ($D_6\times S_4$), $C_8$ ($D_6\times S_4$)

Order 6: $S_3$ ($\SD_{16}\times S_4$) x 2, $C_6$ ($\GL(2,\mathbb{Z}/4):C_2^2$)

Order 4: $C_2^2$ ($\SD_{16}\times S_3^2$), $C_4$ ($C_2^2:D_6^2$)

Order 3: $C_3$ ($C_2\times \SD_{16}\times S_4$)

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

Order 1: $C_1$ ($S_3\times \SD_{16}\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2304: $S_3\times \SD_{16}\times S_4$ ($C_1$)

Order 1152: $C_2^3.D_6^2$ ($C_2$), $\GL(2,\mathbb{Z}/4):D_6$ ($C_2$), $S_3\times A_4\times \SD_{16}$ ($C_2$), $C_{24}:(C_2\times S_4)$ ($C_2$), $C_3\times \SD_{16}\times S_4$ ($C_2$), $Q_8:(S_3\times S_4)$ ($C_2$), $Q_8:S_3\times S_4$ ($C_2$), $D_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_3:(\SD_{16}\times S_4)$ ($C_2$), $C_3:\SD_{16}\times S_4$ ($C_2$), $D_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_{24}:(C_2\times S_4)$ ($C_2$), $C_{24}:C_2\times S_4$ ($C_2$), $S_3\times C_8:S_4$ ($C_2$), $S_3\times C_8\times S_4$ ($C_2$)

Order 576: $S_3\times Q_8\times A_4$ ($C_2^2$), $S_3\times D_4\times A_4$ ($C_2^2$), $Q_8\times C_3:S_4$ ($C_2^2$), $D_4\times C_3:S_4$ ($C_2^2$), $C_3\times Q_8\times S_4$ ($C_2^2$), $\GL(2,\mathbb{Z}/4):C_6$ ($C_2^2$), $D_{12}:S_4$ ($C_2^2$), $S_4\times D_{12}$ ($C_2^2$), $S_3\times C_4:S_4$ ($C_2^2$), $C_4\times S_3\times S_4$ ($C_2^2$), $S_3\times A_4:Q_8$ ($C_2^2$), $C_3:(Q_8\times S_4)$ ($C_2^2$), $C_3:Q_8\times S_4$ ($C_2^2$), $C_3\times A_4\times \SD_{16}$ ($C_2^2$), $A_4\times Q_8:S_3$ ($C_2^2$), $A_4\times C_3:\SD_{16}$ ($C_2^2$), $A_4\times C_{24}:C_2$ ($C_2^2$), $S_3\times C_8\times A_4$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_{24}:S_4$ ($C_2^2$), $C_{24}:S_4$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_{24}:S_4$ ($C_2^2$), $C_{24}\times S_4$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $S_3\times A_4:C_8$ ($C_2^2$), $C_{12}.(C_2\times S_4)$ ($C_2^2$), $C_{12}.(C_2\times S_4)$ ($C_2^2$), $C_3:(C_8\times S_4)$ ($C_2^2$), $C_3:C_8\times S_4$ ($C_2^2$)

Order 384: $C_{24}:C_2^4$ ($S_3$), $\SD_{16}\times S_4$ ($S_3$)

Order 288: $C_2^3.C_6^2$ ($C_2^3$), $C_2^3.C_6^2$ ($C_2^3$), $A_4\times D_{12}$ ($C_2^3$), $C_4\times S_3\times A_4$ ($C_2^3$), $A_4\times C_3:Q_8$ ($C_2^3$), $C_{12}:S_4$ ($C_2^3$), $C_{12}:S_4$ ($C_2^3$), $C_{12}.S_4$ ($C_2^3$), $C_{12}:S_4$ ($C_2^3$), $C_{12}\times S_4$ ($C_2^3$), $C_{12}.S_4$ ($C_2^3$), $D_6.S_4$ ($D_4$), $C_2^3.S_3^2$ ($D_4$), $C_2^3.S_3^2$ ($D_4$), $A_4\times C_{24}$ ($C_2^3$), $A_4\times C_3:C_8$ ($C_2^3$), $C_{12}.S_4$ ($C_2^3$), $A_4:C_{24}$ ($C_2^3$), $D_6\times S_4$ ($D_4$)

Order 192: $Q_8:S_4$ ($D_6$), $A_4:\SD_{16}$ ($D_6$), $C_8:S_4$ ($D_6$), $C_8\times S_4$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $C_{12}:C_2^4$ ($D_6$), $Q_8\times S_4$ ($D_6$), $D_4\times S_4$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $A_4\times \SD_{16}$ ($D_6$)

Order 144: $A_4\times D_6$ ($C_2\times D_4$), $C_6:S_4$ ($C_2\times D_4$), $C_6\times S_4$ ($C_2\times D_4$), $S_3\times S_4$ ($\SD_{16}$), $C_{12}\times A_4$ ($C_2^4$), $A_4\times C_3:C_4$ ($C_2\times D_4$), $C_6.S_4$ ($C_2\times D_4$), $A_4:C_{12}$ ($C_2\times D_4$)

Order 96: $C_8\times A_4$ ($C_2\times D_6$), $A_4:C_8$ ($C_2\times D_6$), $C_6.C_2^4$ ($C_2\times D_6$), $C_{12}:C_2^3$ ($C_2\times D_6$), $C_2^2\times D_{12}$ ($C_2\times D_6$), $C_{12}:C_2^3$ ($C_2\times D_6$), $C_6.C_2^4$ ($C_2\times D_6$), $Q_8\times A_4$ ($C_2\times D_6$), $D_4\times A_4$ ($C_2\times D_6$), $C_4:S_4$ ($C_2\times D_6$), $C_4\times S_4$ ($C_2\times D_6$), $A_4:Q_8$ ($C_2\times D_6$), $C_2^2\times C_{24}$ ($C_2\times D_6$), $C_{12}.C_2^3$ ($C_2\times D_6$), $S_3\times \SD_{16}$ ($S_4$)

Order 72: $C_6\times A_4$ ($C_2^2\times D_4$), $S_3\times A_4$ ($C_2\times \SD_{16}$), $C_3:S_4$ ($C_2\times \SD_{16}$), $C_3\times S_4$ ($C_2\times \SD_{16}$)

Order 64: $C_2^2\times \SD_{16}$ ($S_3^2$)

Order 48: $C_{24}:C_2$ ($C_2\times S_4$), $C_2^2\times D_6$ ($S_3\times D_4$), $C_2\times S_4$ ($S_3\times D_4$), $C_2^2\times C_{12}$ ($C_2^2\times D_6$), $C_6.C_2^3$ ($S_3\times D_4$), $S_3\times Q_8$ ($C_2\times S_4$), $S_3\times C_8$ ($C_2\times S_4$), $S_3\times D_4$ ($C_2\times S_4$), $C_4\times A_4$ ($C_2^2\times D_6$), $A_4:C_4$ ($S_3\times D_4$), $C_3\times \SD_{16}$ ($C_2\times S_4$), $Q_8:S_3$ ($C_2\times S_4$), $C_3:\SD_{16}$ ($C_2\times S_4$)

Order 36: $C_3\times A_4$ ($C_2^2\times \SD_{16}$)

Order 32: $C_2^2\times Q_8$ ($S_3\times D_6$), $C_2^2\times D_4$ ($S_3\times D_6$), $C_2^2\times C_8$ ($S_3\times D_6$)

Order 24: $D_{12}$ ($C_2^2\times S_4$), $C_4\times S_3$ ($C_2^2\times S_4$), $C_3:Q_8$ ($C_2^2\times S_4$), $C_{24}$ ($C_2^2\times S_4$), $C_2^2\times C_6$ ($D_4\times D_6$), $C_2\times D_6$ ($S_3\times \SD_{16}$), $C_2\times A_4$ ($D_4\times D_6$), $S_4$ ($S_3\times \SD_{16}$), $C_3\times Q_8$ ($C_2^2\times S_4$), $C_3\times D_4$ ($C_2^2\times S_4$), $C_3:C_8$ ($C_2^2\times S_4$)

Order 16: $\SD_{16}$ ($S_3\times S_4$), $C_2^2\times C_4$ ($D_6^2$)

Order 12: $C_2\times C_6$ ($D_6\times \SD_{16}$), $D_6$ ($D_4\times S_4$), $A_4$ ($D_6\times \SD_{16}$), $C_{12}$ ($C_2^3\times S_4$), $C_3:C_4$ ($D_4\times S_4$)

Order 8: $C_2^3$ ($D_4\times S_3^2$), $Q_8$ ($D_6\times S_4$), $D_4$ ($D_6\times S_4$), $C_8$ ($D_6\times S_4$)

Order 6: $C_6$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $S_3$ ($\SD_{16}\times S_4$)

Order 4: $C_2^2$ ($\SD_{16}\times S_3^2$), $C_4$ ($C_2^2:D_6^2$)

Order 3: $C_3$ ($C_2\times \SD_{16}\times S_4$)

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

Order 1: $C_1$ ($S_3\times \SD_{16}\times S_4$)

Series

Derived series

$S_3\times \SD_{16}\times S_4$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$S_3\times \SD_{16}\times S_4$

$\rhd$

$\GL(2,\mathbb{Z}/4):D_6$

$\rhd$

$C_4\times S_3\times S_4$

$\rhd$

$C_{12}\times S_4$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_3\times \SD_{16}\times S_4$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$\SD_{16}$

Supergroups

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

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

Character theory

Complex character table

See the $105 \times 105$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.