Abstract group 2304.of: $C_2^4.D_6^2$

• Label: 2304.of
• Order: \( 2^{8} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $5$

Group information

Description:	$C_2^4.D_6^2$
Order:	\(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_5\times C_{15}^2):C_6$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
Composition factors:	$C_2$ x 8, $C_3$ x 2
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	407	26	616	646	608	2304
Conjugacy classes	1	32	3	35	34	24	129
Divisions	1	32	3	35	34	24	129
Autjugacy classes	1	13	3	12	15	8	52

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	32	32	32	10	16	4	2	1	129
Irr. rational chars.	32	32	32	10	16	4	2	1	129

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$5$
Inequivalent generating 5-tuples:	$509967360000$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{6}=d^{4}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(5,7,6)(8,13)(9,12)(10,11)(14,15), (4,7)(5,6), (1,2)(5,7), (1,2)(4,6)(8,14) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_4:S_4)$ $\,\rtimes\,$ $D_6$	$(D_4\times S_4)$ $\,\rtimes\,$ $D_6$	$(D_4:D_6)$ $\,\rtimes\,$ $S_4$	$D_4$ $\,\rtimes\,$ $(D_6\times S_4)$	all 121
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $D_6^2$ (2)	$(D_6\times S_4)$ . $C_2^3$	$D_6$ . $(C_2^3\times S_4)$	$(C_6\times S_4)$ . $C_2^4$	all 34
Aut. group:	$\Aut(C_9\times D_{794})$	$\Aut(C_{193}:C_{192})$

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 110592 subgroups in 12370 conjugacy classes, 615 normal (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2.D_6^2.C_2$
Commutator:	$G' \simeq$ $C_6\times A_4$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2.D_6^2.C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}:C_2^4$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_2^4.D_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2^2$
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: $C_2^4.D_6^2$

Order 1152: $(C_6\times A_4).C_2^4$ x 4, $(C_6\times S_4).C_2^3$ x 4, $(C_6\times S_4).C_2^3$ x 4, $(C_6\times A_4).C_2^4$ x 4, $(C_6\times S_4).C_2^3$ x 4, $(C_6\times S_4).C_2^3$ x 4, $(C_6\times S_4).C_2^3$ x 2, $(C_6\times S_4).C_2^3$ x 2, $(C_6\times A_4).C_2^4$, $(C_6\times A_4).C_2^4$, $(C_6\times A_4).C_2^4$

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$, $D_4^2:D_6$

Order 576: $\GL(2,\mathbb{Z}/4):S_3$ x 28, $D_4\times C_3:S_4$ x 8, $C_2^4:S_3^2$ x 8, $S_3\times \GL(2,\mathbb{Z}/4)$ x 8, $C_2^4.S_3^2$ x 8, $\GL(2,\mathbb{Z}/4):S_3$ x 8, $C_2^2:D_6^2$ x 4, $C_6:D_4\times A_4$ x 4, $A_4\times D_4:S_3$ x 4, $S_3\times D_4\times A_4$ x 4, $C_6\times \GL(2,\mathbb{Z}/4)$ x 4, $\GL(2,\mathbb{Z}/4):C_6$ x 4, $\GL(2,\mathbb{Z}/4):C_6$ x 4, $C_6:\GL(2,\mathbb{Z}/4)$ x 4, $\GL(2,\mathbb{Z}/4):S_3$ x 4, $C_2^4.S_3^2$ x 4, $C_2^4.S_3^2$ x 4, $S_4\times D_{12}$ x 4, $S_3\times C_4:S_4$ x 4, $C_4\times S_3\times S_4$ x 4, $(C_4\times S_3):S_4$ x 4, $(C_4\times S_3):S_4$ x 4, $(C_{12}\times S_4):C_2$ x 4, $C_2^2\times C_6:S_4$ x 2, $A_4\times D_{12}:C_2$ x 2, $C_6:\GL(2,\mathbb{Z}/4)$ x 2, $\GL(2,\mathbb{Z}/4):C_6$ x 2, $D_{12}:S_4$ x 2, $D_{12}:S_4$ x 2, $C_3:(Q_8\times S_4)$ x 2, $C_{12}.(C_2\times S_4)$ x 2, $C_2^4:C_6^2$, $D_6^2:C_2^2$, $C_2\times C_{12}:S_4$, $C_2\times C_{12}:S_4$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 10, $\GL(2,\mathbb{Z}/4):C_2^2$ x 9, $\GL(2,\mathbb{Z}/4):C_2^2$ x 6, $\GL(2,\mathbb{Z}/4):C_2^2$ x 6, $D_4:(S_3\times D_4)$ x 4, $D_4:(S_3\times D_4)$ x 4, $D_4:(S_3\times D_4)$ x 4, $D_4:(S_3\times D_4)$ x 4, $D_4^2:S_3$ x 4, $S_3\times D_4^2$ x 4, $D_4^2:S_3$ x 4, $D_4^2:S_3$ x 4, $C_2^3:(S_3\times D_4)$ x 4, $C_2^3:(S_3\times D_4)$ x 4, $C_2^5:D_6$ x 4, $C_6:D_4^2$ x 4, $D_{12}:(C_2\times D_4)$ x 2, $D_{12}:(C_2\times D_4)$ x 2, $(C_4\times D_4):D_6$ x 2, $(C_4\times D_4):D_6$ x 2, $D_{12}:C_2^4$, $D_{12}:C_2^4$, $A_4\times D_4:C_2^2$, $D_4^2:C_6$, $(C_6\times D_4):D_4$, $C_6:D_4^2$, $(C_2\times D_4):D_{12}$, $D_4:(C_4\times D_6)$

Order 288: $D_6\times S_4$ x 20, $C_2\times C_6:S_4$ x 15, $A_4\times C_3:D_4$ x 12, $C_3\times \GL(2,\mathbb{Z}/4)$ x 12, $D_6:S_4$ x 12, $C_2^3.S_3^2$ x 12, $C_2^3.S_3^2$ x 12, $D_{12}:D_6$ x 8, $C_3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^3.C_6^2$ x 4, $C_2^3:S_3^2$ x 4, $D_6^2:C_2$ x 4, $D_4:S_3^2$ x 4, $D_4\times S_3^2$ x 4, $C_6:C_4\times A_4$ x 4, $C_4\times S_3\times A_4$ x 4, $C_{12}:S_4$ x 4, $C_{12}:S_4$ x 4, $C_2\times A_4:C_{12}$ x 4, $C_{12}\times S_4$ x 4, $D_6:S_4$ x 4, $D_6.S_4$ x 4, $C_2^3.S_3^2$ x 4, $C_2^3.S_3^2$ x 4, $C_2\times A_4\times D_6$ x 4, $C_2\times C_6\times S_4$ x 4, $C_6^2.C_2^3$ x 2, $D_{12}:D_6$ x 2, $D_{12}:D_6$ x 2, $A_4\times D_{12}$ x 2, $A_4\times C_3:Q_8$ x 2, $C_{12}:S_4$ x 2, $C_{12}.S_4$ x 2, $C_2^3:C_6^2$ x 2, $C_2^3.C_6^2$, $C_2\times C_6.S_4$, $C_6^2:C_2^3$

Order 256: $D_4^2:C_2^2$

Order 192: $C_2^4:D_6$ x 76, $D_4\times S_4$ x 62, $C_2^4:D_6$ x 36, $D_{12}:D_4$ x 26, $D_{12}:C_2^3$ x 24, $C_2\times \GL(2,\mathbb{Z}/4)$ x 20, $(D_4\times D_6):C_2$ x 20, $C_2^4.D_6$ x 20, $D_4:S_4$ x 18, $\GL(2,\mathbb{Z}/4):C_2$ x 18, $D_{12}:C_2^3$ x 17, $C_{12}:C_2^4$ x 17, $(C_2\times D_4):D_6$ x 16, $S_3\times C_4:D_4$ x 16, $C_2^4.D_6$ x 16, $C_2^4:D_6$ x 16, $C_2^4.D_6$ x 16, $C_2.(D_4\times D_6)$ x 12, $D_4\times D_{12}$ x 12, $C_2^5:C_6$ x 10, $C_2^3:D_{12}$ x 10, $C_2^4.D_6$ x 10, $C_2^4.D_6$ x 10, $C_{12}.(C_2\times D_4)$ x 10, $C_2^3\times S_4$ x 8, $C_3\times D_4:D_4$ x 8, $C_3\times D_4^2$ x 8, $(C_4\times D_6):C_2^2$ x 8, $(C_2\times D_6):D_4$ x 8, $(C_4\times S_3):D_4$ x 8, $(C_2\times D_4):D_6$ x 8, $C_2.(D_4\times D_6)$ x 8, $C_2.(D_4\times D_6)$ x 8, $C_2^4:D_6$ x 8, $C_2^4.D_6$ x 8, $D_4:D_{12}$ x 8, $C_4^2:D_6$ x 8, $C_4^2:D_6$ x 8, $C_2^4.D_6$ x 8, $C_2^3:D_{12}$ x 8, $C_2^4.D_6$ x 8, $D_6:C_2^4$ x 6, $D_{12}:C_2^3$ x 6, $D_{12}:C_2^3$ x 6, $A_4\times D_4:C_2$ x 6, $Q_8:S_4$ x 6, $C_2^4:D_6$ x 6, $C_2^4.D_6$ x 5, $C_{12}.C_2^4$ x 4, $C_3\times C_2^3:D_4$ x 4, $C_2^5:C_6$ x 4, $(C_2\times D_4):D_6$ x 4, $(C_3\times D_4):D_4$ x 4, $C_2^4.D_6$ x 4, $C_2^4:D_6$ x 4, $C_4^2.D_6$ x 4, $C_3:D_4^2$ x 4, $C_4^2:D_6$ x 4, $C_4^2:D_6$ x 4, $C_4^2.D_6$ x 4, $D_{12}:D_4$ x 4, $C_4^2:D_6$ x 4, $C_4^2.D_6$ x 4, $(C_4\times S_3):D_4$ x 4, $(C_4\times S_3):D_4$ x 4, $(C_4\times S_3):D_4$ x 4, $(C_4\times S_3):D_4$ x 4, $C_4^2.D_6$ x 4, $C_4^2.D_6$ x 4, $D_{12}:D_4$ x 4, $C_4^2.D_6$ x 4, $C_4^2.D_6$ x 4, $C_4^2:D_6$ x 4, $C_4^2.D_6$ x 4, $C_2^3:D_{12}$ x 4, $C_2^4.D_6$ x 4, $D_{12}:C_2^3$ x 2, $Q_8\times S_4$ x 2, $C_3\times C_4^2:C_2^2$ x 2, $C_3\times C_4^2:C_2^2$ x 2, $C_2^4:D_6$ x 2, $(C_2\times C_{12}):D_4$ x 2, $(C_2\times C_{12}):D_4$ x 2, $(Q_8\times D_6):C_2$ x 2, $C_3:(D_4\times Q_8)$ x 2, $D_{12}:D_4$ x 2, $D_{12}:D_4$ x 2, $D_{12}:D_4$ x 2, $C_{12}.(C_2\times D_4)$ x 2, $C_4^2:D_6$ x 2, $C_4^2:D_6$ x 2, $C_4^2.D_6$ x 2, $C_4^2.D_6$ x 2, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $C_3\times C_4^2:C_2^2$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 144: $D_6:D_6$ x 28, $C_6:S_4$ x 22, $S_3\times S_4$ x 16, $D_6:D_6$ x 16, $D_6.D_6$ x 16, $A_4\times D_6$ x 8, $C_6\times S_4$ x 8, $C_{12}:D_6$ x 8, $C_6^2:C_2^2$ x 8, $C_{12}.D_6$ x 8, $C_{12}:D_6$ x 8, $C_6:D_{12}$ x 8, $S_3\times D_{12}$ x 8, $C_{12}.D_6$ x 8, $C_2^2:C_6^2$ x 7, $D_6^2$ x 4, $D_{12}:C_6$ x 4, $C_2^2.S_3^2$ x 4, $D_6.D_6$ x 4, $C_4\times S_3^2$ x 4, $C_{12}.D_6$ x 4, $D_{12}:S_3$ x 4, $A_4\times C_3:C_4$ x 4, $A_4:C_{12}$ x 4, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}\times A_4$ x 2, $C_{12}:D_6$ x 2, $C_{12}.D_6$ x 2, $C_6.S_4$ x 2, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$

Order 128: $D_4^2:C_2$ x 18, $D_4^2:C_2$ x 12, $D_4^2:C_2$ x 9, $C_2\times D_4^2$ x 9, $D_4^2:C_2$ x 6, $D_4^2:C_2$ x 6, $D_4:C_2^4$ x 2, $C_4^2.C_2^3$

Order 96: $D_4\times D_6$ x 216, $D_4:D_6$ x 124, $C_2^3.D_6$ x 96, $C_2^3:D_6$ x 95, $D_6:D_4$ x 88, $C_2^2:D_{12}$ x 84, $C_2^3.D_6$ x 72, $D_6:D_4$ x 72, $C_2^2\times S_4$ x 58, $D_4:D_6$ x 54, $D_4:D_6$ x 54, $D_4:D_6$ x 46, $C_{12}:D_4$ x 46, $\GL(2,\mathbb{Z}/4)$ x 44, $C_2^4:C_6$ x 36, $D_6.D_4$ x 32, $D_{12}:C_2^2$ x 32, $C_{12}:D_4$ x 32, $C_{12}:C_2^3$ x 28, $C_4\times S_4$ x 28, $C_{12}:D_4$ x 28, $C_{12}:D_4$ x 28, $C_2^3.D_6$ x 24, $C_2^4:S_3$ x 24, $C_{12}:D_4$ x 24, $C_4:D_{12}$ x 24, $D_4\times A_4$ x 22, $C_4:S_4$ x 22, $C_2^3.D_6$ x 20, $C_2^2\times D_{12}$ x 19, $C_2^2.D_{12}$ x 19, $C_{12}:C_2^3$ x 17, $D_6.D_4$ x 16, $C_2^3.D_6$ x 16, $D_4\times C_{12}$ x 16, $D_6:Q_8$ x 16, $D_6.D_4$ x 16, $D_{12}:C_4$ x 16, $C_2^2.D_{12}$ x 12, $C_4\times D_{12}$ x 12, $(C_2^2\times C_{12}):C_2$ x 12, $C_2^3:C_{12}$ x 12, $C_2^3\times D_6$ x 10, $C_2^3:C_{12}$ x 10, $C_2^2.S_4$ x 10, $C_6.C_2^4$ x 9, $C_2^3.D_6$ x 9, $C_2^3.D_6$ x 9, $D_6.D_4$ x 8, $C_2^3.D_6$ x 8, $C_2^3\times A_4$ x 8, $C_2^3.D_6$ x 8, $C_2^3.D_6$ x 8, $C_6.C_2^4$ x 6, $C_6.C_2^4$ x 6, $Q_8:D_6$ x 6, $A_4:Q_8$ x 6, $C_6:C_4^2$ x 5, $(C_4\times S_3):C_4$ x 4, $C_{12}:Q_8$ x 4, $C_3:(C_4\times Q_8)$ x 4, $C_4^2:S_3$ x 4, $C_4:D_{12}$ x 4, $C_4^2:S_3$ x 4, $S_3\times C_4^2$ x 4, $C_{12}:Q_8$ x 4, $C_{12}:D_4$ x 4, $C_4^2:C_6$ x 4, $C_{12}.D_4$ x 4, $D_6:Q_8$ x 4, $C_{12}.D_4$ x 4, $D_6:Q_8$ x 4, $C_2^4\times C_6$ x 2, $Q_8\times D_6$ x 2, $C_6.C_2^4$ x 2, $Q_8\times A_4$ x 2, $C_4^2:C_6$ x 2, $C_{12}.D_4$ x 2, $C_{12}.D_4$ x 2, $C_{12}.D_4$ x 2, $C_2^3\times C_{12}$, $C_2^2.D_{12}$

Order 72: $C_6\wr C_2$ x 24, $C_3:D_{12}$ x 24, $S_3\times D_6$ x 20, $C_6:D_6$ x 15, $C_6.D_6$ x 12, $C_6\times D_6$ x 8, $C_6^2:C_2$ x 8, $C_6:C_{12}$ x 8, $S_3\times C_{12}$ x 8, $C_6.D_6$ x 8, $C_3:S_4$ x 7, $C_6\times A_4$ x 6, $S_3\times A_4$ x 4, $C_3\times S_4$ x 4, $D_4\times C_3^2$ x 4, $C_3:D_{12}$ x 4, $C_{12}:S_3$ x 4, $C_3\times D_{12}$ x 4, $C_3^2:Q_8$ x 4, $C_3^2:Q_8$ x 4, $D_6:S_3$ x 4, $C_2\times C_6^2$ x 2, $C_6\times C_{12}$, $C_6.D_6$

Order 64: $D_4^2$ x 162, $D_4:D_4$ x 108, $D_4:C_2^3$ x 47, $C_4^2:C_2^2$ x 36, $C_2^3:D_4$ x 36, $C_4^2:C_2^2$ x 36, $C_2^3:D_4$ x 36, $C_2^3:D_4$ x 36, $D_4\times C_2^3$ x 24, $Q_8:D_4$ x 18, $D_4:D_4$ x 18, $C_2^2.C_2^4$ x 18, $C_4^2.C_2^2$ x 18, $C_4^2:C_2^2$ x 18, $D_4:C_2^3$ x 12, $Q_8:D_4$ x 12, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 9, $C_4^2.C_2^2$ x 6, $C_4^2.C_2^2$ x 6, $D_4\times Q_8$ x 2

Order 48: $S_3\times D_4$ x 394, $C_6:D_4$ x 326, $D_4:S_3$ x 134, $C_6\times D_4$ x 123, $D_{12}:C_2$ x 112, $C_2\times D_{12}$ x 106, $C_4\times D_6$ x 106, $C_2^2\times D_6$ x 101, $C_2\times S_4$ x 76, $D_6:C_4$ x 76, $C_2^2:C_{12}$ x 44, $C_6.C_2^3$ x 41, $C_6.D_4$ x 36, $C_6.D_4$ x 36, $D_4:C_6$ x 30, $D_{12}:C_2$ x 30, $C_2^2\times A_4$ x 24, $C_2^2\times C_{12}$ x 22, $C_2^3\times C_6$ x 21, $C_{12}:C_4$ x 20, $C_6:Q_8$ x 12, $C_4:C_{12}$ x 12, $C_{12}:C_4$ x 12, $S_3\times Q_8$ x 10, $C_4\times A_4$ x 8, $A_4:C_4$ x 8, $C_4\times C_{12}$ x 4, $C_6\times Q_8$ x 2

Order 36: $C_6:S_3$ x 22, $C_6\times S_3$ x 16, $S_3^2$ x 16, $C_3:C_{12}$ x 8, $C_6^2$ x 7, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 348, $C_4:D_4$ x 288, $C_2^2\wr C_2$ x 216, $D_4:C_2^2$ x 171, $C_4\times D_4$ x 168, $D_4:C_2^2$ x 138, $C_2^2.D_4$ x 72, $C_2^3:C_4$ x 72, $C_4:D_4$ x 54, $C_2^2:Q_8$ x 48, $C_4^2:C_2$ x 36, $C_2^3\times C_4$ x 24, $C_4^2:C_2$ x 18, $C_2^2.D_4$ x 18, $C_2^5$ x 12, $C_2\times C_4^2$ x 9, $C_4\times Q_8$ x 8, $C_4:Q_8$ x 6, $C_2^2\times Q_8$ x 4

Order 24: $C_2\times D_6$ x 281, $C_3:D_4$ x 226, $C_4\times S_3$ x 108, $C_3\times D_4$ x 98, $D_{12}$ x 92, $C_6:C_4$ x 75, $C_2^2\times C_6$ x 75, $C_2\times C_{12}$ x 55, $C_3:Q_8$ x 24, $S_4$ x 19, $C_2\times A_4$ x 17, $C_3\times Q_8$ x 6

Order 18: $C_3\times S_3$ x 8, $C_3:S_3$ x 7, $C_3\times C_6$ x 6

Order 16: $C_2\times D_4$ x 847, $D_4:C_2$ x 258, $C_2^2\times C_4$ x 159, $C_2^2:C_4$ x 156, $C_2^4$ x 114, $C_4:C_4$ x 60, $C_4^2$ x 24, $C_2\times Q_8$ x 22

Order 12: $D_6$ x 203, $C_2\times C_6$ x 87, $C_3:C_4$ x 27, $C_{12}$ x 24, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 328, $C_2^3$ x 290, $C_2\times C_4$ x 190, $Q_8$ x 22

Order 6: $C_6$ x 34, $S_3$ x 34

Order 4: $C_2^2$ x 190, $C_4$ x 35

Order 3: $C_3$ x 3

Order 2: $C_2$ x 32

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2304: $C_2^4.D_6^2$

Order 1152: $(C_6\times A_4).C_2^4$, $(C_6\times A_4).C_2^4$, $(C_6\times S_4).C_2^3$, $(C_6\times S_4).C_2^3$, $(C_6\times A_4).C_2^4$, $(C_6\times S_4).C_2^3$, $(C_6\times S_4).C_2^3$, $(C_6\times A_4).C_2^4$, $(C_6\times S_4).C_2^3$, $(C_6\times S_4).C_2^3$, $(C_6\times A_4).C_2^4$

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$, $D_4^2:D_6$

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

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 4, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $D_{12}:C_2^4$, $D_{12}:C_2^4$, $A_4\times D_4:C_2^2$, $D_4^2:C_6$, $(C_6\times D_4):D_4$, $C_6:D_4^2$, $D_4:(S_3\times D_4)$, $D_4:(S_3\times D_4)$, $D_4:(S_3\times D_4)$, $D_4:(S_3\times D_4)$, $D_4^2:S_3$, $S_3\times D_4^2$, $D_4^2:S_3$, $D_4^2:S_3$, $D_{12}:(C_2\times D_4)$, $D_{12}:(C_2\times D_4)$, $C_2^3:(S_3\times D_4)$, $C_2^3:(S_3\times D_4)$, $(C_4\times D_4):D_6$, $(C_4\times D_4):D_6$, $C_2^5:D_6$, $C_6:D_4^2$, $(C_2\times D_4):D_{12}$, $D_4:(C_4\times D_6)$

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

Order 256: $D_4^2:C_2^2$

Order 192: $D_4\times S_4$ x 12, $C_2^4:D_6$ x 10, $D_{12}:C_2^3$ x 8, $D_{12}:C_2^3$ x 7, $C_2^4:D_6$ x 7, $C_{12}:C_2^4$ x 5, $C_2\times \GL(2,\mathbb{Z}/4)$ x 5, $D_{12}:D_4$ x 5, $C_2^5:C_6$ x 4, $D_4:S_4$ x 4, $\GL(2,\mathbb{Z}/4):C_2$ x 4, $C_2^3:D_{12}$ x 4, $C_2^4.D_6$ x 4, $C_2^3\times S_4$ x 3, $C_2^4.D_6$ x 3, $(D_4\times D_6):C_2$ x 3, $S_3\times C_4:D_4$ x 3, $C_{12}.(C_2\times D_4)$ x 3, $C_2^4.D_6$ x 3, $D_4\times D_{12}$ x 3, $D_6:C_2^4$ x 2, $D_{12}:C_2^3$ x 2, $D_{12}:C_2^3$ x 2, $A_4\times D_4:C_2$ x 2, $Q_8:S_4$ x 2, $C_3\times D_4:D_4$ x 2, $C_3\times D_4^2$ x 2, $C_2^4.D_6$ x 2, $C_2^4:D_6$ x 2, $C_2.(D_4\times D_6)$ x 2, $(C_2\times D_4):D_6$ x 2, $(C_4\times S_3):D_4$ x 2, $C_2^4.D_6$ x 2, $C_2^4:D_6$ x 2, $D_4:D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_4^2:D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_{12}.C_2^4$, $C_{12}:C_2^4$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $Q_8\times S_4$, $C_3\times C_4^2:C_2^2$, $C_3\times C_2^3:D_4$, $C_3\times C_4^2:C_2^2$, $C_2^5:C_6$, $C_3\times C_4^2:C_2^2$, $C_2^4:D_6$, $(C_2\times C_{12}):D_4$, $(C_2\times C_{12}):D_4$, $(C_2\times D_4):D_6$, $(C_3\times D_4):D_4$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_4^2.D_6$, $C_3:D_4^2$, $C_4^2:D_6$, $C_4^2:D_6$, $C_4^2.D_6$, $D_{12}:D_4$, $C_4^2:D_6$, $C_4^2.D_6$, $(C_4\times D_6):C_2^2$, $(C_2\times D_6):D_4$, $(Q_8\times D_6):C_2$, $C_3:(D_4\times Q_8)$, $D_{12}:D_4$, $D_{12}:D_4$, $(C_4\times S_3):D_4$, $(C_4\times S_3):D_4$, $D_{12}:D_4$, $(C_4\times S_3):D_4$, $(C_4\times S_3):D_4$, $(C_2\times D_4):D_6$, $C_{12}.(C_2\times D_4)$, $C_2.(D_4\times D_6)$, $C_2.(D_4\times D_6)$, $C_2^4:D_6$, $C_2^4.D_6$, $C_4^2.D_6$, $C_4^2.D_6$, $D_{12}:D_4$, $C_4^2.D_6$, $C_4^2.D_6$, $C_4^2:D_6$, $C_4^2.D_6$, $C_4^2:D_6$, $C_4^2:D_6$, $C_4^2.D_6$, $C_4^2.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 144: $C_6:S_4$ x 7, $D_6:D_6$ x 4, $C_2^2:C_6^2$ x 3, $A_4\times D_6$ x 2, $C_6\times S_4$ x 2, $S_3\times S_4$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$ x 2, $D_{12}:C_6$ x 2, $D_6:D_6$ x 2, $C_6:D_{12}$ x 2, $D_6.D_6$ x 2, $S_3\times D_{12}$ x 2, $C_{12}.D_6$ x 2, $D_{12}:S_3$ x 2, $C_6^2:C_2^2$, $D_6^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_{12}:D_6$, $C_{12}\times A_4$, $C_2^2.S_3^2$, $D_6.D_6$, $C_{12}:D_6$, $C_4\times S_3^2$, $C_{12}.D_6$, $C_{12}.D_6$, $A_4\times C_3:C_4$, $C_6.S_4$, $A_4:C_{12}$

Order 128: $D_4^2:C_2$ x 6, $D_4^2:C_2$ x 4, $D_4^2:C_2$ x 3, $C_2\times D_4^2$ x 3, $D_4:C_2^4$ x 2, $D_4^2:C_2$ x 2, $D_4^2:C_2$ x 2, $C_4^2.C_2^3$

Order 96: $D_4\times D_6$ x 42, $D_4:D_6$ x 31, $C_2^3:D_6$ x 23, $C_2^2\times S_4$ x 14, $D_4:D_6$ x 14, $D_4:D_6$ x 14, $C_2^3.D_6$ x 14, $D_6:D_4$ x 13, $C_2^2:D_{12}$ x 12, $C_2^3.D_6$ x 11, $D_6:D_4$ x 10, $C_{12}:C_2^3$ x 9, $D_4:D_6$ x 9, $D_{12}:C_2^2$ x 9, $C_4\times S_4$ x 8, $C_{12}:D_4$ x 8, $C_{12}:D_4$ x 8, $\GL(2,\mathbb{Z}/4)$ x 7, $C_2^2\times D_{12}$ x 6, $C_{12}:D_4$ x 6, $C_2^2.D_{12}$ x 6, $C_4:D_{12}$ x 6, $D_6.D_4$ x 5, $C_6.C_2^4$ x 5, $C_{12}:C_2^3$ x 5, $D_4\times A_4$ x 5, $C_4:S_4$ x 5, $C_2^4:C_6$ x 5, $C_{12}:D_4$ x 5, $C_{12}:D_4$ x 5, $C_2^3.D_6$ x 4, $C_2^3:C_{12}$ x 4, $C_2^2.S_4$ x 4, $D_4\times C_{12}$ x 4, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 4, $D_{12}:C_4$ x 4, $D_6.D_4$ x 3, $C_2^3.D_6$ x 3, $C_4\times D_{12}$ x 3, $C_2^3\times D_6$ x 3, $C_2^3\times A_4$ x 3, $C_2^3:C_{12}$ x 3, $C_2^4:S_3$ x 3, $C_2^3.D_6$ x 3, $D_6:Q_8$ x 3, $D_6.D_4$ x 3, $D_6.D_4$ x 2, $C_3:(C_4\times Q_8)$ x 2, $C_2^2.D_{12}$ x 2, $C_6.C_2^4$ x 2, $C_6.C_2^4$ x 2, $Q_8:D_6$ x 2, $A_4:Q_8$ x 2, $(C_2^2\times C_{12}):C_2$ x 2, $C_{12}.D_4$ x 2, $C_6:C_4^2$ x 2, $D_6:Q_8$ x 2, $(C_4\times S_3):C_4$, $C_{12}:Q_8$, $C_2^3.D_6$, $C_4^2:S_3$, $C_4:D_{12}$, $C_4^2:S_3$, $S_3\times C_4^2$, $C_{12}:Q_8$, $C_2^4\times C_6$, $C_2^3\times C_{12}$, $Q_8\times D_6$, $C_6.C_2^4$, $Q_8\times A_4$, $C_{12}:D_4$, $C_4^2:C_6$, $C_4^2:C_6$, $C_{12}.D_4$, $D_6:Q_8$, $C_{12}.D_4$, $C_{12}.D_4$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^2.D_{12}$, $C_{12}.D_4$

Order 72: $C_6:D_6$ x 5, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $C_6\times A_4$ x 3, $S_3\times D_6$ x 3, $C_3:S_4$ x 3, $C_6\times D_6$ x 2, $C_{12}:S_3$ x 2, $C_6:C_{12}$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_6.D_6$ x 2, $C_6.D_6$ x 2, $C_2\times C_6^2$, $S_3\times A_4$, $C_3\times S_4$, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_6^2:C_2$, $C_6.D_6$, $C_3:D_{12}$, $C_3^2:Q_8$, $D_6:S_3$

Order 64: $D_4^2$ x 28, $D_4:D_4$ x 20, $D_4:C_2^3$ x 17, $C_2^3:D_4$ x 10, $D_4\times C_2^3$ x 8, $C_4^2:C_2^2$ x 8, $C_4^2:C_2^2$ x 8, $C_2^3:D_4$ x 8, $C_2^3:D_4$ x 6, $C_4^2:C_2^2$ x 6, $Q_8:D_4$ x 5, $D_4:C_2^3$ x 4, $Q_8:D_4$ x 4, $D_4:D_4$ x 4, $C_2^2.C_2^4$ x 4, $C_4^2.C_2^2$ x 4, $C_4^2:C_2^2$ x 3, $C_4^2:C_2^2$ x 3, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $D_4\times Q_8$

Order 48: $C_6:D_4$ x 66, $S_3\times D_4$ x 63, $C_6\times D_4$ x 31, $C_4\times D_6$ x 27, $D_4:S_3$ x 26, $D_{12}:C_2$ x 26, $C_2^2\times D_6$ x 25, $C_2\times D_{12}$ x 25, $C_2\times S_4$ x 18, $D_6:C_4$ x 15, $C_6.C_2^3$ x 12, $D_4:C_6$ x 9, $D_{12}:C_2$ x 9, $C_2^3\times C_6$ x 8, $C_2^2\times A_4$ x 8, $C_2^2\times C_{12}$ x 8, $C_2^2:C_{12}$ x 8, $C_6.D_4$ x 8, $C_6.D_4$ x 7, $C_{12}:C_4$ x 5, $S_3\times Q_8$ x 4, $C_6:Q_8$ x 4, $C_4:C_{12}$ x 4, $C_4\times A_4$ x 3, $A_4:C_4$ x 3, $C_{12}:C_4$ x 3, $C_6\times Q_8$, $C_4\times C_{12}$

Order 36: $C_6:S_3$ x 7, $C_6\times S_3$ x 4, $C_6^2$ x 3, $C_3:C_{12}$ x 2, $S_3^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 76, $C_4:D_4$ x 49, $D_4:C_2^2$ x 44, $D_4:C_2^2$ x 34, $C_4\times D_4$ x 34, $C_2^2\wr C_2$ x 30, $C_2^3:C_4$ x 18, $C_2^2.D_4$ x 12, $C_2^2:Q_8$ x 12, $C_4:D_4$ x 10, $C_2^3\times C_4$ x 8, $C_4^2:C_2$ x 8, $C_2^2.D_4$ x 6, $C_2^5$ x 4, $C_4^2:C_2$ x 4, $C_4\times Q_8$ x 3, $C_2\times C_4^2$ x 3, $C_2^2\times Q_8$ x 2, $C_4:Q_8$ x 2

Order 24: $C_2\times D_6$ x 61, $C_3:D_4$ x 42, $C_2^2\times C_6$ x 27, $C_4\times S_3$ x 26, $C_6:C_4$ x 24, $C_3\times D_4$ x 20, $C_2\times C_{12}$ x 19, $D_{12}$ x 19, $C_3:Q_8$ x 8, $C_2\times A_4$ x 7, $S_4$ x 7, $C_3\times Q_8$ x 3

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

Order 16: $C_2\times D_4$ x 160, $D_4:C_2$ x 58, $C_2^2\times C_4$ x 43, $C_2^2:C_4$ x 30, $C_2^4$ x 30, $C_4:C_4$ x 15, $C_2\times Q_8$ x 8, $C_4^2$ x 6

Order 12: $D_6$ x 49, $C_2\times C_6$ x 32, $C_3:C_4$ x 10, $C_{12}$ x 8, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 71, $D_4$ x 64, $C_2\times C_4$ x 53, $Q_8$ x 8

Order 6: $C_6$ x 15, $S_3$ x 12

Order 4: $C_2^2$ x 55, $C_4$ x 12

Order 3: $C_3$ x 3

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2304: $C_2^4.D_6^2$ ($C_1$)

Order 1152: $(C_6\times A_4).C_2^4$ ($C_2$) x 4, $(C_6\times S_4).C_2^3$ ($C_2$) x 4, $(C_6\times S_4).C_2^3$ ($C_2$) x 4, $(C_6\times A_4).C_2^4$ ($C_2$) x 4, $(C_6\times S_4).C_2^3$ ($C_2$) x 4, $(C_6\times S_4).C_2^3$ ($C_2$) x 4, $(C_6\times S_4).C_2^3$ ($C_2$) x 2, $(C_6\times S_4).C_2^3$ ($C_2$) x 2, $(C_6\times A_4).C_2^4$ ($C_2$), $(C_6\times A_4).C_2^4$ ($C_2$), $(C_6\times A_4).C_2^4$ ($C_2$)

Order 576: $\GL(2,\mathbb{Z}/4):S_3$ ($C_2^2$) x 28, $D_4\times C_3:S_4$ ($C_2^2$) x 8, $C_2^4:S_3^2$ ($C_2^2$) x 8, $S_3\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 8, $C_2^4.S_3^2$ ($C_2^2$) x 8, $\GL(2,\mathbb{Z}/4):S_3$ ($C_2^2$) x 8, $C_2^2:D_6^2$ ($C_2^2$) x 4, $C_6:D_4\times A_4$ ($C_2^2$) x 4, $A_4\times D_4:S_3$ ($C_2^2$) x 4, $S_3\times D_4\times A_4$ ($C_2^2$) x 4, $C_6\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 4, $\GL(2,\mathbb{Z}/4):C_6$ ($C_2^2$) x 4, $\GL(2,\mathbb{Z}/4):C_6$ ($C_2^2$) x 4, $C_6:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 4, $\GL(2,\mathbb{Z}/4):S_3$ ($C_2^2$) x 4, $C_2^4.S_3^2$ ($C_2^2$) x 4, $C_2^4.S_3^2$ ($C_2^2$) x 4, $S_4\times D_{12}$ ($C_2^2$) x 4, $S_3\times C_4:S_4$ ($C_2^2$) x 4, $C_4\times S_3\times S_4$ ($C_2^2$) x 4, $(C_4\times S_3):S_4$ ($C_2^2$) x 4, $(C_4\times S_3):S_4$ ($C_2^2$) x 4, $(C_{12}\times S_4):C_2$ ($C_2^2$) x 4, $C_2^2\times C_6:S_4$ ($C_2^2$) x 2, $A_4\times D_{12}:C_2$ ($C_2^2$) x 2, $C_6:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $\GL(2,\mathbb{Z}/4):C_6$ ($C_2^2$) x 2, $D_{12}:S_4$ ($C_2^2$) x 2, $D_{12}:S_4$ ($C_2^2$) x 2, $C_3:(Q_8\times S_4)$ ($C_2^2$) x 2, $C_{12}.(C_2\times S_4)$ ($C_2^2$) x 2, $C_2^4:C_6^2$ ($C_2^2$), $C_2\times C_{12}:S_4$ ($C_2^2$), $C_2\times C_{12}:S_4$ ($C_2^2$)

Order 384: $D_{12}:C_2^4$ ($S_3$), $\GL(2,\mathbb{Z}/4):C_2^2$ ($S_3$)

Order 288: $A_4\times C_3:D_4$ ($C_2^3$) x 12, $C_3\times \GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 12, $D_6:S_4$ ($C_2^3$) x 12, $C_2^3.S_3^2$ ($C_2^3$) x 12, $C_2^3.S_3^2$ ($C_2^3$) x 12, $D_6\times S_4$ ($C_2^3$) x 12, $C_2\times C_6:S_4$ ($C_2^3$) x 11, $C_3:\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 8, $C_2^3.C_6^2$ ($C_2^3$) x 4, $C_6:C_4\times A_4$ ($C_2^3$) x 4, $C_4\times S_3\times A_4$ ($C_2^3$) x 4, $C_{12}:S_4$ ($C_2^3$) x 4, $C_{12}:S_4$ ($C_2^3$) x 4, $C_2\times A_4:C_{12}$ ($C_2^3$) x 4, $C_{12}\times S_4$ ($C_2^3$) x 4, $D_6:S_4$ ($C_2^3$) x 4, $D_6.S_4$ ($C_2^3$) x 4, $C_2^3.S_3^2$ ($C_2^3$) x 4, $C_2^3.S_3^2$ ($C_2^3$) x 4, $C_2\times A_4\times D_6$ ($C_2^3$) x 4, $C_2\times C_6\times S_4$ ($C_2^3$) x 4, $A_4\times D_{12}$ ($C_2^3$) x 2, $A_4\times C_3:Q_8$ ($C_2^3$) x 2, $C_{12}:S_4$ ($C_2^3$) x 2, $C_{12}.S_4$ ($C_2^3$) x 2, $C_2^3:C_6^2$ ($C_2^3$) x 2, $C_2^3.C_6^2$ ($C_2^3$), $C_2\times C_6.S_4$ ($C_2^3$)

Order 192: $D_6:C_2^4$ ($D_6$) x 4, $C_{12}.C_2^4$ ($D_6$) x 4, $C_{12}:C_2^4$ ($D_6$) x 4, $C_2\times \GL(2,\mathbb{Z}/4)$ ($D_6$) x 4, $D_4:S_4$ ($D_6$) x 4, $D_4\times S_4$ ($D_6$) x 4, $D_{12}:C_2^3$ ($D_6$) x 2, $\GL(2,\mathbb{Z}/4):C_2$ ($D_6$) x 2, $C_{12}:C_2^4$ ($D_6$), $C_2^5:C_6$ ($D_6$)

Order 144: $C_6:S_4$ ($C_2^4$) x 6, $C_2^2:C_6^2$ ($C_2^4$) x 5, $A_4\times D_6$ ($C_2^4$) x 4, $C_6\times S_4$ ($C_2^4$) x 4, $A_4\times C_3:C_4$ ($C_2^4$) x 4, $A_4:C_{12}$ ($C_2^4$) x 4, $C_{12}\times A_4$ ($C_2^4$) x 2, $C_6.S_4$ ($C_2^4$) x 2

Order 96: $C_2^3:D_6$ ($C_2\times D_6$) x 12, $\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$) x 12, $C_2^3\times D_6$ ($C_2\times D_6$) x 4, $C_2^2\times S_4$ ($C_2\times D_6$) x 4, $C_{12}:C_2^3$ ($C_2\times D_6$) x 4, $C_6.C_2^4$ ($C_2\times D_6$) x 4, $C_{12}:C_2^3$ ($C_2\times D_6$) x 4, $D_4\times A_4$ ($C_2\times D_6$) x 4, $C_2^2.S_4$ ($C_2\times D_6$) x 4, $C_4\times S_4$ ($C_2\times D_6$) x 4, $C_2^4\times C_6$ ($C_2\times D_6$) x 2, $C_2^3\times A_4$ ($C_2\times D_6$) x 2, $C_2^2\times D_{12}$ ($C_2\times D_6$) x 2, $C_6.C_2^4$ ($C_2\times D_6$) x 2, $C_4:S_4$ ($C_2\times D_6$) x 2, $A_4:Q_8$ ($C_2\times D_6$) x 2, $C_2^3\times C_{12}$ ($C_2\times D_6$), $D_4:D_6$ ($S_4$), $C_2^3:C_{12}$ ($C_2\times D_6$)

Order 72: $C_3:S_4$ ($D_4:C_2^2$) x 2, $C_6\times A_4$ ($C_2^5$)

Order 64: $D_4\times C_2^3$ ($S_3^2$)

Order 48: $C_2^3\times C_6$ ($C_2^2\times D_6$) x 5, $C_2^2\times A_4$ ($C_2^2\times D_6$) x 5, $C_2^2\times D_6$ ($C_2^2\times D_6$) x 4, $C_2\times S_4$ ($C_2^2\times D_6$) x 4, $C_6:D_4$ ($C_2\times S_4$) x 4, $C_6.C_2^3$ ($C_2^2\times D_6$) x 4, $D_4:S_3$ ($C_2\times S_4$) x 4, $S_3\times D_4$ ($C_2\times S_4$) x 4, $A_4:C_4$ ($C_2^2\times D_6$) x 4, $C_2^2\times C_{12}$ ($C_2^2\times D_6$) x 2, $D_{12}:C_2$ ($C_2\times S_4$) x 2, $C_4\times A_4$ ($C_2^2\times D_6$) x 2, $C_6\times D_4$ ($C_2\times S_4$)

Order 36: $C_3\times A_4$ ($D_4:C_2^3$)

Order 32: $C_2^2\times D_4$ ($S_3\times D_6$) x 4, $C_2^5$ ($S_3\times D_6$) x 2, $C_2^3\times C_4$ ($S_3\times D_6$)

Order 24: $C_3:D_4$ ($C_2^2\times S_4$) x 12, $C_6:C_4$ ($C_2^2\times S_4$) x 4, $C_4\times S_3$ ($C_2^2\times S_4$) x 4, $C_2\times D_6$ ($C_2^2\times S_4$) x 4, $C_3\times D_4$ ($C_2^2\times S_4$) x 4, $D_{12}$ ($C_2^2\times S_4$) x 2, $C_3:Q_8$ ($C_2^2\times S_4$) x 2, $C_2^2\times C_6$ ($C_2^2\times S_4$) x 2, $C_2\times C_{12}$ ($C_2^2\times S_4$), $C_2^2\times C_6$ ($C_2^3\times D_6$), $C_2\times A_4$ ($C_2^3\times D_6$)

Order 16: $C_2^4$ ($D_6^2$) x 5, $C_2^2\times C_4$ ($D_6^2$) x 2, $C_2\times D_4$ ($S_3\times S_4$)

Order 12: $C_2\times C_6$ ($C_2^3\times S_4$) x 5, $D_6$ ($C_2^3\times S_4$) x 4, $C_3:C_4$ ($C_2^3\times S_4$) x 4, $C_{12}$ ($C_2^3\times S_4$) x 2, $C_2\times C_6$ ($D_{12}:C_2^3$), $A_4$ ($D_{12}:C_2^3$)

Order 8: $D_4$ ($D_6\times S_4$) x 4, $C_2^3$ ($D_6\times S_4$) x 2, $C_2^3$ ($C_2\times D_6^2$), $C_2\times C_4$ ($D_6\times S_4$)

Order 6: $C_6$ ($C_2^4\times S_4$)

Order 4: $C_2^2$ ($C_2^2:D_6^2$) x 5, $C_4$ ($C_2^2:D_6^2$) x 2, $C_2^2$ ($D_6^2:C_2^2$)

Order 3: $C_3$ ($\GL(2,\mathbb{Z}/4):C_2^3$)

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

Order 1: $C_1$ ($C_2^4.D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2304: $C_2^4.D_6^2$ ($C_1$)

Order 1152: $(C_6\times A_4).C_2^4$ ($C_2$), $(C_6\times A_4).C_2^4$ ($C_2$), $(C_6\times S_4).C_2^3$ ($C_2$), $(C_6\times S_4).C_2^3$ ($C_2$), $(C_6\times A_4).C_2^4$ ($C_2$), $(C_6\times S_4).C_2^3$ ($C_2$), $(C_6\times S_4).C_2^3$ ($C_2$), $(C_6\times A_4).C_2^4$ ($C_2$), $(C_6\times S_4).C_2^3$ ($C_2$), $(C_6\times S_4).C_2^3$ ($C_2$), $(C_6\times A_4).C_2^4$ ($C_2$)

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

Order 384: $D_{12}:C_2^4$ ($S_3$), $\GL(2,\mathbb{Z}/4):C_2^2$ ($S_3$)

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

Order 192: $C_{12}:C_2^4$ ($D_6$), $D_6:C_2^4$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $C_{12}:C_2^4$ ($D_6$), $D_{12}:C_2^3$ ($D_6$), $C_2^5:C_6$ ($D_6$), $C_2\times \GL(2,\mathbb{Z}/4)$ ($D_6$), $D_4:S_4$ ($D_6$), $D_4\times S_4$ ($D_6$), $\GL(2,\mathbb{Z}/4):C_2$ ($D_6$)

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

Order 96: $C_2^3:D_6$ ($C_2\times D_6$) x 2, $\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$) x 2, $C_2^4\times C_6$ ($C_2\times D_6$), $C_2^3\times D_6$ ($C_2\times D_6$), $C_2^3\times A_4$ ($C_2\times D_6$), $C_2^2\times S_4$ ($C_2\times D_6$), $C_{12}:C_2^3$ ($C_2\times D_6$), $C_2^3\times C_{12}$ ($C_2\times D_6$), $C_6.C_2^4$ ($C_2\times D_6$), $D_4:D_6$ ($S_4$), $C_2^2\times D_{12}$ ($C_2\times D_6$), $C_{12}:C_2^3$ ($C_2\times D_6$), $C_6.C_2^4$ ($C_2\times D_6$), $D_4\times A_4$ ($C_2\times D_6$), $C_2^3:C_{12}$ ($C_2\times D_6$), $C_2^2.S_4$ ($C_2\times D_6$), $C_4:S_4$ ($C_2\times D_6$), $C_4\times S_4$ ($C_2\times D_6$), $A_4:Q_8$ ($C_2\times D_6$)

Order 72: $C_6\times A_4$ ($C_2^5$), $C_3:S_4$ ($D_4:C_2^2$)

Order 64: $D_4\times C_2^3$ ($S_3^2$)

Order 48: $C_2^3\times C_6$ ($C_2^2\times D_6$) x 2, $C_2^2\times A_4$ ($C_2^2\times D_6$) x 2, $C_2^2\times D_6$ ($C_2^2\times D_6$), $C_2\times S_4$ ($C_2^2\times D_6$), $C_6\times D_4$ ($C_2\times S_4$), $C_2^2\times C_{12}$ ($C_2^2\times D_6$), $C_6:D_4$ ($C_2\times S_4$), $C_6.C_2^3$ ($C_2^2\times D_6$), $D_4:S_3$ ($C_2\times S_4$), $S_3\times D_4$ ($C_2\times S_4$), $D_{12}:C_2$ ($C_2\times S_4$), $C_4\times A_4$ ($C_2^2\times D_6$), $A_4:C_4$ ($C_2^2\times D_6$)

Order 36: $C_3\times A_4$ ($D_4:C_2^3$)

Order 32: $C_2^5$ ($S_3\times D_6$), $C_2^2\times D_4$ ($S_3\times D_6$), $C_2^3\times C_4$ ($S_3\times D_6$)

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

Order 16: $C_2^4$ ($D_6^2$) x 2, $C_2\times D_4$ ($S_3\times S_4$), $C_2^2\times C_4$ ($D_6^2$)

Order 12: $C_2\times C_6$ ($C_2^3\times S_4$) x 2, $C_2\times C_6$ ($D_{12}:C_2^3$), $D_6$ ($C_2^3\times S_4$), $A_4$ ($D_{12}:C_2^3$), $C_{12}$ ($C_2^3\times S_4$), $C_3:C_4$ ($C_2^3\times S_4$)

Order 8: $C_2^3$ ($C_2\times D_6^2$), $C_2^3$ ($D_6\times S_4$), $D_4$ ($D_6\times S_4$), $C_2\times C_4$ ($D_6\times S_4$)

Order 6: $C_6$ ($C_2^4\times S_4$)

Order 4: $C_2^2$ ($C_2^2:D_6^2$) x 2, $C_2^2$ ($D_6^2:C_2^2$), $C_4$ ($C_2^2:D_6^2$)

Order 3: $C_3$ ($\GL(2,\mathbb{Z}/4):C_2^3$)

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

Order 1: $C_1$ ($C_2^4.D_6^2$)

Series

Derived series

$C_2^4.D_6^2$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^4.D_6^2$

$\rhd$

$(C_6\times A_4).C_2^4$

$\rhd$

$C_6\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2\times C_6\times S_4$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^4.D_6^2$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $129 \times 129$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.