Abstract group 1536.408613558: $D_8:C_4\times S_4$

• Label: 1536.408613558
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $16$
• Trans deg.: $96$
• Rank: $4$

Group information

Description:	$D_8:C_4\times S_4$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$(C_2\times A_4).C_2^6.C_2^5$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	199	8	568	152	256	224	128	1536
Conjugacy classes	1	23	1	48	7	16	10	4	110
Divisions	1	23	1	32	7	12	6	3	85
Autjugacy classes	1	9	1	16	3	8	4	2	44

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	32	24	32	8	8	2	4	110
Irr. rational chars.	16	20	16	12	12	3	6	85

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$96$
Rank:	$4$
Inequivalent generating quadruples:	$33022080$

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

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{12}=d^{2}=e^{2}=f^{8}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2)(3,6)(4,5)(7,8)(14,16), (14,15,16), (1,3)(2,6)(4,8)(5,7)(9,10)(11,12) \!\cdots\! \rangle$
Direct product:	$S_4$ $\, \times\, $ $(D_8:C_4)$
Semidirect product:	$(D_8\times S_4)$ $\,\rtimes\,$ $C_4$	$D_8$ $\,\rtimes\,$ $(C_4\times S_4)$	$A_4$ $\,\rtimes\,$ $(C_4^2.D_4)$	$(C_2^3.D_8)$ $\,\rtimes\,$ $D_6$	all 51
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(A_4:C_4^2)$ . $D_4$	$(C_2^4.D_6)$ . $D_4$	$(C_2^3\times D_8)$ . $D_6$	$C_2^4$ . $(D_4\times D_6)$	all 57
Aut. group:	$\Aut(D_{16}:C_2^2)$	$\Aut(C_{16}:S_4)$	$\Aut(C_{16}.S_4)$	$\Aut(D_8:S_4)$	all 13

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 26076 subgroups in 3546 conjugacy classes, 218 normal (60 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $\GL(2,\mathbb{Z}/4):C_2^2$
Commutator:	$G' \simeq$ $C_4\times A_4$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2^3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2.C_2^4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $D_8:C_4\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $D_4\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^2\times D_8).C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 1536: $D_8:C_4\times S_4$

Order 768: $C_4\times D_4\times S_4$ x 2, $A_4.D_8:C_4$ x 2, $S_4\times D_4:C_4$ x 2, $A_4.D_8:C_4$ x 2, $C_2\times D_8\times S_4$, $A_4.D_8:C_4$, $S_4\times C_8:C_4$, $A_4.D_8:C_4$, $A_4.D_8:C_4$, $S_4\times C_2.\OD_{16}$, $C_2^2.C_2^5.C_6$

Order 512: $(C_2^2\times D_8).C_2^3$

Order 384: $D_8\times S_4$ x 8, $A_4\times D_4:C_4$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $(C_8\times A_4):C_4$ x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^5.D_6$ x 2, $C_4\times D_4\times A_4$ x 2, $C_2^5.D_6$ x 2, $C_4.\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_8\times S_4$ x 2, $C_4:(C_4\times S_4)$ x 2, $C_4:C_4\times S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_4$ x 2, $C_2^5.D_6$ x 2, $C_4^2:S_4$ x 2, $A_4\times C_8:C_4$, $A_4\times C_8:C_4$, $C_2^4.D_{12}$, $C_4^2.S_4$, $C_2\times A_4\times D_8$, $C_2^3:D_{24}$, $C_4^2\times S_4$, $D_8:C_4\times S_3$

Order 256: $C_2^2.D_4^2$ x 4, $D_4:(C_4\times D_4)$ x 4, $C_4^2.C_2^4$ x 2, $C_4\times D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $D_4^2:C_4$ x 2, $C_8:(C_4\times D_4)$ x 2, $C_4^3.C_2^2$ x 2, $D_4^2:C_2^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_4^3.C_2^2$, $D_8:C_4^2$, $C_4^3.C_2^2$

Order 192: $C_2^4.D_6$ x 17, $D_4\times S_4$ x 12, $D_4:S_4$ x 8, $C_8\times S_4$ x 6, $C_2.\GL(2,\mathbb{Z}/4)$ x 4, $C_8:S_4$ x 4, $A_4\times D_8$ x 4, $A_4:C_4^2$ x 3, $A_4\times C_4:C_4$ x 2, $C_2^4:C_{12}$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.D_{12}$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times A_4:C_8$ x 2, $C_4^2.D_6$ x 2, $D_4:(C_4\times S_3)$ x 2, $D_6.D_8$ x 2, $C_2^3\times S_4$ x 2, $C_2^5:C_6$ x 2, $C_2^4:C_{12}$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.S_4$ x 2, $C_2^3:D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_2^3:C_{24}$ x 2, $A_4\times C_4^2$, $D_8:C_{12}$, $C_{24}:(C_2\times C_4)$, $D_{24}:C_4$, $D_6.\SD_{16}$, $D_{24}:C_4$, $D_6.C_4^2$, $D_6\times D_8$

Order 128: $C_4^2.D_4$ x 26, $D_4\times D_8$ x 16, $(C_2\times C_4):D_8$ x 12, $C_4.(C_4\times D_4)$ x 8, $D_4:C_4^2$ x 8, $(C_2\times D_8):C_4$ x 6, $C_4^2:C_2^3$ x 6, $C_2.D_4^2$ x 6, $C_2.D_4^2$ x 6, $C_4^2:D_4$ x 6, $C_4.(C_4\times D_4)$ x 4, $C_4.(C_4\times D_4)$ x 4, $C_2^3.D_8$ x 4, $C_2^3.D_8$ x 4, $C_2^4.D_4$ x 4, $C_2^3.D_8$ x 4, $C_2^3.C_4^2$ x 4, $C_2^3:D_8$ x 4, $C_2^3.D_8$ x 4, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2^3.C_2^4$ x 4, $D_4\times C_4^2$ x 3, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.Q_8$ x 2, $C_4.(C_4\times Q_8)$ x 2, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_8:C_4^2$ x 2, $(C_4\times C_8):C_4$ x 2, $D_4:C_4^2$ x 2, $C_4^2.D_4$ x 2, $C_2^3\times D_8$ x 2, $C_2\times D_4^2$ x 2, $C_2^3:D_8$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2.D_4^2$ x 2, $C_4^3:C_2$ x 2, $C_4^2.Q_8$, $C_8:C_4^2$, $C_4^3.C_2$, $C_8:C_4^2$, $C_4^2.D_4$, $C_4^2.D_4$

Order 96: $C_4\times S_4$ x 20, $C_2^2\times S_4$ x 19, $C_2^3:C_{12}$ x 9, $C_2^2.S_4$ x 9, $\GL(2,\mathbb{Z}/4)$ x 8, $S_3\times D_8$ x 8, $D_4\times A_4$ x 6, $C_4:S_4$ x 6, $C_8\times A_4$ x 3, $A_4:C_8$ x 3, $D_6.D_4$ x 2, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_4\times D_{12}$ x 2, $D_4:C_{12}$ x 2, $C_6.D_8$ x 2, $D_{12}:C_4$ x 2, $C_2^3\times A_4$ x 2, $C_{24}:C_4$ x 2, $D_4\times D_6$ x 2, $C_{12}:C_2^3$ x 2, $D_4\times C_{12}$ x 2, $D_{12}:C_4$ x 2, $C_{12}.Q_8$ x 2, $C_2^3.D_6$ x 2, $C_6:D_8$ x 2, $C_{12}:D_4$ x 2, $C_8\times D_6$ x 2, $D_{12}:C_4$ x 2, $S_3\times C_4^2$, $C_8:C_{12}$, $C_8:C_{12}$, $C_{24}:C_4$, $C_6\times D_8$, $C_2\times D_{24}$, $C_6.C_4^2$

Order 64: $C_4^2:C_2^2$ x 73, $D_8:C_4$ x 51, $C_2^2.D_8$ x 38, $C_2^2\times D_8$ x 33, $D_4:D_4$ x 32, $D_4^2$ x 24, $C_2^3.D_4$ x 20, $C_2^2:C_4^2$ x 16, $C_8:D_4$ x 16, $C_4:D_8$ x 16, $C_4.C_4^2$ x 14, $C_2^3.C_2^3$ x 14, $C_4.C_4^2$ x 14, $C_2^3.D_4$ x 14, $C_2^3.D_4$ x 12, $C_8\times D_4$ x 12, $C_4:C_4^2$ x 9, $C_2^3.C_2^3$ x 8, $C_2^4:C_4$ x 8, $C_4:D_8$ x 8, $C_4\times D_8$ x 8, $C_2^3.C_2^3$ x 6, $C_2^3.D_4$ x 6, $D_4\times C_2^3$ x 6, $C_2^3:D_4$ x 6, $C_2^3.D_4$ x 6, $C_2^2.C_4^2$ x 6, $C_2^3:C_8$ x 4, $C_2^4:C_4$ x 4, $C_2^4\times C_4$ x 4, $C_2^3\times C_8$ x 4, $C_2^3:D_4$ x 4, $C_2^2\times C_4^2$ x 4, $C_2\times C_4\times C_8$ x 2, $C_4^2:C_4$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2.C_4$ x 2, $C_4^3$, $C_2^2.D_8$

Order 48: $C_2\times S_4$ x 30, $C_4\times D_6$ x 17, $S_3\times D_4$ x 12, $C_2^2\times A_4$ x 9, $C_3:D_8$ x 8, $S_3\times C_8$ x 6, $C_4\times A_4$ x 6, $A_4:C_4$ x 6, $D_{24}$ x 4, $C_3\times D_8$ x 4, $D_6:C_4$ x 4, $C_{12}:C_4$ x 3, $C_6:C_8$ x 2, $C_2^2\times D_6$ x 2, $C_6\times D_4$ x 2, $C_2^2\times C_{12}$ x 2, $C_6:D_4$ x 2, $C_6.C_2^3$ x 2, $C_2\times D_{12}$ x 2, $C_2\times C_{24}$ x 2, $C_4:C_{12}$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_6.D_4$ x 2, $C_4\times C_{12}$

Order 32: $C_4\times D_4$ x 152, $C_2\times D_8$ x 99, $C_2^2\times D_4$ x 87, $C_2^3:C_4$ x 72, $D_4:C_4$ x 58, $C_2^3\times C_4$ x 56, $C_4:D_4$ x 40, $C_2^2.D_4$ x 39, $C_2^2\wr C_2$ x 32, $C_2\times C_4^2$ x 31, $C_2^2\times C_8$ x 27, $C_2.C_4^2$ x 20, $C_8:C_4$ x 16, $C_8:C_4$ x 15, $C_2^2:C_8$ x 12, $C_4:D_4$ x 12, $C_4\times C_8$ x 6, $C_4:C_8$ x 6, $C_2^5$ x 4, $C_8:C_4$ x 4

Order 24: $C_4\times S_3$ x 20, $C_2\times D_6$ x 19, $C_2\times C_{12}$ x 9, $C_6:C_4$ x 9, $C_3:D_4$ x 8, $S_4$ x 8, $C_2\times A_4$ x 7, $D_{12}$ x 6, $C_3\times D_4$ x 6, $C_{24}$ x 3, $C_3:C_8$ x 3, $C_2^2\times C_6$ x 2

Order 16: $C_2\times D_4$ x 248, $C_2^2\times C_4$ x 191, $C_2^2:C_4$ x 98, $D_8$ x 52, $C_2^4$ x 52, $C_4:C_4$ x 48, $C_2\times C_8$ x 41, $C_4^2$ x 35

Order 12: $D_6$ x 30, $C_2\times C_6$ x 9, $C_{12}$ x 6, $C_3:C_4$ x 6, $A_4$

Order 8: $C_2\times C_4$ x 171, $C_2^3$ x 159, $D_4$ x 126, $C_8$ x 12

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$ x 124, $C_4$ x 32

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $D_8:C_4\times S_4$

Order 768: $C_2\times D_8\times S_4$, $C_4\times D_4\times S_4$, $A_4.D_8:C_4$, $A_4.D_8:C_4$, $S_4\times C_8:C_4$, $A_4.D_8:C_4$, $S_4\times D_4:C_4$, $A_4.D_8:C_4$, $A_4.D_8:C_4$, $S_4\times C_2.\OD_{16}$, $C_2^2.C_2^5.C_6$

Order 512: $(C_2^2\times D_8).C_2^3$

Order 384: $(C_8\times A_4):C_4$ x 2, $C_2\times C_8\times S_4$ x 2, $A_4\times C_8:C_4$, $A_4\times D_4:C_4$, $A_4\times C_8:C_4$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_{12}$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_4^2.S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^5.D_6$, $C_2\times A_4\times D_8$, $C_4\times D_4\times A_4$, $C_2^5.D_6$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4\times \GL(2,\mathbb{Z}/4)$, $D_8\times S_4$, $C_2^3:D_{24}$, $C_4:(C_4\times S_4)$, $C_4:C_4\times S_4$, $\GL(2,\mathbb{Z}/4):C_4$, $C_2^5.D_6$, $C_4^2:S_4$, $C_4^2\times S_4$, $D_8:C_4\times S_3$

Order 256: $C_4^2.C_2^4$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_8:(C_4\times D_4)$ x 2, $D_4:(C_4\times D_4)$ x 2, $D_4^2:C_2^2$, $C_4\times D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $D_4^2:C_4$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $D_8:C_4^2$, $C_4^3.C_2^2$

Order 192: $C_2^4.D_6$ x 6, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $A_4:C_4^2$ x 2, $C_2\times A_4:C_8$ x 2, $C_8\times S_4$ x 2, $D_4\times S_4$ x 2, $C_2^3:C_{24}$ x 2, $A_4\times C_4:C_4$, $C_2^4:C_{12}$, $A_4\times C_4^2$, $C_2.\GL(2,\mathbb{Z}/4)$, $D_4:S_4$, $C_2^3.D_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_8:S_4$, $D_8:C_{12}$, $C_{24}:(C_2\times C_4)$, $C_4^2.D_6$, $D_{24}:C_4$, $D_6.\SD_{16}$, $D_4:(C_4\times S_3)$, $D_6.D_8$, $D_{24}:C_4$, $D_6.C_4^2$, $C_2^3\times S_4$, $C_2^5:C_6$, $C_2^4:C_{12}$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3.S_4$, $C_2^3:D_{12}$, $D_6\times D_8$, $C_4^2:D_6$, $A_4\times D_8$

Order 128: $C_4^2.D_4$ x 19, $(C_2\times C_4):D_8$ x 6, $(C_2\times D_8):C_4$ x 5, $C_4.(C_4\times D_4)$ x 4, $D_4:C_4^2$ x 4, $C_2^3.C_4^2$ x 4, $C_4^2:C_2^3$ x 3, $C_2.D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_4^2:D_4$ x 3, $C_4.(C_4\times Q_8)$ x 2, $C_4.(C_4\times D_4)$ x 2, $C_4.(C_4\times D_4)$ x 2, $C_2^3.D_8$ x 2, $C_2^3.D_8$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3.D_8$ x 2, $C_8:C_4^2$ x 2, $(C_4\times C_8):C_4$ x 2, $C_4^2.D_4$ x 2, $C_2^3\times D_8$ x 2, $D_4\times D_8$ x 2, $C_2^3:D_8$ x 2, $C_2^3:D_8$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^3.D_8$ x 2, $C_2^3.C_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $D_4\times C_4^2$ x 2, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.Q_8$, $C_4^2.Q_8$, $C_4^2.D_4$, $C_8:C_4^2$, $C_4^3.C_2$, $D_4:C_4^2$, $C_8:C_4^2$, $C_2\times D_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_2.D_4^2$, $C_4^3:C_2$

Order 96: $C_4\times S_4$ x 6, $C_2^2\times S_4$ x 5, $C_2^3:C_{12}$ x 4, $C_2^2.S_4$ x 4, $C_8\times A_4$ x 2, $A_4:C_8$ x 2, $C_{24}:C_4$ x 2, $D_4\times A_4$ x 2, $C_4:S_4$ x 2, $C_8\times D_6$ x 2, $D_6.D_4$, $C_2^3.D_6$, $D_6.D_4$, $C_4\times D_{12}$, $S_3\times C_4^2$, $C_8:C_{12}$, $D_4:C_{12}$, $C_8:C_{12}$, $C_6.D_8$, $D_{12}:C_4$, $C_{24}:C_4$, $C_2^3\times A_4$, $D_4\times D_6$, $C_{12}:C_2^3$, $\GL(2,\mathbb{Z}/4)$, $C_6\times D_8$, $D_4\times C_{12}$, $D_{12}:C_4$, $C_{12}.Q_8$, $C_2^3.D_6$, $C_6:D_8$, $C_{12}:D_4$, $S_3\times D_8$, $C_2\times D_{24}$, $D_{12}:C_4$, $C_6.C_4^2$

Order 64: $D_8:C_4$ x 32, $C_4^2:C_2^2$ x 31, $C_2^2.D_8$ x 19, $C_4.C_4^2$ x 14, $C_2^2\times D_8$ x 13, $C_2^3.D_4$ x 10, $C_2^3.D_4$ x 10, $C_2^2:C_4^2$ x 9, $C_2^3.C_2^3$ x 7, $C_4.C_4^2$ x 7, $C_2^3.D_4$ x 6, $C_2^2.C_4^2$ x 6, $C_4:C_4^2$ x 5, $C_2^3:C_8$ x 4, $C_2^3.C_2^3$ x 4, $C_2^3\times C_8$ x 4, $D_4^2$ x 4, $C_2^4:C_4$ x 4, $C_8\times D_4$ x 4, $C_2^3.C_2^3$ x 3, $C_2^3.D_4$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_2^3.D_4$ x 3, $C_2^2\times C_4^2$ x 3, $C_8:D_4$ x 3, $D_4:D_4$ x 3, $C_2\times C_4\times C_8$ x 2, $C_2^4:C_4$ x 2, $C_2^4\times C_4$ x 2, $C_2^3:D_4$ x 2, $C_4^2.C_4$ x 2, $C_4^2:C_4$, $C_4^3$, $C_4^2:C_2^2$, $C_4:D_8$, $C_4:D_8$, $C_4\times D_8$, $C_2^2.D_8$

Order 48: $C_2\times S_4$ x 6, $C_4\times D_6$ x 6, $C_2^2\times A_4$ x 4, $C_4\times A_4$ x 4, $A_4:C_4$ x 3, $C_6:C_8$ x 2, $S_3\times C_8$ x 2, $S_3\times D_4$ x 2, $C_2\times C_{24}$ x 2, $D_6:C_4$ x 2, $C_{12}:C_4$ x 2, $D_{24}$, $C_2^2\times D_6$, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_6.C_2^3$, $C_2\times D_{12}$, $C_3\times D_8$, $C_4:C_{12}$, $C_2^2:C_{12}$, $C_4\times C_{12}$, $C_6.D_4$, $C_3:D_8$, $C_{12}:C_4$, $C_6.D_4$

Order 32: $C_4\times D_4$ x 50, $C_2^3:C_4$ x 32, $C_2^2\times D_4$ x 27, $C_2\times D_8$ x 24, $C_2^3\times C_4$ x 23, $C_2^2\times C_8$ x 20, $D_4:C_4$ x 19, $C_2^2.D_4$ x 17, $C_2\times C_4^2$ x 16, $C_8:C_4$ x 14, $C_2.C_4^2$ x 10, $C_8:C_4$ x 10, $C_4:D_4$ x 8, $C_2^2:C_8$ x 6, $C_2^2\wr C_2$ x 4, $C_2^5$ x 2, $C_4:D_4$ x 2, $C_4\times C_8$ x 2, $C_4:C_8$ x 2, $C_8:C_4$

Order 24: $C_4\times S_3$ x 6, $C_2\times D_6$ x 5, $C_2\times C_{12}$ x 4, $C_6:C_4$ x 4, $C_2\times A_4$ x 3, $D_{12}$ x 2, $C_{24}$ x 2, $S_4$ x 2, $C_3\times D_4$ x 2, $C_3:C_8$ x 2, $C_3:D_4$, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 71, $C_2\times D_4$ x 54, $C_2^2:C_4$ x 34, $C_2\times C_8$ x 28, $C_4:C_4$ x 19, $C_2^4$ x 19, $C_4^2$ x 17, $D_8$ x 9

Order 12: $D_6$ x 6, $C_2\times C_6$ x 4, $C_{12}$ x 4, $C_3:C_4$ x 3, $A_4$

Order 8: $C_2\times C_4$ x 64, $C_2^3$ x 46, $D_4$ x 24, $C_8$ x 8

Order 6: $C_6$ x 3, $S_3$ x 2

Order 4: $C_2^2$ x 37, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $D_8:C_4\times S_4$ ($C_1$)

Order 768: $C_4\times D_4\times S_4$ ($C_2$) x 2, $A_4.D_8:C_4$ ($C_2$) x 2, $S_4\times D_4:C_4$ ($C_2$) x 2, $A_4.D_8:C_4$ ($C_2$) x 2, $C_2\times D_8\times S_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $S_4\times C_8:C_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $S_4\times C_2.\OD_{16}$ ($C_2$), $C_2^2.C_2^5.C_6$ ($C_2$)

Order 384: $D_8\times S_4$ ($C_4$) x 8, $A_4\times D_4:C_4$ ($C_2^2$) x 2, $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $(C_8\times A_4):C_4$ ($C_2^2$) x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ ($C_2^2$) x 2, $C_4\times D_4\times A_4$ ($C_2^2$) x 2, $C_2^5.D_6$ ($C_2^2$) x 2, $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $C_2\times C_8\times S_4$ ($C_2^2$) x 2, $C_4:(C_4\times S_4)$ ($C_2^2$) x 2, $C_4:C_4\times S_4$ ($C_2^2$) x 2, $C_4^2:S_4$ ($C_2^2$) x 2, $A_4\times C_8:C_4$ ($C_2^2$), $A_4\times C_8:C_4$ ($C_2^2$), $C_2^4.D_{12}$ ($C_2^2$), $C_4^2.S_4$ ($C_2^2$), $C_2\times A_4\times D_8$ ($C_2^2$), $C_2^3:D_{24}$ ($C_2^2$), $C_4^2\times S_4$ ($C_2^2$)

Order 256: $C_4^2.C_2^4$ ($S_3$)

Order 192: $D_4:S_4$ ($C_2\times C_4$) x 8, $D_4\times S_4$ ($C_2\times C_4$) x 8, $C_8:S_4$ ($C_2\times C_4$) x 4, $C_8\times S_4$ ($C_2\times C_4$) x 4, $A_4\times D_8$ ($C_2\times C_4$) x 4, $A_4\times C_4:C_4$ ($C_2^3$) x 2, $C_2^3.D_{12}$ ($C_2^3$) x 2, $A_4:C_4^2$ ($D_4$) x 2, $C_2\times A_4:C_8$ ($C_2^3$) x 2, $C_2^5:C_6$ ($C_2^3$) x 2, $C_2^3:D_{12}$ ($C_2^3$) x 2, $C_2^4.D_6$ ($D_4$) x 2, $C_2^3:C_{24}$ ($C_2^3$) x 2, $A_4\times C_4^2$ ($C_2^3$), $A_4:C_4^2$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$)

Order 128: $C_4^2:C_2^3$ ($D_6$) x 2, $C_2^3.D_8$ ($D_6$) x 2, $C_2^3\times D_8$ ($D_6$), $C_2^4.D_4$ ($D_6$), $C_2^3.C_4^2$ ($D_6$)

Order 96: $D_4\times A_4$ ($C_2^2\times C_4$) x 4, $C_4:S_4$ ($C_2^2\times C_4$) x 4, $C_2^2.S_4$ ($C_2\times D_4$) x 3, $C_8\times A_4$ ($C_2^2\times C_4$) x 2, $A_4:C_8$ ($C_2^2\times C_4$) x 2, $C_2^3:C_{12}$ ($C_2\times D_4$) x 2, $C_4\times S_4$ ($D_4:C_2$) x 2, $C_4\times S_4$ ($C_2^2\times C_4$) x 2, $C_2^2\times S_4$ ($C_2\times D_4$), $C_2^3:C_{12}$ ($C_2^4$)

Order 64: $C_2^2\times D_8$ ($C_4\times S_3$) x 4, $D_4\times C_2^3$ ($C_2\times D_6$) x 2, $C_2^3\times C_8$ ($C_2\times D_6$) x 2, $C_2^3.D_4$ ($C_2\times D_6$) x 2, $C_2^2\times C_4^2$ ($C_2\times D_6$), $D_8:C_4$ ($S_4$)

Order 48: $C_2\times S_4$ ($D_8:C_2$) x 4, $C_2\times S_4$ ($C_4\times D_4$) x 2, $A_4:C_4$ ($C_4\times D_4$) x 2, $C_2^2\times A_4$ ($C_2^2\times D_4$), $C_4\times A_4$ ($D_4:C_2^2$), $C_4\times A_4$ ($C_2^3\times C_4$)

Order 32: $C_2^2\times D_4$ ($C_4\times D_6$) x 4, $D_4:C_4$ ($C_2\times S_4$) x 2, $C_2^3\times C_4$ ($S_3\times D_4$) x 2, $C_2^2\times C_8$ ($C_4\times D_6$) x 2, $C_4\times D_4$ ($C_2\times S_4$) x 2, $C_2^3\times C_4$ ($C_2^2\times D_6$), $C_8:C_4$ ($C_2\times S_4$), $C_2\times D_8$ ($C_2\times S_4$), $C_8:C_4$ ($C_2\times S_4$)

Order 24: $S_4$ ($D_8:C_4$) x 4, $C_2\times A_4$ ($C_8:C_2^3$) x 2, $C_2\times A_4$ ($C_4^2:C_2^2$)

Order 16: $D_8$ ($C_4\times S_4$) x 4, $C_2\times C_8$ ($C_2^2\times S_4$) x 2, $C_4:C_4$ ($C_2^2\times S_4$) x 2, $C_2\times D_4$ ($C_2^2\times S_4$) x 2, $C_4^2$ ($C_2^2\times S_4$), $C_2^4$ ($D_4\times D_6$), $C_2^2\times C_4$ ($D_4:D_6$), $C_2^2\times C_4$ ($C_{12}:C_2^3$)

Order 12: $A_4$ ($C_4^2.D_4$)

Order 8: $D_4$ ($C_2^4.D_6$) x 4, $C_2^3$ ($D_8:D_6$) x 2, $C_2\times C_4$ ($D_4\times S_4$) x 2, $C_8$ ($C_2^4.D_6$) x 2, $C_2^3$ ($C_4^2:D_6$), $C_2\times C_4$ ($C_2^3\times S_4$)

Order 4: $C_2^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_2^2$ ($D_8:C_4\times S_3$), $C_4$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_4$ ($C_2^5.D_6$)

Order 2: $C_2$ ($D_8:C_2\times S_4$) x 2, $C_2$ ($C_4\times D_4\times S_4$)

Order 1: $C_1$ ($D_8:C_4\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $D_8:C_4\times S_4$ ($C_1$)

Order 768: $C_2\times D_8\times S_4$ ($C_2$), $C_4\times D_4\times S_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $S_4\times C_8:C_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $S_4\times D_4:C_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $A_4.D_8:C_4$ ($C_2$), $S_4\times C_2.\OD_{16}$ ($C_2$), $C_2^2.C_2^5.C_6$ ($C_2$)

Order 384: $(C_8\times A_4):C_4$ ($C_2^2$) x 2, $C_2\times C_8\times S_4$ ($C_2^2$) x 2, $A_4\times C_8:C_4$ ($C_2^2$), $A_4\times D_4:C_4$ ($C_2^2$), $A_4\times C_8:C_4$ ($C_2^2$), $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_2^4.D_{12}$ ($C_2^2$), $C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_4^2.S_4$ ($C_2^2$), $\GL(2,\mathbb{Z}/4):C_2^2$ ($C_2^2$), $C_2\times A_4\times D_8$ ($C_2^2$), $C_4\times D_4\times A_4$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $C_4.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $D_8\times S_4$ ($C_4$), $C_2^3:D_{24}$ ($C_2^2$), $C_4:(C_4\times S_4)$ ($C_2^2$), $C_4:C_4\times S_4$ ($C_2^2$), $C_4^2:S_4$ ($C_2^2$), $C_4^2\times S_4$ ($C_2^2$)

Order 256: $C_4^2.C_2^4$ ($S_3$)

Order 192: $C_2\times A_4:C_8$ ($C_2^3$) x 2, $C_2^3:C_{24}$ ($C_2^3$) x 2, $A_4\times C_4:C_4$ ($C_2^3$), $A_4\times C_4^2$ ($C_2^3$), $D_4:S_4$ ($C_2\times C_4$), $C_2^3.D_{12}$ ($C_2^3$), $A_4:C_4^2$ ($C_2^3$), $A_4:C_4^2$ ($D_4$), $C_8:S_4$ ($C_2\times C_4$), $C_8\times S_4$ ($C_2\times C_4$), $C_2^5:C_6$ ($C_2^3$), $D_4\times S_4$ ($C_2\times C_4$), $C_2^3:D_{12}$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($D_4$), $A_4\times D_8$ ($C_2\times C_4$)

Order 128: $C_2^3\times D_8$ ($D_6$), $C_4^2:C_2^3$ ($D_6$), $C_2^4.D_4$ ($D_6$), $C_2^3.D_8$ ($D_6$), $C_2^3.C_4^2$ ($D_6$)

Order 96: $C_2^2.S_4$ ($C_2\times D_4$) x 2, $C_8\times A_4$ ($C_2^2\times C_4$), $A_4:C_8$ ($C_2^2\times C_4$), $C_2^2\times S_4$ ($C_2\times D_4$), $D_4\times A_4$ ($C_2^2\times C_4$), $C_2^3:C_{12}$ ($C_2^4$), $C_2^3:C_{12}$ ($C_2\times D_4$), $C_4:S_4$ ($C_2^2\times C_4$), $C_4\times S_4$ ($D_4:C_2$), $C_4\times S_4$ ($C_2^2\times C_4$)

Order 64: $C_2^3\times C_8$ ($C_2\times D_6$) x 2, $D_4\times C_2^3$ ($C_2\times D_6$), $C_2^2\times D_8$ ($C_4\times S_3$), $C_2^3.D_4$ ($C_2\times D_6$), $C_2^2\times C_4^2$ ($C_2\times D_6$), $D_8:C_4$ ($S_4$)

Order 48: $C_2^2\times A_4$ ($C_2^2\times D_4$), $C_2\times S_4$ ($D_8:C_2$), $C_2\times S_4$ ($C_4\times D_4$), $C_4\times A_4$ ($D_4:C_2^2$), $C_4\times A_4$ ($C_2^3\times C_4$), $A_4:C_4$ ($C_4\times D_4$)

Order 32: $D_4:C_4$ ($C_2\times S_4$), $C_2^2\times D_4$ ($C_4\times D_6$), $C_2^3\times C_4$ ($C_2^2\times D_6$), $C_2^3\times C_4$ ($S_3\times D_4$), $C_8:C_4$ ($C_2\times S_4$), $C_2\times D_8$ ($C_2\times S_4$), $C_2^2\times C_8$ ($C_4\times D_6$), $C_4\times D_4$ ($C_2\times S_4$), $C_8:C_4$ ($C_2\times S_4$)

Order 24: $C_2\times A_4$ ($C_8:C_2^3$), $C_2\times A_4$ ($C_4^2:C_2^2$), $S_4$ ($D_8:C_4$)

Order 16: $C_2\times C_8$ ($C_2^2\times S_4$) x 2, $D_8$ ($C_4\times S_4$), $C_4:C_4$ ($C_2^2\times S_4$), $C_4^2$ ($C_2^2\times S_4$), $C_2^4$ ($D_4\times D_6$), $C_2\times D_4$ ($C_2^2\times S_4$), $C_2^2\times C_4$ ($D_4:D_6$), $C_2^2\times C_4$ ($C_{12}:C_2^3$)

Order 12: $A_4$ ($C_4^2.D_4$)

Order 8: $C_2^3$ ($D_8:D_6$), $C_2^3$ ($C_4^2:D_6$), $D_4$ ($C_2^4.D_6$), $C_2\times C_4$ ($C_2^3\times S_4$), $C_2\times C_4$ ($D_4\times S_4$), $C_8$ ($C_2^4.D_6$)

Order 4: $C_2^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_2^2$ ($D_8:C_4\times S_3$), $C_4$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_4$ ($C_2^5.D_6$)

Order 2: $C_2$ ($D_8:C_2\times S_4$), $C_2$ ($C_4\times D_4\times S_4$)

Order 1: $C_1$ ($D_8:C_4\times S_4$)

Series

Derived series

$D_8:C_4\times S_4$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$D_8:C_4\times S_4$

$\rhd$

$C_2^2.C_2^5.C_6$

$\rhd$

$C_2\times A_4\times D_8$

$\rhd$

$C_2^5:C_6$

$\rhd$

$C_2^3:C_{12}$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$D_8:C_4\times S_4$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4^2$

$\lhd$

$D_8:C_4$

Character theory

Complex character table

See the $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $85 \times 85$ rational character table.