Abstract group 1536.10766179: $(C_2\times C_{16}).D_{24}$

• Label: 1536.10766179
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $39$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$(C_2\times C_{16}).D_{24}$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_3:(C_2^2.C_2^6.C_2^6)$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	55	2	168	14	288	48	512	192	256	1536
Conjugacy classes	1	6	1	17	7	56	24	72	96	128	408
Divisions	1	6	1	11	5	18	8	10	16	8	84
Autjugacy classes	1	5	1	8	4	11	6	6	9	4	55

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	376	0	0	0	0	408
Irr. rational chars.	8	12	20	20	16	8	84

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$21504$

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:	$\langle a, b, c, d \mid b^{8}=c^{2}=d^{48}=[a,c]=[b,c]=[b,d]=[c,d]=1, a^{2}=d^{30}, b^{a}=b^{7}d^{30}, d^{a}=cd^{17} \rangle$
Permutation group:	Degree $39$ $\langle(1,2,5,12,8,13,17,28,3,14,16,9,7,23,6,11)(4,18,19,26,22,32,30,27,15,24,29,31,21,25,20,10) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.Q_{32})$ $\,\rtimes\,$ $C_8$ (2)	$(D_{24}:C_4)$ $\,\rtimes\,$ $C_8$ (2)	$(C_2\times C_8\times C_{16})$ $\,\rtimes\,$ $S_3$	$(C_2\times C_8\times C_{48})$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_8\times C_{24})$ . $D_4$ (2)	$(C_2\times C_{48})$ . $D_8$ (2)	$C_{24}$ . $(C_4.D_8)$	$C_{24}$ . $(C_8:D_4)$	all 120

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 2234 subgroups in 538 conjugacy classes, 161 normal (131 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $C_2\times D_{24}$
Commutator:	$G' \simeq$ $C_2\times C_{24}$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4\times C_8$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_8\times C_{48}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2\times C_{16}).D_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_8\times D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2\times C_8\times C_{16}).C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1536: $(C_2\times C_{16}).D_{24}$

Order 768: $C_4.(C_8\times D_{12})$, $C_{12}.(C_8\times Q_8)$, $D_{24}.C_4^2$, $C_8^2:D_6$, $(C_8\times C_{24}):C_4$, $C_{24}.(C_4\times C_8)$, $C_2\times C_8\times C_{48}$

Order 512: $(C_2\times C_8\times C_{16}).C_2$

Order 384: $C_2\times C_4\times C_{48}$ x 2, $C_8\times C_{48}$ x 2, $C_2\times C_{24}.C_8$ x 2, $D_{24}:C_8$ x 2, $C_2\times C_8\times C_{24}$, $C_2\times C_4.D_{24}$, $C_2^3.D_{24}$, $C_2\times C_8.D_{12}$, $(C_2^2\times C_{48}):C_2$, $(C_2\times C_{16}).D_6$, $C_{24}.C_4^2$, $C_{24}.C_4^2$, $(C_2\times C_{48}):C_4$, $C_{24}.C_4^2$, $C_2\times D_{24}:C_4$

Order 256: $C_4.(C_8\times D_4)$, $C_4.(C_8\times Q_8)$, $D_8.C_4^2$, $C_8^2:C_2^2$, $C_2\times C_8\times C_{16}$, $C_8^2:C_4$, $C_8.(C_4\times C_8)$

Order 192: $C_2^2\times C_{48}$ x 4, $D_{24}:C_4$ x 4, $C_{24}.C_8$ x 4, $C_4\times C_{48}$ x 4, $C_8\times C_{24}$ x 3, $C_2\times C_4\times C_{24}$ x 2, $C_4.D_{24}$ x 2, $D_{24}:C_4$ x 2, $C_8.D_{12}$ x 2, $C_2\times C_{24}.C_4$ x 2, $D_6:C_{16}$ x 2, $C_6:\OD_{32}$ x 2, $C_6.Q_{32}$ x 2, $C_{24}.D_4$ x 2, $C_{12}.C_4^2$, $C_2^3.D_{12}$, $C_{12}.C_4^2$, $C_4^2:D_6$, $D_{24}:C_2^2$, $C_{12}.C_2^4$

Order 128: $C_2^3.\OD_{16}$ x 2, $C_2\times C_4\times C_{16}$ x 2, $C_8\wr C_2$ x 2, $C_8\times C_{16}$ x 2, $C_8.(C_2\times D_4)$, $C_2^3.\SD_{16}$, $C_2^3.D_8$, $\OD_{32}:C_2^2$, $(C_2^2\times C_{16}):C_2$, $C_8.C_4^2$, $C_2\times C_8^2$, $C_4^2.C_2^3$, $C_8.C_4^2$, $C_8.C_4^2$, $\OD_{32}:C_4$

Order 96: $C_2\times C_{48}$ x 16, $C_4\times C_{24}$ x 8, $C_{24}.C_4$ x 4, $C_3:\OD_{32}$ x 4, $C_2^2\times C_{24}$ x 4, $D_{24}:C_2$ x 4, $C_8.D_6$ x 4, $C_{24}:C_4$ x 2, $C_6:C_{16}$ x 2, $C_6:\OD_{16}$ x 2, $D_{12}:C_4$ x 2, $C_{24}:C_4$, $C_{24}:C_4$, $D_{12}:C_2^2$, $C_{24}:C_4$, $C_2\times C_4\times C_{12}$, $C_2^3.D_6$, $C_6:Q_{16}$, $C_2\times D_{24}$, $C_8:D_6$, $C_8:D_6$, $C_8\times D_6$

Order 64: $C_8.C_8$ x 4, $C_4\times C_{16}$ x 4, $C_2^2\times C_{16}$ x 4, $D_8:C_4$ x 4, $C_8^2$ x 3, $C_2\times C_4\times C_8$ x 2, $C_4:C_{16}$ x 2, $C_4.D_8$ x 2, $Q_{16}:C_4$ x 2, $D_8:C_4$ x 2, $\OD_{32}:C_2$ x 2, $C_2^2:C_{16}$ x 2, $C_2\times \OD_{32}$ x 2, $C_2^3.Q_8$ x 2, $\OD_{16}:C_4$, $D_8:C_2^2$, $C_8.C_2^3$, $C_4^2:C_4$, $C_2^3.D_4$, $C_4^2:C_2^2$

Order 48: $C_2\times C_{24}$ x 24, $C_{48}$ x 8, $D_{12}:C_2$ x 4, $C_3:\OD_{16}$ x 4, $C_4\times C_{12}$ x 3, $C_6:C_8$ x 2, $C_3:Q_{16}$ x 2, $D_{24}$ x 2, $C_{24}:C_2$ x 2, $C_{24}:C_2$ x 2, $C_2^2\times C_{12}$ x 2, $S_3\times C_8$ x 2, $C_{12}:C_4$ x 2, $C_3:C_{16}$ x 2, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_6:Q_8$, $C_6.D_4$, $C_{12}:C_4$

Order 32: $C_2\times C_{16}$ x 18, $C_4\times C_8$ x 9, $C_2^2\times C_8$ x 5, $D_8:C_2$ x 4, $\OD_{16}:C_2$ x 4, $\OD_{32}$ x 4, $C_8.C_4$ x 4, $C_2\times \OD_{16}$ x 3, $C_8:C_4$ x 2, $C_4\wr C_2$ x 2, $D_4:C_2^2$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_8:C_4$, $C_2\times D_8$, $C_4^2:C_2$, $C_2\times C_4^2$, $C_8:C_4$

Order 24: $C_{24}$ x 16, $C_2\times C_{12}$ x 12, $C_6:C_4$ x 3, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_3:C_8$ x 2, $C_2^2\times C_6$, $C_2\times D_6$

Order 16: $C_2\times C_8$ x 28, $C_{16}$ x 10, $\OD_{16}$ x 6, $C_4^2$ x 4, $D_4:C_2$ x 4, $C_2^2\times C_4$ x 3, $Q_{16}$ x 2, $\SD_{16}$ x 2, $D_8$ x 2, $C_4:C_4$ x 2, $C_2\times D_4$ x 2, $C_2^2:C_4$, $C_2\times Q_8$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5, $D_6$ x 3, $C_3:C_4$ x 3

Order 8: $C_8$ x 18, $C_2\times C_4$ x 17, $D_4$ x 4, $C_2^3$ x 2, $Q_8$ x 2

Order 6: $C_6$ x 5, $S_3$

Order 4: $C_4$ x 11, $C_2^2$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $(C_2\times C_{16}).D_{24}$

Order 768: $C_4.(C_8\times D_{12})$, $C_{12}.(C_8\times Q_8)$, $D_{24}.C_4^2$, $C_8^2:D_6$, $(C_8\times C_{24}):C_4$, $C_{24}.(C_4\times C_8)$, $C_2\times C_8\times C_{48}$

Order 512: $(C_2\times C_8\times C_{16}).C_2$

Order 384: $C_2\times C_4\times C_{48}$ x 2, $C_2\times C_{24}.C_8$ x 2, $C_2\times C_8\times C_{24}$, $C_8\times C_{48}$, $C_2\times C_4.D_{24}$, $C_2^3.D_{24}$, $C_2\times C_8.D_{12}$, $(C_2^2\times C_{48}):C_2$, $(C_2\times C_{16}).D_6$, $C_{24}.C_4^2$, $C_{24}.C_4^2$, $(C_2\times C_{48}):C_4$, $C_{24}.C_4^2$, $D_{24}:C_8$, $C_2\times D_{24}:C_4$

Order 256: $C_4.(C_8\times D_4)$, $C_4.(C_8\times Q_8)$, $D_8.C_4^2$, $C_8^2:C_2^2$, $C_2\times C_8\times C_{16}$, $C_8^2:C_4$, $C_8.(C_4\times C_8)$

Order 192: $C_2^2\times C_{48}$ x 4, $D_{24}:C_4$ x 4, $C_{24}.C_8$ x 4, $C_4\times C_{48}$ x 4, $C_2\times C_4\times C_{24}$ x 2, $C_2\times C_{24}.C_4$ x 2, $C_6:\OD_{32}$ x 2, $C_8\times C_{24}$ x 2, $C_{12}.C_4^2$, $C_4.D_{24}$, $D_{24}:C_4$, $C_8.D_{12}$, $C_2^3.D_{12}$, $C_{12}.C_4^2$, $D_6:C_{16}$, $C_6.Q_{32}$, $C_{24}.D_4$, $C_4^2:D_6$, $D_{24}:C_2^2$, $C_{12}.C_2^4$

Order 128: $C_2^3.\OD_{16}$ x 2, $C_2\times C_4\times C_{16}$ x 2, $C_8.(C_2\times D_4)$, $C_2^3.\SD_{16}$, $C_2^3.D_8$, $\OD_{32}:C_2^2$, $(C_2^2\times C_{16}):C_2$, $C_8\wr C_2$, $C_8.C_4^2$, $C_8\times C_{16}$, $C_2\times C_8^2$, $C_4^2.C_2^3$, $C_8.C_4^2$, $C_8.C_4^2$, $\OD_{32}:C_4$

Order 96: $C_2\times C_{48}$ x 12, $C_4\times C_{24}$ x 6, $C_{24}.C_4$ x 4, $C_3:\OD_{32}$ x 4, $D_{24}:C_2$ x 4, $C_8.D_6$ x 4, $C_2^2\times C_{24}$ x 3, $C_6:C_{16}$ x 2, $C_6:\OD_{16}$ x 2, $D_{12}:C_4$ x 2, $C_{24}:C_4$, $C_{24}:C_4$, $C_{24}:C_4$, $D_{12}:C_2^2$, $C_{24}:C_4$, $C_2\times C_4\times C_{12}$, $C_2^3.D_6$, $C_6:Q_{16}$, $C_2\times D_{24}$, $C_8:D_6$, $C_8:D_6$, $C_8\times D_6$

Order 64: $C_8.C_8$ x 4, $C_4\times C_{16}$ x 4, $C_2^2\times C_{16}$ x 4, $D_8:C_4$ x 4, $C_2\times C_4\times C_8$ x 2, $C_8^2$ x 2, $C_2\times \OD_{32}$ x 2, $C_2^3.Q_8$ x 2, $\OD_{16}:C_4$, $C_4:C_{16}$, $C_4.D_8$, $Q_{16}:C_4$, $D_8:C_4$, $\OD_{32}:C_2$, $C_2^2:C_{16}$, $D_8:C_2^2$, $C_8.C_2^3$, $C_4^2:C_4$, $C_2^3.D_4$, $C_4^2:C_2^2$

Order 48: $C_2\times C_{24}$ x 14, $D_{12}:C_2$ x 4, $C_{48}$ x 4, $C_3:\OD_{16}$ x 4, $C_4\times C_{12}$ x 3, $C_6:C_8$ x 2, $C_3:Q_{16}$ x 2, $D_{24}$ x 2, $C_{24}:C_2$ x 2, $C_{24}:C_2$ x 2, $C_2^2\times C_{12}$ x 2, $S_3\times C_8$ x 2, $C_3:C_{16}$ x 2, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_6:Q_8$, $C_6.D_4$, $C_{12}:C_4$, $C_{12}:C_4$

Order 32: $C_2\times C_{16}$ x 14, $C_4\times C_8$ x 7, $D_8:C_2$ x 4, $\OD_{16}:C_2$ x 4, $C_2^2\times C_8$ x 4, $\OD_{32}$ x 4, $C_8.C_4$ x 4, $C_2\times \OD_{16}$ x 3, $C_4\wr C_2$ x 2, $D_4:C_2^2$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_8:C_4$, $C_2\times D_8$, $C_4^2:C_2$, $C_2\times C_4^2$, $C_8:C_4$, $C_8:C_4$

Order 24: $C_2\times C_{12}$ x 9, $C_{24}$ x 9, $C_3:D_4$ x 2, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_3:C_8$ x 2, $C_2^2\times C_6$, $C_2\times D_6$

Order 16: $C_2\times C_8$ x 18, $\OD_{16}$ x 6, $C_{16}$ x 6, $C_4^2$ x 4, $D_4:C_2$ x 4, $C_2^2\times C_4$ x 3, $Q_{16}$ x 2, $\SD_{16}$ x 2, $D_8$ x 2, $C_2\times D_4$ x 2, $C_4:C_4$, $C_2^2:C_4$, $C_2\times Q_8$

Order 12: $C_{12}$ x 6, $C_2\times C_6$ x 4, $D_6$ x 2, $C_3:C_4$ x 2

Order 8: $C_2\times C_4$ x 13, $C_8$ x 11, $D_4$ x 4, $C_2^3$ x 2, $Q_8$ x 2

Order 6: $C_6$ x 4, $S_3$

Order 4: $C_4$ x 8, $C_2^2$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $(C_2\times C_{16}).D_{24}$ ($C_1$)

Order 768: $C_4.(C_8\times D_{12})$ ($C_2$), $C_{12}.(C_8\times Q_8)$ ($C_2$), $D_{24}.C_4^2$ ($C_2$), $C_8^2:D_6$ ($C_2$), $(C_8\times C_{24}):C_4$ ($C_2$), $C_{24}.(C_4\times C_8)$ ($C_2$), $C_2\times C_8\times C_{48}$ ($C_2$)

Order 384: $C_2\times C_4\times C_{48}$ ($C_2^2$) x 2, $C_2\times C_{24}.C_8$ ($C_2^2$) x 2, $C_2\times C_8\times C_{24}$ ($C_2^2$), $C_2\times C_4.D_{24}$ ($C_4$), $C_2^3.D_{24}$ ($C_4$), $C_{24}.C_4^2$ ($C_4$), $C_{24}.C_4^2$ ($C_4$), $C_{24}.C_4^2$ ($C_2^2$), $C_2\times D_{24}:C_4$ ($C_2^2$)

Order 256: $C_2\times C_8\times C_{16}$ ($S_3$)

Order 192: $C_2^2\times C_{48}$ ($D_4$) x 2, $C_2^2\times C_{48}$ ($C_2\times C_4$) x 2, $D_{24}:C_4$ ($C_8$) x 2, $C_2\times C_{24}.C_4$ ($C_2\times C_4$) x 2, $C_6.Q_{32}$ ($C_8$) x 2, $C_8\times C_{24}$ ($D_4$) x 2, $C_2\times C_4\times C_{24}$ ($C_2^3$), $C_2^3.D_{12}$ ($C_2\times C_4$), $D_{24}:C_2^2$ ($C_2\times C_4$)

Order 128: $C_2\times C_4\times C_{16}$ ($D_6$) x 2, $C_2\times C_8^2$ ($D_6$)

Order 96: $C_4\times C_{24}$ ($D_4:C_2$) x 3, $C_2\times C_{48}$ ($\SD_{16}$) x 2, $C_2\times C_{48}$ ($D_8$) x 2, $C_2\times C_{48}$ ($\OD_{16}$) x 2, $C_2\times C_{48}$ ($C_2\times C_8$) x 2, $C_{24}:C_4$ ($C_2\times C_8$) x 2, $C_4\times C_{24}$ ($C_2\times D_4$), $C_2^2\times C_{24}$ ($C_2\times D_4$), $C_2^2\times C_{24}$ ($C_2^2\times C_4$), $C_2\times C_4\times C_{12}$ ($D_4:C_2$), $C_6:Q_{16}$ ($C_2\times C_8$), $C_2\times D_{24}$ ($C_2\times C_8$)

Order 64: $C_8^2$ ($C_3:D_4$) x 2, $C_2^2\times C_{16}$ ($D_{12}$) x 2, $C_2^2\times C_{16}$ ($C_4\times S_3$) x 2, $C_2\times C_4\times C_8$ ($C_2\times D_6$)

Order 48: $C_2\times C_{24}$ ($D_8:C_2$) x 2, $C_2\times C_{24}$ ($\OD_{16}:C_2$) x 2, $C_2^2\times C_{12}$ ($C_4\times D_4$), $C_2\times C_{24}$ ($C_2\times \SD_{16}$), $C_2\times C_{24}$ ($C_2\times D_8$), $C_2\times C_{24}$ ($C_2\times \OD_{16}$), $C_2\times C_{24}$ ($C_2^2\times C_8$), $C_2\times C_{24}$ ($C_4^2:C_2$), $C_2\times C_{24}$ ($C_4:D_4$), $C_2\times C_{24}$ ($C_4\times D_4$), $C_2\times C_{24}$ ($C_4^2:C_2$), $C_4\times C_{12}$ ($C_4^2:C_2$), $C_4\times C_{12}$ ($C_2^2.D_4$)

Order 32: $C_4\times C_8$ ($D_{12}:C_2$) x 3, $C_2\times C_{16}$ ($D_{24}$) x 2, $C_2\times C_{16}$ ($C_{24}:C_2$) x 2, $C_2\times C_{16}$ ($C_{24}:C_2$) x 2, $C_2\times C_{16}$ ($S_3\times C_8$) x 2, $C_2^2\times C_8$ ($C_2\times D_{12}$), $C_2^2\times C_8$ ($C_4\times D_6$), $C_4\times C_8$ ($C_6:D_4$), $C_2\times C_4^2$ ($D_{12}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_2^3.C_2^3$), $C_2\times C_{12}$ ($D_8:C_4$), $C_2\times C_{12}$ ($C_4\times \SD_{16}$), $C_2\times C_{12}$ ($C_4\times D_8$), $C_2\times C_{12}$ ($C_8:D_4$), $C_2\times C_{12}$ ($C_8\times D_4$), $C_{24}$ ($C_4^2.C_2^2$), $C_{24}$ ($C_4.D_8$), $C_{24}$ ($C_8:D_4$), $C_{24}$ ($C_8:D_4$), $C_{24}$ ($C_8:D_4$), $C_{24}$ ($C_8\times D_4$), $C_{24}$ ($C_4^2.C_2^2$), $C_{24}$ ($C_4^2.C_4$), $C_2^2\times C_6$ ($D_8:C_4$)

Order 16: $C_2\times C_8$ ($D_{24}:C_2$) x 2, $C_2\times C_8$ ($C_8.D_6$) x 2, $C_2\times C_8$ ($C_4^2:S_3$), $C_2\times C_8$ ($C_4^2:S_3$), $C_2\times C_8$ ($C_{12}:D_4$), $C_2\times C_8$ ($C_{12}:D_4$), $C_2\times C_8$ ($C_2\times D_{24}$), $C_2\times C_8$ ($C_8:D_6$), $C_2\times C_8$ ($C_8:D_6$), $C_2\times C_8$ ($C_8\times D_6$), $C_4^2$ ($C_4^2:S_3$), $C_4^2$ ($C_2^3.D_6$), $C_2^2\times C_4$ ($C_4\times D_{12}$)

Order 12: $C_2\times C_6$ ($C_4^2.D_4$), $C_2\times C_6$ ($C_8^2:C_2$), $C_2\times C_6$ ($C_8\times D_8$), $C_{12}$ ($(C_2\times C_4).D_8$), $C_{12}$ ($C_2.(C_8\times D_4)$), $C_{12}$ ($C_8:D_8$), $C_{12}$ ($C_8\times \SD_{16}$)

Order 8: $C_2^3$ ($D_{24}:C_4$), $C_2\times C_4$ ($(C_2\times C_4^2):S_3$), $C_2\times C_4$ ($D_{24}:C_4$), $C_2\times C_4$ ($C_4\times D_{24}$), $C_2\times C_4$ ($C_{12}:\SD_{16}$), $C_2\times C_4$ ($C_8:D_{12}$), $C_2\times C_4$ ($C_8\times D_{12}$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_4^2.D_6$), $C_8$ ($C_4.D_{24}$), $C_8$ ($C_4^2.D_6$), $C_8$ ($C_4^2.D_6$)

Order 6: $C_6$ ($D_{16}:C_8$) x 2, $C_6$ ($(C_2\times C_8^2):C_2$)

Order 4: $C_2^2$ ($C_8^2:S_3$), $C_2^2$ ($C_8\times D_{24}$), $C_2^2$ ($(C_2\times C_8).D_{12}$), $C_4$ ($C_8^2:S_3$), $C_4$ ($C_8:D_{24}$), $C_4$ ($(C_2\times C_8).D_{12}$), $C_4$ ($(C_2\times C_4).D_{24}$)

Order 3: $C_3$ ($(C_2\times C_8\times C_{16}).C_2$)

Order 2: $C_2$ ($D_{48}:C_8$) x 2, $C_2$ ($C_3:(C_4^2.D_4:C_2)$)

Order 1: $C_1$ ($(C_2\times C_{16}).D_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $(C_2\times C_{16}).D_{24}$ ($C_1$)

Order 768: $C_4.(C_8\times D_{12})$ ($C_2$), $C_{12}.(C_8\times Q_8)$ ($C_2$), $D_{24}.C_4^2$ ($C_2$), $C_8^2:D_6$ ($C_2$), $(C_8\times C_{24}):C_4$ ($C_2$), $C_{24}.(C_4\times C_8)$ ($C_2$), $C_2\times C_8\times C_{48}$ ($C_2$)

Order 384: $C_2\times C_4\times C_{48}$ ($C_2^2$) x 2, $C_2\times C_{24}.C_8$ ($C_2^2$) x 2, $C_2\times C_8\times C_{24}$ ($C_2^2$), $C_2\times C_4.D_{24}$ ($C_4$), $C_2^3.D_{24}$ ($C_4$), $C_{24}.C_4^2$ ($C_4$), $C_{24}.C_4^2$ ($C_4$), $C_{24}.C_4^2$ ($C_2^2$), $C_2\times D_{24}:C_4$ ($C_2^2$)

Order 256: $C_2\times C_8\times C_{16}$ ($S_3$)

Order 192: $C_2^2\times C_{48}$ ($D_4$) x 2, $C_2^2\times C_{48}$ ($C_2\times C_4$) x 2, $C_2\times C_{24}.C_4$ ($C_2\times C_4$) x 2, $C_2\times C_4\times C_{24}$ ($C_2^3$), $D_{24}:C_4$ ($C_8$), $C_2^3.D_{12}$ ($C_2\times C_4$), $C_6.Q_{32}$ ($C_8$), $D_{24}:C_2^2$ ($C_2\times C_4$), $C_8\times C_{24}$ ($D_4$)

Order 128: $C_2\times C_4\times C_{16}$ ($D_6$) x 2, $C_2\times C_8^2$ ($D_6$)

Order 96: $C_4\times C_{24}$ ($D_4:C_2$) x 3, $C_2\times C_{48}$ ($\SD_{16}$), $C_2\times C_{48}$ ($D_8$), $C_2\times C_{48}$ ($\OD_{16}$), $C_2\times C_{48}$ ($C_2\times C_8$), $C_4\times C_{24}$ ($C_2\times D_4$), $C_{24}:C_4$ ($C_2\times C_8$), $C_2^2\times C_{24}$ ($C_2\times D_4$), $C_2^2\times C_{24}$ ($C_2^2\times C_4$), $C_2\times C_4\times C_{12}$ ($D_4:C_2$), $C_6:Q_{16}$ ($C_2\times C_8$), $C_2\times D_{24}$ ($C_2\times C_8$)

Order 64: $C_2^2\times C_{16}$ ($D_{12}$) x 2, $C_2^2\times C_{16}$ ($C_4\times S_3$) x 2, $C_2\times C_4\times C_8$ ($C_2\times D_6$), $C_8^2$ ($C_3:D_4$)

Order 48: $C_2\times C_{24}$ ($D_8:C_2$) x 2, $C_2\times C_{24}$ ($\OD_{16}:C_2$) x 2, $C_2^2\times C_{12}$ ($C_4\times D_4$), $C_2\times C_{24}$ ($C_2\times \SD_{16}$), $C_2\times C_{24}$ ($C_2\times D_8$), $C_2\times C_{24}$ ($C_2\times \OD_{16}$), $C_2\times C_{24}$ ($C_2^2\times C_8$), $C_2\times C_{24}$ ($C_4^2:C_2$), $C_2\times C_{24}$ ($C_4:D_4$), $C_2\times C_{24}$ ($C_4\times D_4$), $C_2\times C_{24}$ ($C_4^2:C_2$), $C_4\times C_{12}$ ($C_4^2:C_2$), $C_4\times C_{12}$ ($C_2^2.D_4$)

Order 32: $C_4\times C_8$ ($D_{12}:C_2$) x 3, $C_2^2\times C_8$ ($C_2\times D_{12}$), $C_2^2\times C_8$ ($C_4\times D_6$), $C_4\times C_8$ ($C_6:D_4$), $C_2\times C_4^2$ ($D_{12}:C_2$), $C_2\times C_{16}$ ($D_{24}$), $C_2\times C_{16}$ ($C_{24}:C_2$), $C_2\times C_{16}$ ($C_{24}:C_2$), $C_2\times C_{16}$ ($S_3\times C_8$)

Order 24: $C_2\times C_{12}$ ($C_2^3.C_2^3$), $C_2\times C_{12}$ ($D_8:C_4$), $C_2\times C_{12}$ ($C_4\times \SD_{16}$), $C_2\times C_{12}$ ($C_4\times D_8$), $C_2\times C_{12}$ ($C_8:D_4$), $C_2\times C_{12}$ ($C_8\times D_4$), $C_{24}$ ($C_4^2.C_2^2$), $C_{24}$ ($C_4.D_8$), $C_{24}$ ($C_8:D_4$), $C_{24}$ ($C_8:D_4$), $C_{24}$ ($C_8:D_4$), $C_{24}$ ($C_8\times D_4$), $C_{24}$ ($C_4^2.C_2^2$), $C_{24}$ ($C_4^2.C_4$), $C_2^2\times C_6$ ($D_8:C_4$)

Order 16: $C_2\times C_8$ ($D_{24}:C_2$) x 2, $C_2\times C_8$ ($C_8.D_6$) x 2, $C_2\times C_8$ ($C_4^2:S_3$), $C_2\times C_8$ ($C_4^2:S_3$), $C_2\times C_8$ ($C_{12}:D_4$), $C_2\times C_8$ ($C_{12}:D_4$), $C_2\times C_8$ ($C_2\times D_{24}$), $C_2\times C_8$ ($C_8:D_6$), $C_2\times C_8$ ($C_8:D_6$), $C_2\times C_8$ ($C_8\times D_6$), $C_4^2$ ($C_4^2:S_3$), $C_4^2$ ($C_2^3.D_6$), $C_2^2\times C_4$ ($C_4\times D_{12}$)

Order 12: $C_2\times C_6$ ($C_4^2.D_4$), $C_2\times C_6$ ($C_8^2:C_2$), $C_2\times C_6$ ($C_8\times D_8$), $C_{12}$ ($(C_2\times C_4).D_8$), $C_{12}$ ($C_2.(C_8\times D_4)$), $C_{12}$ ($C_8:D_8$), $C_{12}$ ($C_8\times \SD_{16}$)

Order 8: $C_2^3$ ($D_{24}:C_4$), $C_2\times C_4$ ($(C_2\times C_4^2):S_3$), $C_2\times C_4$ ($D_{24}:C_4$), $C_2\times C_4$ ($C_4\times D_{24}$), $C_2\times C_4$ ($C_{12}:\SD_{16}$), $C_2\times C_4$ ($C_8:D_{12}$), $C_2\times C_4$ ($C_8\times D_{12}$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_{24}:D_4$), $C_8$ ($C_4^2.D_6$), $C_8$ ($C_4.D_{24}$), $C_8$ ($C_4^2.D_6$), $C_8$ ($C_4^2.D_6$)

Order 6: $C_6$ ($D_{16}:C_8$), $C_6$ ($(C_2\times C_8^2):C_2$)

Order 4: $C_2^2$ ($C_8^2:S_3$), $C_2^2$ ($C_8\times D_{24}$), $C_2^2$ ($(C_2\times C_8).D_{12}$), $C_4$ ($C_8^2:S_3$), $C_4$ ($C_8:D_{24}$), $C_4$ ($(C_2\times C_8).D_{12}$), $C_4$ ($(C_2\times C_4).D_{24}$)

Order 3: $C_3$ ($(C_2\times C_8\times C_{16}).C_2$)

Order 2: $C_2$ ($D_{48}:C_8$), $C_2$ ($C_3:(C_4^2.D_4:C_2)$)

Order 1: $C_1$ ($(C_2\times C_{16}).D_{24}$)

Series

Derived series

$(C_2\times C_{16}).D_{24}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_1$

Chief series

$(C_2\times C_{16}).D_{24}$

$\rhd$

$C_8^2:D_6$

$\rhd$

$C_2\times C_{24}.C_8$

$\rhd$

$C_2\times C_4\times C_{24}$

$\rhd$

$C_2^2\times C_{24}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_{16}).D_{24}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2^2\times C_{16}$

$\lhd$

$C_2\times C_4\times C_{16}$

$\lhd$

$C_2\times C_8\times C_{16}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $408 \times 408$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $84 \times 84$ rational character table.