Abstract group 4096.buz: $C_8^2.C_8:C_2^3$

• Label: 4096.buz
• Order: \( 2^{12} \)
• Exponent: \( 2^{5} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{16} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $64$
• Trans deg.: not computed
• Rank: $5$

Group information

Description:	$C_8^2.C_8:C_2^3$
Order:	\(4096\)\(\medspace = 2^{12} \)
Exponent:	\(32\)\(\medspace = 2^{5} \)
Automorphism group:	$C_2^2\times (C_2\times C_4).C_2^5.C_2^6$, of order \(65536\)\(\medspace = 2^{16} \)
Composition factors:	$C_2$ x 12
Nilpotency class:	$5$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	2	4	8	16	32
Elements	1	287	608	1152	1024	1024	4096
Conjugacy classes	1	19	34	62	44	24	184
Divisions	1	19	30	34	24	14	122
Autjugacy classes	1	11	23	41	30	8	114

Dimension	1	2	4	8	16	32
Irr. complex chars.	64	80	24	12	2	2	184
Irr. rational chars.	32	40	24	18	6	2	122

Minimal presentations

Permutation degree:	$64$
Transitive degree:	not computed
Rank:	$5$
Inequivalent generating 5-tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	32	32	32
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid e^{4}=f^{4}=[a,e]=[a,f]=[b,f]=[d,f]=1, a^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $64$ $\langle(1,2)(3,4)(5,14)(6,13)(7,16)(8,15)(9,26)(10,25)(11,28)(12,27)(17,50)(18,49) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{16}$ . $(D_4^2.C_4)$ (2)	$(C_4^2.C_2^5)$ . $D_4$	$(C_4^2.D_8)$ . $C_2^4$ (2)	$(C_4^2.D_8)$ . $C_2^4$ (2)	all 430
Aut. group:	$\Aut(C_2\times D_{32})$	$\Aut(C_2\times Q_{64})$

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

Homology

Abelianization:	$C_{2}^{4} \times C_{4} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 1185 normal subgroups (431 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $C_2^2\times C_{16}$
Frattini:	a subgroup isomorphic to $C_8.C_4^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8^2.C_8:C_2^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 4 (order at least 1024) are shown, as well as all normal subgroups of any index.

Order 4096: $C_8^2.C_8:C_2^3$

Order 2048: $(C_2\times C_8).C_2^4.C_2^3$ x 20, $C_2.\OD_{32}.C_2^5$ x 2, $(C_2\times C_8).C_2^5.C_2^2$ x 2, $D_{16}.(C_8\times D_4)$ x 2, $(C_2^2\times C_{16}).C_2^5$, $(C_2^2\times C_8).C_2^5.C_2$, $(C_2\times C_8).C_2^6.C_2$, $(C_2\times C_4\times C_8).C_2^5$, $(C_4\times C_8).C_2^6$

Order 1024: $C_2.\OD_{32}.C_2^4$ x 48, $(C_2\times C_8).C_2^3.C_2^3$ x 36, $(C_4\times C_8).C_2^5$ x 21, $(C_2^2\times C_{16}).C_2^4$ x 16, $(C_2\times C_8).C_2^4.C_2^2$ x 16, $(C_2\times C_4\times C_8).C_2^4$ x 15, $(C_2\times C_8).C_2^5.C_2$ x 10, $C_8.C_2^4.C_2^3$ x 8, $(C_2^2\times C_8).C_2^4.C_2$ x 8, $C_2\times D_{32}:C_8$ x 6, $(C_2\times D_{32}):C_8$ x 4, $\OD_{64}:(C_2\times C_8)$ x 4, $C_{16}.(C_8\times D_4)$ x 4, $(D_4\times D_{16}).C_4$ x 4, $(C_2^2\times C_8).C_2^5$ x 2, $(C_2\times C_{16}).C_2^5$ x 2, $C_{16}.(C_8\times D_4)$ x 2, $(C_2^2\times D_{16}).C_8$ x 2, $C_{16}.(C_8\times D_4)$ x 2, $D_{16}.(C_4\times D_4)$ x 2, $C_{16}.(C_8\times D_4)$ x 2, $C_{16}.(C_8\times D_4)$ x 2, $C_4^2.C_2^6$, $(C_2\times C_8).C_2^6$, $C_2^2\times D_{16}:C_8$

Order 512: $C_2.\OD_{32}.C_2^3$ x 8, $C_2\times D_{16}.C_8$ x 8, $C_2\times D_{16}:C_8$ x 6, $(C_2\times C_{16}).D_4:C_2$ x 4, $C_8.(C_2.C_4:D_4)$ x 4, $C_4^2.D_{16}$ x 4, $C_4^2.\SD_{32}$ x 4, $C_{16}.(C_4\times D_4)$ x 4, $D_{16}.C_4^2$ x 4, $C_{16}.(C_4\times D_4)$ x 4, $C_4^2.\OD_{32}$ x 4, $C_2.\OD_{32}.C_2^3$ x 4, $\OD_{64}:C_8$ x 4, $C_2.\OD_{32}.C_2^3$ x 4, $C_2\times D_{16}:C_8$ x 4, $C_2.\OD_{32}.C_2^3$ x 4, $\OD_{64}:C_8$ x 4, $(C_2\times C_{16}).C_2^4$ x 4, $\OD_{64}.C_2^3$ x 4, $C_4^2.C_2^4.C_2$ x 4, $C_4^2.C_2^4.C_2$ x 4, $C_4^2.C_2^4.C_2$ x 4, $C_{16}.(C_4\times C_8)$ x 4, $C_2.\OD_{32}.D_4$ x 4, $C_2.\OD_{32}.D_4$ x 4, $\OD_{64}:C_8$ x 4, $\OD_{64}:C_8$ x 4, $(C_2\times C_8).C_2^5$ x 2, $C_4.C_2^5.C_2^2$ x 2, $(C_2\times C_8).C_2^5$ x 2, $(C_2\times C_8).C_2^5$ x 2, $(C_2\times C_8).C_2^5$ x 2, $(C_2\times C_{16}).D_4:C_2$ x 2, $(C_2\times C_{16}).D_4:C_2$ x 2, $(C_2\times C_{16}).D_4:C_2$ x 2, $(C_2\times C_{16}).D_4:C_2$ x 2, $(C_2\times C_{16}).D_4:C_2$ x 2, $(C_2\times C_{16}).D_4:C_2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $(C_2\times C_8).C_4^2.C_2$ x 2, $C_{16}.C_4^2.C_2$ x 2, $(C_2\times C_8).C_4:D_4$ x 2, $C_{16}.C_4:D_4$ x 2, $C_{16}.C_2^2:D_4$ x 2, $C_{16}.C_2^2:D_4$ x 2, $C_{16}.C_2^2:D_4$ x 2, $C_2^4.D_{16}$ x 2, $C_{16}.C_2^2:D_4$ x 2, $C_{16}.C_2^2:D_4$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_2^4.\SD_{32}$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^5.C_2$ x 2, $C_2^4.\SD_{32}$ x 2, $(C_2\times D_{16}):C_8$ x 2, $(C_2\times C_8).C_4:D_4$ x 2, $(C_4\times C_8).C_2^3.C_2$ x 2, $C_8.(C_8\times Q_8)$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_8.C_2^5.C_2$ x 2, $C_2^4.D_{16}$ x 2, $(C_2\times C_{16}).C_2^4$ x 2, $(C_2\times C_{16}).C_2^4$ x 2, $(C_2\times C_{16}).C_2^4$ x 2, $C_8.C_2^4.C_2^2$ x 2, $C_4.(C_2^3\times C_8).C_2$ x 2, $(C_2^2\times C_{16}).C_8$ x 2, $(C_2^2\times C_8).C_2^4$ x 2, $(C_2^2\times C_8).C_2^4$ x 2, $(C_2^2\times C_8).C_2^4$ x 2, $C_{16}.(C_4\times D_4)$ x 2, $C_4^2.C_2^4.C_2$ x 2, $C_4^2.C_2^4.C_2$ x 2, $C_4^2.C_2^4.C_2$ x 2, $C_4^2.C_2^4.C_2$ x 2, $C_4^2.C_2^4.C_2$ x 2, $C_4^2.C_2^4.C_2$ x 2, $C_4^2.C_2^4.C_2$ x 2, $(C_4\times C_8).C_2^4$ x 2, $C_2\times D_{16}:C_8$ x 2, $(C_4\times C_8).C_2^4$ x 2, $(C_4\times C_8).C_2^4$ x 2, $(C_4\times C_8).C_2^4$ x 2, $(C_4\times C_8).C_2^4$ x 2, $(C_4\times C_8).C_2^4$ x 2, $(C_4\times C_8).C_2^4$ x 2, $C_4^2.C_2^4.C_2$ x 2, $(C_2\times C_4\times C_8).C_2^3$ x 2, $C_{16}.(C_4\times C_8)$ x 2, $C_2.\OD_{32}.D_4$ x 2, $C_2.\OD_{32}.D_4$ x 2, $C_{16}.C_2^5$, $(C_2\times C_8).C_2^5$, $(C_2\times C_8).C_2^5$, $(C_2\times C_8).C_2^5$, $C_4.C_2^5.C_2^2$, $(C_2\times C_8).C_2^5$, $(C_2\times C_8).C_2^5$, $C_2^2\times D_{16}:C_4$, $C_4^2.C_2^5$, $C_8.C_2^4.C_2^2$, $C_8.C_2^4.C_2^2$, $C_8.C_2^4.C_2^2$, $C_8.C_2^4.C_2^2$, $(C_2\times C_{16}).D_4:C_2$, $(C_2\times C_{16}).D_4:C_2$, $C_8.C_2^5.C_2$, $C_8.C_2^5.C_2$, $(C_2\times C_{16}).C_2^4$, $(C_2\times C_{16}).C_2^4$, $C_8.C_2^5.C_2$, $C_8.C_2^5.C_2$, $C_8.C_2^4.C_2^2$, $C_8.C_2^5.C_2$, $C_8.C_2^5.C_2$, $(C_2^2\times C_8).C_2^4$, $(C_2^2\times C_8).C_2^4$, $(C_2^2\times C_8).C_2^4$, $(C_2^2\times C_8).C_2^4$, $(C_2^2\times C_8).C_2^4$, $(C_4\times C_8).C_2^4$, $(C_4\times C_8).C_2^4$, $(C_4\times C_8).C_2^4$, $C_2^2\times C_{16}:C_8$, $(C_4\times C_8).C_2^4$, $C_2^2\times C_{16}:C_8$

Order 256: $D_{16}:C_8$ x 18, $\OD_{64}:C_2^2$ x 8, $\OD_{64}:C_2^2$ x 8, $\OD_{64}:C_4$ x 8, $C_{16}:\OD_{16}$ x 6, $Q_{32}:C_8$ x 6, $\OD_{32}:C_8$ x 6, $C_8^2.C_4$ x 6, $\OD_{32}:C_8$ x 6, $C_4^2.C_4^2$ x 6, $C_4^2.D_8$ x 6, $\OD_{64}:C_4$ x 4, $\OD_{64}:C_4$ x 4, $C_2^3.\SD_{32}$ x 4, $\OD_{64}:C_2^2$ x 4, $C_2^3.\SD_{32}$ x 4, $C_2^3.D_{16}$ x 4, $\OD_{64}:C_2^2$ x 4, $C_2^2:\OD_{64}$ x 4, $C_2\times C_{16}.C_8$ x 4, $C_4^2.\SD_{16}$ x 4, $C_4^2.\SD_{16}$ x 4, $\OD_{64}:C_4$ x 4, $\OD_{64}:C_4$ x 4, $C_{16}.C_4^2$ x 4, $C_{16}.C_4^2$ x 4, $\OD_{64}:C_4$ x 4, $C_4^2.C_{16}$ x 4, $D_{32}:C_2^2$ x 4, $C_{32}:C_2^3$ x 4, $C_2^4.D_8$ x 4, $D_{16}:(C_2\times C_4)$ x 4, $(C_2\times D_{16}):C_4$ x 4, $C_4^2.D_8$ x 4, $(C_2\times C_{16}):D_4$ x 4, $(C_4\times \OD_{32}):C_2$ x 4, $\OD_{16}.C_4^2$ x 4, $\OD_{16}.C_{16}$ x 2, $\OD_{64}.C_2^2$ x 2, $\OD_{64}:C_4$ x 2, $\OD_{16}.D_8$ x 2, $C_4^2.D_8$ x 2, $\OD_{16}.D_8$ x 2, $C_4^2.Q_{16}$ x 2, $\OD_{32}.D_4$ x 2, $(C_2\times C_{16}).D_4$ x 2, $C_2^4.Q_{16}$ x 2, $C_{16}.C_4^2$ x 2, $D_{16}:C_2^3$ x 2, $C_{16}.\OD_{16}$ x 2, $C_{16}.\OD_{16}$ x 2, $C_{16}:\OD_{16}$ x 2, $\OD_{64}:C_2^2$ x 2, $C_4^2.D_8$ x 2, $C_4^2.D_8$ x 2, $(C_4\times \SD_{32}):C_2$ x 2, $(C_4\times D_{16}):C_2$ x 2, $\OD_{32}:Q_8$ x 2, $\OD_{32}:D_4$ x 2, $\OD_{32}.C_2^3$ x 2, $(C_2\times C_{16}):D_4$ x 2, $(C_2\times C_{16}):D_4$ x 2, $(C_2\times C_{16}):Q_8$ x 2, $\OD_{16}.D_8$ x 2, $C_2\times D_{16}:C_4$ x 2, $(C_2\times C_{16}):D_4$ x 2, $(C_4\times C_{16}):C_2^2$ x 2, $\OD_{32}:(C_2\times C_4)$ x 2, $C_2^3.\OD_{32}$ x 2, $C_8^2.C_2^2$ x 2, $C_8^2.C_2^2$ x 2, $(C_2\times C_{16}).D_4$, $(C_2^2\times C_{16}):C_4$, $C_{16}.C_2^4$, $C_2^3\times D_{16}$, $C_4^2.C_2^4$, $C_4^2.D_8$, $C_4^2.D_8$, $C_{16}.(C_2^2\times C_4)$, $C_2^4.D_8$, $C_8^2:C_2^2$, $C_4^2.C_4^2$

Order 128: $D_{16}:C_4$ x 18, $C_{16}:C_8$ x 16, $C_8.C_4^2$ x 10, $C_2\times \OD_{64}$ x 8, $\OD_{32}:C_4$ x 8, $C_4\times \OD_{32}$ x 8, $C_{16}:C_8$ x 8, $C_{16}:D_4$ x 6, $Q_{32}:C_4$ x 6, $C_2^3.D_8$ x 6, $D_{16}:C_2^2$ x 6, $C_2\times \SD_{64}$ x 4, $C_2\times D_{32}$ x 4, $C_2^3.Q_{16}$ x 4, $C_2^3.D_8$ x 4, $C_8.(C_2\times D_4)$ x 4, $(C_2^2\times C_{16}):C_2$ x 4, $C_2^2\times D_{16}$ x 4, $\OD_{32}:C_2^2$ x 4, $C_8^2:C_2$ x 4, $C_2^3.C_4^2$ x 4, $C_4^2.C_2^3$ x 4, $C_{16}:Q_8$ x 2, $C_8.Q_{16}$ x 2, $C_8.D_8$ x 2, $C_4:D_{16}$ x 2, $C_{16}.D_4$ x 2, $C_4:\OD_{32}$ x 2, $C_2^2:\OD_{32}$ x 2, $C_2^3\times C_{16}$ x 2, $C_8^2:C_2$ x 2, $C_8^2:C_2$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $D_4.C_4^2$ x 2, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_2^2\times C_4\times C_8$

Order 64: $C_{16}:C_4$ x 14, $C_2\times D_{16}$ x 12, $\OD_{16}:C_4$ x 8, $C_{16}:C_4$ x 8, $C_2\times \OD_{32}$ x 8, $C_2^2\times C_{16}$ x 8, $C_2\times C_4\times C_8$ x 6, $C_8:C_8$ x 6, $C_4\times C_{16}$ x 6, $(C_2^2\times C_8):C_2$ x 4, $C_4\times \OD_{16}$ x 4, $C_2\times C_{32}$ x 4, $C_8.C_2^3$ x 4, $C_2\times Q_{32}$ x 4, $C_4^2.C_2^2$ x 4, $C_2^2:\OD_{16}$ x 2, $C_2^3\times C_8$ x 2, $C_4^2:C_2^2$ x 2, $C_8^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4:\OD_{16}$ x 2, $C_4^2.C_2^2$, $C_4^2.C_2^2$, $C_2^2\times C_4^2$

Order 32: $C_2\times C_{16}$ x 12, $C_4\times C_8$ x 10, $C_2\times \OD_{16}$ x 8, $C_2^2\times C_8$ x 8, $C_8:C_4$ x 6, $C_4^2:C_2$ x 4, $C_2\times C_4^2$ x 4, $D_{16}$ x 4, $D_4:C_2^2$ x 2, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_2^3\times C_4$, $C_4:Q_8$, $C_4:D_4$

Order 16: $C_2\times C_8$ x 12, $C_4^2$ x 6, $C_2\times D_4$ x 4, $C_2^2\times C_4$ x 4, $C_{16}$ x 4, $C_2^2:C_4$ x 2, $C_2\times Q_8$ x 2, $C_2^4$

Order 8: $C_2\times C_4$ x 8, $C_8$ x 4, $C_2^3$ x 3

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 4096: $C_8^2.C_8:C_2^3$ ($C_1$)

Order 2048: $(C_2\times C_8).C_2^4.C_2^3$ ($C_2$) x 20, $C_2.\OD_{32}.C_2^5$ ($C_2$) x 2, $(C_2\times C_8).C_2^5.C_2^2$ ($C_2$) x 2, $D_{16}.(C_8\times D_4)$ ($C_2$) x 2, $(C_2^2\times C_{16}).C_2^5$ ($C_2$), $(C_2^2\times C_8).C_2^5.C_2$ ($C_2$), $(C_2\times C_8).C_2^6.C_2$ ($C_2$), $(C_2\times C_4\times C_8).C_2^5$ ($C_2$), $(C_4\times C_8).C_2^6$ ($C_2$)

Order 1024: $(C_2\times C_8).C_2^3.C_2^3$ ($C_2^2$) x 36, $C_2.\OD_{32}.C_2^4$ ($C_2^2$) x 22, $(C_2\times C_8).C_2^4.C_2^2$ ($C_2^2$) x 16, $(C_4\times C_8).C_2^5$ ($C_2^2$) x 14, $(C_2\times C_8).C_2^5.C_2$ ($C_2^2$) x 10, $C_8.C_2^4.C_2^3$ ($C_4$) x 8, $(C_2^2\times C_{16}).C_2^4$ ($C_2^2$) x 8, $(C_2^2\times C_{16}).C_2^4$ ($C_4$) x 8, $(C_2^2\times C_8).C_2^4.C_2$ ($C_2^2$) x 8, $(C_2\times C_4\times C_8).C_2^4$ ($C_2^2$) x 8, $(C_2\times D_{32}):C_8$ ($C_2^2$) x 4, $\OD_{64}:(C_2\times C_8)$ ($C_2^2$) x 4, $C_{16}.(C_8\times D_4)$ ($C_2^2$) x 4, $C_2\times D_{32}:C_8$ ($C_2^2$) x 4, $(C_2\times C_{16}).C_2^5$ ($C_2^2$) x 2, $C_{16}.(C_8\times D_4)$ ($C_2^2$) x 2, $(C_2^2\times D_{16}).C_8$ ($C_2^2$) x 2, $C_{16}.(C_8\times D_4)$ ($C_2^2$) x 2, $D_{16}.(C_4\times D_4)$ ($C_2^2$) x 2, $C_{16}.(C_8\times D_4)$ ($C_2^2$) x 2, $C_{16}.(C_8\times D_4)$ ($C_2^2$) x 2, $(C_2^2\times C_8).C_2^5$ ($C_2^2$), $(C_2\times C_8).C_2^6$ ($C_2^2$), $C_2^2\times D_{16}:C_8$ ($C_2^2$)

Order 512: $C_2.\OD_{32}.C_2^3$ ($C_2^3$) x 8, $C_2\times D_{16}.C_8$ ($C_2^3$) x 8, $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$) x 4, $C_8.(C_2.C_4:D_4)$ ($C_2\times C_4$) x 4, $C_4^2.D_{16}$ ($C_2\times C_4$) x 4, $C_4^2.\SD_{32}$ ($C_2\times C_4$) x 4, $C_{16}.(C_4\times D_4)$ ($C_2^3$) x 4, $D_{16}.C_4^2$ ($C_2^3$) x 4, $C_{16}.(C_4\times D_4)$ ($C_2^3$) x 4, $C_4^2.\OD_{32}$ ($C_2^3$) x 4, $C_2.\OD_{32}.C_2^3$ ($C_2^3$) x 4, $\OD_{64}:C_8$ ($C_2^3$) x 4, $C_2.\OD_{32}.C_2^3$ ($C_2^3$) x 4, $C_2\times D_{16}:C_8$ ($C_2^3$) x 4, $C_2.\OD_{32}.C_2^3$ ($C_2^3$) x 4, $\OD_{64}:C_8$ ($C_2^3$) x 4, $(C_2\times C_{16}).C_2^4$ ($C_2^3$) x 4, $\OD_{64}.C_2^3$ ($C_2^3$) x 4, $C_2\times D_{16}:C_8$ ($C_2^3$) x 4, $C_{16}.(C_4\times C_8)$ ($C_2^3$) x 4, $C_2.\OD_{32}.D_4$ ($C_2^3$) x 4, $C_2.\OD_{32}.D_4$ ($C_2^3$) x 4, $\OD_{64}:C_8$ ($C_2^3$) x 4, $\OD_{64}:C_8$ ($C_2^3$) x 4, $(C_2\times C_8).C_2^5$ ($C_2\times C_4$) x 2, $C_4.C_2^5.C_2^2$ ($C_2\times C_4$) x 2, $(C_2\times C_8).C_2^5$ ($C_2^3$) x 2, $(C_2\times C_8).C_2^5$ ($C_2^3$) x 2, $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $(C_2\times C_8).C_4^2.C_2$ ($C_2\times C_4$) x 2, $C_{16}.C_4^2.C_2$ ($C_2\times C_4$) x 2, $(C_2\times C_8).C_4:D_4$ ($C_2\times C_4$) x 2, $C_{16}.C_4:D_4$ ($C_2\times C_4$) x 2, $C_{16}.C_2^2:D_4$ ($C_2\times C_4$) x 2, $C_{16}.C_2^2:D_4$ ($C_2\times C_4$) x 2, $C_{16}.C_2^2:D_4$ ($C_2\times C_4$) x 2, $C_2^4.D_{16}$ ($C_2\times C_4$) x 2, $C_{16}.C_2^2:D_4$ ($C_2\times C_4$) x 2, $C_{16}.C_2^2:D_4$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_2^4.\SD_{32}$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^5.C_2$ ($C_2\times C_4$) x 2, $C_2^4.\SD_{32}$ ($C_2\times C_4$) x 2, $(C_2\times D_{16}):C_8$ ($C_2^3$) x 2, $(C_2\times C_8).C_4:D_4$ ($C_2^3$) x 2, $(C_4\times C_8).C_2^3.C_2$ ($C_2^3$) x 2, $C_8.(C_8\times Q_8)$ ($C_2^3$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^4.C_2^2$ ($C_2\times C_4$) x 2, $C_8.C_2^5.C_2$ ($C_2\times C_4$) x 2, $C_2^4.D_{16}$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).C_2^4$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).C_2^4$ ($C_2\times C_4$) x 2, $(C_2\times C_{16}).C_2^4$ ($C_2^3$) x 2, $C_8.C_2^4.C_2^2$ ($C_2^3$) x 2, $C_4.(C_2^3\times C_8).C_2$ ($C_2^3$) x 2, $(C_2^2\times C_{16}).C_8$ ($C_2^3$) x 2, $(C_2^2\times C_8).C_2^4$ ($C_2\times C_4$) x 2, $(C_2^2\times C_8).C_2^4$ ($C_2\times C_4$) x 2, $(C_2^2\times C_8).C_2^4$ ($C_2^3$) x 2, $C_{16}.(C_4\times D_4)$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($D_4$) x 2, $C_4^2.C_2^4.C_2$ ($D_4$) x 2, $C_4^2.C_2^4.C_2$ ($D_4$) x 2, $C_4^2.C_2^4.C_2$ ($D_4$) x 2, $C_4^2.C_2^4.C_2$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($C_2^3$) x 2, $(C_4\times C_8).C_2^4$ ($C_2^3$) x 2, $C_2\times D_{16}:C_8$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($D_4$) x 2, $C_4^2.C_2^4.C_2$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($D_4$) x 2, $C_4^2.C_2^4.C_2$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($D_4$) x 2, $C_2\times D_{16}:C_8$ ($D_4$) x 2, $(C_4\times C_8).C_2^4$ ($C_2^3$) x 2, $(C_4\times C_8).C_2^4$ ($C_2^3$) x 2, $(C_4\times C_8).C_2^4$ ($D_4$) x 2, $(C_4\times C_8).C_2^4$ ($D_4$) x 2, $(C_4\times C_8).C_2^4$ ($C_2^3$) x 2, $C_4^2.C_2^4.C_2$ ($C_2^3$) x 2, $(C_2\times C_4\times C_8).C_2^3$ ($C_2^3$) x 2, $C_{16}.(C_4\times C_8)$ ($C_2^3$) x 2, $C_2.\OD_{32}.D_4$ ($C_2^3$) x 2, $C_2.\OD_{32}.D_4$ ($C_2^3$) x 2, $C_{16}.C_2^5$ ($C_2\times C_4$), $(C_2\times C_8).C_2^5$ ($C_2\times C_4$), $(C_2\times C_8).C_2^5$ ($C_2\times C_4$), $(C_2\times C_8).C_2^5$ ($C_2\times C_4$), $C_4.C_2^5.C_2^2$ ($C_2\times C_4$), $(C_2\times C_8).C_2^5$ ($C_2^3$), $(C_2\times C_8).C_2^5$ ($C_2^3$), $(C_2\times C_8).C_2^5$ ($D_4$), $(C_2\times C_8).C_2^5$ ($C_2^3$), $C_2^2\times D_{16}:C_4$ ($C_2^3$), $C_4^2.C_2^5$ ($D_4$), $C_8.C_2^4.C_2^2$ ($C_2\times C_4$), $C_8.C_2^4.C_2^2$ ($C_2\times C_4$), $C_8.C_2^4.C_2^2$ ($C_2\times C_4$), $C_8.C_2^4.C_2^2$ ($C_2\times C_4$), $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$), $(C_2\times C_{16}).D_4:C_2$ ($C_2\times C_4$), $C_8.C_2^5.C_2$ ($C_2\times C_4$), $C_8.C_2^5.C_2$ ($C_2\times C_4$), $(C_2\times C_{16}).C_2^4$ ($C_2\times C_4$), $(C_2\times C_{16}).C_2^4$ ($C_2\times C_4$), $C_8.C_2^5.C_2$ ($C_2\times C_4$), $C_8.C_2^5.C_2$ ($C_2\times C_4$), $C_8.C_2^4.C_2^2$ ($C_2\times C_4$), $C_8.C_2^5.C_2$ ($C_2\times C_4$), $C_8.C_2^5.C_2$ ($C_2\times C_4$), $(C_2^2\times C_8).C_2^4$ ($C_2\times C_4$), $(C_2^2\times C_8).C_2^4$ ($C_2\times C_4$), $(C_2^2\times C_8).C_2^4$ ($C_2\times C_4$), $(C_2^2\times C_8).C_2^4$ ($C_2\times C_4$), $(C_2^2\times C_8).C_2^4$ ($C_2^3$), $(C_4\times C_8).C_2^4$ ($C_2^3$), $(C_4\times C_8).C_2^4$ ($C_2^3$), $(C_4\times C_8).C_2^4$ ($C_2^3$), $C_2^2\times C_{16}:C_8$ ($C_2^3$), $(C_4\times C_8).C_2^4$ ($D_4$), $(C_4\times C_8).C_2^4$ ($C_2^3$), $(C_4\times C_8).C_2^4$ ($D_4$), $C_2^2\times C_{16}:C_8$ ($C_2^3$)

Order 256: $D_{16}:C_8$ ($C_2\times D_4$) x 18, $\OD_{64}:C_2^2$ ($C_2^2\times C_4$) x 8, $\OD_{64}:C_2^2$ ($C_2^2\times C_4$) x 8, $\OD_{64}:C_4$ ($C_2^4$) x 8, $C_{16}:\OD_{16}$ ($C_2\times D_4$) x 6, $Q_{32}:C_8$ ($C_2\times D_4$) x 6, $\OD_{32}:C_8$ ($C_2\times D_4$) x 6, $\OD_{32}:C_8$ ($C_2\times D_4$) x 6, $\OD_{64}:C_4$ ($C_2^2\times C_4$) x 4, $\OD_{64}:C_4$ ($C_2^2\times C_4$) x 4, $C_2^3.\SD_{32}$ ($C_2^2\times C_4$) x 4, $\OD_{64}:C_2^2$ ($C_2^2\times C_4$) x 4, $C_2^3.\SD_{32}$ ($C_2^2\times C_4$) x 4, $C_2^3.D_{16}$ ($C_2^2\times C_4$) x 4, $\OD_{64}:C_2^2$ ($C_2^2\times C_4$) x 4, $C_2^2:\OD_{64}$ ($C_2^2\times C_4$) x 4, $C_2\times C_{16}.C_8$ ($C_2^4$) x 4, $\OD_{64}:C_4$ ($C_2^2\times C_4$) x 4, $\OD_{64}:C_4$ ($C_2^2\times C_4$) x 4, $C_{16}.C_4^2$ ($C_2^2\times C_4$) x 4, $C_{16}.C_4^2$ ($C_2^2\times C_4$) x 4, $\OD_{64}:C_4$ ($C_2^2\times C_4$) x 4, $C_4^2.C_{16}$ ($C_2^2\times C_4$) x 4, $C_8^2.C_4$ ($C_2\times D_4$) x 4, $C_4^2.C_4^2$ ($C_2\times D_4$) x 4, $D_{32}:C_2^2$ ($C_2^2\times C_4$) x 4, $C_{32}:C_2^3$ ($C_2^2\times C_4$) x 4, $C_2^4.D_8$ ($C_2^2\times C_4$) x 4, $(C_4\times \OD_{32}):C_2$ ($C_2\times D_4$) x 4, $\OD_{16}.C_{16}$ ($C_2^2\times C_4$) x 2, $\OD_{64}.C_2^2$ ($C_2^2\times C_4$) x 2, $\OD_{64}:C_4$ ($C_2^2\times C_4$) x 2, $\OD_{16}.D_8$ ($C_2^2\times C_4$) x 2, $C_4^2.D_8$ ($C_2^2\times C_4$) x 2, $\OD_{16}.D_8$ ($C_2^2\times C_4$) x 2, $C_4^2.Q_{16}$ ($C_2^2\times C_4$) x 2, $\OD_{32}.D_4$ ($C_2^2\times C_4$) x 2, $(C_2\times C_{16}).D_4$ ($C_2^2\times C_4$) x 2, $C_2^4.Q_{16}$ ($C_2^2\times C_4$) x 2, $C_{16}.C_4^2$ ($C_2^4$) x 2, $D_{16}:C_2^3$ ($C_2^2\times C_4$) x 2, $C_{16}.\OD_{16}$ ($C_2\times D_4$) x 2, $C_{16}.\OD_{16}$ ($C_2\times D_4$) x 2, $C_{16}:\OD_{16}$ ($C_2\times D_4$) x 2, $C_4^2.\SD_{16}$ ($C_2^4$) x 2, $C_4^2.\SD_{16}$ ($C_2\times D_4$) x 2, $C_4^2.\SD_{16}$ ($C_2^4$) x 2, $C_4^2.\SD_{16}$ ($C_2\times D_4$) x 2, $C_8^2.C_4$ ($C_2^4$) x 2, $C_4^2.C_4^2$ ($C_2^4$) x 2, $\OD_{64}:C_2^2$ ($C_2^2\times C_4$) x 2, $C_4^2.D_8$ ($C_2\times D_4$) x 2, $C_4^2.D_8$ ($C_2\times D_4$) x 2, $(C_4\times \SD_{32}):C_2$ ($D_4:C_2$) x 2, $(C_4\times D_{16}):C_2$ ($D_4:C_2$) x 2, $\OD_{32}:Q_8$ ($C_2^2\times C_4$) x 2, $\OD_{32}:D_4$ ($C_2^2\times C_4$) x 2, $\OD_{32}.C_2^3$ ($C_2^2\times C_4$) x 2, $(C_2\times C_{16}):D_4$ ($C_2^2\times C_4$) x 2, $(C_2\times C_{16}):D_4$ ($C_2^2\times C_4$) x 2, $\OD_{16}.D_8$ ($C_2^2\times C_4$) x 2, $C_2\times D_{16}:C_4$ ($C_2^4$) x 2, $D_{16}:(C_2\times C_4)$ ($D_4:C_2$) x 2, $D_{16}:(C_2\times C_4)$ ($C_2^2\times C_4$) x 2, $(C_2\times D_{16}):C_4$ ($D_4:C_2$) x 2, $(C_2\times D_{16}):C_4$ ($C_2^2\times C_4$) x 2, $C_4^2.D_8$ ($C_2^4$) x 2, $C_4^2.D_8$ ($C_2\times D_4$) x 2, $C_4^2.D_8$ ($C_2^4$) x 2, $C_4^2.D_8$ ($C_2\times D_4$) x 2, $C_4^2.D_8$ ($C_2^2\times C_4$) x 2, $(C_2\times C_{16}):D_4$ ($D_4:C_2$) x 2, $(C_2\times C_{16}):D_4$ ($C_2^2\times C_4$) x 2, $(C_4\times C_{16}):C_2^2$ ($C_2\times D_4$) x 2, $\OD_{32}:(C_2\times C_4)$ ($C_2^2\times C_4$) x 2, $\OD_{16}.C_4^2$ ($C_2^4$) x 2, $\OD_{16}.C_4^2$ ($C_2\times D_4$) x 2, $C_8^2.C_2^2$ ($C_2\times D_4$) x 2, $C_8^2.C_2^2$ ($C_2\times D_4$) x 2, $(C_2\times C_{16}).D_4$ ($C_2^2\times C_4$), $(C_2^2\times C_{16}):C_4$ ($C_2^2\times C_4$), $C_{16}.C_2^4$ ($C_2^2\times C_4$), $C_2^3\times D_{16}$ ($C_2^2\times C_4$), $C_4^2.C_2^4$ ($C_2\times D_4$), $C_4^2.D_8$ ($C_2^2\times C_4$), $C_4^2.D_8$ ($C_2^2\times C_4$), $(C_2\times C_{16}):Q_8$ ($D_4:C_2$), $(C_2\times C_{16}):Q_8$ ($C_2^2\times C_4$), $(C_2\times C_{16}):D_4$ ($D_4:C_2$), $(C_2\times C_{16}):D_4$ ($C_2^2\times C_4$), $C_{16}.(C_2^2\times C_4)$ ($C_2^2\times C_4$), $C_2^4.D_8$ ($C_2^2\times C_4$), $C_2^3.\OD_{32}$ ($C_2^4$), $C_2^3.\OD_{32}$ ($C_2\times D_4$), $C_8^2:C_2^2$ ($C_2\times D_4$), $C_4^2.C_4^2$ ($C_2\times D_4$)

Order 128: $C_2\times \OD_{64}$ ($C_2^3\times C_4$) x 8, $C_{16}:C_8$ ($C_2^2\times D_4$) x 8, $C_{16}:C_8$ ($C_2^2\wr C_2$) x 8, $C_{16}:C_8$ ($C_2^2\times D_4$) x 8, $C_{16}:D_4$ ($C_4\times D_4$) x 6, $D_{16}:C_4$ ($D_4:C_2^2$) x 6, $D_{16}:C_4$ ($C_2^2\times D_4$) x 6, $D_{16}:C_4$ ($C_4\times D_4$) x 6, $\OD_{32}:C_4$ ($D_4:C_2^2$) x 6, $C_4\times \OD_{32}$ ($C_2^2\times D_4$) x 6, $C_8.C_4^2$ ($C_2^2\times D_4$) x 5, $C_2\times \SD_{64}$ ($\OD_{16}:C_2$) x 4, $C_2\times D_{32}$ ($\OD_{16}:C_2$) x 4, $C_2^3.Q_{16}$ ($C_2^3\times C_4$) x 4, $C_2^3.D_8$ ($C_2^3\times C_4$) x 4, $C_8.(C_2\times D_4)$ ($C_4\times D_4$) x 4, $(C_2^2\times C_{16}):C_2$ ($C_4\times D_4$) x 4, $D_{16}:C_2^2$ ($C_2^3\times C_4$) x 4, $C_{16}:Q_8$ ($C_4\times D_4$) x 2, $C_8.Q_{16}$ ($C_4\times D_4$) x 2, $C_8.D_8$ ($C_4\times D_4$) x 2, $C_4:D_{16}$ ($C_4\times D_4$) x 2, $C_{16}.D_4$ ($C_4\times D_4$) x 2, $Q_{32}:C_4$ ($D_4:C_2^2$) x 2, $Q_{32}:C_4$ ($C_2^2\times D_4$) x 2, $Q_{32}:C_4$ ($C_4\times D_4$) x 2, $C_2^3.D_8$ ($C_4\times D_4$) x 2, $C_2^3.D_8$ ($C_2^3\times C_4$) x 2, $C_2^3.D_8$ ($C_4\times D_4$) x 2, $C_4:\OD_{32}$ ($C_4\times D_4$) x 2, $C_2^2:\OD_{32}$ ($C_4\times D_4$) x 2, $\OD_{32}:C_4$ ($C_2^3\times C_4$) x 2, $C_4\times \OD_{32}$ ($C_2^2\wr C_2$) x 2, $C_8.C_4^2$ ($C_2^2\wr C_2$) x 2, $D_{16}:C_2^2$ ($C_4\times D_4$) x 2, $C_2^2\times D_{16}$ ($C_2^3\times C_4$) x 2, $C_2^2\times D_{16}$ ($C_4\times D_4$) x 2, $\OD_{32}:C_2^2$ ($D_4:C_2^2$) x 2, $\OD_{32}:C_2^2$ ($C_2^3\times C_4$) x 2, $C_8^2:C_2$ ($D_8:C_2$) x 2, $C_8^2:C_2$ ($D_8:C_2$) x 2, $C_8^2:C_2$ ($C_2^2\times D_4$) x 2, $C_8^2:C_2$ ($C_2^2\wr C_2$) x 2, $C_2^3.C_4^2$ ($C_2^2\times D_4$) x 2, $C_2^3.C_4^2$ ($C_2^2\wr C_2$) x 2, $C_4^2.C_2^3$ ($D_8:C_2$) x 2, $C_4^2.C_2^3$ ($C_4\times D_4$) x 2, $C_4^2.C_2^3$ ($D_8:C_2$) x 2, $D_4.C_4^2$ ($C_2^2\times D_4$) x 2, $C_8.C_4^2$ ($C_2^5$), $C_8.C_4^2$ ($D_4:C_2^2$), $C_8.C_4^2$ ($C_2^3\times C_4$), $C_4^2.C_2^3$ ($D_8:C_2$), $C_2^3\times C_{16}$ ($D_4:C_2^2$), $C_2^3\times C_{16}$ ($C_2^3\times C_4$), $C_4^2.C_2^3$ ($D_8:C_2$), $C_4^2.C_2^3$ ($C_4\times D_4$), $C_4^2.C_2^3$ ($D_8:C_2$), $C_4^2.C_2^3$ ($C_4\times D_4$), $C_4^2.C_2^3$ ($D_8:C_2$), $C_2^2\times C_4\times C_8$ ($C_2^2\times D_4$)

Order 64: $C_{16}:C_4$ ($C_4^2:C_2^2$) x 8, $C_{16}:C_4$ ($C_2^3:D_4$) x 6, $C_2\times D_{16}$ ($C_8.C_2^3$) x 6, $C_2\times D_{16}$ ($C_4^2:C_2^2$) x 6, $C_2\times \OD_{32}$ ($C_4^2:C_2^2$) x 6, $(C_2^2\times C_8):C_2$ ($D_8:C_4$) x 4, $C_4\times \OD_{16}$ ($C_8:C_2^3$) x 4, $C_2\times C_{32}$ ($C_8.C_2^3$) x 4, $C_8:C_8$ ($C_2^3:D_4$) x 4, $C_4\times C_{16}$ ($C_4^2:C_2^2$) x 4, $C_2^2\times C_{16}$ ($C_4^2:C_2^2$) x 4, $C_4^2.C_2^2$ ($D_8:C_4$) x 4, $C_2^2:\OD_{16}$ ($D_8:C_4$) x 2, $\OD_{16}:C_4$ ($D_8:C_2^2$) x 2, $\OD_{16}:C_4$ ($C_8:C_2^3$) x 2, $\OD_{16}:C_4$ ($C_4^2:C_2^2$) x 2, $\OD_{16}:C_4$ ($C_4^2:C_2^2$) x 2, $C_2\times C_4\times C_8$ ($C_8:C_2^3$) x 2, $C_2\times C_4\times C_8$ ($C_2^3:D_4$) x 2, $C_8:C_8$ ($C_8:C_2^3$) x 2, $C_{16}:C_4$ ($D_4:C_2^3$) x 2, $C_{16}:C_4$ ($D_4\times C_2^3$) x 2, $C_{16}:C_4$ ($C_4^2:C_2^2$) x 2, $C_{16}:C_4$ ($C_4^2:C_2^2$) x 2, $C_4\times C_{16}$ ($C_4^2:C_2^2$) x 2, $C_8.C_2^3$ ($C_8:C_2^3$) x 2, $C_8.C_2^3$ ($C_4^2:C_2^2$) x 2, $C_4^2:C_2^2$ ($D_8:C_4$) x 2, $C_8^2$ ($C_8:C_2^3$) x 2, $C_4^2.C_2^2$ ($C_8:C_2^3$) x 2, $C_2\times Q_{32}$ ($C_8.C_2^3$) x 2, $C_2\times Q_{32}$ ($C_4^2:C_2^2$) x 2, $C_2\times \OD_{32}$ ($C_4^2:C_2^2$) x 2, $C_2^2\times C_{16}$ ($C_4^2:C_2^2$) x 2, $C_4:\OD_{16}$ ($D_8:C_4$) x 2, $C_2\times C_4\times C_8$ ($D_4\times C_2^3$), $C_2\times C_4\times C_8$ ($C_4^2:C_2^2$), $C_2^3\times C_8$ ($C_8:C_2^3$), $C_2^3\times C_8$ ($C_4^2:C_2^2$), $C_4^2.C_2^2$ ($D_8:C_4$), $C_4^2.C_2^2$ ($D_8:C_4$), $C_2^2\times C_4^2$ ($C_8:C_2^3$), $C_2^2\times C_{16}$ ($D_4:C_2^3$), $C_2^2\times C_{16}$ ($C_2^4\times C_4$)

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

Order 16: $C_2\times C_8$ ($D_4:(C_4\times D_4)$) x 4, $C_2\times D_4$ ($\OD_{32}.C_2^3$) x 4, $C_2\times C_8$ ($C_4^2.C_2^4$) x 2, $C_2\times C_8$ ($D_4^2.C_4$) x 2, $C_2^2:C_4$ ($\OD_{32}.C_2^3$) x 2, $C_4^2$ ($\OD_{32}.C_2^3$) x 2, $C_4^2$ ($D_4:(C_4\times D_4)$) x 2, $C_2\times Q_8$ ($\OD_{32}.C_2^3$) x 2, $C_2^2\times C_4$ ($D_4:(C_4\times D_4)$) x 2, $C_{16}$ ($D_4^2.C_4$) x 2, $C_2\times C_8$ ($C_4^2.C_2^4$), $C_2\times C_8$ ($C_2^5.D_4$), $C_2\times C_8$ ($C_4^2.C_2^4$), $C_2\times C_8$ ($C_2^3.C_2^5$), $C_4^2$ ($C_4^2.C_2^4$), $C_4^2$ ($C_2^5.D_4$), $C_2^4$ ($\OD_{32}.C_2^3$), $C_2^2\times C_4$ ($C_4^2.C_2^4$), $C_2^2\times C_4$ ($\OD_{32}.C_2^3$), $C_{16}$ ($C_4^2.C_2^4$), $C_{16}$ ($C_4^2.C_2^4$)

Order 8: $C_2^3$ ($D_8.(C_4\times D_4)$) x 2, $C_2\times C_4$ ($(C_2\times C_8).C_2^5$) x 2, $C_2\times C_4$ ($C_4^2.C_2^5$) x 2, $C_2\times C_4$ ($D_8.(C_4\times D_4)$) x 2, $C_8$ ($C_4^2.C_2^5$) x 2, $C_2^3$ ($\OD_{32}.C_2^4$), $C_2\times C_4$ ($\OD_{32}.C_2^4$), $C_2\times C_4$ ($C_2^4.(C_4\times D_4)$), $C_8$ ($C_4.C_2^6.C_2$), $C_8$ ($(C_2^4\times C_4).C_2^3$)

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

Order 2: $C_2$ ($(C_2\times C_4\times C_8).C_2^5$)

Order 1: $C_1$ ($C_8^2.C_8:C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

See the $184 \times 184$ 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 $122 \times 122$ rational character table.