Abstract group 1024.bp: $C_4^3.C_2^4$

• Label: 1024.bp
• Order: \( 2^{10} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{25} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{20} \)
• Perm deg.: not computed
• Trans deg.: $128$
• Rank: $6$

Group information

Description:	$C_4^3.C_2^4$
Order:	\(1024\)\(\medspace = 2^{10} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^9.C_2^6.C_2^6.C_2^4$, of order \(33554432\)\(\medspace = 2^{25} \)
Composition factors:	$C_2$ x 10
Nilpotency class:	$3$
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
Elements	1	95	416	512	1024
Conjugacy classes	1	27	116	112	256
Divisions	1	27	76	64	168
Autjugacy classes	1	7	12	6	26

Dimension	1	2	4	8
Irr. complex chars.	128	96	32	0	256
Irr. rational chars.	64	48	40	16	168

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$128$
Rank:	$6$
Inequivalent generating 6-tuples:	$10079354880$

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:	${\langle a, b, c, d, e, f \mid c^{8}=d^{4}=e^{2}=f^{2}=[c,d]=[c,e]=[c,f]= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 3 & 9 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_4^3.C_2^3)$
Semidirect product:	$(D_8:C_4^2)$ $\,\rtimes\,$ $C_2^2$	$(C_4^2.C_2^4)$ $\,\rtimes\,$ $C_4$	$(C_4^2.C_2^4)$ $\,\rtimes\,$ $C_4$	$(C_4^2.C_2^4)$ $\,\rtimes\,$ $C_4$	all 53
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^3$ . $C_2^4$ (2)	$(C_2^5.Q_8)$ . $C_2^2$	$(C_2^4.Q_8)$ . $C_2^3$	$(C_2^4.Q_8)$ . $C_2^3$	all 235

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{16}\Z)$.

Homology

Abelianization:	$C_{2}^{5} \times C_{4} $
Schur multiplier:	$C_{2}^{13}$
Commutator length:	$1$

Subgroups

There are 26165 subgroups in 14729 conjugacy classes, 6285 normal (45 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4^2$	$G/Z \simeq$ $C_2^2\times D_4$
Commutator:	$G' \simeq$ $C_2\times C_4$	$G/G' \simeq$ $C_2^5\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2^6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^3.C_2^4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4^3.C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_4^2:C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^3.C_2^4$

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 1024: $C_4^3.C_2^4$

Order 512: $(C_2^3\times C_4).C_2^4$ x 32, $(C_2^4\times C_8).C_2^2$ x 8, $(C_2^3\times C_4).C_2^4$ x 4, $(C_2^3\times C_4).C_2^4$ x 4, $(C_2^3\times C_4).C_2^4$ x 4, $(C_2^3\times C_4).C_2^4$ x 2, $C_2^2.C_2^6.C_2$ x 2, $(C_2^3\times C_4).C_2^4$ x 2, $C_2^2.C_2^6.C_2$ x 2, $(C_2^4\times C_4).C_2^3$, $(C_2\times C_4).C_2^6$, $C_2^2\times C_4\times C_2.\OD_{16}$

Order 256: $C_8.(C_4\times D_4)$ x 128, $C_4^3.C_2^2$ x 64, $C_4^3.C_2^2$ x 64, $D_8:C_4^2$ x 64, $C_4^2.C_2^4$ x 48, $C_4^2.C_2^4$ x 32, $C_4^3.C_2^2$ x 32, $C_4^3.C_2^2$ x 32, $C_4^3.C_2^2$ x 32, $C_4.C_4^3$ x 24, $C_2\times C_4^2.D_4$ x 16, $C_4^2.(C_2\times D_4)$ x 16, $C_2\times D_4.C_4^2$ x 16, $C_4^2.C_2^4$ x 8, $C_4^2.C_2^4$ x 8, $C_4^2.C_2^4$ x 8, $C_4^2.C_2^4$ x 8, $(C_2^2\times D_8):C_4$ x 8, $C_2\times C_4^2.Q_8$ x 8, $C_2^5.D_4$ x 8, $C_2\times C_8.C_4^2$ x 8, $C_4^3:C_2^2$ x 8, $C_4^2.C_2^4$ x 4, $C_2^5.D_4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^3.C_2^2$ x 4, $C_4^3:C_2^2$ x 4, $C_2^5.Q_8$ x 4, $C_2\times C_8.C_4^2$ x 4, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^3\times C_4\times C_8$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_2^2\times C_4^3$

Order 128: $C_4^2.D_4$ x 128, $D_4.C_4^2$ x 128, $C_4^2.D_4$ x 128, $C_4^2.C_2^3$ x 128, $C_4^2.C_2^3$ x 128, $C_4^2.C_2^3$ x 96, $C_2^3.C_4^2$ x 80, $C_4^2.D_4$ x 64, $C_4^2.Q_8$ x 64, $C_4^2:D_4$ x 64, $C_8.C_4^2$ x 64, $D_4:C_4^2$ x 64, $C_8:C_4^2$ x 64, $C_4^2.D_4$ x 64, $C_2^4.D_4$ x 64, $C_4^2.C_2^3$ x 64, $C_4^2.C_2^3$ x 64, $C_4^2.D_4$ x 64, $C_2^2\times C_4\times C_8$ x 60, $C_4^2.C_2^3$ x 48, $C_4^2.C_2^3$ x 48, $C_4^2.C_2^3$ x 48, $C_4^2.C_2^3$ x 48, $D_4.C_4^2$ x 48, $C_4^2:Q_8$ x 32, $C_4^3:C_2$ x 32, $C_2^4.Q_8$ x 32, $C_8.C_4^2$ x 32, $C_4^2.D_4$ x 32, $C_4^2.D_4$ x 32, $C_4^2.D_4$ x 32, $C_4^2.C_2^3$ x 24, $C_4^2.C_2^3$ x 24, $C_2\times C_4^3$ x 18, $C_4^2.D_4$ x 16, $C_4^2.D_4$ x 16, $C_2^4.D_4$ x 16, $C_4^2.C_2^3$ x 16, $C_4^2.D_4$ x 16, $C_4^2.D_4$ x 16, $C_2^3:\SD_{16}$ x 16, $C_4^2.D_4$ x 16, $C_2^3:\OD_{16}$ x 16, $C_4^2:C_2^3$ x 16, $C_2^3.C_4^2$ x 16, $C_4^2.C_2^3$ x 12, $C_2^4.Q_8$ x 8, $C_8.C_2^4$ x 8, $C_4^2:C_2^3$ x 8, $C_4^2.C_2^3$ x 8, $C_4^2.D_4$ x 8, $C_2^3:Q_{16}$ x 8, $C_2^3:D_8$ x 8, $C_4^2.C_2^3$ x 8, $C_4^2.C_2^3$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2.C_2^3$ x 8, $C_4^2.C_2^3$ x 8, $C_2^4.Q_8$ x 8, $C_4^2.C_2^3$ x 8, $C_2^4.D_4$ x 8, $(C_2^3\times C_8):C_2$ x 8, $C_2^3.C_4^2$ x 8, $C_4^2.C_2^3$ x 6, $C_4^2.C_2^3$ x 6, $C_2^3\times C_4^2$ x 5, $D_8:C_2^3$ x 4, $C_2^4\times C_8$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.C_2^3$ x 4, $C_2^4.D_4$ x 4, $C_4^2.C_2^3$ x 4, $C_2^3:\OD_{16}$ x 4, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$, $C_4^2.D_4$

Order 64: $C_4.C_4^2$ x 284, $C_2\times C_4\times C_8$ x 268, $D_8:C_4$ x 256, $C_4^2.C_2^2$ x 224, $C_8\times D_4$ x 192, $C_8:D_4$ x 128, $C_4\times \SD_{16}$ x 128, $C_8:D_4$ x 128, $C_4^2:C_2^2$ x 128, $\OD_{16}:C_4$ x 112, $C_8.C_2^3$ x 112, $C_4^2:C_2^2$ x 112, $C_4^2.C_2^2$ x 112, $C_2^2\times C_4^2$ x 106, $C_2^3\times C_8$ x 96, $C_4^2.C_4$ x 96, $C_4^2.C_2^2$ x 72, $C_4^2.C_2^2$ x 72, $C_2^3.D_4$ x 64, $D_8:C_2^2$ x 64, $C_4^2:C_4$ x 64, $C_4^2.C_2^2$ x 64, $C_2^2:Q_{16}$ x 64, $C_8:D_4$ x 64, $C_8:Q_8$ x 64, $C_8\times Q_8$ x 64, $C_4\times Q_{16}$ x 64, $C_4\times D_8$ x 64, $C_8:D_4$ x 64, $C_4^2.C_2^2$ x 64, $C_2^3.Q_8$ x 64, $C_4\times \OD_{16}$ x 56, $(C_2^2\times C_8):C_2$ x 48, $C_4^2.C_2^2$ x 48, $C_4^3$ x 40, $C_4^2:C_2^2$ x 40, $C_4.C_4^2$ x 32, $C_8.Q_8$ x 32, $C_8.D_4$ x 32, $C_4.Q_{16}$ x 32, $C_4.D_8$ x 32, $C_4^2.C_4$ x 32, $C_2^3.D_4$ x 32, $C_2^2:\OD_{16}$ x 24, $C_2^3:C_8$ x 24, $C_4:\OD_{16}$ x 24, $C_4^2.C_4$ x 24, $C_4^2:C_2^2$ x 20, $C_2^2.Q_{16}$ x 16, $C_2^2.D_8$ x 16, $C_2^2\times \OD_{16}$ x 16, $C_2^3:Q_8$ x 16, $C_2^3:D_4$ x 16, $C_4^2.C_2^2$ x 16, $C_8:Q_8$ x 16, $C_8:Q_8$ x 16, $C_8:D_4$ x 16, $C_2^3.D_4$ x 12, $D_4:C_2^3$ x 8, $C_2^2\times \SD_{16}$ x 8, $C_4^2:C_2^2$ x 8, $C_4^2.C_2^2$ x 8, $C_4^2:C_2^2$ x 8, $C_4:Q_{16}$ x 8, $C_4:D_8$ x 8, $C_2^2.D_8$ x 8, $C_2^3.D_4$ x 8, $C_4^2.C_2^2$ x 6, $C_2^4\times C_4$ x 5, $C_2^2\times Q_{16}$ x 4, $C_2^2\times D_8$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 2

Order 32: $C_2^2\times C_8$ x 424, $C_2\times C_4^2$ x 408, $C_4\times D_4$ x 304, $C_8:C_4$ x 208, $\OD_{16}:C_2$ x 192, $C_4\times C_8$ x 176, $C_4^2:C_2$ x 152, $D_8:C_2$ x 128, $C_4\wr C_2$ x 128, $C_2^3\times C_4$ x 122, $D_4:C_2^2$ x 120, $C_2\times \OD_{16}$ x 120, $C_2^2:Q_8$ x 112, $C_4:D_4$ x 112, $C_4\times Q_8$ x 112, $C_2^2:C_8$ x 96, $C_2^2.D_4$ x 96, $C_4:C_8$ x 96, $D_4:C_4$ x 64, $C_2\times \SD_{16}$ x 64, $C_4^2:C_2$ x 64, $C_8.C_4$ x 64, $Q_8:C_4$ x 64, $C_4.Q_8$ x 56, $C_4^2:C_2$ x 56, $C_2^2.D_4$ x 40, $C_4:Q_8$ x 36, $C_2^3:C_4$ x 36, $C_2\times Q_{16}$ x 32, $C_2\times D_8$ x 32, $C_2^2\wr C_2$ x 32, $C_8:C_4$ x 32, $C_8:C_4$ x 32, $C_2^2\times D_4$ x 20, $C_4:D_4$ x 12, $C_2^2\times Q_8$ x 8, $C_2^5$

Order 16: $C_2^2\times C_4$ x 540, $C_2\times C_8$ x 400, $C_4^2$ x 248, $D_4:C_2$ x 224, $C_2\times D_4$ x 156, $C_4:C_4$ x 152, $C_2^2:C_4$ x 144, $\OD_{16}$ x 112, $\SD_{16}$ x 64, $C_2\times Q_8$ x 60, $Q_{16}$ x 32, $D_8$ x 32, $C_2^4$ x 23

Order 8: $C_2\times C_4$ x 500, $D_4$ x 152, $C_2^3$ x 119, $C_8$ x 64, $Q_8$ x 56

Order 4: $C_2^2$ x 127, $C_4$ x 76

Order 2: $C_2$ x 27

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1024: $C_4^3.C_2^4$

Order 512: $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $(C_2^4\times C_4).C_2^3$, $C_2^2.C_2^6.C_2$, $(C_2\times C_4).C_2^6$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $(C_2^4\times C_8).C_2^2$, $C_2^2\times C_4\times C_2.\OD_{16}$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $C_2^2.C_2^6.C_2$

Order 256: $C_4.C_4^3$ x 5, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^3\times C_4\times C_8$ x 2, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_2^5.D_4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_2^2\times C_4^3$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_8.(C_4\times D_4)$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $(C_2^2\times D_8):C_4$, $C_4^3:C_2^2$, $C_2\times C_4^2.Q_8$, $C_2\times C_4^2.D_4$, $C_2^5.D_4$, $C_4^2.(C_2\times D_4)$, $C_2\times C_4^2.D_4$, $C_2\times C_4^2.D_4$, $C_2^5.Q_8$, $D_8:C_4^2$, $C_2\times C_8.C_4^2$, $C_2\times C_8.C_4^2$, $C_2\times D_4.C_4^2$, $C_4^3:C_2^2$

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

Order 64: $C_2\times C_4\times C_8$ x 32, $C_2^2\times C_4^2$ x 21, $C_2^3\times C_8$ x 14, $C_4.C_4^2$ x 11, $C_4^2.C_2^2$ x 6, $C_4^2.C_2^2$ x 6, $C_4^2.C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2.C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $C_4^3$ x 5, $C_4^2:C_2^2$ x 4, $\OD_{16}:C_4$ x 3, $C_4\times \OD_{16}$ x 3, $C_2^4\times C_4$ x 3, $D_8:C_2^2$ x 3, $C_8.C_2^3$ x 3, $C_2^3:Q_8$ x 3, $C_2^3:D_4$ x 3, $C_4^2.C_2^2$ x 3, $C_2^3.D_4$ x 2, $(C_2^2\times C_8):C_2$ x 2, $C_2^2:\OD_{16}$ x 2, $C_2^3:C_8$ x 2, $D_4:C_2^3$ x 2, $C_2^2\times \OD_{16}$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_8\times D_4$ x 2, $C_4^2.C_4$ x 2, $C_2^3.Q_8$ x 2, $C_2^3.D_4$ x 2, $C_4^2.C_2^2$ x 2, $C_4:\OD_{16}$ x 2, $C_4^2.C_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2.Q_{16}$, $C_2^2.D_8$, $C_2^2\times Q_{16}$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_4.C_4^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_4$, $C_8:Q_8$, $C_8.Q_8$, $C_8:Q_8$, $C_8.D_4$, $C_4:Q_{16}$, $C_4:D_8$, $C_8:D_4$, $C_4^2.C_2^2$, $C_4.Q_{16}$, $C_4.D_8$, $C_2^2:Q_{16}$, $C_8:D_4$, $C_8:D_4$, $C_8:Q_8$, $C_8\times Q_8$, $D_8:C_4$, $C_4\times Q_{16}$, $C_4\times \SD_{16}$, $C_4\times D_8$, $C_8:D_4$, $C_8:D_4$, $C_4^2.C_2^2$, $C_4^2.C_4$, $C_2^2.D_8$, $C_2^3.D_4$

Order 32: $C_2\times C_4^2$ x 45, $C_2^2\times C_8$ x 30, $C_2^3\times C_4$ x 24, $C_4\times C_8$ x 14, $C_4\times D_4$ x 10, $C_4^2:C_2$ x 8, $D_4:C_2^2$ x 7, $C_8:C_4$ x 6, $C_2^2.D_4$ x 6, $C_2\times \OD_{16}$ x 5, $C_2^2:Q_8$ x 5, $C_4:D_4$ x 5, $C_4\times Q_8$ x 5, $C_2^3:C_4$ x 5, $C_2^2\times D_4$ x 4, $C_4:Q_8$ x 4, $C_4.Q_8$ x 3, $C_4^2:C_2$ x 3, $C_2^2.D_4$ x 3, $C_2^2:C_8$ x 2, $C_2^2\times Q_8$ x 2, $D_8:C_2$ x 2, $C_2\times Q_{16}$ x 2, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $\OD_{16}:C_2$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_4:C_8$ x 2, $D_4:C_4$, $C_2^5$, $C_4^2:C_2$, $C_8.C_4$, $C_8:C_4$, $C_8:C_4$, $C_4\wr C_2$, $Q_8:C_4$

Order 16: $C_2^2\times C_4$ x 54, $C_4^2$ x 25, $C_2\times C_8$ x 24, $C_2\times D_4$ x 10, $C_4:C_4$ x 7, $C_2^4$ x 7, $C_2^2:C_4$ x 6, $D_4:C_2$ x 5, $C_2\times Q_8$ x 5, $\OD_{16}$ x 3, $Q_{16}$, $\SD_{16}$, $D_8$

Order 8: $C_2\times C_4$ x 45, $C_2^3$ x 17, $D_4$ x 6, $C_8$ x 6, $Q_8$ x 3

Order 4: $C_2^2$ x 17, $C_4$ x 12

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1024: $C_4^3.C_2^4$ ($C_1$)

Order 512: $(C_2^3\times C_4).C_2^4$ ($C_2$) x 32, $(C_2^4\times C_8).C_2^2$ ($C_2$) x 8, $(C_2^3\times C_4).C_2^4$ ($C_2$) x 4, $(C_2^3\times C_4).C_2^4$ ($C_2$) x 4, $(C_2^3\times C_4).C_2^4$ ($C_2$) x 4, $(C_2^3\times C_4).C_2^4$ ($C_2$) x 2, $C_2^2.C_2^6.C_2$ ($C_2$) x 2, $(C_2^3\times C_4).C_2^4$ ($C_2$) x 2, $C_2^2.C_2^6.C_2$ ($C_2$) x 2, $(C_2^4\times C_4).C_2^3$ ($C_2$), $(C_2\times C_4).C_2^6$ ($C_2$), $C_2^2\times C_4\times C_2.\OD_{16}$ ($C_2$)

Order 256: $C_8.(C_4\times D_4)$ ($C_2^2$) x 128, $C_4^3.C_2^2$ ($C_2^2$) x 64, $C_4^3.C_2^2$ ($C_2^2$) x 64, $D_8:C_4^2$ ($C_2^2$) x 64, $C_4^2.C_2^4$ ($C_2^2$) x 32, $C_4^3.C_2^2$ ($C_2^2$) x 32, $C_4^3.C_2^2$ ($C_2^2$) x 32, $C_4^3.C_2^2$ ($C_2^2$) x 32, $C_4.C_4^3$ ($C_2^2$) x 24, $C_4^2.C_2^4$ ($C_2^2$) x 16, $C_4^2.C_2^4$ ($C_4$) x 16, $C_2\times C_4^2.D_4$ ($C_2^2$) x 16, $C_4^2.(C_2\times D_4)$ ($C_2^2$) x 16, $C_2\times D_4.C_4^2$ ($C_2^2$) x 16, $C_4^2.C_2^4$ ($C_4$) x 8, $C_4^2.C_2^4$ ($C_2^2$) x 8, $C_4^2.C_2^4$ ($C_2^2$) x 8, $(C_2^2\times D_8):C_4$ ($C_2^2$) x 8, $C_2\times C_4^2.Q_8$ ($C_2^2$) x 8, $C_2^5.D_4$ ($C_2^2$) x 8, $C_2\times C_8.C_4^2$ ($C_2^2$) x 8, $C_4^3:C_2^2$ ($C_2^2$) x 8, $C_4^2.C_2^4$ ($C_4$) x 4, $C_2^5.D_4$ ($C_2^2$) x 4, $C_4^2.C_2^4$ ($C_2^2$) x 4, $C_4^2.C_2^4$ ($C_2^2$) x 4, $C_4^2.C_2^4$ ($C_2^2$) x 4, $C_4^2.C_2^4$ ($C_2^2$) x 4, $C_4^2.C_2^4$ ($C_2^2$) x 4, $C_4^2.C_2^4$ ($C_2^2$) x 4, $C_4^3.C_2^2$ ($C_2^2$) x 4, $C_4^3:C_2^2$ ($C_2^2$) x 4, $C_2^5.Q_8$ ($C_2^2$) x 4, $C_2\times C_8.C_4^2$ ($C_2^2$) x 4, $C_4^2.C_2^4$ ($C_2^2$) x 2, $C_4^2.C_2^4$ ($C_4$) x 2, $C_4^2.C_2^4$ ($C_4$) x 2, $C_4^2.C_2^4$ ($C_2^2$) x 2, $C_4^2.C_2^4$ ($C_2^2$) x 2, $C_2^3\times C_4\times C_8$ ($C_2^2$) x 2, $C_2\times C_4^2.D_4$ ($C_2^2$) x 2, $C_2\times C_4^2.D_4$ ($C_2^2$) x 2, $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_2^2\times C_4^3$ ($C_2^2$)

Order 128: $C_4^2.D_4$ ($C_2^3$) x 128, $D_4.C_4^2$ ($C_2^3$) x 128, $C_4^2.D_4$ ($C_2\times C_4$) x 128, $C_4^2.D_4$ ($C_2^3$) x 64, $C_4^2.Q_8$ ($C_2^3$) x 64, $C_4^2:D_4$ ($C_2^3$) x 64, $C_8.C_4^2$ ($C_2^3$) x 64, $D_4:C_4^2$ ($C_2^3$) x 64, $C_8:C_4^2$ ($C_2^3$) x 64, $C_4^2.D_4$ ($C_2\times C_4$) x 64, $C_4^2.C_2^3$ ($C_2^3$) x 64, $C_4^2.C_2^3$ ($C_2^3$) x 64, $C_2^3.C_4^2$ ($C_2^3$) x 48, $C_4^2:Q_8$ ($C_2^3$) x 32, $C_4^3:C_2$ ($C_2^3$) x 32, $C_2^4.Q_8$ ($C_2^3$) x 32, $C_8.C_4^2$ ($C_2^3$) x 32, $C_4^2.D_4$ ($C_2\times C_4$) x 32, $C_4^2.D_4$ ($C_2\times C_4$) x 32, $C_2^4.D_4$ ($C_2^3$) x 32, $C_2^4.D_4$ ($C_2\times C_4$) x 32, $C_4^2.C_2^3$ ($C_2^3$) x 32, $C_4^2.D_4$ ($C_2^3$) x 32, $C_4^2.D_4$ ($C_2\times C_4$) x 32, $C_4^2.C_2^3$ ($C_2^3$) x 32, $C_4^2.C_2^3$ ($C_2^3$) x 32, $D_4.C_4^2$ ($C_2^3$) x 32, $C_2^2\times C_4\times C_8$ ($C_2^3$) x 24, $C_4^2.D_4$ ($C_2^3$) x 16, $C_4^2.D_4$ ($C_2^3$) x 16, $C_2^4.D_4$ ($C_2^3$) x 16, $C_4^2.C_2^3$ ($C_2^3$) x 16, $C_4^2.C_2^3$ ($C_2\times C_4$) x 16, $C_4^2.D_4$ ($C_2^3$) x 16, $C_4^2.D_4$ ($C_2\times C_4$) x 16, $C_2^3:\SD_{16}$ ($C_2\times C_4$) x 16, $C_4^2.D_4$ ($C_2\times C_4$) x 16, $C_4^2.C_2^3$ ($C_2^3$) x 16, $C_4^2:C_2^3$ ($C_2^3$) x 16, $C_2^2\times C_4\times C_8$ ($D_4$) x 16, $C_2^2\times C_4\times C_8$ ($C_2\times C_4$) x 16, $C_2\times C_4^3$ ($C_2^3$) x 12, $C_2^4.Q_8$ ($C_2^3$) x 8, $C_8.C_2^4$ ($C_2^3$) x 8, $C_4^2.C_2^3$ ($C_2\times C_4$) x 8, $C_4^2.D_4$ ($C_2^3$) x 8, $C_4^2.D_4$ ($C_2\times C_4$) x 8, $C_4^2.D_4$ ($C_2^3$) x 8, $C_4^2.D_4$ ($C_2\times C_4$) x 8, $C_4^2.D_4$ ($C_2\times C_4$) x 8, $C_2^3:Q_{16}$ ($C_2\times C_4$) x 8, $C_2^3:D_8$ ($C_2\times C_4$) x 8, $C_4^2.D_4$ ($C_2\times C_4$) x 8, $C_4^2.D_4$ ($C_2\times C_4$) x 8, $C_2^4.Q_8$ ($C_2^3$) x 8, $C_4^2.C_2^3$ ($C_2^3$) x 8, $C_2^4.D_4$ ($C_2^3$) x 8, $(C_2^3\times C_8):C_2$ ($C_2^3$) x 8, $C_2^3.C_4^2$ ($C_2^3$) x 8, $D_8:C_2^3$ ($C_2^3$) x 4, $C_2^4\times C_8$ ($C_2^3$) x 4, $C_4^2.C_2^3$ ($C_2\times C_4$) x 4, $C_4^2.C_2^3$ ($C_2\times C_4$) x 4, $C_4^2:C_2^3$ ($C_2^3$) x 4, $C_4^2.C_2^3$ ($C_2^3$) x 4, $C_4^2.D_4$ ($C_2\times C_4$) x 4, $C_4^2.D_4$ ($C_2\times C_4$) x 4, $C_4^2.D_4$ ($C_2\times C_4$) x 4, $C_4^2.D_4$ ($C_2\times C_4$) x 4, $C_2^4.D_4$ ($C_2^3$) x 4, $C_4^2.C_2^3$ ($C_2^3$) x 4, $C_2^3:\OD_{16}$ ($C_2^3$) x 4, $C_2^3.C_4^2$ ($C_2^3$) x 4, $C_2^3\times C_4^2$ ($C_2^3$) x 3, $C_4^2.C_2^3$ ($C_2^3$) x 2, $C_4^2.C_2^3$ ($C_2^3$) x 2, $C_4^2.D_4$ ($C_2\times C_4$) x 2, $C_4^2.D_4$ ($C_2\times C_4$) x 2, $C_4^2.D_4$ ($C_2\times C_4$) x 2, $C_4^2.D_4$ ($C_2\times C_4$), $C_4^2.D_4$ ($C_2\times C_4$)

Order 64: $C_8:D_4$ ($C_2^2\times C_4$) x 128, $C_4\times \SD_{16}$ ($C_2^2\times C_4$) x 128, $C_2\times C_4\times C_8$ ($C_2\times D_4$) x 96, $C_2\times C_4\times C_8$ ($C_2^2\times C_4$) x 96, $C_4.C_4^2$ ($C_2^4$) x 64, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 64, $C_4^2:C_4$ ($C_2^4$) x 64, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 64, $C_2^2:Q_{16}$ ($C_2^2\times C_4$) x 64, $C_8:D_4$ ($C_2^2\times C_4$) x 64, $C_4\times Q_{16}$ ($C_2^2\times C_4$) x 64, $C_4\times D_8$ ($C_2^2\times C_4$) x 64, $C_4^2:C_2^2$ ($C_2^4$) x 64, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 40, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 40, $C_2^3.D_4$ ($C_2^4$) x 32, $C_2^3.D_4$ ($C_2^2\times C_4$) x 32, $(C_2^2\times C_8):C_2$ ($C_2^4$) x 32, $\OD_{16}:C_4$ ($C_2^4$) x 32, $C_4.C_4^2$ ($D_4:C_2$) x 32, $C_2\times C_4\times C_8$ ($C_2^4$) x 32, $C_8.C_2^3$ ($C_2^4$) x 32, $C_4.C_4^2$ ($C_2^4$) x 32, $C_8.Q_8$ ($C_2^2\times C_4$) x 32, $C_8.D_4$ ($C_2^2\times C_4$) x 32, $C_4.Q_{16}$ ($C_2^2\times C_4$) x 32, $C_4.D_8$ ($C_2^2\times C_4$) x 32, $C_2^3.Q_8$ ($C_2^4$) x 32, $C_4^2.C_2^2$ ($C_2^4$) x 32, $C_2^3\times C_8$ ($C_2^4$) x 24, $C_2^2\times C_4^2$ ($C_2^4$) x 18, $C_2^2.Q_{16}$ ($C_2^2\times C_4$) x 16, $C_2^2.D_8$ ($C_2^2\times C_4$) x 16, $C_2^2:\OD_{16}$ ($C_2^4$) x 16, $C_4\times \OD_{16}$ ($C_2^4$) x 16, $C_4^3$ ($C_2^4$) x 16, $D_8:C_2^2$ ($C_2^4$) x 16, $D_8:C_2^2$ ($C_2^2\times C_4$) x 16, $C_2^3\times C_8$ ($C_2\times D_4$) x 16, $C_2^3\times C_8$ ($C_2^2\times C_4$) x 16, $C_4^2:C_2^2$ ($C_2^4$) x 16, $C_4^2:C_2^2$ ($C_2^2\times C_4$) x 16, $C_4^2.C_2^2$ ($C_2^4$) x 16, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 16, $C_8:Q_8$ ($C_2^2\times C_4$) x 16, $C_8:Q_8$ ($C_2^2\times C_4$) x 16, $C_8:D_4$ ($C_2^2\times C_4$) x 16, $C_2^3.D_4$ ($C_2^4$) x 16, $C_2^3.D_4$ ($C_2^2\times C_4$) x 16, $C_4:\OD_{16}$ ($C_2^4$) x 16, $C_2^2\times \SD_{16}$ ($C_2^2\times C_4$) x 8, $C_2^2\times \OD_{16}$ ($C_2^4$) x 8, $C_4^2.C_2^2$ ($C_2^4$) x 8, $C_4^2.C_2^2$ ($C_2^4$) x 8, $C_2^3:Q_8$ ($C_2^2\times C_4$) x 8, $C_2^3:D_4$ ($C_2^2\times C_4$) x 8, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 8, $C_4^2:C_2^2$ ($C_2^2\times C_4$) x 8, $C_2^2\times C_4^2$ ($C_2\times D_4$) x 8, $C_2^2\times C_4^2$ ($C_2^2\times C_4$) x 8, $C_4:Q_{16}$ ($C_2^2\times C_4$) x 8, $C_4:D_8$ ($C_2^2\times C_4$) x 8, $C_2^2.D_8$ ($C_2^2\times C_4$) x 8, $C_2^3.D_4$ ($C_2^2\times C_4$) x 8, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 6, $C_2^2\times C_4^2$ ($D_4:C_2$) x 6, $D_4:C_2^3$ ($C_2^4$) x 4, $C_2^2\times Q_{16}$ ($C_2^2\times C_4$) x 4, $C_2^2\times D_8$ ($C_2^2\times C_4$) x 4, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 4, $C_4^2:C_2^2$ ($C_2^2\times C_4$) x 4, $C_4^2:C_2^2$ ($C_2^4$) x 4, $C_2^4\times C_4$ ($D_4:C_2$) x 2, $C_4^2:C_2^2$ ($C_2^2\times C_4$) x 2, $C_2^4\times C_4$ ($C_2^4$)

Order 32: $D_4:C_4$ ($C_2^3\times C_4$) x 64, $C_8:C_4$ ($D_4:C_2^2$) x 64, $C_4\times C_8$ ($C_2^2\times D_4$) x 64, $C_4\times C_8$ ($C_2^3\times C_4$) x 64, $Q_8:C_4$ ($C_2^3\times C_4$) x 64, $C_2^2\times C_8$ ($C_2^2\times D_4$) x 48, $C_2^2\times C_8$ ($C_2^3\times C_4$) x 48, $C_2\times C_4^2$ ($C_4\times D_4$) x 48, $C_2\times \SD_{16}$ ($C_2^3\times C_4$) x 32, $C_2^2\times C_8$ ($D_4:C_2^2$) x 32, $C_2^2:Q_8$ ($C_2^3\times C_4$) x 32, $C_4:D_4$ ($C_2^3\times C_4$) x 32, $C_4\times Q_8$ ($C_2^3\times C_4$) x 32, $C_4\times D_4$ ($C_2^3\times C_4$) x 32, $C_8:C_4$ ($C_2^3\times C_4$) x 32, $C_8:C_4$ ($C_2^3\times C_4$) x 32, $C_2\times C_4^2$ ($D_4:C_2^2$) x 28, $C_4:Q_8$ ($C_2^3\times C_4$) x 24, $C_2\times C_4^2$ ($C_2^2\times D_4$) x 24, $C_2\times C_4^2$ ($C_2^3\times C_4$) x 24, $C_2^3\times C_4$ ($C_4\times D_4$) x 16, $C_2\times Q_{16}$ ($C_2^3\times C_4$) x 16, $C_2\times D_8$ ($C_2^3\times C_4$) x 16, $C_2\times \OD_{16}$ ($C_2^5$) x 16, $C_2^2\times C_8$ ($C_2^5$) x 16, $C_4.Q_8$ ($C_2^3\times C_4$) x 16, $C_4^2:C_2$ ($C_2^3\times C_4$) x 16, $C_2^3\times C_4$ ($D_4:C_2^2$) x 15, $C_2\times C_4^2$ ($C_2^5$) x 12, $D_4:C_2^2$ ($C_2^5$) x 8, $D_4:C_2^2$ ($C_2^3\times C_4$) x 8, $C_4:D_4$ ($C_2^3\times C_4$) x 8, $C_4^2:C_2$ ($C_2^5$) x 8, $C_4^2:C_2$ ($C_2^3\times C_4$) x 8, $C_2^2.D_4$ ($C_2^3\times C_4$) x 8, $C_2^2\times Q_8$ ($C_2^3\times C_4$) x 4, $C_2^2\times D_4$ ($C_2^3\times C_4$) x 4, $C_2^3\times C_4$ ($C_2^2\times D_4$) x 4, $C_2^3\times C_4$ ($C_2^3\times C_4$) x 4, $C_2^3\times C_4$ ($C_2^5$) x 3, $C_2^5$ ($D_4:C_2^2$)

Order 16: $C_2^2\times C_4$ ($C_4^2:C_2^2$) x 60, $C_4^2$ ($C_4^2:C_2^2$) x 48, $C_2\times C_8$ ($D_4:C_2^3$) x 32, $C_2\times C_8$ ($C_4^2.C_2^2$) x 32, $C_2\times C_8$ ($C_4^2.C_2^2$) x 32, $C_2^2\times C_4$ ($D_4:C_2^3$) x 19, $C_2\times C_8$ ($D_4\times C_2^3$) x 16, $C_2\times C_8$ ($C_2^4\times C_4$) x 16, $C_4:C_4$ ($C_2^4\times C_4$) x 16, $C_4^2$ ($D_4:C_2^3$) x 16, $C_2^2\times C_4$ ($D_8:C_4$) x 16, $C_4^2$ ($D_4\times C_2^3$) x 8, $C_4^2$ ($C_2^4\times C_4$) x 8, $C_2\times Q_8$ ($C_2^4\times C_4$) x 8, $C_2\times D_4$ ($C_2^4\times C_4$) x 8, $C_2^2\times C_4$ ($D_4\times C_2^3$) x 6, $C_2^2\times C_4$ ($C_2^4\times C_4$) x 6, $C_2^4$ ($C_4^2:C_2^2$) x 4, $C_2^4$ ($D_4:C_2^3$) x 3, $C_2^2\times C_4$ ($C_2^6$)

Order 8: $C_2\times C_4$ ($C_4^2.C_2^3$) x 48, $C_2\times C_4$ ($C_4^2:C_2^3$) x 22, $C_8$ ($C_4^2.C_2^3$) x 16, $C_8$ ($C_4^2.C_2^3$) x 16, $C_2^3$ ($C_4^2.C_2^3$) x 8, $C_2\times C_4$ ($C_4^2.C_2^3$) x 8, $C_2\times C_4$ ($C_4^2.C_2^3$) x 8, $C_2^3$ ($C_4^2:C_2^3$) x 6, $C_2^3$ ($C_4^2.C_2^3$) x 4, $C_2\times C_4$ ($D_4:C_2^4$) x 4, $C_2\times C_4$ ($C_4^2.C_2^3$) x 4, $C_2^3$ ($D_4:C_2^4$), $C_2\times C_4$ ($D_4\times C_2^4$), $C_2\times C_4$ ($C_2^5\times C_4$)

Order 4: $C_4$ ($C_4^2.C_2^4$) x 16, $C_2^2$ ($C_4^2.C_2^4$) x 12, $C_2^2$ ($C_4^3.C_2^2$) x 4, $C_4$ ($C_4^3.C_2^2$) x 4, $C_2^2$ ($C_4^2.C_2^4$) x 2, $C_4$ ($C_4^2.C_2^4$) x 2, $C_4$ ($C_4^2.C_2^4$) x 2, $C_2^2$ ($C_4^2:C_2^4$)

Order 2: $C_2$ ($(C_2^3\times C_4).C_2^4$) x 4, $C_2$ ($C_4.C_2^6.C_2$) x 2, $C_2$ ($C_2\times C_2^4.C_2^4$)

Order 1: $C_1$ ($C_4^3.C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1024: $C_4^3.C_2^4$ ($C_1$)

Order 512: $(C_2^3\times C_4).C_2^4$ ($C_2$), $(C_2^3\times C_4).C_2^4$ ($C_2$), $(C_2^4\times C_4).C_2^3$ ($C_2$), $C_2^2.C_2^6.C_2$ ($C_2$), $(C_2\times C_4).C_2^6$ ($C_2$), $(C_2^3\times C_4).C_2^4$ ($C_2$), $(C_2^3\times C_4).C_2^4$ ($C_2$), $(C_2^4\times C_8).C_2^2$ ($C_2$), $C_2^2\times C_4\times C_2.\OD_{16}$ ($C_2$), $(C_2^3\times C_4).C_2^4$ ($C_2$), $(C_2^3\times C_4).C_2^4$ ($C_2$), $C_2^2.C_2^6.C_2$ ($C_2$)

Order 256: $C_4.C_4^3$ ($C_2^2$) x 5, $C_2^3\times C_4\times C_8$ ($C_2^2$) x 2, $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_4$), $C_4^2.C_2^4$ ($C_4$), $C_4^2.C_2^4$ ($C_4$), $C_4^2.C_2^4$ ($C_4$), $C_4^2.C_2^4$ ($C_4$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_2^5.D_4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_4^2.C_2^4$ ($C_2^2$), $C_2^2\times C_4^3$ ($C_2^2$), $C_4^3.C_2^2$ ($C_2^2$), $C_4^3.C_2^2$ ($C_2^2$), $C_4^3.C_2^2$ ($C_2^2$), $C_8.(C_4\times D_4)$ ($C_2^2$), $C_4^3.C_2^2$ ($C_2^2$), $C_4^3.C_2^2$ ($C_2^2$), $C_4^3.C_2^2$ ($C_2^2$), $(C_2^2\times D_8):C_4$ ($C_2^2$), $C_4^3:C_2^2$ ($C_2^2$), $C_2\times C_4^2.Q_8$ ($C_2^2$), $C_2\times C_4^2.D_4$ ($C_2^2$), $C_2^5.D_4$ ($C_2^2$), $C_4^2.(C_2\times D_4)$ ($C_2^2$), $C_2\times C_4^2.D_4$ ($C_2^2$), $C_2\times C_4^2.D_4$ ($C_2^2$), $C_2^5.Q_8$ ($C_2^2$), $D_8:C_4^2$ ($C_2^2$), $C_2\times C_8.C_4^2$ ($C_2^2$), $C_2\times C_8.C_4^2$ ($C_2^2$), $C_2\times D_4.C_4^2$ ($C_2^2$), $C_4^3:C_2^2$ ($C_2^2$)

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

Order 64: $C_2\times C_4\times C_8$ ($C_2^2\times C_4$) x 13, $C_2\times C_4\times C_8$ ($C_2\times D_4$) x 10, $C_2\times C_4\times C_8$ ($C_2^4$) x 6, $C_2^3\times C_8$ ($C_2^4$) x 6, $C_2^2\times C_4^2$ ($C_2^4$) x 5, $C_4.C_4^2$ ($C_2^4$) x 4, $C_2^3\times C_8$ ($C_2^2\times C_4$) x 3, $C_2^2\times C_4^2$ ($C_2\times D_4$) x 3, $C_2^2\times C_4^2$ ($C_2^2\times C_4$) x 3, $C_4^3$ ($C_2^4$) x 2, $C_2^3\times C_8$ ($C_2\times D_4$) x 2, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 2, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 2, $C_4^2.C_2^2$ ($C_2^2\times C_4$) x 2, $C_2^2\times C_4^2$ ($D_4:C_2$) x 2, $C_2^3.D_4$ ($C_2^4$), $C_2^3.D_4$ ($C_2^2\times C_4$), $C_2^2.Q_{16}$ ($C_2^2\times C_4$), $C_2^2.D_8$ ($C_2^2\times C_4$), $(C_2^2\times C_8):C_2$ ($C_2^4$), $C_2^2:\OD_{16}$ ($C_2^4$), $\OD_{16}:C_4$ ($C_2^4$), $C_4\times \OD_{16}$ ($C_2^4$), $C_4.C_4^2$ ($D_4:C_2$), $D_4:C_2^3$ ($C_2^4$), $C_2^4\times C_4$ ($C_2^4$), $C_2^4\times C_4$ ($D_4:C_2$), $D_8:C_2^2$ ($C_2^4$), $D_8:C_2^2$ ($C_2^2\times C_4$), $C_2^2\times Q_{16}$ ($C_2^2\times C_4$), $C_2^2\times \SD_{16}$ ($C_2^2\times C_4$), $C_2^2\times D_8$ ($C_2^2\times C_4$), $C_8.C_2^3$ ($C_2^4$), $C_2^2\times \OD_{16}$ ($C_2^4$), $C_4.C_4^2$ ($C_2^4$), $C_4^2.C_2^2$ ($C_2^4$), $C_4^2.C_2^2$ ($C_2^4$), $C_4^2:C_2^2$ ($C_2^2\times C_4$), $C_4^2.C_2^2$ ($C_2^2\times C_4$), $C_4^2.C_2^2$ ($C_2^2\times C_4$), $C_4^2:C_2^2$ ($C_2^2\times C_4$), $C_4^2:C_2^2$ ($C_2^4$), $C_4^2:C_2^2$ ($C_2^2\times C_4$), $C_2^3:Q_8$ ($C_2^2\times C_4$), $C_2^3:D_4$ ($C_2^2\times C_4$), $C_4^2:C_4$ ($C_2^4$), $C_4^2.C_2^2$ ($C_2^4$), $C_4^2.C_2^2$ ($C_2^2\times C_4$), $C_4^2.C_2^2$ ($C_2^2\times C_4$), $C_4^2:C_2^2$ ($C_2^2\times C_4$), $C_4^2:C_2^2$ ($C_2^4$), $C_8:Q_8$ ($C_2^2\times C_4$), $C_8.Q_8$ ($C_2^2\times C_4$), $C_8:Q_8$ ($C_2^2\times C_4$), $C_8.D_4$ ($C_2^2\times C_4$), $C_4:Q_{16}$ ($C_2^2\times C_4$), $C_4:D_8$ ($C_2^2\times C_4$), $C_8:D_4$ ($C_2^2\times C_4$), $C_4^2.C_2^2$ ($C_2^2\times C_4$), $C_4.Q_{16}$ ($C_2^2\times C_4$), $C_4.D_8$ ($C_2^2\times C_4$), $C_2^2:Q_{16}$ ($C_2^2\times C_4$), $C_8:D_4$ ($C_2^2\times C_4$), $C_8:D_4$ ($C_2^2\times C_4$), $C_4\times Q_{16}$ ($C_2^2\times C_4$), $C_4\times \SD_{16}$ ($C_2^2\times C_4$), $C_4\times D_8$ ($C_2^2\times C_4$), $C_2^3.Q_8$ ($C_2^4$), $C_2^3.D_4$ ($C_2^4$), $C_2^3.D_4$ ($C_2^2\times C_4$), $C_2^2.D_8$ ($C_2^2\times C_4$), $C_2^3.D_4$ ($C_2^2\times C_4$), $C_4^2.C_2^2$ ($C_2^4$), $C_4:\OD_{16}$ ($C_2^4$), $C_4^2:C_2^2$ ($C_2^4$)

Order 32: $C_2\times C_4^2$ ($C_4\times D_4$) x 9, $C_2^2\times C_8$ ($C_2^3\times C_4$) x 7, $C_4\times C_8$ ($C_2^3\times C_4$) x 7, $C_2\times C_4^2$ ($C_2^2\times D_4$) x 7, $C_2\times C_4^2$ ($C_2^3\times C_4$) x 7, $C_2^3\times C_4$ ($D_4:C_2^2$) x 5, $C_2^3\times C_4$ ($C_4\times D_4$) x 5, $C_2^2\times C_8$ ($C_2^2\times D_4$) x 5, $C_4\times C_8$ ($C_2^2\times D_4$) x 5, $C_2^2\times C_8$ ($C_2^5$) x 4, $C_2\times C_4^2$ ($D_4:C_2^2$) x 4, $C_2\times C_4^2$ ($C_2^5$) x 3, $C_2^3\times C_4$ ($C_2^5$) x 2, $C_2^3\times C_4$ ($C_2^2\times D_4$) x 2, $C_2^3\times C_4$ ($C_2^3\times C_4$) x 2, $C_2^2\times C_8$ ($D_4:C_2^2$) x 2, $C_4:Q_8$ ($C_2^3\times C_4$) x 2, $D_4:C_4$ ($C_2^3\times C_4$), $C_2^5$ ($D_4:C_2^2$), $D_4:C_2^2$ ($C_2^5$), $D_4:C_2^2$ ($C_2^3\times C_4$), $C_2^2\times Q_8$ ($C_2^3\times C_4$), $C_2^2\times D_4$ ($C_2^3\times C_4$), $C_2\times Q_{16}$ ($C_2^3\times C_4$), $C_2\times \SD_{16}$ ($C_2^3\times C_4$), $C_8:C_4$ ($D_4:C_2^2$), $C_2\times D_8$ ($C_2^3\times C_4$), $C_2\times \OD_{16}$ ($C_2^5$), $C_4:D_4$ ($C_2^3\times C_4$), $C_4.Q_8$ ($C_2^3\times C_4$), $C_4^2:C_2$ ($C_2^3\times C_4$), $C_2^2:Q_8$ ($C_2^3\times C_4$), $C_4:D_4$ ($C_2^3\times C_4$), $C_4\times Q_8$ ($C_2^3\times C_4$), $C_4\times D_4$ ($C_2^3\times C_4$), $C_4^2:C_2$ ($C_2^5$), $C_4^2:C_2$ ($C_2^3\times C_4$), $C_2^2.D_4$ ($C_2^3\times C_4$), $C_8:C_4$ ($C_2^3\times C_4$), $C_8:C_4$ ($C_2^3\times C_4$), $Q_8:C_4$ ($C_2^3\times C_4$)

Order 16: $C_2^2\times C_4$ ($C_4^2:C_2^2$) x 14, $C_4^2$ ($C_4^2:C_2^2$) x 10, $C_2^2\times C_4$ ($D_4:C_2^3$) x 5, $C_2\times C_8$ ($C_2^4\times C_4$) x 3, $C_2\times C_8$ ($C_4^2.C_2^2$) x 3, $C_4^2$ ($D_4\times C_2^3$) x 3, $C_4^2$ ($C_2^4\times C_4$) x 3, $C_2^2\times C_4$ ($D_4\times C_2^3$) x 3, $C_2^2\times C_4$ ($C_2^4\times C_4$) x 3, $C_2\times C_8$ ($D_4:C_2^3$) x 2, $C_2\times C_8$ ($D_4\times C_2^3$) x 2, $C_2\times C_8$ ($C_4^2.C_2^2$) x 2, $C_2^4$ ($D_4:C_2^3$) x 2, $C_2^4$ ($C_4^2:C_2^2$) x 2, $C_4:C_4$ ($C_2^4\times C_4$), $C_4^2$ ($D_4:C_2^3$), $C_2\times Q_8$ ($C_2^4\times C_4$), $C_2\times D_4$ ($C_2^4\times C_4$), $C_2^2\times C_4$ ($C_2^6$), $C_2^2\times C_4$ ($D_8:C_4$)

Order 8: $C_2\times C_4$ ($C_4^2:C_2^3$) x 7, $C_2^3$ ($C_4^2:C_2^3$) x 3, $C_2\times C_4$ ($C_4^2.C_2^3$) x 3, $C_2\times C_4$ ($C_4^2.C_2^3$) x 2, $C_2\times C_4$ ($C_4^2.C_2^3$) x 2, $C_8$ ($C_4^2.C_2^3$) x 2, $C_2^3$ ($D_4:C_2^4$), $C_2^3$ ($C_4^2.C_2^3$), $C_2^3$ ($C_4^2.C_2^3$), $C_2\times C_4$ ($D_4:C_2^4$), $C_2\times C_4$ ($D_4\times C_2^4$), $C_2\times C_4$ ($C_2^5\times C_4$), $C_2\times C_4$ ($C_4^2.C_2^3$), $C_8$ ($C_4^2.C_2^3$)

Order 4: $C_2^2$ ($C_4^2.C_2^4$) x 2, $C_2^2$ ($C_4^3.C_2^2$) x 2, $C_4$ ($C_4^2.C_2^4$) x 2, $C_4$ ($C_4^3.C_2^2$) x 2, $C_2^2$ ($C_4^2.C_2^4$), $C_2^2$ ($C_4^2:C_2^4$), $C_4$ ($C_4^2.C_2^4$), $C_4$ ($C_4^2.C_2^4$)

Order 2: $C_2$ ($C_4.C_2^6.C_2$), $C_2$ ($(C_2^3\times C_4).C_2^4$), $C_2$ ($C_2\times C_2^4.C_2^4$)

Order 1: $C_1$ ($C_4^3.C_2^4$)

Series

Derived series

$C_4^3.C_2^4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_1$

Chief series

$C_4^3.C_2^4$

$\rhd$

$(C_2\times C_4).C_2^6$

$\rhd$

$C_2^3\times C_4\times C_8$

$\rhd$

$C_2^3\times C_4^2$

$\rhd$

$C_2^4\times C_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^3.C_2^4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4^2$

$\lhd$

$C_2^3\times C_4\times C_8$

$\lhd$

$C_4^3.C_2^4$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

See the $256 \times 256$ 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 $168 \times 168$ rational character table (warning: may be slow to load).