Abstract group 2048.tj: $C_4^3.(C_4\times D_4)$

• Label: 2048.tj
• Order: \( 2^{11} \)
• Exponent: \( 2^{4} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{20} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \)
• Perm deg.: $42$
• Trans deg.: $256$
• Rank: $4$

Group information

Description:	$C_4^3.(C_4\times D_4)$
Order:	\(2048\)\(\medspace = 2^{11} \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Automorphism group:	$(C_2^5\times C_4^3).C_2^6.C_2^3$, of order \(1048576\)\(\medspace = 2^{20} \)
Composition factors:	$C_2$ x 11
Nilpotency class:	$4$
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
Elements	1	47	208	768	1024	2048
Conjugacy classes	1	13	58	104	96	272
Divisions	1	13	34	48	24	120
Autjugacy classes	1	7	19	25	8	60

Dimension	1	2	4	8	16
Irr. complex chars.	128	96	32	16	0	272
Irr. rational chars.	16	32	36	28	8	120

Minimal presentations

Permutation degree:	$42$
Transitive degree:	$256$
Rank:	$4$
Inequivalent generating quadruples:	$5160960$

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:	${\langle a, b, c, d, e \mid d^{8}=e^{2}=[a,e]=[b,d]=[b,e]=[c,e]=[d,e]=1, a^{8}= \!\cdots\! \rangle}$
Permutation group:	Degree $42$ $\langle(1,4)(2,5,12,25)(6,9)(7,10)(8,26)(11,31,28,27)(13,14,15,16)(17,30,19,29) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 26 \\ 16 & 31 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 22 \\ 16 & 27 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_4^2.(C_8\times D_4))$
Semidirect product:	$(C_4^2.D_8)$ $\,\rtimes\,$ $C_8$	$(C_4^2.D_8)$ $\,\rtimes\,$ $C_8$	$(C_4^2.D_8)$ $\,\rtimes\,$ $C_8$	$(D_8:C_8)$ $\,\rtimes\,$ $(C_2\times C_8)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^3$ . $(C_4\times D_4)$ (2)	$C_4^2$ . $(D_8:C_8)$ (2)	$(C_{16}:C_4^2)$ . $D_4$ (2)	$(C_2^4.Q_8)$ . $C_4^2$ (2)	all 233

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

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

Subgroups

There are 8552 subgroups in 2990 conjugacy classes, 880 normal (186 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4^2$	$G/Z \simeq$ $D_8:C_4$
Commutator:	$G' \simeq$ $C_2\times C_8$	$G/G' \simeq$ $C_2^2\times C_4\times C_8$
Frattini:	$\Phi \simeq$ $C_4^2\times C_8$	$G/\Phi \simeq$ $C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^3.(C_4\times D_4)$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4^3.(C_4\times D_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $D_4:(C_4\times C_8)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^3.(C_4\times D_4)$

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 2048: $C_4^3.(C_4\times D_4)$

Order 1024: $(C_4^2\times C_8).C_2^3$ x 8, $C_2\times C_4.C_4^3.C_2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$

Order 512: $(C_4^2\times C_8).C_2^2$ x 12, $(C_4^2\times C_8).C_2^2$ x 8, $(C_4^2\times C_8).C_2^2$ x 8, $C_4.C_4^3.C_2$ x 6, $(C_4^2\times C_8).C_2^2$ x 6, $(C_4^2\times C_8).C_2^2$ x 4, $(C_4^2\times C_8).C_2^2$ x 4, $C_2\times (C_2^2\times C_4).C_4^2$ x 2, $C_2\times C_4.C_4^3$ x 2, $C_2\times (C_4\times C_8).C_2^3$ x 2, $C_2^2\times C_2.D_4:C_8$ x 2, $C_2\times (C_2^2\times C_8).C_2^3$ x 2, $C_2\times C_4.(C_2\times C_4^2).C_2$ x 2, $C_2\times C_4.C_4:(C_4:C_4)$ x 2, $C_2\times C_4.(C_2\times C_4^2).C_2$ x 2, $C_2\times (C_4^2\times C_8).C_2$ x 2, $C_2\times (C_2^2\times C_8).C_2^3$ x 2, $C_2\times (C_2^2\times C_4).C_4^2$, $C_2\times C_4^2.C_2^3.C_2$, $C_2\times (C_4^2\times C_8).C_2$, $C_2\times (C_2\times C_8).C_2^4$, $C_2\times (C_4^2\times C_8).C_2$, $C_2\times (C_2^2\times C_8).C_2^3$, $C_2\times C_2^2.C_4^3$

Order 256: $C_4^2.\OD_{16}$ x 16, $C_2\times D_8:C_8$ x 16, $C_4^2.D_8$ x 16, $C_{16}:C_4^2$ x 14, $C_4.(C_8\times D_4)$ x 12, $C_4.(C_8\times Q_8)$ x 12, $C_4^3.C_4$ x 12, $(C_4\times \OD_{32}):C_2$ x 12, $D_8:C_4^2$ x 12, $\OD_{16}.C_4^2$ x 8, $C_8.(C_4\times C_8)$ x 8, $C_4^2.D_8$ x 8, $C_4.C_8^2$ x 8, $C_2^2.C_4^3$ x 6, $C_4^2.C_4^2$ x 6, $C_2^3.\OD_{32}$ x 6, $C_4^3.C_2^2$ x 6, $C_4^2.D_8$ x 4, $C_4^2.D_8$ x 4, $C_2^4.\OD_{16}$ x 4, $C_2^2\times C_4\times C_{16}$ x 4, $\OD_{32}:C_8$ x 4, $C_4^2.C_4^2$ x 3, $C_2^4.\OD_{16}$ x 2, $C_2^4.\OD_{16}$ x 2, $C_4^2.C_2^4$ x 2, $C_8^2.C_4$ x 2, $(C_4\times C_{16}):C_2^2$ x 2, $(C_2^2\times C_4):C_{16}$ x 2, $C_4^2.(C_2\times C_8)$ x 2, $\OD_{32}:C_2^3$ x 2, $(C_2^3\times C_{16}):C_2$ x 2, $C_2\times C_4\times \OD_{32}$ x 2, $C_2\times C_8.C_4^2$ x 2, $C_2\times D_4.C_4^2$ x 2, $C_2\times C_4^2\times C_8$ x 2, $C_4^2.C_2^4$ x 2, $C_4^3.C_4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_2\times C_8.C_4^2$, $C_4^3:C_2^2$, $C_8^2:C_2^2$

Order 128: $C_8.C_4^2$ x 46, $C_2\times C_4\times C_{16}$ x 36, $C_2^3.\OD_{16}$ x 28, $C_4^2.C_2^3$ x 28, $C_2^3.C_4^2$ x 27, $C_8.\OD_{16}$ x 16, $C_4^2.C_8$ x 16, $\OD_{32}:C_2^2$ x 16, $D_8:C_8$ x 16, $Q_{16}:C_8$ x 16, $D_8:C_8$ x 16, $D_4.C_4^2$ x 16, $C_4^2\times C_8$ x 14, $C_8.(C_2\times D_4)$ x 12, $(C_2^2\times C_{16}):C_2$ x 12, $C_4\times \OD_{32}$ x 12, $C_8.C_4^2$ x 12, $C_4^2.D_4$ x 12, $C_4^2.D_4$ x 12, $D_4.C_4^2$ x 12, $C_2^2\times C_4\times C_8$ x 11, $D_4:C_4^2$ x 8, $C_8:C_{16}$ x 8, $\OD_{32}:C_4$ x 8, $\OD_{32}:C_4$ x 8, $C_8.C_4^2$ x 6, $C_2^3\times C_{16}$ x 6, $C_8^2:C_2$ x 6, $C_4^2.C_8$ x 4, $C_2^3:C_{16}$ x 4, $C_4^2:C_8$ x 4, $C_4^2.D_4$ x 4, $C_2^2\times \OD_{32}$ x 4, $C_2^3.C_4^2$ x 4, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_8.C_2^4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.Q_8$ x 2, $C_4^2:C_2^3$ x 2, $C_2^3.C_4^2$ x 2, $C_2\times C_4^3$, $C_2^4.Q_8$, $D_8:C_2^3$, $C_4^2.C_2^3$, $C_2\times C_8^2$, $C_2^4.D_4$, $C_2^4.D_4$

Order 64: $C_2\times C_4\times C_8$ x 111, $C_2^2\times C_{16}$ x 56, $C_{16}:C_4$ x 48, $D_8:C_4$ x 48, $C_4\times C_{16}$ x 44, $C_8:C_8$ x 32, $\OD_{16}:C_4$ x 28, $C_8.C_2^3$ x 28, $C_2\times \OD_{32}$ x 28, $C_8.C_8$ x 24, $C_4:C_{16}$ x 16, $\OD_{32}:C_2$ x 16, $C_2^2:C_{16}$ x 16, $C_4\times \SD_{16}$ x 16, $C_4^2:C_2^2$ x 16, $C_4\times \OD_{16}$ x 14, $D_8:C_2^2$ x 14, $C_2^3.Q_8$ x 14, $C_4^2.C_2^2$ x 12, $C_8:C_8$ x 12, $C_4\times Q_{16}$ x 12, $C_4\times D_8$ x 12, $C_4^2.C_4$ x 12, $C_2^3.D_4$ x 8, $C_2^3\times C_8$ x 8, $C_4^2:C_4$ x 8, $C_8:C_8$ x 8, $C_4^3$ x 6, $C_2^2\times C_4^2$ x 6, $C_2^3.D_4$ x 6, $C_2^2\times \OD_{16}$ x 4, $C_4.C_4^2$ x 4, $C_8^2$ x 4, $C_4^2:C_2^2$ x 4, $C_4^2.C_4$ x 4, $C_4.C_4^2$ x 3, $C_2^2.Q_{16}$ x 2, $C_2^2.D_8$ x 2, $C_2^3:C_8$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2.D_8$ x 2, $D_4:C_2^3$, $C_2^2\times Q_{16}$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^3.D_4$

Order 32: $C_4\times C_8$ x 136, $C_2^2\times C_8$ x 94, $C_2\times C_{16}$ x 88, $C_2\times C_4^2$ x 57, $\OD_{16}:C_2$ x 48, $C_2\times \OD_{16}$ x 30, $C_4\times D_4$ x 28, $D_8:C_2$ x 24, $\OD_{32}$ x 24, $C_4\wr C_2$ x 16, $D_4:C_2^2$ x 14, $C_4^2:C_2$ x 14, $C_8:C_4$ x 12, $C_4\times Q_8$ x 12, $C_8.C_4$ x 12, $C_4:C_8$ x 12, $D_4:C_4$ x 8, $C_2^2:C_8$ x 8, $C_2^3\times C_4$ x 8, $C_2\times \SD_{16}$ x 8, $Q_8:C_4$ x 8, $C_2\times Q_{16}$ x 7, $C_2\times D_8$ x 7, $C_8:C_4$ x 6, $C_2^2.D_4$ x 4, $C_8:C_4$ x 4, $C_2^3:C_4$ x 3, $C_2^2\times D_4$ x 2, $C_2^2\times Q_8$

Order 16: $C_2\times C_8$ x 172, $C_2^2\times C_4$ x 81, $C_4^2$ x 64, $\OD_{16}$ x 28, $D_4:C_2$ x 24, $C_{16}$ x 24, $C_2\times D_4$ x 15, $C_4:C_4$ x 14, $C_2^2:C_4$ x 12, $\SD_{16}$ x 8, $C_2\times Q_8$ x 7, $Q_{16}$ x 6, $D_8$ x 6, $C_2^4$ x 2

Order 8: $C_2\times C_4$ x 132, $C_8$ x 48, $C_2^3$ x 19, $D_4$ x 14, $Q_8$ x 6

Order 4: $C_4$ x 34, $C_2^2$ x 33

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2048: $C_4^3.(C_4\times D_4)$

Order 1024: $(C_4^2\times C_8).C_2^3$, $C_2\times C_4.C_4^3.C_2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$, $C_2\times (C_4^2\times C_8).C_2^2$

Order 512: $(C_4^2\times C_8).C_2^2$ x 4, $C_2\times C_4.C_4^3$ x 2, $(C_4^2\times C_8).C_2^2$ x 2, $C_4.C_4^3.C_2$ x 2, $(C_4^2\times C_8).C_2^2$ x 2, $(C_4^2\times C_8).C_2^2$ x 2, $C_2\times (C_4^2\times C_8).C_2$ x 2, $C_2\times (C_2^2\times C_4).C_4^2$, $C_2\times (C_2^2\times C_4).C_4^2$, $C_2\times (C_4\times C_8).C_2^3$, $C_2^2\times C_2.D_4:C_8$, $C_2\times C_4^2.C_2^3.C_2$, $(C_4^2\times C_8).C_2^2$, $C_2\times (C_2^2\times C_8).C_2^3$, $C_2\times C_4.(C_2\times C_4^2).C_2$, $C_2\times (C_4^2\times C_8).C_2$, $C_2\times C_4.C_4:(C_4:C_4)$, $C_2\times (C_2\times C_8).C_2^4$, $C_2\times C_4.(C_2\times C_4^2).C_2$, $C_2\times (C_4^2\times C_8).C_2$, $C_2\times (C_2^2\times C_8).C_2^3$, $(C_4^2\times C_8).C_2^2$, $C_2\times C_2^2.C_4^3$, $C_2\times (C_2^2\times C_8).C_2^3$

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

Order 128: $C_2\times C_4\times C_{16}$ x 20, $C_8.C_4^2$ x 15, $C_4^2.C_2^3$ x 12, $C_4^2\times C_8$ x 11, $C_2^2\times C_4\times C_8$ x 9, $C_2^3.\OD_{16}$ x 8, $C_4\times \OD_{32}$ x 8, $C_8.C_4^2$ x 8, $D_4.C_4^2$ x 8, $C_2^3.C_4^2$ x 8, $D_4.C_4^2$ x 8, $C_4^2.D_4$ x 6, $C_8.C_4^2$ x 4, $D_4:C_4^2$ x 4, $C_4^2.D_4$ x 4, $C_8^2:C_2$ x 4, $C_2^3.C_4^2$ x 4, $\OD_{32}:C_4$ x 4, $C_2^3\times C_{16}$ x 3, $C_8.\OD_{16}$ x 2, $C_4^2.C_8$ x 2, $C_8.(C_2\times D_4)$ x 2, $\OD_{32}:C_2^2$ x 2, $(C_2^2\times C_{16}):C_2$ x 2, $C_4^2:C_8$ x 2, $C_2^4.D_4$ x 2, $C_8.C_2^4$ x 2, $C_2^2\times \OD_{32}$ x 2, $C_2^4.Q_8$ x 2, $C_4^2:C_2^3$ x 2, $C_2^3.C_4^2$ x 2, $C_2\times C_4^3$, $C_4^2.C_8$, $C_2^3:C_{16}$, $D_8:C_8$, $Q_{16}:C_8$, $D_8:C_8$, $C_2^4.Q_8$, $C_8:C_{16}$, $C_4^2.D_4$, $C_4^2.D_4$, $D_8:C_2^3$, $C_4^2.C_2^3$, $C_2\times C_8^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$, $C_2^4.D_4$, $C_2^4.D_4$, $\OD_{32}:C_4$

Order 64: $C_2\times C_4\times C_8$ x 57, $C_2^2\times C_{16}$ x 17, $\OD_{16}:C_4$ x 16, $C_4\times C_{16}$ x 16, $C_{16}:C_4$ x 13, $C_8.C_2^3$ x 12, $D_8:C_4$ x 12, $C_4\times \OD_{16}$ x 8, $D_8:C_2^2$ x 8, $C_2\times \OD_{32}$ x 8, $C_2^3.Q_8$ x 8, $C_8:C_8$ x 7, $C_2^3\times C_8$ x 7, $C_8.C_8$ x 6, $C_4^2.C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $C_4^3$ x 5, $C_2^2\times C_4^2$ x 5, $C_2^2\times \OD_{16}$ x 4, $C_4^2:C_4$ x 4, $C_4\times Q_{16}$ x 4, $C_4\times \SD_{16}$ x 4, $C_4\times D_8$ x 4, $C_4^2.C_4$ x 4, $C_2^3.D_4$ x 4, $C_2^3.D_4$ x 3, $C_4.C_4^2$ x 3, $C_4.C_4^2$ x 2, $C_8^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_8:C_8$ x 2, $C_8:C_8$ x 2, $C_2^2.Q_{16}$, $C_2^2.D_8$, $C_2^3:C_8$, $C_4:C_{16}$, $\OD_{32}:C_2$, $C_2^2:C_{16}$, $D_4:C_2^3$, $C_2^2\times Q_{16}$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_4^2.C_2^2$, $C_2^2.D_8$, $C_2^3.D_4$, $C_4^2.C_4$

Order 32: $C_4\times C_8$ x 57, $C_2^2\times C_8$ x 44, $C_2\times C_4^2$ x 29, $C_2\times C_{16}$ x 19, $C_2\times \OD_{16}$ x 14, $\OD_{16}:C_2$ x 12, $D_8:C_2$ x 10, $D_4:C_2^2$ x 8, $C_4\times D_4$ x 8, $C_4^2:C_2$ x 7, $C_2^3\times C_4$ x 6, $C_8:C_4$ x 6, $\OD_{32}$ x 6, $C_8.C_4$ x 6, $C_2\times Q_{16}$ x 4, $C_2\times \SD_{16}$ x 4, $C_2\times D_8$ x 4, $C_4\times Q_8$ x 4, $C_4\wr C_2$ x 4, $D_4:C_4$ x 2, $C_2^2:C_8$ x 2, $C_2^2\times D_4$ x 2, $C_2^2.D_4$ x 2, $C_2^3:C_4$ x 2, $C_8:C_4$ x 2, $C_8:C_4$ x 2, $C_4:C_8$ x 2, $Q_8:C_4$ x 2, $C_2^2\times Q_8$

Order 16: $C_2\times C_8$ x 63, $C_2^2\times C_4$ x 34, $C_4^2$ x 28, $\OD_{16}$ x 10, $D_4:C_2$ x 10, $C_2\times D_4$ x 8, $C_{16}$ x 5, $C_4:C_4$ x 4, $C_2^2:C_4$ x 4, $C_2\times Q_8$ x 4, $Q_{16}$ x 3, $\SD_{16}$ x 3, $D_8$ x 3, $C_2^4$ x 2

Order 8: $C_2\times C_4$ x 47, $C_8$ x 20, $C_2^3$ x 9, $D_4$ x 6, $Q_8$ x 3

Order 4: $C_4$ x 16, $C_2^2$ x 13

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2048: $C_4^3.(C_4\times D_4)$ ($C_1$)

Order 1024: $(C_4^2\times C_8).C_2^3$ ($C_2$) x 8, $C_2\times C_4.C_4^3.C_2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$)

Order 512: $(C_4^2\times C_8).C_2^2$ ($C_2^2$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_4$) x 4, $C_4.C_4^3.C_2$ ($C_2^2$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_2^2$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_2^2$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_4$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_2^2$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_2^2$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_4$) x 4, $(C_4^2\times C_8).C_2^2$ ($C_2^2$) x 4, $C_2\times C_4.C_4^3$ ($C_2^2$) x 2, $C_2\times (C_4\times C_8).C_2^3$ ($C_4$) x 2, $C_2^2\times C_2.D_4:C_8$ ($C_4$) x 2, $C_2\times C_4.(C_2\times C_4^2).C_2$ ($C_4$) x 2, $C_2\times C_4.C_4:(C_4:C_4)$ ($C_4$) x 2, $C_2\times C_4.(C_2\times C_4^2).C_2$ ($C_4$) x 2, $C_2\times (C_4^2\times C_8).C_2$ ($C_2^2$) x 2, $C_2\times (C_2^2\times C_8).C_2^3$ ($C_4$) x 2, $C_2\times (C_4^2\times C_8).C_2$ ($C_2^2$), $C_2\times (C_4^2\times C_8).C_2$ ($C_2^2$), $C_2\times C_2^2.C_4^3$ ($C_2^2$)

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

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

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

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

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

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

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

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

Order 1: $C_1$ ($C_4^3.(C_4\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2048: $C_4^3.(C_4\times D_4)$ ($C_1$)

Order 1024: $(C_4^2\times C_8).C_2^3$ ($C_2$), $C_2\times C_4.C_4^3.C_2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$), $C_2\times (C_4^2\times C_8).C_2^2$ ($C_2$)

Order 512: $C_2\times C_4.C_4^3$ ($C_2^2$) x 2, $C_2\times (C_4^2\times C_8).C_2$ ($C_2^2$) x 2, $(C_4^2\times C_8).C_2^2$ ($C_2^2$), $(C_4^2\times C_8).C_2^2$ ($C_4$), $C_2\times (C_4\times C_8).C_2^3$ ($C_4$), $C_2^2\times C_2.D_4:C_8$ ($C_4$), $C_4.C_4^3.C_2$ ($C_2^2$), $(C_4^2\times C_8).C_2^2$ ($C_2^2$), $(C_4^2\times C_8).C_2^2$ ($C_2^2$), $(C_4^2\times C_8).C_2^2$ ($C_4$), $C_2\times C_4.(C_2\times C_4^2).C_2$ ($C_4$), $(C_4^2\times C_8).C_2^2$ ($C_2^2$), $C_2\times (C_4^2\times C_8).C_2$ ($C_2^2$), $C_2\times C_4.C_4:(C_4:C_4)$ ($C_4$), $(C_4^2\times C_8).C_2^2$ ($C_2^2$), $(C_4^2\times C_8).C_2^2$ ($C_4$), $C_2\times C_4.(C_2\times C_4^2).C_2$ ($C_4$), $C_2\times (C_4^2\times C_8).C_2$ ($C_2^2$), $(C_4^2\times C_8).C_2^2$ ($C_2^2$), $C_2\times C_2^2.C_4^3$ ($C_2^2$), $C_2\times (C_2^2\times C_8).C_2^3$ ($C_4$)

Order 256: $D_8:C_4^2$ ($C_2\times C_4$) x 3, $C_4^3.C_4$ ($C_2^3$) x 2, $C_4^3.C_4$ ($C_2\times C_4$) x 2, $C_{16}:C_4^2$ ($C_2^3$) x 2, $C_{16}:C_4^2$ ($D_4$) x 2, $C_{16}:C_4^2$ ($C_2\times C_4$) x 2, $C_4^2.C_2^4$ ($C_2\times C_4$) x 2, $C_2^4.\OD_{16}$ ($C_2\times C_4$) x 2, $C_2^2\times C_4\times C_{16}$ ($D_4$) x 2, $C_2\times C_8.C_4^2$ ($C_2\times C_4$) x 2, $C_2^2.C_4^3$ ($C_2^3$), $C_2^2.C_4^3$ ($C_2\times C_4$), $C_4.(C_8\times D_4)$ ($C_2\times C_4$), $C_4.(C_8\times Q_8)$ ($C_2\times C_4$), $C_4^2.C_2^4$ ($C_2\times C_4$), $C_2\times D_8:C_8$ ($C_2\times C_4$), $C_4^2.D_8$ ($C_2\times C_4$), $C_4^2.D_8$ ($C_8$), $C_4^2.D_8$ ($C_8$), $C_4^2.D_8$ ($C_8$), $C_2^3.\OD_{32}$ ($D_4$), $C_2^3.\OD_{32}$ ($C_2\times C_4$), $C_2^2\times C_4\times C_{16}$ ($C_2\times C_4$), $C_8.(C_4\times C_8)$ ($C_2\times C_4$), $C_4^2.D_8$ ($C_2\times C_4$), $D_8:C_4^2$ ($C_2^3$), $C_2\times C_8.C_4^2$ ($C_2\times C_4$), $C_2\times C_4^2\times C_8$ ($C_2^3$), $C_4^3.C_2^2$ ($C_2^3$), $C_4^3.C_2^2$ ($C_2\times C_4$), $C_4^2.C_2^4$ ($C_2\times C_4$), $C_4^2.C_4^2$ ($C_2\times C_4$)

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

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

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

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

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

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

Order 2: $C_2$ ($(C_4^2\times C_8).C_2^3$), $C_2$ ($C_2\times (C_2\times C_4\times C_8).C_2^3$), $C_2$ ($C_2\times C_4^3.C_2^3$)

Order 1: $C_1$ ($C_4^3.(C_4\times D_4)$)

Series

Derived series

$C_4^3.(C_4\times D_4)$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_1$

Chief series

$C_4^3.(C_4\times D_4)$

$\rhd$

$C_2\times (C_4^2\times C_8).C_2^2$

$\rhd$

$C_2\times (C_4^2\times C_8).C_2$

$\rhd$

$C_2\times C_4^2\times C_8$

$\rhd$

$C_4^2\times C_8$

$\rhd$

$C_2\times C_4\times C_8$

$\rhd$

$C_2^2\times C_8$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^3.(C_4\times D_4)$

$\rhd$

$C_2\times C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4^2$

$\lhd$

$C_2^2\times C_4\times C_8$

$\lhd$

$C_2\times C_4.C_4^3$

$\lhd$

$C_4^3.(C_4\times D_4)$

Character theory

Complex character table

See the $272 \times 272$ 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 $120 \times 120$ rational character table.