Group #513264 — subgroups

Subgroups — #513264

Order 512.

304 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33962	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₃₂	C₂	1	1	yes	maximal, characteristic
3	#13720	C₂	1	3	yes	maximal
4	#13706	C₂	1	3	yes	maximal
5	#13688	C₂	1	3	yes	maximal
6	Q₈×C₃₂	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₆₄	C₂	1	3	yes	maximal
8	#3395	—	2	1	no	—
9	Q₈×Q₁₆	—	2	1	no	—
10	C₂×Q₆₄	C₂²	1	3	yes	—
11	#2252	C₂²	1	3	yes	—
12	#2247	C₂²	1	3	yes	—
13	#2227	—	2	3	no	—
14	#2209	—	2	3	no	—
15	Q₈×C₁₆	C₂²	1	1	yes	characteristic
16	C₄×Q₃₂	C₂²	1	3	yes	—
17	C₄×Q₃₂	C₂²	1	3	yes	—
18	#1423	C₂²	1	3	yes	—
19	C₄⋊C₃₂	C₂²	1	3	yes	—
20	#1416	C₂²	1	1	yes	characteristic
21	#1416	C₂²	1	3	yes	—
22	#1416	C₂²	1	6	yes	—
23	C₄×C₃₂	C₂²	1	3	yes	—
24	Q₈²	—	4	1	no	—
25	#557	—	4	1	no	—
26	C₂×Q₃₂	—	2	3	no	—
27	C₂×Q₃₂	C₂³	1	3	yes	—
28	C₂×Q₃₂	C₂³	1	1	yes	characteristic
29	#500	—	2	3	no	—
30	#494	—	2	3	no	—
31	#477	—	4	3	no	—
32	#462	—	4	3	no	—
33	C₈×Q₈	D₄	1	1	yes	characteristic
34	C₄×Q₁₆	—	2	3	no	—
35	C₄×Q₁₆	—	2	3	no	—
36	Q₆₄	—	2	6	no	—
37	C₂×C₃₂	C₂³	1	1	yes	characteristic
38	C₂×C₃₂	C₂³	1	3	yes	—
39	#366	—	2	3	no	—
40	#366	C₂³	1	1	yes	characteristic
41	#366	C₂³	1	3	yes	—
42	C₄⋊C₁₆	D₄	1	3	yes	—
43	#358	—	2	3	no	—
44	#358	—	2	3	no	—
45	C₄×C₁₆	C₂³	1	3	yes	—
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×Q₁₆	—	2	3	no	—
48	C₂×Q₁₆	—	4	3	no	—
49	C₄⋊Q₈	—	4	3	no	—
50	C₄⋊Q₈	—	4	3	no	—
51	C₄⋊Q₈	—	8	3	no	—
52	#125	—	8	3	no	—
53	C₄×Q₈	—	4	3	no	—
54	C₄×Q₈	—	4	3	no	—
55	C₄×Q₈	—	8	1	no	—
56	C₄×Q₈	—	8	1	no	—
57	C₄×Q₈	D₈	1	1	yes	characteristic
58	Q₃₂	—	2	3	no	—
59	Q₃₂	—	2	1	no	—
60	Q₃₂	—	2	3	no	—
61	Q₃₂	—	8	3	no	—
62	Q₃₂	#41	1	2	yes	—
63	C₂×C₁₆	C₂×D₄	1	3	yes	—
64	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
65	#107	—	2	1	no	—
66	#107	—	2	3	no	—
67	#107	—	4	3	no	—
68	C₄⋊C₈	D₈	1	3	yes	—
69	#103	—	4	3	no	—
70	#103	—	4	3	no	—
71	C₄×C₈	C₂×D₄	1	3	yes	—
72	C₃₂	—	2	3	no	—
73	C₃₂	—	2	1	no	—
74	C₂×Q₈	—	4	1	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×Q₈	—	8	3	no	—
77	C₂×Q₈	D₁₆	1	1	yes	characteristic
78	Q₁₆	—	2	2	no	—
79	Q₁₆	—	4	3	no	—
80	Q₁₆	—	4	1	no	—
81	Q₁₆	—	4	3	no	—
82	Q₁₆	—	16	3	no	—
83	C₂×C₈	C₂×D₈	1	3	yes	—
84	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
85	C₄⋊C₄	—	4	1	no	—
86	C₄⋊C₄	—	4	3	no	—
87	C₄⋊C₄	—	8	3	no	—
88	C₄⋊C₄	—	8	3	no	—
89	C₄⋊C₄	—	8	3	no	—
90	C₄⋊C₄	—	8	3	no	—
91	C₄⋊C₄	—	8	3	no	—
92	C₄⋊C₄	D₁₆	1	3	yes	—
93	C₄²	—	8	3	no	—
94	C₄²	—	8	3	no	—
95	C₄²	C₂×D₈	1	3	yes	—
96	C₁₆	—	2	3	no	—
97	C₁₆	C₂×#41	1	1	yes	characteristic
98	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
99	Q₈	—	4	2	no	—
100	Q₈	—	8	1	no	—
101	Q₈	—	8	3	no	—
102	Q₈	—	8	3	no	—
103	Q₈	—	32	3	no	—
104	Q₈	Q₆₄	1	4	yes	complemented
105	C₂×C₄	—	8	1	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	—	8	3	no	—
108	C₂×C₄	—	8	3	no	—
109	C₂×C₄	C₂×D₁₆	1	3	yes	—
110	C₂×C₄	C₂×D₁₆	1	3	yes	—
111	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
112	C₈	—	2	3	no	—
113	C₈	#548	1	1	yes	characteristic
114	C₈	#577	1	1	yes	characteristic
115	C₂²	C₂²×D₁₆	1	1	yes	characteristic
116	C₄	—	2	3	no	—
117	C₄	—	8	2	no	—
118	C₄	—	16	3	no	—
119	C₄	—	16	1	no	—
120	C₄	—	16	3	no	—
121	C₄	C₂×Q₆₄	1	6	yes	—
122	C₄	#3358	1	1	yes	characteristic
123	C₄	#3416	1	1	yes	characteristic
124	C₂	#33925	1	1	yes	characteristic
125	C₂	C₂²×Q₆₄	1	1	yes	characteristic
126	C₂	#33983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33962

C₂

yes

maximal, characteristic

Q₈×Q₃₂

C₂

yes

maximal, characteristic

#13720

C₂

yes

maximal

#13706

C₂

yes

maximal

#13688

C₂

yes

maximal

Q₈×C₃₂

C₂

yes

maximal, characteristic

C₄×Q₆₄

C₂

yes

maximal

#3395

—

Q₈×Q₁₆

—

C₂×Q₆₄

C₂²

yes

—

#2252

C₂²

yes

—

#2247

C₂²

yes

—

#2227

—

#2209

—

Q₈×C₁₆

C₂²

yes

characteristic

C₄×Q₃₂

C₂²

yes

—

C₄×Q₃₂

C₂²

yes

—

#1423

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

—

#1416

C₂²

yes

characteristic

#1416

C₂²

yes

—

#1416

C₂²

yes

—

C₄×C₃₂

C₂²

yes

—

Q₈²

—

#557

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂³

yes

—

C₂×Q₃₂

C₂³

yes

—

—

—

—

C₈×Q₈

D₄

yes

characteristic

C₄×Q₁₆

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

#366

—

#366

C₂³

yes

characteristic

#366

C₂³

yes

—

C₄⋊C₁₆

D₄

yes

—

#358

—

#358

—

C₄×C₁₆

C₂³

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

#41

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

C₄⋊C₈

D₈

yes

—

#103

—

#103

—

C₄×C₈

C₂×D₄

yes

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₆

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂²×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂×D₈

yes

—

C₁₆

—

C₁₆

C₂×#41

yes

characteristic

C₁₆

2₋¹⁺⁴

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

Q₈

Q₆₄

yes

complemented

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×D₁₆

yes

—

110

C₂×C₄

C₂×D₁₆

yes

—

111

C₂×C₄

C₂²×D₈

yes

characteristic

112

C₈

—

113

C₈

#548

yes

characteristic

114

C₈

#577

yes

characteristic

115

C₂²

C₂²×D₁₆

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×Q₆₄

yes

—

122

C₄

#3358

yes

characteristic

123

C₄

#3416

yes

characteristic

124

C₂

#33925

yes

characteristic

125

C₂

C₂²×Q₆₄

yes

characteristic

126

C₂

#33983

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

C₂²×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂⁴

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
104. Q₈	Q₆₄	semidirect

H ≅ G/N

type

104. Q₈

Q₆₄

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#33962	2
Q₈×Q₃₂	2
#13720	2
#13706	2
#13688	2
Q₈×C₃₂	2
C₄×Q₆₄	2

subgroup

index

Q₈×Q₃₂

Q₈×C₃₂

C₄×Q₆₄

Subgroups — #513264

Conjugacy classes of subgroups

Complements

Maximal subgroups