Group #153003 — subgroups

Subgroups — #153003

Order 512.

207 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 96 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	#33475	C₂	1	1	yes	maximal, characteristic
2	#13630	C₂	1	1	yes	maximal, characteristic
3	#13551	C₂	1	1	yes	maximal, characteristic
4	#7469	C₂	1	2	yes	maximal
5	#7466	C₂	1	2	yes	maximal
6	#2960	C₄	1	1	yes	characteristic
7	C₂×M₆₄	C₂²	1	2	yes	—
8	C₂×M₆₄	C₂²	1	2	yes	—
9	#2183	—	2	2	no	—
10	#2183	—	2	2	no	—
11	#2151	C₄	1	1	yes	characteristic
12	#2151	C₄	1	1	yes	characteristic
13	#2150	C₄	1	1	yes	characteristic
14	#2108	C₂²	1	1	yes	characteristic
15	#2108	C₂²	1	1	yes	characteristic
16	C₂×#346	C₂²	1	1	yes	characteristic
17	#1422	—	2	2	no	—
18	#1398	—	2	2	no	—
19	#533	C₈	1	1	yes	characteristic
20	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂³	1	1	yes	characteristic
24	#446	—	2	2	no	—
25	C₈×Q₈	—	2	2	no	—
26	#424	C₈	1	1	yes	characteristic
27	#424	C₈	1	1	yes	characteristic
28	#423	C₈	1	1	yes	characteristic
29	#405	C₂×C₄	1	1	yes	characteristic
30	#405	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	M₆₄	—	2	2	no	—
33	M₆₄	—	2	2	no	—
34	M₆₄	—	2	2	no	—
35	M₆₄	—	2	2	no	—
36	C₂×C₃₂	—	2	2	no	—
37	C₂×C₃₂	Q₈	1	4	yes	complemented
38	C₄⋊C₁₆	—	2	2	no	—
39	C₄⋊C₁₆	—	2	2	no	—
40	C₄⋊C₁₆	—	2	2	no	—
41	#346	—	2	1	no	—
42	#346	—	2	1	no	—
43	#346	—	2	1	no	—
44	#346	D₄	1	2	yes	—
45	C₄×C₁₆	—	2	1	no	—
46	C₄×C₁₆	D₄	1	2	yes	—
47	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
48	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	1	no	—
52	#125	—	2	1	no	—
53	C₂²⋊Q₈	—	2	2	no	—
54	C₄×Q₈	—	2	2	no	—
55	#117	C₂×C₈	1	1	yes	characteristic
56	#117	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
58	M₃₂	—	2	2	no	—
59	M₃₂	—	2	2	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	—	2	1	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
64	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
65	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
66	C₂×C₁₆	C₂×Q₈	1	2	yes	—
67	C₂×C₁₆	#41	1	2	yes	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	2	no	—
70	C₄⋊C₈	—	2	2	no	—
71	#97	—	2	1	no	—
72	#97	M₁₆	1	2	yes	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	C₂²⋊C₄	1	2	yes	—
77	C₃₂	—	2	4	no	—
78	C₃₂	—	4	2	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	C₂²⋊C₈	1	2	yes	—
88	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈	C₄×Q₈	1	2	yes	—
90	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₈	#131	1	2	yes	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₂²⋊C₄	M₃₂	1	2	yes	—
98	C₄²	—	2	1	no	—
99	C₄²	—	2	1	no	—
100	C₄²	C₂²⋊C₈	1	2	yes	—
101	C₄²	M₃₂	1	1	yes	characteristic
102	C₄²	M₃₂	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	—	4	1	no	—
106	C₁₆	C₂²⋊Q₈	1	2	yes	—
107	C₁₆	#124	1	1	yes	characteristic
108	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
109	C₂³	C₂×M₃₂	1	1	yes	characteristic
110	Q₈	—	8	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄	C₈×Q₈	1	2	yes	—
118	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
119	C₂×C₄	#504	1	2	yes	—
120	C₈	—	2	2	no	—
121	C₈	—	4	1	no	—
122	C₈	—	4	1	no	—
123	C₈	#391	1	1	yes	characteristic
124	C₈	#408	1	1	yes	characteristic
125	C₈	#446	1	2	yes	—
126	C₂²	—	4	1	no	—
127	C₂²	#2112	1	1	yes	characteristic
128	C₂²	#2183	1	2	yes	—
129	C₄	—	4	1	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	4	2	no	—
133	C₄	#1984	1	1	yes	characteristic
134	C₄	#2113	1	1	yes	characteristic
135	C₄	#2183	1	2	yes	—
136	C₂	—	2	1	no	—
137	C₂	—	2	2	no	—
138	C₂	#13264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33475

C₂

yes

maximal, characteristic

#13630

C₂

yes

maximal, characteristic

#13551

C₂

yes

maximal, characteristic

#7469

C₂

yes

maximal

#7466

C₂

yes

maximal

#2960

C₄

yes

characteristic

C₂×M₆₄

C₂²

yes

—

C₂×M₆₄

C₂²

yes

—

#2183

—

#2183

—

#2151

C₄

yes

characteristic

#2151

C₄

yes

characteristic

#2150

C₄

yes

characteristic

#2108

C₂²

yes

characteristic

#2108

C₂²

yes

characteristic

C₂×#346

C₂²

yes

—

—

C₈

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₈

yes

characteristic

#424

C₈

yes

characteristic

#423

C₈

yes

characteristic

#405

C₂×C₄

yes

characteristic

#405

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

M₆₄

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

Q₈

yes

complemented

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

D₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₂×M₁₆

C₂×C₈

yes

characteristic

C₂×M₁₆

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×C₈

yes

characteristic

#117

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

M₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

#131

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

M₃₂

yes

—

C₄²

—

C₄²

—

100

C₄²

C₂²⋊C₈

yes

—

101

C₄²

M₃₂

yes

characteristic

102

C₄²

M₃₂

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

C₁₆

C₂²⋊Q₈

yes

—

107

C₁₆

#124

yes

characteristic

108

C₁₆

C₄⋊Q₈

yes

characteristic

109

C₂³

C₂×M₃₂

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

117

C₂×C₄

C₈×Q₈

yes

—

118

C₂×C₄

C₂×M₃₂

yes

characteristic

119

C₂×C₄

#504

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

#391

yes

characteristic

124

C₈

#408

yes

characteristic

125

C₈

#446

yes

—

126

C₂²

—

127

C₂²

#2112

yes

characteristic

128

C₂²

#2183

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#1984

yes

characteristic

134

C₄

#2113

yes

characteristic

135

C₄

#2183

yes

—

136

C₂

—

137

C₂

—

138

C₂

#13264

yes

characteristic

	subgroup	G/H
center	C₈	#391
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13264
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#391

commutator

C₂²×C₄

C₂²×C₈

Frattini, cosocle

C₂²×C₁₆

C₂³

socle

C₂

#13264

prime radical

C₂³

C₂×M₃₂

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
37. C₂×C₃₂	Q₈	semidirect

H ≅ G/N

type

37. C₂×C₃₂

Q₈

semidirect

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

subgroup	index
#33475	2
#13630	2
#13551	2
#7469	2
#7466	2

subgroup

index

Subgroups — #153003

Conjugacy classes of subgroups

Complements

Maximal subgroups