Group #129168 — subgroups

Subgroups — #129168

Order 512.

278 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 154 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	#12662	C₂	1	2	yes	maximal
2	C₂×#1370	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	C₈×#106	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#346	—	2	2	no	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×#106	C₄	1	1	yes	characteristic
11	C₈×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#334	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1370	C₄	1	4	yes	complemented
16	#1312	C₄	1	4	yes	complemented
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	2	no	—
19	C₂²×C₁₆	—	2	1	no	—
20	C₂²×C₁₆	D₄	1	2	yes	—
21	C₂×#106	C₈	1	2	yes	—
22	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₈	—	2	2	no	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₄⋊C₁₆	—	2	2	no	—
29	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
30	#346	—	4	4	no	—
31	C₄×C₁₆	—	2	1	no	—
32	C₄×C₁₆	C₂×C₄	1	2	yes	—
33	C₄×C₁₆	C₂×C₄	1	2	yes	—
34	C₄×C₁₆	Q₈	1	2	yes	—
35	#340	C₈	1	2	yes	—
36	#336	—	2	2	no	—
37	#334	C₈	1	4	yes	—
38	C₈²	C₂×C₄	1	2	yes	—
39	C₈²	C₂×C₄	1	2	yes	—
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₈	#41	1	1	yes	characteristic
46	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
47	C₂×C₄²	—	2	2	no	—
48	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	—	4	2	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
57	#106	—	2	4	no	—
58	C₄⋊C₈	—	2	2	no	—
59	C₄⋊C₈	C₂×C₈	1	4	yes	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	4	2	no	—
62	C₄×C₈	—	4	2	no	—
63	C₄×C₈	C₄²	1	2	yes	—
64	C₄×C₈	C₄²	1	2	yes	—
65	C₄×C₈	C₂×C₈	1	2	yes	—
66	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₄×C₈	#41	1	1	yes	characteristic
70	#95	—	2	2	no	—
71	C₂²×C₄	—	2	4	no	—
72	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
74	C₂²×C₄	#131	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	C₄×C₈	1	2	yes	—
83	C₂×C₈	C₄×C₈	1	2	yes	—
84	C₂×C₈	M₃₂	1	4	yes	—
85	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈	#117	1	1	yes	characteristic
87	C₂×C₈	C₄×D₄	1	1	yes	characteristic
88	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₈	#137	1	2	yes	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	—	4	2	no	—
94	C₄²	—	4	2	no	—
95	C₄²	C₂×C₄²	1	1	yes	characteristic
96	C₄²	C₄×Q₈	1	1	yes	characteristic
97	C₄²	C₂²×C₈	1	1	yes	characteristic
98	C₄²	#131	1	1	yes	characteristic
99	C₁₆	—	2	4	no	—
100	C₁₆	—	4	2	no	—
101	C₁₆	—	4	4	no	—
102	C₂³	C₈×D₄	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	4	4	no	—
107	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
109	C₂×C₄	#405	1	1	yes	characteristic
110	C₂×C₄	C₈×D₄	1	1	yes	characteristic
111	C₂×C₄	#440	1	2	yes	—
112	C₂×C₄	#444	1	2	yes	—
113	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
114	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
115	C₂×C₄	C₂×M₃₂	1	2	yes	—
116	C₂×C₄	#504	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	4	2	no	—
121	C₈	—	4	4	no	—
122	C₈	#346	1	4	yes	complemented
123	C₈	#428	1	2	yes	—
124	C₈	#441	1	1	yes	characteristic
125	C₈	#442	1	1	yes	characteristic
126	C₂²	SD₁₆⋊C₈	1	2	yes	—
127	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
128	C₂²	#2169	1	2	yes	—
129	C₂²	#2171	1	2	yes	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	8	4	no	—
133	C₄	SD₁₆⋊C₈	1	2	yes	—
134	C₄	#1773	1	1	yes	characteristic
135	C₄	#1776	1	1	yes	characteristic
136	C₄	C₂×#346	1	1	yes	characteristic
137	C₄	#2108	1	1	yes	characteristic
138	C₄	#2183	1	2	yes	—
139	C₂	#8285	1	1	yes	characteristic
140	C₂	#12058	1	4	yes	—
141	C₂	#12662	1	1	yes	characteristic
142	C₂	#12665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12662

C₂

yes

maximal

C₂×#1370

C₂

yes

maximal, characteristic

C₂×#1312

C₂

yes

maximal, characteristic

C₈×#106

C₂

yes

maximal, characteristic

#7167

C₂

yes

maximal

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×#346

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#106

C₄

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

C₂×#334

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1374

—

#1370

C₄

yes

complemented

#1312

C₄

yes

complemented

#1279

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

C₂×#106

C₈

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#340

C₈

yes

—

#336

—

#334

C₈

yes

—

C₈²

C₂×C₄

yes

—

C₈²

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

#131

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×C₄²

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

C₄²

#131

yes

characteristic

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₂³

C₈×D₄

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

108

C₂×C₄

C₂×C₄×C₈

yes

characteristic

109

C₂×C₄

#405

yes

characteristic

110

C₂×C₄

C₈×D₄

yes

characteristic

111

C₂×C₄

#440

yes

—

112

C₂×C₄

#444

yes

—

113

C₂×C₄

C₈×Q₈

yes

characteristic

114

C₂×C₄

C₈×Q₈

yes

characteristic

115

C₂×C₄

C₂×M₃₂

yes

—

116

C₂×C₄

#504

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

#346

yes

complemented

123

C₈

#428

yes

—

124

C₈

#441

yes

characteristic

125

C₈

#442

yes

characteristic

126

C₂²

SD₁₆⋊C₈

yes

—

127

C₂²

C₈×C₄⋊C₄

yes

characteristic

128

C₂²

#2169

yes

—

129

C₂²

#2171

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

SD₁₆⋊C₈

yes

—

134

C₄

#1773

yes

characteristic

135

C₄

#1776

yes

characteristic

136

C₄

C₂×#346

yes

characteristic

137

C₄

#2108

yes

characteristic

138

C₄

#2183

yes

—

139

C₂

#8285

yes

characteristic

140

C₂

#12058

yes

—

141

C₂

#12662

yes

characteristic

142

C₂

#12665

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂×D₄

commutator

C₂×C₄

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₈×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
15. #1370	C₄	semidirect
16. #1312	C₄	semidirect
122. C₈	#346	semidirect

H ≅ G/N

type

15. #1370

C₄

semidirect

16. #1312

C₄

semidirect

122. C₈

#346

semidirect

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

subgroup	index
#12662	2
C₂×#1370	2
C₂×#1312	2
C₈×#106	2
#7167	2

subgroup

index

Subgroups — #129168

Conjugacy classes of subgroups

Complements

Maximal subgroups