Group #140358 — subgroups

Subgroups — #140358

Order 512.

262 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 134 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	#13159	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	—	2	2	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
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	—	2	2	no	—
16	#1312	—	2	2	no	—
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂×#106	C₈	1	1	yes	characteristic
23	C₂×#106	C₈	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	—	2	2	no	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₄⋊C₁₆	—	2	2	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	—	4	4	no	—
33	C₄×C₁₆	—	2	1	no	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	—	2	1	no	—
36	C₄×C₁₆	Q₈	1	2	yes	—
37	#340	C₈	1	2	yes	—
38	#336	—	2	2	no	—
39	#334	—	2	2	no	—
40	C₈²	D₄	1	2	yes	—
41	C₈²	Q₈	1	2	yes	—
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₈	#41	1	1	yes	characteristic
48	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
49	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
50	C₂×C₄²	—	2	2	no	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	2	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	4	2	no	—
58	C₂×C₁₆	—	4	2	no	—
59	C₂×C₁₆	—	4	2	no	—
60	C₂×C₁₆	—	4	2	no	—
61	#106	—	2	2	no	—
62	#106	C₁₆	1	4	yes	complemented
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	—	4	2	no	—
68	C₄×C₈	C₄⋊C₄	1	2	yes	—
69	C₄×C₈	C₄⋊C₄	1	2	yes	—
70	C₄×C₈	M₁₆	1	2	yes	—
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	#95	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
80	C₂²×C₄	#131	1	1	yes	characteristic
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	C₄⋊C₈	1	2	yes	—
90	C₂×C₈	C₄⋊C₈	1	2	yes	—
91	C₂×C₈	C₂×C₁₆	1	2	yes	—
92	C₂×C₈	M₃₂	1	2	yes	—
93	C₂×C₈	#121	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₈	#125	1	1	yes	characteristic
96	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
98	C₂×C₈	#137	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
103	C₄²	—	4	2	no	—
104	C₄²	—	4	2	no	—
105	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₄²	C₄×Q₈	1	1	yes	characteristic
107	C₄²	C₂×M₁₆	1	1	yes	characteristic
108	C₄²	#131	1	1	yes	characteristic
109	C₁₆	—	4	2	no	—
110	C₁₆	—	4	1	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	8	2	no	—
113	C₂³	C₈×D₄	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	#389	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂×C₄	#424	1	1	yes	characteristic
124	C₂×C₄	#436	1	1	yes	characteristic
125	C₂×C₄	#440	1	2	yes	—
126	C₂×C₄	#444	1	2	yes	—
127	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
128	C₂×C₄	#446	1	1	yes	characteristic
129	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
130	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
131	C₂×C₄	#504	1	1	yes	characteristic
132	C₂×C₄	#504	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	4	2	no	—
137	C₈	—	8	2	no	—
138	C₈	C₄⋊C₁₆	1	4	yes	complemented
139	C₈	#469	1	1	yes	characteristic
140	C₈	#470	1	1	yes	characteristic
141	C₈	#501	1	2	yes	—
142	C₂²	SD₁₆⋊C₈	1	2	yes	—
143	C₂²	#1939	1	1	yes	characteristic
144	C₂²	D₄×C₁₆	1	1	yes	characteristic
145	C₂²	#2169	1	1	yes	characteristic
146	C₂²	#2171	1	2	yes	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	4	no	—
150	C₄	—	8	2	no	—
151	C₄	#1592	1	2	yes	—
152	C₄	#1944	1	1	yes	characteristic
153	C₄	#1946	1	1	yes	characteristic
154	C₄	C₂×C₄⋊C₁₆	1	1	yes	characteristic
155	C₄	#2151	1	1	yes	characteristic
156	C₄	Q₈×C₁₆	1	1	yes	characteristic
157	C₄	#2183	1	1	yes	characteristic
158	C₂	#9596	1	1	yes	characteristic
159	C₂	SD₁₆⋊C₁₆	1	2	yes	—
160	C₂	#12058	1	2	yes	—
161	C₂	#13159	1	1	yes	characteristic
162	C₂	#13162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13159

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

—

C₂×C₄⋊C₁₆

C₂²

yes

—

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

—

—

—

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#106

C₈

yes

characteristic

C₂×#106

C₈

yes

characteristic

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

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

Q₈

yes

—

#340

C₈

yes

—

#336

—

#334

—

C₈²

D₄

yes

—

C₈²

Q₈

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×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

—

C₂×C₁₆

—

#106

—

#106

C₁₆

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

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

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂×C₁₆

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×C₄⋊C₄

yes

characteristic

106

C₄²

C₄×Q₈

yes

characteristic

107

C₄²

C₂×M₁₆

yes

characteristic

108

C₄²

#131

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₂³

C₈×D₄

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#389

yes

characteristic

122

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

123

C₂×C₄

#424

yes

characteristic

124

C₂×C₄

#436

yes

characteristic

125

C₂×C₄

#440

yes

—

126

C₂×C₄

#444

yes

—

127

C₂×C₄

C₈×Q₈

yes

characteristic

128

C₂×C₄

#446

yes

characteristic

129

C₂×C₄

C₂²×C₁₆

yes

characteristic

130

C₂×C₄

C₂×M₃₂

yes

characteristic

131

C₂×C₄

#504

yes

characteristic

132

C₂×C₄

#504

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

C₄⋊C₁₆

yes

complemented

139

C₈

#469

yes

characteristic

140

C₈

#470

yes

characteristic

141

C₈

#501

yes

—

142

C₂²

SD₁₆⋊C₈

yes

—

143

C₂²

#1939

yes

characteristic

144

C₂²

D₄×C₁₆

yes

characteristic

145

C₂²

#2169

yes

characteristic

146

C₂²

#2171

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#1592

yes

—

152

C₄

#1944

yes

characteristic

153

C₄

#1946

yes

characteristic

154

C₄

C₂×C₄⋊C₁₆

yes

characteristic

155

C₄

#2151

yes

characteristic

156

C₄

Q₈×C₁₆

yes

characteristic

157

C₄

#2183

yes

characteristic

158

C₂

#9596

yes

characteristic

159

C₂

SD₁₆⋊C₁₆

yes

—

160

C₂

#12058

yes

—

161

C₂

#13159

yes

characteristic

162

C₂

#13162

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
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

C₂²×C₁₆

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
62. #106	C₁₆	semidirect
138. C₈	C₄⋊C₁₆	semidirect

H ≅ G/N

type

62. #106

C₁₆

semidirect

138. C₈

C₄⋊C₁₆

semidirect

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

subgroup	index
#13159	2
C₂×#1370	2
C₂×#1312	2
C₈×#106	2
#7167	2

subgroup

index

Subgroups — #140358

Conjugacy classes of subgroups

Complements

Maximal subgroups