Group #140711 — subgroups

Subgroups — #140711

Order 512.

336 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₂×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#13263	C₂	1	2	yes	maximal
3	C₂×#1370	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Q₈	C₂²	1	2	yes	—
7	C₂×C₄×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Q₁₆	—	2	2	no	—
12	C₈×Q₁₆	—	2	2	no	—
13	C₂×#335	C₄	1	1	yes	characteristic
14	C₂×Q₈⋊C₈	C₄	1	2	yes	—
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1374	—	2	4	no	—
17	#1370	—	2	2	no	—
18	Q₈⋊C₁₆	—	2	4	no	—
19	#1311	—	2	2	no	—
20	C₂²×Q₁₆	C₈	1	1	yes	characteristic
21	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
22	C₂²×C₁₆	—	2	1	no	—
23	C₂²×C₁₆	—	2	2	no	—
24	C₂²×C₁₆	D₄	1	2	yes	—
25	C₈×Q₈	—	2	2	no	—
26	C₈×Q₈	—	2	2	no	—
27	C₈×Q₈	—	2	2	no	—
28	C₈×Q₈	—	2	2	no	—
29	C₄×Q₁₆	—	2	2	no	—
30	C₄×Q₁₆	—	2	2	no	—
31	C₂×#107	C₈	1	1	yes	characteristic
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×#103	C₈	1	2	yes	—
35	C₂×C₄×C₈	—	2	2	no	—
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊C₁₆	—	2	2	no	—
40	C₄⋊C₁₆	—	2	2	no	—
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	D₄	1	2	yes	—
45	#335	—	2	2	no	—
46	Q₈⋊C₈	—	2	4	no	—
47	C₈²	D₄	1	2	yes	—
48	C₈²	D₄	1	2	yes	—
49	C₂²×Q₈	C₂×C₈	1	2	yes	—
50	C₂×Q₁₆	—	2	4	no	—
51	C₂×Q₁₆	—	2	2	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	—	2	2	no	—
54	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₈	#41	1	1	yes	characteristic
58	C₄×Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	—	2	2	no	—
62	C₂×C₄⋊C₄	—	2	2	no	—
63	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
64	C₂×C₄²	—	2	2	no	—
65	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₁₆	—	2	4	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	—	4	2	no	—
71	C₂×C₁₆	—	4	4	no	—
72	C₂×C₁₆	Q₁₆	1	4	yes	complemented
73	#107	—	2	2	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	#103	—	2	4	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	4	2	no	—
81	C₄×C₈	—	4	2	no	—
82	C₄×C₈	C₂²⋊C₄	1	2	yes	—
83	C₄×C₈	C₂²⋊C₄	1	2	yes	—
84	C₄×C₈	M₁₆	1	2	yes	—
85	C₄×C₈	C₂×D₄	1	1	yes	characteristic
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	C₂×D₄	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	M₃₂	1	4	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
98	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
99	C₂²×C₄	#131	1	1	yes	characteristic
100	Q₁₆	—	2	8	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	C₂²⋊C₈	1	2	yes	—
109	C₂×C₈	C₂²⋊C₈	1	2	yes	—
110	C₂×C₈	#121	1	1	yes	characteristic
111	C₂×C₈	#121	1	1	yes	characteristic
112	C₂×C₈	#124	1	1	yes	characteristic
113	C₂×C₈	#127	1	1	yes	characteristic
114	C₂×C₈	C₂×Q₁₆	1	2	yes	—
115	C₂×C₈	#135	1	2	yes	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄²	—	4	2	no	—
121	C₄²	—	4	2	no	—
122	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₄²	C₄×D₄	1	1	yes	characteristic
124	C₄²	C₂×M₁₆	1	1	yes	characteristic
125	C₄²	#131	1	1	yes	characteristic
126	C₁₆	—	2	4	no	—
127	C₁₆	—	2	4	no	—
128	C₁₆	—	8	2	no	—
129	C₂³	C₈×D₄	1	1	yes	characteristic
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	4	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	4	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	#390	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₂×C₄	#408	1	1	yes	characteristic
143	C₂×C₄	C₈×D₄	1	1	yes	characteristic
144	C₂×C₄	#436	1	1	yes	characteristic
145	C₂×C₄	#436	1	1	yes	characteristic
146	C₂×C₄	C₄×Q₁₆	1	2	yes	—
147	C₂×C₄	#443	1	2	yes	—
148	C₂×C₄	C₂×M₃₂	1	2	yes	—
149	C₂×C₄	#504	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	4	2	no	—
155	C₈	—	8	2	no	—
156	C₈	#467	1	2	yes	—
157	C₈	#494	1	1	yes	characteristic
158	C₈	#495	1	1	yes	characteristic
159	C₂²	C₈×Q₁₆	1	2	yes	—
160	C₂²	#1954	1	1	yes	characteristic
161	C₂²	#2168	1	2	yes	—
162	C₂²	#2171	1	2	yes	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	4	no	—
166	C₄	—	8	2	no	—
167	C₄	#1591	1	2	yes	—
168	C₄	#1969	1	1	yes	characteristic
169	C₄	#1974	1	1	yes	characteristic
170	C₄	#2112	1	1	yes	characteristic
171	C₄	#2113	1	1	yes	characteristic
172	C₄	#2168	1	2	yes	—
173	C₂	#9656	1	1	yes	characteristic
174	C₂	#12053	1	4	yes	—
175	C₂	#13190	1	1	yes	characteristic
176	C₂	#13196	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₁₆

C₂

yes

maximal, characteristic

#13263

C₂

yes

maximal

C₂×#1370

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₁₆

C₂

yes

maximal

C₂×#1311

C₂

yes

maximal, characteristic

C₂×C₈×Q₈

C₂²

yes

—

C₂×C₄×Q₁₆

C₄

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₈×Q₁₆

—

C₈×Q₁₆

—

C₂×#335

C₄

yes

characteristic

C₂×Q₈⋊C₈

C₄

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1374

—

#1370

—

Q₈⋊C₁₆

—

#1311

—

C₂²×Q₁₆

C₈

yes

characteristic

C₂×C₄×Q₈

C₂×C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#107

C₈

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×#103

C₈

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⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

#335

—

Q₈⋊C₈

—

C₈²

D₄

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

C₂×C₈

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

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

Q₁₆

yes

complemented

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

M₃₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

#131

yes

characteristic

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊C₈

yes

—

109

C₂×C₈

C₂²⋊C₈

yes

—

110

C₂×C₈

#121

yes

characteristic

111

C₂×C₈

#121

yes

characteristic

112

C₂×C₈

#124

yes

characteristic

113

C₂×C₈

#127

yes

characteristic

114

C₂×C₈

C₂×Q₁₆

yes

—

115

C₂×C₈

#135

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×C₂²⋊C₄

yes

characteristic

123

C₄²

C₄×D₄

yes

characteristic

124

C₄²

C₂×M₁₆

yes

characteristic

125

C₄²

#131

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

129

C₂³

C₈×D₄

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#390

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

142

C₂×C₄

#408

yes

characteristic

143

C₂×C₄

C₈×D₄

yes

characteristic

144

C₂×C₄

#436

yes

characteristic

145

C₂×C₄

#436

yes

characteristic

146

C₂×C₄

C₄×Q₁₆

yes

—

147

C₂×C₄

#443

yes

—

148

C₂×C₄

C₂×M₃₂

yes

—

149

C₂×C₄

#504

yes

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

#467

yes

—

157

C₈

#494

yes

characteristic

158

C₈

#495

yes

characteristic

159

C₂²

C₈×Q₁₆

yes

—

160

C₂²

#1954

yes

characteristic

161

C₂²

#2168

yes

—

162

C₂²

#2171

yes

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#1591

yes

—

168

C₄

#1969

yes

characteristic

169

C₄

#1974

yes

characteristic

170

C₄

#2112

yes

characteristic

171

C₄

#2113

yes

characteristic

172

C₄

#2168

yes

—

173

C₂

#9656

yes

characteristic

174

C₂

#12053

yes

—

175

C₂

#13190

yes

characteristic

176

C₂

#13196

yes

characteristic

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

C₂²×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₈×D₄

LCS

G ▷ C₂²×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
72. C₂×C₁₆	Q₁₆	semidirect

H ≅ G/N

type

72. C₂×C₁₆

Q₁₆

semidirect

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

subgroup	index
C₂×C₈×Q₁₆	2
#13263	2
C₂×#1370	2
C₂×Q₈⋊C₁₆	2
C₂×#1311	2

subgroup

index

C₂×C₈×Q₁₆

#13263

C₂×#1370

C₂×Q₈⋊C₁₆

C₂×#1311

Subgroups — #140711

Conjugacy classes of subgroups

Complements

Maximal subgroups