Group #140738 — subgroups

Subgroups — #140738

Order 512.

285 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 106 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	#17958	C₂	1	1	yes	maximal, characteristic
2	#13265	C₂	1	2	yes	maximal
3	C₂×#1367	C₂	1	1	yes	maximal, characteristic
4	#11610	C₂	1	2	yes	maximal
5	#11402	C₂	1	1	yes	maximal, characteristic
6	#3170	C₄	1	1	yes	characteristic
7	#2960	C₂²	1	2	yes	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₄×M₃₂	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	#1591	—	2	1	no	—
12	#1591	—	2	1	no	—
13	C₈×Q₁₆	—	2	1	no	—
14	C₈×Q₁₆	—	2	1	no	—
15	#1567	C₄	1	1	yes	characteristic
16	#1493	C₄	1	2	yes	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1379	—	2	4	no	—
19	#1367	—	2	1	no	—
20	#1367	—	2	1	no	—
21	#1338	—	2	4	no	—
22	#1312	—	2	1	no	—
23	#1312	—	2	1	no	—
24	#533	C₂×C₄	1	2	yes	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	D₄	1	2	yes	—
27	C₂²×C₁₆	—	2	2	no	—
28	#500	C₈	1	1	yes	characteristic
29	#494	C₈	1	1	yes	characteristic
30	#487	C₈	1	2	yes	—
31	#467	—	2	1	no	—
32	#467	—	2	1	no	—
33	#446	—	2	2	no	—
34	#446	—	2	2	no	—
35	C₈×Q₈	—	2	2	no	—
36	C₈×Q₈	—	2	2	no	—
37	C₄×Q₁₆	—	2	1	no	—
38	C₄×Q₁₆	—	2	1	no	—
39	#424	—	2	2	no	—
40	#423	C₂×C₄	1	2	yes	—
41	#405	—	2	2	no	—
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	2	no	—
46	C₄⋊C₁₆	—	2	2	no	—
47	#346	—	2	1	no	—
48	#346	D₄	1	2	yes	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	—	2	1	no	—
51	#335	—	2	1	no	—
52	#335	—	2	1	no	—
53	Q₈⋊C₈	—	2	2	no	—
54	Q₈⋊C₈	—	2	2	no	—
55	C₈²	D₄	1	1	yes	characteristic
56	C₈²	D₄	1	1	yes	characteristic
57	C₈²	D₄	1	1	yes	characteristic
58	C₈²	D₄	1	1	yes	characteristic
59	C₂×Q₁₆	—	2	1	no	—
60	C₂×Q₁₆	—	2	1	no	—
61	C₂×M₁₆	—	2	2	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	M₁₆	1	1	yes	characteristic
64	C₂²×C₈	M₁₆	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₈	#41	1	1	yes	characteristic
67	C₄⋊Q₈	C₂×C₈	1	2	yes	—
68	C₄⋊Q₈	C₂×C₈	1	2	yes	—
69	#125	—	2	2	no	—
70	C₂²⋊Q₈	—	2	2	no	—
71	C₂²⋊Q₈	—	2	2	no	—
72	C₄×Q₈	—	2	2	no	—
73	C₄×Q₈	—	2	2	no	—
74	#117	—	2	2	no	—
75	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
76	M₃₂	—	2	4	no	—
77	M₃₂	—	4	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	4	2	no	—
81	C₂×C₁₆	—	4	4	no	—
82	#107	—	2	1	no	—
83	#107	—	2	1	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	2	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	#103	—	2	2	no	—
89	#103	—	2	2	no	—
90	#97	—	4	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	4	2	no	—
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₈	C₂×C₈	1	1	yes	characteristic
98	C₄×C₈	C₂×C₈	1	1	yes	characteristic
99	C₄×C₈	C₂×D₄	1	1	yes	characteristic
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
106	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
107	C₂²×C₄	#131	1	1	yes	characteristic
108	Q₁₆	—	4	2	no	—
109	Q₁₆	—	4	2	no	—
110	M₁₆	—	4	4	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₈	#121	1	1	yes	characteristic
123	C₂×C₈	#121	1	1	yes	characteristic
124	C₂×C₈	#124	1	1	yes	characteristic
125	C₂×C₈	#127	1	1	yes	characteristic
126	C₂×C₈	#137	1	4	yes	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₄²	—	4	2	no	—
135	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₄²	C₄×D₄	1	1	yes	characteristic
137	C₄²	C₂²×C₈	1	1	yes	characteristic
138	C₄²	#131	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	C₁₆	—	4	2	no	—
141	C₁₆	—	8	2	no	—
142	C₂³	#436	1	1	yes	characteristic
143	Q₈	—	4	2	no	—
144	Q₈	—	4	2	no	—
145	Q₈	—	8	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	#390	1	1	yes	characteristic
152	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₂×C₄	#408	1	1	yes	characteristic
154	C₂×C₄	C₈×D₄	1	1	yes	characteristic
155	C₂×C₄	C₈×D₄	1	1	yes	characteristic
156	C₂×C₄	#436	1	1	yes	characteristic
157	C₂×C₄	#441	1	2	yes	—
158	C₂×C₄	#444	1	2	yes	—
159	C₈	—	2	2	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	4	1	no	—
166	C₈	—	8	2	no	—
167	C₈	#350	1	2	yes	—
168	C₈	#470	1	2	yes	—
169	C₈	#497	1	2	yes	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	#1593	1	2	yes	—
173	C₂²	#1954	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	8	2	no	—
178	C₄	—	8	2	no	—
179	C₄	#1579	1	2	yes	—
180	C₄	#1971	1	1	yes	characteristic
181	C₄	#1975	1	1	yes	characteristic
182	C₄	C₂×#350	1	1	yes	characteristic
183	C₄	#2117	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	#9658	1	1	yes	characteristic
186	C₂	#13193	1	1	yes	characteristic
187	C₂	#13197	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17958

C₂

yes

maximal, characteristic

#13265

C₂

yes

maximal

C₂×#1367

C₂

yes

maximal, characteristic

#11610

C₂

yes

maximal

#11402

C₂

yes

maximal, characteristic

#3170

C₄

yes

characteristic

#2960

C₂²

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1591

—

#1591

—

C₈×Q₁₆

—

C₈×Q₁₆

—

#1567

C₄

yes

characteristic

#1493

C₄

yes

—

C₂×C₈²

C₂²

yes

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

—

#500

C₈

yes

characteristic

#494

C₈

yes

characteristic

#487

C₈

yes

—

—

—

—

—

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#424

—

#423

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

D₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

#335

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

C₂×C₈

yes

—

C₄⋊Q₈

C₂×C₈

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

100

C₄×C₈

C₂×D₄

yes

characteristic

101

C₄×C₈

C₂×D₄

yes

characteristic

102

C₄×C₈

#41

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

C₄×D₄

yes

characteristic

106

C₂²×C₄

C₂×M₁₆

yes

characteristic

107

C₂²×C₄

#131

yes

characteristic

108

Q₁₆

—

109

Q₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²⋊C₈

yes

characteristic

119

C₂×C₈

C₂²⋊C₈

yes

characteristic

120

C₂×C₈

C₂²⋊C₈

yes

characteristic

121

C₂×C₈

C₂²⋊C₈

yes

characteristic

122

C₂×C₈

#121

yes

characteristic

123

C₂×C₈

#121

yes

characteristic

124

C₂×C₈

#124

yes

characteristic

125

C₂×C₈

#127

yes

characteristic

126

C₂×C₈

#137

yes

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

C₂×C₂²⋊C₄

yes

characteristic

136

C₄²

C₄×D₄

yes

characteristic

137

C₄²

C₂²×C₈

yes

characteristic

138

C₄²

#131

yes

characteristic

139

C₁₆

—

140

C₁₆

—

141

C₁₆

—

142

C₂³

#436

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#390

yes

characteristic

152

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

153

C₂×C₄

#408

yes

characteristic

154

C₂×C₄

C₈×D₄

yes

characteristic

155

C₂×C₄

C₈×D₄

yes

characteristic

156

C₂×C₄

#436

yes

characteristic

157

C₂×C₄

#441

yes

—

158

C₂×C₄

#444

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

#350

yes

—

168

C₈

#470

yes

—

169

C₈

#497

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

#1593

yes

—

173

C₂²

#1954

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#1579

yes

—

180

C₄

#1971

yes

characteristic

181

C₄

#1975

yes

characteristic

182

C₄

C₂×#350

yes

characteristic

183

C₄

#2117

yes

characteristic

184

C₂

—

185

C₂

#9658

yes

characteristic

186

C₂

#13193

yes

characteristic

187

C₂

#13197

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	#436
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#121

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#17958	2
#13265	2
C₂×#1367	2
#11610	2
#11402	2

subgroup

index

Subgroups — #140738

Conjugacy classes of subgroups

Maximal subgroups