Group #141080 — subgroups

Subgroups — #141080

Order 512.

340 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 110 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	#33311	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	C₂×#1363	C₂	1	4	yes	maximal
4	#8273	C₂	1	1	yes	maximal, characteristic
5	C₂×#499	—	2	1	no	—
6	C₂×#499	C₂²	1	2	yes	—
7	#2859	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	#1846	—	2	1	no	—
10	#1846	C₄	1	2	yes	—
11	#1845	C₄	1	1	yes	characteristic
12	#1845	C₄	1	1	yes	characteristic
13	#1844	—	2	1	no	—
14	#1767	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#334	C₂²	1	2	yes	—
17	#1374	—	2	1	no	—
18	#1374	—	2	1	no	—
19	#1363	—	2	8	no	—
20	#1277	—	2	1	no	—
21	C₂×#125	—	2	2	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	#499	—	2	4	no	—
24	#499	—	2	4	no	—
25	#499	—	4	4	no	—
26	C₂×#107	—	2	1	no	—
27	C₂×#107	—	2	1	no	—
28	C₂×#107	C₂×C₄	1	2	yes	—
29	C₂×#107	C₂×C₄	1	2	yes	—
30	C₂×#106	—	2	2	no	—
31	C₂×#106	—	2	2	no	—
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×#97	—	4	1	no	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#389	—	4	1	no	—
41	#389	—	4	1	no	—
42	#389	—	4	1	no	—
43	#389	—	4	1	no	—
44	#384	—	2	1	no	—
45	#384	—	2	1	no	—
46	#382	—	2	2	no	—
47	C₄³	D₄	1	1	yes	characteristic
48	C₄⋊C₁₆	—	2	4	no	—
49	#336	—	4	1	no	—
50	#334	—	2	2	no	—
51	#334	D₄	1	4	yes	—
52	C₂²×C₈	—	4	2	no	—
53	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	#125	—	4	4	no	—
58	#125	—	4	4	no	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	4	1	no	—
62	C₂×C₄⋊C₄	—	4	1	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄⋊C₄	—	4	2	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄²	D₈	1	1	yes	characteristic
70	C₂×C₄²	D₈	1	1	yes	characteristic
71	C₂×C₄²	SD₁₆	1	1	yes	characteristic
72	C₂×C₄²	SD₁₆	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₁₆	—	4	2	no	—
75	C₂×C₁₆	—	4	2	no	—
76	C₂×C₁₆	—	4	2	no	—
77	#107	—	2	4	no	—
78	#107	—	2	4	no	—
79	#107	—	4	2	no	—
80	#107	—	4	2	no	—
81	#106	—	4	4	no	—
82	#106	—	4	2	no	—
83	#106	—	4	2	no	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	4	2	no	—
86	#97	—	8	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	C₂×D₄	1	2	yes	—
93	C₄×C₈	#41	1	2	yes	—
94	#95	—	4	2	no	—
95	#95	—	4	2	no	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	#102	1	1	yes	characteristic
99	C₂²×C₄	#102	1	1	yes	characteristic
100	C₂²×C₄	#102	1	1	yes	characteristic
101	C₂²×C₄	#102	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
104	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	4	4	no	—
112	C₂×C₈	—	8	2	no	—
113	C₂×C₈	—	8	2	no	—
114	C₂×C₈	#101	1	2	yes	—
115	C₂×C₈	#101	1	2	yes	—
116	C₂×C₈	C₄×D₄	1	2	yes	—
117	C₂×C₈	#124	1	1	yes	characteristic
118	C₂×C₈	#127	1	1	yes	characteristic
119	C₂×C₈	#137	1	2	yes	—
120	C₂×C₈	#137	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	8	2	no	—
126	C₄⋊C₄	—	8	2	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₄⋊C₄	—	8	2	no	—
129	C₄⋊C₄	—	8	4	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	4	1	no	—
138	C₄²	—	4	1	no	—
139	C₄²	#121	1	2	yes	—
140	C₄²	AGL(1,ℤ/8)	1	2	yes	—
141	C₁₆	—	8	2	no	—
142	C₂³	C₂×#102	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	#359	1	2	yes	—
164	C₂×C₄	#360	1	2	yes	—
165	C₂×C₄	#390	1	1	yes	characteristic
166	C₂×C₄	C₂×#101	1	1	yes	characteristic
167	C₂×C₄	C₂×#101	1	1	yes	characteristic
168	C₂×C₄	#418	1	1	yes	characteristic
169	C₂×C₄	#419	1	1	yes	characteristic
170	C₂×C₄	#419	1	1	yes	characteristic
171	C₂×C₄	#459	1	2	yes	—
172	C₂×C₄	#460	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	—	2	2	no	—
176	C₈	—	8	4	no	—
177	C₈	#491	1	2	yes	—
178	C₈	#497	1	2	yes	—
179	C₂²	#1546	1	1	yes	characteristic
180	C₂²	#1546	1	1	yes	characteristic
181	C₂²	#1982	1	1	yes	characteristic
182	C₂²	C₂×#359	1	1	yes	characteristic
183	C₂²	#2143	1	1	yes	characteristic
184	C₂²	C₂×#360	1	1	yes	characteristic
185	C₂²	#2145	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	1	no	—
192	C₄	—	16	2	no	—
193	C₄	—	16	2	no	—
194	C₄	#1556	1	1	yes	characteristic
195	C₄	#1556	1	1	yes	characteristic
196	C₄	#1960	1	1	yes	characteristic
197	C₄	#1966	1	1	yes	characteristic
198	C₂	#9705	1	1	yes	characteristic
199	C₂	#11967	1	1	yes	characteristic
200	C₂	#11968	1	1	yes	characteristic
201	C₂	#11969	1	1	yes	characteristic
202	C₂	#11970	1	1	yes	characteristic
203	C₂	#13251	1	1	yes	characteristic
204	C₂	#13255	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#12879

C₂

yes

maximal, characteristic

C₂×#1363

C₂

yes

maximal

#8273

C₂

yes

maximal, characteristic

C₂×#499

—

C₂×#499

C₂²

yes

—

#2859

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1846

—

#1846

C₄

yes

—

#1845

C₄

yes

characteristic

#1845

C₄

yes

characteristic

#1844

—

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

C₂²

yes

—

—

—

—

—

—

C₂²×C₁₆

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

#336

—

#334

—

#334

D₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

100

C₂²×C₄

#102

yes

characteristic

101

C₂²×C₄

#102

yes

characteristic

102

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

103

C₂²×C₄

C₂×D₈

yes

characteristic

104

C₂²×C₄

C₂×SD₁₆

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

#101

yes

—

115

C₂×C₈

#101

yes

—

116

C₂×C₈

C₄×D₄

yes

—

117

C₂×C₈

#124

yes

characteristic

118

C₂×C₈

#127

yes

characteristic

119

C₂×C₈

#137

yes

—

120

C₂×C₈

#137

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

#121

yes

—

140

C₄²

AGL(1,ℤ/8)

yes

—

141

C₁₆

—

142

C₂³

C₂×#102

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

#359

yes

—

164

C₂×C₄

#360

yes

—

165

C₂×C₄

#390

yes

characteristic

166

C₂×C₄

C₂×#101

yes

characteristic

167

C₂×C₄

C₂×#101

yes

characteristic

168

C₂×C₄

#418

yes

characteristic

169

C₂×C₄

#419

yes

characteristic

170

C₂×C₄

#419

yes

characteristic

171

C₂×C₄

#459

yes

—

172

C₂×C₄

#460

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

#491

yes

—

178

C₈

#497

yes

—

179

C₂²

#1546

yes

characteristic

180

C₂²

#1546

yes

characteristic

181

C₂²

#1982

yes

characteristic

182

C₂²

C₂×#359

yes

characteristic

183

C₂²

#2143

yes

characteristic

184

C₂²

C₂×#360

yes

characteristic

185

C₂²

#2145

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#1556

yes

characteristic

195

C₄

#1556

yes

characteristic

196

C₄

#1960

yes

characteristic

197

C₄

#1966

yes

characteristic

198

C₂

#9705

yes

characteristic

199

C₂

#11967

yes

characteristic

200

C₂

#11968

yes

characteristic

201

C₂

#11969

yes

characteristic

202

C₂

#11970

yes

characteristic

203

C₂

#13251

yes

characteristic

204

C₂

#13255

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#102

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×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
#33311	2
#12879	2
C₂×#1363	2
#8273	2

subgroup

index

Subgroups — #141080

Conjugacy classes of subgroups

Maximal subgroups