Group #102373 — subgroups

Subgroups — #102373

Order 512.

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

normal subgroups only

This group has 364 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	#8785	C₂	1	1	yes	maximal, characteristic
3	#8261	C₂	1	1	yes	maximal, characteristic
4	#7663	C₂	1	2	yes	maximal
5	#7663	C₂	1	2	yes	maximal
6	C₂×#499	—	2	1	no	—
7	C₂×#499	—	2	1	no	—
8	#2859	C₂²	1	2	yes	—
9	#1846	—	2	1	no	—
10	#1846	—	2	1	no	—
11	#1845	C₄	1	1	yes	characteristic
12	#1845	C₄	1	1	yes	characteristic
13	#1844	C₄	1	2	yes	—
14	#1831	C₂²	1	2	yes	—
15	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#322	—	2	1	no	—
19	#1277	—	2	1	no	—
20	#1277	—	2	1	no	—
21	#1273	—	2	1	no	—
22	#1271	—	2	4	no	—
23	#1271	—	2	4	no	—
24	C₂×#125	—	2	2	no	—
25	#499	—	4	2	no	—
26	#499	—	4	2	no	—
27	#499	—	4	2	no	—
28	#499	—	4	2	no	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	—	2	1	no	—
32	C₂×#107	—	2	1	no	—
33	C₂×#106	—	2	2	no	—
34	C₂×#106	—	2	2	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	#389	—	2	2	no	—
47	#389	—	2	2	no	—
48	#389	—	2	2	no	—
49	#389	—	2	2	no	—
50	#384	C₂×C₄	1	2	yes	—
51	#384	C₂×C₄	1	2	yes	—
52	#382	—	2	2	no	—
53	C₄³	C₂³	1	1	yes	characteristic
54	#336	—	2	2	no	—
55	#336	—	2	2	no	—
56	#336	D₄	1	2	yes	—
57	#336	D₄	1	2	yes	—
58	#322	—	4	2	no	—
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	4	2	no	—
62	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	#125	—	4	2	no	—
67	#125	—	4	2	no	—
68	#125	—	4	2	no	—
69	#125	—	4	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	4	2	no	—
77	C₂×C₄⋊C₄	—	4	2	no	—
78	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄²	#41	1	1	yes	characteristic
84	C₂×C₄²	#41	1	1	yes	characteristic
85	#107	—	4	2	no	—
86	#107	—	4	2	no	—
87	#107	—	4	2	no	—
88	#107	—	4	2	no	—
89	#106	—	4	4	no	—
90	#106	—	4	4	no	—
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	2	no	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	—	4	1	no	—
105	C₄×C₈	D₈	1	2	yes	—
106	C₄×C₈	SD₁₆	1	2	yes	—
107	#95	—	4	2	no	—
108	#95	—	4	2	no	—
109	C₂²×C₄	—	4	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₄	#121	1	1	yes	characteristic
115	C₂²×C₄	#121	1	1	yes	characteristic
116	C₂²×C₄	#124	1	1	yes	characteristic
117	C₂²×C₄	#127	1	1	yes	characteristic
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	8	2	no	—
133	C₂×C₈	—	8	2	no	—
134	C₂×C₈	—	8	2	no	—
135	C₂×C₈	#101	1	2	yes	—
136	C₂×C₈	#101	1	2	yes	—
137	C₂×C₈	#102	1	2	yes	—
138	C₂×C₈	#102	1	2	yes	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	8	2	no	—
146	C₄⋊C₄	—	8	2	no	—
147	C₄⋊C₄	—	8	2	no	—
148	C₄⋊C₄	—	8	2	no	—
149	C₄⋊C₄	—	8	2	no	—
150	C₄⋊C₄	—	8	2	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	4	1	no	—
156	C₄²	—	4	1	no	—
157	C₄²	C₂×D₈	1	1	yes	characteristic
158	C₄²	C₂×SD₁₆	1	1	yes	characteristic
159	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₄²	#137	1	2	yes	—
162	C₄²	#137	1	2	yes	—
163	C₂³	#390	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	C₂×#101	1	1	yes	characteristic
183	C₂×C₄	C₂×#101	1	1	yes	characteristic
184	C₂×C₄	C₂×#102	1	1	yes	characteristic
185	C₂×C₄	#418	1	1	yes	characteristic
186	C₂×C₄	#419	1	1	yes	characteristic
187	C₂×C₄	#419	1	1	yes	characteristic
188	C₂×C₄	#443	1	2	yes	—
189	C₂×C₄	#444	1	2	yes	—
190	C₂×C₄	#459	1	1	yes	characteristic
191	C₂×C₄	#459	1	1	yes	characteristic
192	C₂×C₄	#460	1	1	yes	characteristic
193	C₂×C₄	#460	1	1	yes	characteristic
194	C₂×C₄	#491	1	2	yes	—
195	C₂×C₄	#497	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	4	4	no	—
201	C₈	—	8	2	no	—
202	C₈	—	16	2	no	—
203	C₂²	#1960	1	1	yes	characteristic
204	C₂²	#1966	1	1	yes	characteristic
205	C₂²	#1974	1	1	yes	characteristic
206	C₂²	#1975	1	1	yes	characteristic
207	C₂²	#1976	1	1	yes	characteristic
208	C₂²	#1976	1	1	yes	characteristic
209	C₂²	#1982	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	16	2	no	—
219	C₄	—	16	2	no	—
220	C₄	#1546	1	1	yes	characteristic
221	C₄	#1546	1	1	yes	characteristic
222	C₄	#1556	1	1	yes	characteristic
223	C₄	#1556	1	1	yes	characteristic
224	C₂	#9705	1	1	yes	characteristic
225	C₂	#9750	1	1	yes	characteristic
226	C₂	#9753	1	1	yes	characteristic
227	C₂	#9759	1	1	yes	characteristic
228	C₂	#9788	1	1	yes	characteristic
229	C₂	#9792	1	1	yes	characteristic
230	C₂	#9859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#8785

C₂

yes

maximal, characteristic

#8261

C₂

yes

maximal, characteristic

#7663

C₂

yes

maximal

#7663

C₂

yes

—

—

C₂²

yes

—

#1846

—

#1846

—

#1845

C₄

yes

characteristic

#1845

C₄

yes

characteristic

#1844

C₄

yes

—

#1831

C₂²

yes

—

C₄×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₄⋊C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

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

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#384

C₂×C₄

yes

—

#382

—

C₄³

C₂³

yes

—

—

D₄

yes

—

#336

D₄

yes

—

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

—

—

—

—

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

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

C₂×C₄²

yes

C₂×C₄²

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

D₈

yes

—

106

C₄×C₈

SD₁₆

yes

—

107

#95

—

108

#95

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

112

C₂²×C₄

C₄×D₄

yes

characteristic

113

C₂²×C₄

C₄×D₄

yes

characteristic

114

C₂²×C₄

#121

yes

characteristic

115

C₂²×C₄

#121

yes

characteristic

116

C₂²×C₄

#124

yes

characteristic

117

C₂²×C₄

#127

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

#101

yes

—

136

C₂×C₈

#101

yes

—

137

C₂×C₈

#102

yes

—

138

C₂×C₈

#102

yes

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

C₂×D₈

yes

characteristic

158

C₄²

C₂×SD₁₆

yes

characteristic

159

C₄²

AGL(1,ℤ/8)

yes

characteristic

160

C₄²

AGL(1,ℤ/8)

yes

characteristic

161

C₄²

#137

yes

—

162

C₄²

#137

yes

—

163

C₂³

#390

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

C₂×#101

yes

characteristic

183

C₂×C₄

C₂×#101

yes

characteristic

184

C₂×C₄

C₂×#102

yes

characteristic

185

C₂×C₄

#418

yes

characteristic

186

C₂×C₄

#419

yes

characteristic

187

C₂×C₄

#419

yes

characteristic

188

C₂×C₄

#443

yes

—

189

C₂×C₄

#444

yes

—

190

C₂×C₄

#459

yes

characteristic

191

C₂×C₄

#459

yes

characteristic

192

C₂×C₄

#460

yes

characteristic

193

C₂×C₄

#460

yes

characteristic

194

C₂×C₄

#491

yes

—

195

C₂×C₄

#497

yes

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₂²

#1960

yes

characteristic

204

C₂²

#1966

yes

characteristic

205

C₂²

#1974

yes

characteristic

206

C₂²

#1975

yes

characteristic

207

C₂²

#1976

yes

characteristic

208

C₂²

#1976

yes

characteristic

209

C₂²

#1982

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

#1546

yes

characteristic

221

C₄

#1546

yes

characteristic

222

C₄

#1556

yes

characteristic

223

C₄

#1556

yes

characteristic

224

C₂

#9705

yes

characteristic

225

C₂

#9750

yes

characteristic

226

C₂

#9753

yes

characteristic

227

C₂

#9759

yes

characteristic

228

C₂

#9788

yes

characteristic

229

C₂

#9792

yes

characteristic

230

C₂

#9859

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#390

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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

subgroup	index
#33311	2
#8785	2
#8261	2
#7663	2

subgroup

index

Subgroups — #102373

Conjugacy classes of subgroups

Maximal subgroups