Group #102017 — subgroups

Subgroups — #102017

Order 512.

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

normal subgroups only

This group has 428 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	C₄×#500	C₂	1	1	yes	maximal, characteristic
2	#9673	C₂	1	2	yes	maximal
3	C₄×#335	C₂	1	1	yes	maximal, characteristic
4	#8261	C₂	1	1	yes	maximal, characteristic
5	#7570	C₂	1	2	yes	maximal
6	C₂×#500	C₄	1	1	yes	characteristic
7	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
8	#1990	C₄	1	2	yes	—
9	#1942	—	2	1	no	—
10	#1942	—	2	1	no	—
11	#1845	C₄	1	1	yes	characteristic
12	C₄×#107	—	2	1	no	—
13	C₄×#107	—	2	1	no	—
14	C₄×C₄⋊C₈	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×#335	—	2	1	no	—
19	C₂×Q₈⋊C₈	—	2	4	no	—
20	C₂×#322	—	2	1	no	—
21	#1277	—	2	1	no	—
22	#1273	—	2	1	no	—
23	#1271	—	2	4	no	—
24	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	#500	—	2	4	no	—
27	C₂×#107	—	2	1	no	—
28	C₂×#107	—	2	1	no	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	D₄	1	2	yes	—
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	—	2	1	no	—
36	C₂×C₄×C₈	—	2	2	no	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	#391	—	2	2	no	—
44	#389	—	2	2	no	—
45	#389	—	2	2	no	—
46	#384	C₂×C₄	1	2	yes	—
47	C₄×C₄⋊C₄	—	2	2	no	—
48	C₄×C₄⋊C₄	—	2	2	no	—
49	C₄³	C₂³	1	1	yes	characteristic
50	#340	—	2	2	no	—
51	#340	—	2	2	no	—
52	#336	—	2	2	no	—
53	#336	—	2	1	no	—
54	#336	D₄	1	2	yes	—
55	#335	—	2	4	no	—
56	Q₈⋊C₈	—	4	8	no	—
57	#322	—	4	2	no	—
58	C₂²×Q₈	—	2	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	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	—	2	2	no	—
68	C₄⋊Q₈	—	2	2	no	—
69	C₄⋊Q₈	M₁₆	1	4	yes	—
70	C₄×Q₈	—	4	4	no	—
71	C₄×Q₈	—	4	4	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	—	2	2	no	—
77	C₂×C₄⋊C₄	—	4	2	no	—
78	C₂×C₄²	—	4	2	no	—
79	C₂×C₄²	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄²	#41	1	1	yes	characteristic
85	C₂×C₄²	#41	1	1	yes	characteristic
86	#107	—	2	4	no	—
87	#107	—	2	4	no	—
88	#107	—	4	2	no	—
89	#107	—	4	2	no	—
90	C₄⋊C₈	—	2	4	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	2	no	—
94	C₄⋊C₈	D₈	1	4	yes	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	4	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	SD₁₆	1	2	yes	—
107	C₄×C₈	Q₁₆	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	4	2	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	—	4	4	no	—
113	C₂×Q₈	—	4	4	no	—
114	C₂²×C₄	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
119	C₂²×C₄	#121	1	1	yes	characteristic
120	C₂²×C₄	#121	1	1	yes	characteristic
121	C₂²×C₄	#124	1	1	yes	characteristic
122	C₂²×C₄	#127	1	1	yes	characteristic
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	—	8	2	no	—
141	C₂×C₈	#103	1	2	yes	—
142	C₂×C₈	#103	1	2	yes	—
143	C₂×C₈	#104	1	2	yes	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	8	2	no	—
152	C₄⋊C₄	—	8	2	no	—
153	C₄⋊C₄	—	8	2	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	—	8	4	no	—
160	C₄²	C₂×M₁₆	1	2	yes	—
161	C₄²	#131	1	2	yes	—
162	C₄²	C₂×D₈	1	2	yes	—
163	C₄²	C₂×SD₁₆	1	1	yes	characteristic
164	C₄²	C₂×Q₁₆	1	1	yes	characteristic
165	C₄²	#135	1	2	yes	—
166	C₄²	#137	1	1	yes	characteristic
167	C₄²	#137	1	1	yes	characteristic
168	C₂³	#390	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	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	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	8	2	no	—
186	C₂×C₄	#407	1	1	yes	characteristic
187	C₂×C₄	#408	1	1	yes	characteristic
188	C₂×C₄	C₂×#103	1	1	yes	characteristic
189	C₂×C₄	#419	1	1	yes	characteristic
190	C₂×C₄	C₂×#104	1	1	yes	characteristic
191	C₂×C₄	#421	1	1	yes	characteristic
192	C₂×C₄	#435	1	2	yes	—
193	C₂×C₄	#435	1	2	yes	—
194	C₂×C₄	C₄×D₈	1	2	yes	—
195	C₂×C₄	#444	1	2	yes	—
196	C₂×C₄	#461	1	1	yes	characteristic
197	C₂×C₄	#461	1	1	yes	characteristic
198	C₂×C₄	#462	1	1	yes	characteristic
199	C₂×C₄	#462	1	1	yes	characteristic
200	C₂×C₄	#466	1	2	yes	—
201	C₂×C₄	#473	1	2	yes	—
202	C₂×C₄	#493	1	1	yes	characteristic
203	C₂×C₄	#495	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	4	4	no	—
208	C₈	—	4	1	no	—
209	C₈	—	8	2	no	—
210	C₈	—	8	4	no	—
211	C₂²	#1955	1	1	yes	characteristic
212	C₂²	#1970	1	1	yes	characteristic
213	C₂²	#1975	1	1	yes	characteristic
214	C₂²	#1976	1	1	yes	characteristic
215	C₂²	#1979	1	1	yes	characteristic
216	C₂²	#1980	1	1	yes	characteristic
217	C₂²	#1981	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	1	no	—
224	C₄	—	8	4	no	—
225	C₄	—	16	2	no	—
226	C₄	#1487	1	1	yes	characteristic
227	C₄	#1491	1	1	yes	characteristic
228	C₄	#1493	1	1	yes	characteristic
229	C₄	#1497	1	1	yes	characteristic
230	C₄	#1581	1	4	yes	—
231	C₄	#1673	1	2	yes	—
232	C₄	#1674	1	2	yes	—
233	C₂	#9663	1	1	yes	characteristic
234	C₂	#9671	1	1	yes	characteristic
235	C₂	#9677	1	1	yes	characteristic
236	C₂	#9719	1	1	yes	characteristic
237	C₂	#9727	1	1	yes	characteristic
238	C₂	#9731	1	1	yes	characteristic
239	C₂	#9752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#500

C₂

yes

maximal, characteristic

#9673

C₂

yes

maximal

C₄×#335

C₂

yes

maximal, characteristic

#8261

C₂

yes

maximal, characteristic

#7570

C₂

yes

maximal

C₂×#500

C₄

yes

characteristic

C₄×C₄⋊Q₈

C₂²

yes

—

#1990

C₄

yes

—

#1942

—

#1942

—

#1845

C₄

yes

characteristic

C₄×#107

—

C₄×#107

—

C₄×C₄⋊C₈

C₂²

yes

—

C₄×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₄⋊C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×Q₈⋊C₈

—

—

—

—

—

C₂×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄×Q₈

—

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

—

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

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂³

yes

—

—

—

—

D₄

yes

—

#335

—

Q₈⋊C₈

—

#322

—

C₂²×Q₈

—

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

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

M₁₆

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

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

D₈

yes

—

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

—

106

C₄×C₈

SD₁₆

yes

—

107

C₄×C₈

Q₁₆

yes

—

108

#95

—

109

#95

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

117

C₂²×C₄

C₄×D₄

yes

characteristic

118

C₂²×C₄

C₄×D₄

yes

characteristic

119

C₂²×C₄

#121

yes

characteristic

120

C₂²×C₄

#121

yes

characteristic

121

C₂²×C₄

#124

yes

characteristic

122

C₂²×C₄

#127

yes

characteristic

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

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

#103

yes

—

142

C₂×C₈

#103

yes

—

143

C₂×C₈

#104

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×M₁₆

yes

—

161

C₄²

#131

yes

—

162

C₄²

C₂×D₈

yes

—

163

C₄²

C₂×SD₁₆

yes

characteristic

164

C₄²

C₂×Q₁₆

yes

characteristic

165

C₄²

#135

yes

—

166

C₄²

#137

yes

characteristic

167

C₄²

#137

yes

characteristic

168

C₂³

#390

yes

characteristic

169

Q₈

—

170

Q₈

—

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

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

#407

yes

characteristic

187

C₂×C₄

#408

yes

characteristic

188

C₂×C₄

C₂×#103

yes

characteristic

189

C₂×C₄

#419

yes

characteristic

190

C₂×C₄

C₂×#104

yes

characteristic

191

C₂×C₄

#421

yes

characteristic

192

C₂×C₄

#435

yes

—

193

C₂×C₄

#435

yes

—

194

C₂×C₄

C₄×D₈

yes

—

195

C₂×C₄

#444

yes

—

196

C₂×C₄

#461

yes

characteristic

197

C₂×C₄

#461

yes

characteristic

198

C₂×C₄

#462

yes

characteristic

199

C₂×C₄

#462

yes

characteristic

200

C₂×C₄

#466

yes

—

201

C₂×C₄

#473

yes

—

202

C₂×C₄

#493

yes

characteristic

203

C₂×C₄

#495

yes

characteristic

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

209

C₈

—

210

C₈

—

211

C₂²

#1955

yes

characteristic

212

C₂²

#1970

yes

characteristic

213

C₂²

#1975

yes

characteristic

214

C₂²

#1976

yes

characteristic

215

C₂²

#1979

yes

characteristic

216

C₂²

#1980

yes

characteristic

217

C₂²

#1981

yes

characteristic

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

#1487

yes

characteristic

227

C₄

#1491

yes

characteristic

228

C₄

#1493

yes

characteristic

229

C₄

#1497

yes

characteristic

230

C₄

#1581

yes

—

231

C₄

#1673

yes

—

232

C₄

#1674

yes

—

233

C₂

#9663

yes

characteristic

234

C₂

#9671

yes

characteristic

235

C₂

#9677

yes

characteristic

236

C₂

#9719

yes

characteristic

237

C₂

#9727

yes

characteristic

238

C₂

#9731

yes

characteristic

239

C₂

#9752

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#121

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

#390

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ 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
C₄×#500	2
#9673	2
C₄×#335	2
#8261	2
#7570	2

subgroup

index

Subgroups — #102017

Conjugacy classes of subgroups

Maximal subgroups