Group #99083 — subgroups

Subgroups — #99083

Order 512.

342 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 122 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	#21712	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	#8807	C₂	1	1	yes	maximal, characteristic
4	#7652	C₂	1	2	yes	maximal
5	#7040	C₂	1	2	yes	maximal
6	C₂×#446	—	2	1	no	—
7	C₄×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	#1984	—	2	1	no	—
9	#1939	—	2	1	no	—
10	#1939	C₄	1	2	yes	—
11	#1842	C₂²	1	1	yes	characteristic
12	#1831	C₂²	1	2	yes	—
13	#1831	C₂²	1	2	yes	—
14	#1770	—	2	1	no	—
15	#1770	C₄	1	2	yes	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#335	—	2	1	no	—
18	C₂×#335	—	2	1	no	—
19	C₂×#334	—	2	1	no	—
20	C₂×#334	—	2	1	no	—
21	#1271	—	2	4	no	—
22	C₂×C₄⋊Q₈	Q₈	1	1	yes	characteristic
23	C₂×C₄×Q₈	—	2	1	no	—
24	#446	—	4	2	no	—
25	#446	—	4	2	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄⋊C₈	—	2	2	no	—
31	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×#97	—	2	2	no	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	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	#391	—	2	1	no	—
40	#389	—	2	1	no	—
41	#389	D₄	1	2	yes	—
42	#384	Q₈	1	1	yes	characteristic
43	C₄×C₄⋊C₄	—	2	1	no	—
44	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₄³	C₂³	1	1	yes	characteristic
46	#336	—	2	2	no	—
47	#336	—	2	2	no	—
48	#336	—	2	2	no	—
49	#336	—	2	2	no	—
50	#336	—	2	2	no	—
51	#336	—	2	2	no	—
52	#335	—	4	2	no	—
53	#335	—	4	2	no	—
54	#334	—	4	2	no	—
55	#334	—	4	2	no	—
56	C₂²×Q₈	—	2	1	no	—
57	C₂²×C₈	—	2	2	no	—
58	C₂²×C₈	—	4	2	no	—
59	C₂²×C₈	—	4	2	no	—
60	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊Q₈	—	4	2	no	—
65	C₄×Q₈	—	4	1	no	—
66	C₄×Q₈	—	4	1	no	—
67	C₄×Q₈	—	4	1	no	—
68	C₄×Q₈	—	4	1	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	2	1	no	—
71	C₂×C₄⋊C₄	—	2	1	no	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
74	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₄²	—	2	2	no	—
76	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄²	#41	1	1	yes	characteristic
82	C₂×C₄²	#41	1	1	yes	characteristic
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	4	no	—
85	C₄⋊C₈	—	4	1	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	4	1	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	1	no	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	4	1	no	—
93	#97	—	4	4	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	D₈	1	2	yes	—
101	C₄×C₈	SD₁₆	1	2	yes	—
102	C₄×C₈	SD₁₆	1	2	yes	—
103	C₄×C₈	Q₁₆	1	2	yes	—
104	#95	—	2	2	no	—
105	#95	—	2	2	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄	#117	1	1	yes	characteristic
115	C₂²×C₄	#121	1	1	yes	characteristic
116	C₂²×C₄	#121	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
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	—	4	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	—	8	2	no	—
131	C₂×C₈	—	8	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	#102	1	2	yes	—
137	C₂×C₈	#102	1	2	yes	—
138	C₂×C₈	#103	1	2	yes	—
139	C₂×C₈	#103	1	2	yes	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	4	1	no	—
152	C₄²	—	4	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	4	2	no	—
156	C₄²	C₂×D₈	1	1	yes	characteristic
157	C₄²	C₂×SD₁₆	1	1	yes	characteristic
158	C₄²	C₂×SD₁₆	1	1	yes	characteristic
159	C₄²	C₂×Q₁₆	1	1	yes	characteristic
160	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₄²	#137	1	1	yes	characteristic
163	C₄²	#137	1	1	yes	characteristic
164	C₂³	#380	1	1	yes	characteristic
165	Q₈	—	8	2	no	—
166	Q₈	—	8	2	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	—	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	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	C₂×#101	1	1	yes	characteristic
187	C₂×C₄	#413	1	1	yes	characteristic
188	C₂×C₄	C₂×#102	1	1	yes	characteristic
189	C₂×C₄	C₂×#103	1	1	yes	characteristic
190	C₂×C₄	#417	1	1	yes	characteristic
191	C₂×C₄	#417	1	1	yes	characteristic
192	C₂×C₄	#430	1	2	yes	—
193	C₂×C₄	#459	1	1	yes	characteristic
194	C₂×C₄	#460	1	1	yes	characteristic
195	C₂×C₄	#461	1	1	yes	characteristic
196	C₂×C₄	#462	1	1	yes	characteristic
197	C₂×C₄	#474	1	1	yes	characteristic
198	C₂×C₄	#475	1	1	yes	characteristic
199	C₂×C₄	#476	1	1	yes	characteristic
200	C₂×C₄	#477	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	4	1	no	—
204	C₈	—	8	2	no	—
205	C₈	—	16	2	no	—
206	C₈	—	16	2	no	—
207	C₈	#332	1	4	yes	—
208	C₂²	#1801	1	1	yes	characteristic
209	C₂²	#1802	1	1	yes	characteristic
210	C₂²	#1803	1	1	yes	characteristic
211	C₂²	#1811	1	1	yes	characteristic
212	C₂²	#1812	1	1	yes	characteristic
213	C₂²	#1814	1	1	yes	characteristic
214	C₂²	#1814	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	8	2	no	—
227	C₄	C₂×#332	1	1	yes	characteristic
228	C₄	#1547	1	1	yes	characteristic
229	C₄	#1550	1	1	yes	characteristic
230	C₄	#1561	1	1	yes	characteristic
231	C₂	#8586	1	1	yes	characteristic
232	C₂	#8625	1	1	yes	characteristic
233	C₂	#8635	1	1	yes	characteristic
234	C₂	#8662	1	1	yes	characteristic
235	C₂	#8663	1	1	yes	characteristic
236	C₂	#8674	1	1	yes	characteristic
237	C₂	#8677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21712

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#7652

C₂

yes

maximal

#7040

C₂

yes

maximal

C₂×#446

—

C₄×C₄⋊Q₈

C₂²

yes

—

—

C₄

yes

—

#1842

C₂²

yes

characteristic

#1831

C₂²

yes

—

#1831

C₂²

yes

—

#1770

—

#1770

C₄

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈

Q₈

yes

characteristic

C₂×C₄×Q₈

—

#446

—

#446

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

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

D₄

yes

—

—

D₄

yes

—

#384

Q₈

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂×C₄

yes

—

C₄³

C₂³

yes

—

—

—

—

—

—

—

—

—

—

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

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄⋊C₄

yes

—

C₂×C₄⋊C₄

C₄⋊C₄

yes

—

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×Q₈

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

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

D₈

yes

—

101

C₄×C₈

SD₁₆

yes

—

102

C₄×C₈

SD₁₆

yes

—

103

C₄×C₈

Q₁₆

yes

—

104

#95

—

105

#95

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

113

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

114

C₂²×C₄

#117

yes

characteristic

115

C₂²×C₄

#121

yes

characteristic

116

C₂²×C₄

#121

yes

characteristic

117

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

118

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

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

#102

yes

—

137

C₂×C₈

#102

yes

—

138

C₂×C₈

#103

yes

—

139

C₂×C₈

#103

yes

—

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

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

C₂×D₈

yes

characteristic

157

C₄²

C₂×SD₁₆

yes

characteristic

158

C₄²

C₂×SD₁₆

yes

characteristic

159

C₄²

C₂×Q₁₆

yes

characteristic

160

C₄²

AGL(1,ℤ/8)

yes

characteristic

161

C₄²

AGL(1,ℤ/8)

yes

characteristic

162

C₄²

#137

yes

characteristic

163

C₄²

#137

yes

characteristic

164

C₂³

#380

yes

characteristic

165

Q₈

—

166

Q₈

—

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

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

C₂×#101

yes

characteristic

187

C₂×C₄

#413

yes

characteristic

188

C₂×C₄

C₂×#102

yes

characteristic

189

C₂×C₄

C₂×#103

yes

characteristic

190

C₂×C₄

#417

yes

characteristic

191

C₂×C₄

#417

yes

characteristic

192

C₂×C₄

#430

yes

—

193

C₂×C₄

#459

yes

characteristic

194

C₂×C₄

#460

yes

characteristic

195

C₂×C₄

#461

yes

characteristic

196

C₂×C₄

#462

yes

characteristic

197

C₂×C₄

#474

yes

characteristic

198

C₂×C₄

#475

yes

characteristic

199

C₂×C₄

#476

yes

characteristic

200

C₂×C₄

#477

yes

characteristic

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

#332

yes

—

208

C₂²

#1801

yes

characteristic

209

C₂²

#1802

yes

characteristic

210

C₂²

#1803

yes

characteristic

211

C₂²

#1811

yes

characteristic

212

C₂²

#1812

yes

characteristic

213

C₂²

#1814

yes

characteristic

214

C₂²

#1814

yes

characteristic

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

C₂×#332

yes

characteristic

228

C₄

#1547

yes

characteristic

229

C₄

#1550

yes

characteristic

230

C₄

#1561

yes

characteristic

231

C₂

#8586

yes

characteristic

232

C₂

#8625

yes

characteristic

233

C₂

#8635

yes

characteristic

234

C₂

#8662

yes

characteristic

235

C₂

#8663

yes

characteristic

236

C₂

#8674

yes

characteristic

237

C₂

#8677

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#380
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

#380

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
#21712	2
#8807	2
#7652	2
#7040	2

subgroup

index

Subgroups — #99083

Conjugacy classes of subgroups

Maximal subgroups