Group #99042 — subgroups

Subgroups — #99042

Order 512.

368 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 130 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	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#8784	C₂	1	1	yes	maximal, characteristic
3	#8784	C₂	1	1	yes	maximal, characteristic
4	#8586	C₂	1	2	yes	maximal
5	#7050	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	—	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	2	yes	—
12	#1842	C₂²	1	2	yes	—
13	#1842	C₂²	1	1	yes	characteristic
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×C₄⋊C₄	C₄	1	2	yes	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#332	—	2	2	no	—
18	#1296	—	2	2	no	—
19	#1277	—	2	1	no	—
20	#1277	—	2	1	no	—
21	#1277	—	2	1	no	—
22	#1277	—	2	1	no	—
23	C₂×C₄⋊Q₈	D₄	1	1	yes	characteristic
24	C₂×C₄×Q₈	—	2	1	no	—
25	C₈×Q₈	—	4	4	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	—	2	2	no	—
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
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	1	no	—
44	#389	—	2	1	no	—
45	#389	Q₈	1	2	yes	—
46	#384	D₄	1	1	yes	characteristic
47	C₄×C₄⋊C₄	—	2	1	no	—
48	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₄³	C₂³	1	1	yes	characteristic
50	#336	—	2	2	no	—
51	#336	—	2	2	no	—
52	#332	—	4	4	no	—
53	C₂²×Q₈	—	2	1	no	—
54	C₂²×C₈	—	2	2	no	—
55	C₂²×C₈	—	4	2	no	—
56	C₂²×C₈	—	4	2	no	—
57	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₄⋊Q₈	—	2	1	no	—
62	C₄⋊Q₈	—	2	1	no	—
63	C₄⋊Q₈	D₈	1	2	yes	—
64	C₄⋊Q₈	Q₁₆	1	2	yes	—
65	C₄×Q₈	—	4	2	no	—
66	C₄×Q₈	—	4	2	no	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
72	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
73	C₂×C₄²	—	2	2	no	—
74	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₄²	#41	1	1	yes	characteristic
80	C₂×C₄²	#41	1	1	yes	characteristic
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	4	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	2	no	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	8	2	no	—
94	C₄⋊C₈	—	8	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	4	no	—
102	#95	—	2	2	no	—
103	#95	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄	#117	1	1	yes	characteristic
113	C₂²×C₄	#121	1	1	yes	characteristic
114	C₂²×C₄	#121	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₈	—	2	1	no	—
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	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	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	—	8	2	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	#101	1	2	yes	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	#107	1	4	yes	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	4	1	no	—
151	C₄²	—	4	2	no	—
152	C₄²	—	4	2	no	—
153	C₄²	C₂×D₈	1	1	yes	characteristic
154	C₄²	C₂×Q₁₆	1	1	yes	characteristic
155	C₄²	#135	1	2	yes	—
156	C₄²	#135	1	2	yes	—
157	C₄²	#135	1	1	yes	characteristic
158	C₄²	#135	1	1	yes	characteristic
159	C₄²	AGL(1,ℤ/8)	1	2	yes	—
160	C₄²	#137	1	2	yes	—
161	C₂³	#380	1	1	yes	characteristic
162	Q₈	—	4	4	no	—
163	Q₈	—	4	4	no	—
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	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂×#101	1	1	yes	characteristic
180	C₂×C₄	#413	1	1	yes	characteristic
181	C₂×C₄	#416	1	1	yes	characteristic
182	C₂×C₄	#416	1	1	yes	characteristic
183	C₂×C₄	#418	1	1	yes	characteristic
184	C₂×C₄	#419	1	1	yes	characteristic
185	C₂×C₄	C₂×#107	1	1	yes	characteristic
186	C₂×C₄	#427	1	1	yes	characteristic
187	C₂×C₄	#463	1	2	yes	—
188	C₂×C₄	#464	1	2	yes	—
189	C₂×C₄	#466	1	1	yes	characteristic
190	C₂×C₄	#466	1	1	yes	characteristic
191	C₂×C₄	#467	1	1	yes	characteristic
192	C₂×C₄	#467	1	1	yes	characteristic
193	C₂×C₄	#471	1	2	yes	—
194	C₂×C₄	#471	1	2	yes	—
195	C₂×C₄	#478	1	2	yes	—
196	C₂×C₄	#479	1	2	yes	—
197	C₈	—	2	2	no	—
198	C₈	—	4	1	no	—
199	C₈	—	4	1	no	—
200	C₈	—	4	1	no	—
201	C₈	—	8	2	no	—
202	C₈	—	16	2	no	—
203	C₈	—	16	2	no	—
204	C₂²	#1801	1	1	yes	characteristic
205	C₂²	#1804	1	1	yes	characteristic
206	C₂²	#1805	1	1	yes	characteristic
207	C₂²	#1808	1	1	yes	characteristic
208	C₂²	#1813	1	1	yes	characteristic
209	C₂²	#1815	1	1	yes	characteristic
210	C₂²	#1815	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	1	no	—
216	C₄	—	4	4	no	—
217	C₄	—	8	2	no	—
218	C₄	#1544	1	1	yes	characteristic
219	C₄	#1553	1	1	yes	characteristic
220	C₄	#1554	1	1	yes	characteristic
221	C₄	#1556	1	1	yes	characteristic
222	C₄	#1681	1	2	yes	—
223	C₄	#1682	1	2	yes	—
224	C₄	#1683	1	2	yes	—
225	C₄	#1684	1	2	yes	—
226	C₂	#8593	1	1	yes	characteristic
227	C₂	#8617	1	1	yes	characteristic
228	C₂	#8637	1	1	yes	characteristic
229	C₂	#8648	1	1	yes	characteristic
230	C₂	#8649	1	1	yes	characteristic
231	C₂	#8686	1	1	yes	characteristic
232	C₂	#8687	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×C₄⋊Q₈

C₂

yes

maximal, characteristic

#8784

C₂

yes

maximal, characteristic

#8784

C₂

yes

maximal, characteristic

#8586

C₂

yes

maximal

#7050

C₂

yes

maximal

C₂×C₈×Q₈

—

C₄×C₄⋊Q₈

C₂²

yes

—

—

C₄

yes

—

#1842

C₂²

yes

—

#1842

C₂²

yes

—

#1842

C₂²

yes

characteristic

C₈×C₄⋊C₄

—

C₈×C₄⋊C₄

C₄

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

—

—

C₂×C₄⋊Q₈

D₄

yes

characteristic

C₂×C₄×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

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

—

—

Q₈

yes

—

#384

D₄

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

D₈

yes

—

C₄⋊Q₈

Q₁₆

yes

—

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

—

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

#95

—

103

#95

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

111

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

112

C₂²×C₄

#117

yes

characteristic

113

C₂²×C₄

#121

yes

characteristic

114

C₂²×C₄

#121

yes

characteristic

115

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

116

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

117

C₂×C₈

—

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

—

136

C₂×C₈

—

137

C₂×C₈

#101

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

#107

yes

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

C₂×D₈

yes

characteristic

154

C₄²

C₂×Q₁₆

yes

characteristic

155

C₄²

#135

yes

—

156

C₄²

#135

yes

—

157

C₄²

#135

yes

characteristic

158

C₄²

#135

yes

characteristic

159

C₄²

AGL(1,ℤ/8)

yes

—

160

C₄²

#137

yes

—

161

C₂³

#380

yes

characteristic

162

Q₈

—

163

Q₈

—

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

C₂×#101

yes

characteristic

180

C₂×C₄

#413

yes

characteristic

181

C₂×C₄

#416

yes

characteristic

182

C₂×C₄

#416

yes

characteristic

183

C₂×C₄

#418

yes

characteristic

184

C₂×C₄

#419

yes

characteristic

185

C₂×C₄

C₂×#107

yes

characteristic

186

C₂×C₄

#427

yes

characteristic

187

C₂×C₄

#463

yes

—

188

C₂×C₄

#464

yes

—

189

C₂×C₄

#466

yes

characteristic

190

C₂×C₄

#466

yes

characteristic

191

C₂×C₄

#467

yes

characteristic

192

C₂×C₄

#467

yes

characteristic

193

C₂×C₄

#471

yes

—

194

C₂×C₄

#471

yes

—

195

C₂×C₄

#478

yes

—

196

C₂×C₄

#479

yes

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₂²

#1801

yes

characteristic

205

C₂²

#1804

yes

characteristic

206

C₂²

#1805

yes

characteristic

207

C₂²

#1808

yes

characteristic

208

C₂²

#1813

yes

characteristic

209

C₂²

#1815

yes

characteristic

210

C₂²

#1815

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#1544

yes

characteristic

219

C₄

#1553

yes

characteristic

220

C₄

#1554

yes

characteristic

221

C₄

#1556

yes

characteristic

222

C₄

#1681

yes

—

223

C₄

#1682

yes

—

224

C₄

#1683

yes

—

225

C₄

#1684

yes

—

226

C₂

#8593

yes

characteristic

227

C₂

#8617

yes

characteristic

228

C₂

#8637

yes

characteristic

229

C₂

#8648

yes

characteristic

230

C₂

#8649

yes

characteristic

231

C₂

#8686

yes

characteristic

232

C₂

#8687

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
C₈×C₄⋊Q₈	2
#8784	2
#8586	2
#7050	2

subgroup

index

C₈×C₄⋊Q₈

#8784

#8586

#7050

Subgroups — #99042

Conjugacy classes of subgroups

Maximal subgroups