Group #99089 — subgroups

Subgroups — #99089

Order 512.

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

normal subgroups only

This group has 332 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	#21712	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	#8785	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×#334	—	2	1	no	—
19	#1277	—	2	1	no	—
20	#1277	—	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	#334	—	4	2	no	—
54	C₂²×Q₈	—	2	1	no	—
55	C₂²×C₈	—	2	2	no	—
56	C₂²×C₈	—	4	2	no	—
57	C₂²×C₈	—	4	2	no	—
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	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₄⋊Q₈	—	4	2	no	—
63	C₄×Q₈	—	4	1	no	—
64	C₄×Q₈	—	4	1	no	—
65	C₄×Q₈	—	4	1	no	—
66	C₄×Q₈	—	4	1	no	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	—	2	1	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	1	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄⋊C₈	—	4	1	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	#97	—	4	4	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	—	4	1	no	—
97	C₄×C₈	D₈	1	2	yes	—
98	C₄×C₈	SD₁₆	1	2	yes	—
99	#95	—	2	2	no	—
100	#95	—	2	2	no	—
101	C₂×Q₈	—	4	2	no	—
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄	#117	1	1	yes	characteristic
110	C₂²×C₄	#121	1	1	yes	characteristic
111	C₂²×C₄	#121	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
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	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	8	2	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	—	8	2	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	—	8	2	no	—
131	C₂×C₈	—	8	2	no	—
132	C₂×C₈	#101	1	2	yes	—
133	C₂×C₈	#102	1	2	yes	—
134	C₂×C₈	#102	1	2	yes	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	2	no	—
151	C₄²	C₂×D₈	1	1	yes	characteristic
152	C₄²	C₂×SD₁₆	1	1	yes	characteristic
153	C₄²	#135	1	2	yes	—
154	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₄²	#137	1	1	yes	characteristic
156	C₄²	#137	1	2	yes	—
157	C₂³	#380	1	1	yes	characteristic
158	Q₈	—	8	2	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	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	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	2	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	C₂×#102	1	1	yes	characteristic
182	C₂×C₄	#416	1	1	yes	characteristic
183	C₂×C₄	#417	1	1	yes	characteristic
184	C₂×C₄	#419	1	1	yes	characteristic
185	C₂×C₄	#430	1	2	yes	—
186	C₂×C₄	#459	1	1	yes	characteristic
187	C₂×C₄	#461	1	1	yes	characteristic
188	C₂×C₄	#464	1	2	yes	—
189	C₂×C₄	#474	1	1	yes	characteristic
190	C₂×C₄	#476	1	1	yes	characteristic
191	C₂×C₄	#479	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	1	no	—
196	C₈	—	8	2	no	—
197	C₈	—	16	2	no	—
198	C₈	—	16	2	no	—
199	C₂²	#1801	1	1	yes	characteristic
200	C₂²	#1802	1	1	yes	characteristic
201	C₂²	#1805	1	1	yes	characteristic
202	C₂²	#1811	1	1	yes	characteristic
203	C₂²	#1812	1	1	yes	characteristic
204	C₂²	#1814	1	1	yes	characteristic
205	C₂²	#1814	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	8	2	no	—
216	C₄	#1540	1	1	yes	characteristic
217	C₄	#1546	1	1	yes	characteristic
218	C₄	#1552	1	1	yes	characteristic
219	C₄	#1559	1	1	yes	characteristic
220	C₂	#8589	1	1	yes	characteristic
221	C₂	#8625	1	1	yes	characteristic
222	C₂	#8631	1	1	yes	characteristic
223	C₂	#8662	1	1	yes	characteristic
224	C₂	#8665	1	1	yes	characteristic
225	C₂	#8676	1	1	yes	characteristic
226	C₂	#8681	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21712

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#8785

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

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

#95

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

108

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

109

C₂²×C₄

#117

yes

characteristic

110

C₂²×C₄

#121

yes

characteristic

111

C₂²×C₄

#121

yes

characteristic

112

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

113

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

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

#101

yes

—

133

C₂×C₈

#102

yes

—

134

C₂×C₈

#102

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×D₈

yes

characteristic

152

C₄²

C₂×SD₁₆

yes

characteristic

153

C₄²

#135

yes

—

154

C₄²

AGL(1,ℤ/8)

yes

characteristic

155

C₄²

#137

yes

characteristic

156

C₄²

#137

yes

—

157

C₂³

#380

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

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

C₂×#102

yes

characteristic

182

C₂×C₄

#416

yes

characteristic

183

C₂×C₄

#417

yes

characteristic

184

C₂×C₄

#419

yes

characteristic

185

C₂×C₄

#430

yes

—

186

C₂×C₄

#459

yes

characteristic

187

C₂×C₄

#461

yes

characteristic

188

C₂×C₄

#464

yes

—

189

C₂×C₄

#474

yes

characteristic

190

C₂×C₄

#476

yes

characteristic

191

C₂×C₄

#479

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₂²

#1801

yes

characteristic

200

C₂²

#1802

yes

characteristic

201

C₂²

#1805

yes

characteristic

202

C₂²

#1811

yes

characteristic

203

C₂²

#1812

yes

characteristic

204

C₂²

#1814

yes

characteristic

205

C₂²

#1814

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

#1540

yes

characteristic

217

C₄

#1546

yes

characteristic

218

C₄

#1552

yes

characteristic

219

C₄

#1559

yes

characteristic

220

C₂

#8589

yes

characteristic

221

C₂

#8625

yes

characteristic

222

C₂

#8631

yes

characteristic

223

C₂

#8662

yes

characteristic

224

C₂

#8665

yes

characteristic

225

C₂

#8676

yes

characteristic

226

C₂

#8681

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
#8785	2
#7652	2
#7040	2

subgroup

index

Subgroups — #99089

Conjugacy classes of subgroups

Maximal subgroups