Group #99072 — subgroups

Subgroups — #99072

Order 512.

348 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 114 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	#8807	C₂	1	1	yes	maximal, characteristic
3	#8785	C₂	1	1	yes	maximal, characteristic
4	#7653	C₂	1	2	yes	maximal
5	#7040	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₄	1	2	yes	—
7	C₄×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	#1984	C₄	1	2	yes	—
9	#1939	—	2	1	no	—
10	#1939	—	2	1	no	—
11	#1842	C₂²	1	1	yes	characteristic
12	#1831	C₂²	1	2	yes	—
13	#1831	C₂²	1	2	yes	—
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×C₄⋊C₄	—	2	1	no	—
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	#1272	—	2	4	no	—
22	C₂×C₄⋊Q₈	Q₈	1	1	yes	characteristic
23	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
24	C₈×Q₈	—	4	4	no	—
25	C₂×C₄⋊C₈	—	2	2	no	—
26	C₂×C₄⋊C₈	—	2	1	no	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	—	2	2	no	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	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	#391	D₄	1	2	yes	—
39	#389	—	2	1	no	—
40	#389	—	2	1	no	—
41	#384	Q₈	1	1	yes	characteristic
42	C₄×C₄⋊C₄	—	2	1	no	—
43	C₄×C₄⋊C₄	—	2	1	no	—
44	C₄³	C₂³	1	1	yes	characteristic
45	#336	—	2	2	no	—
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	#335	—	4	2	no	—
52	#334	—	4	2	no	—
53	C₂²×Q₈	C₄⋊C₄	1	2	yes	—
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	4	no	—
62	C₄×Q₈	—	4	2	no	—
63	C₄×Q₈	—	4	2	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	1	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₄²	#41	1	1	yes	characteristic
77	C₂×C₄²	#41	1	1	yes	characteristic
78	C₄⋊C₈	—	4	2	no	—
79	C₄⋊C₈	—	4	4	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	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	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	4	1	no	—
93	C₄×C₈	—	4	4	no	—
94	C₄×C₈	SD₁₆	1	2	yes	—
95	C₄×C₈	Q₁₆	1	2	yes	—
96	#95	—	2	2	no	—
97	#95	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂×Q₈	#108	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₄	#117	1	1	yes	characteristic
107	C₂²×C₄	#121	1	1	yes	characteristic
108	C₂²×C₄	#121	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
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	—	4	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	—	8	2	no	—
125	C₂×C₈	—	8	2	no	—
126	C₂×C₈	—	8	2	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	—	8	2	no	—
129	C₂×C₈	#100	1	2	yes	—
130	C₂×C₈	#103	1	2	yes	—
131	C₂×C₈	#103	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	4	1	no	—
142	C₄²	—	4	1	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	C₂×SD₁₆	1	1	yes	characteristic
147	C₄²	C₂×Q₁₆	1	1	yes	characteristic
148	C₄²	#135	1	2	yes	—
149	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₄²	#137	1	1	yes	characteristic
151	C₄²	#137	1	2	yes	—
152	C₂³	#380	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	Q₈	—	4	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	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	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂×#100	1	1	yes	characteristic
175	C₂×C₄	#413	1	1	yes	characteristic
176	C₂×C₄	C₂×#103	1	1	yes	characteristic
177	C₂×C₄	#416	1	1	yes	characteristic
178	C₂×C₄	#417	1	1	yes	characteristic
179	C₂×C₄	#419	1	1	yes	characteristic
180	C₂×C₄	C₂×#108	1	2	yes	—
181	C₂×C₄	#460	1	1	yes	characteristic
182	C₂×C₄	#462	1	1	yes	characteristic
183	C₂×C₄	#464	1	2	yes	—
184	C₂×C₄	#475	1	1	yes	characteristic
185	C₂×C₄	#477	1	1	yes	characteristic
186	C₂×C₄	#479	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	2	2	no	—
190	C₈	—	4	1	no	—
191	C₈	—	8	2	no	—
192	C₈	—	16	2	no	—
193	C₈	—	16	2	no	—
194	C₂²	#1800	1	1	yes	characteristic
195	C₂²	#1803	1	1	yes	characteristic
196	C₂²	#1805	1	1	yes	characteristic
197	C₂²	#1810	1	1	yes	characteristic
198	C₂²	#1812	1	1	yes	characteristic
199	C₂²	#1814	1	1	yes	characteristic
200	C₂²	#1814	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	4	4	no	—
209	C₄	—	4	1	no	—
210	C₄	—	8	2	no	—
211	C₄	#1542	1	1	yes	characteristic
212	C₄	#1548	1	1	yes	characteristic
213	C₄	#1551	1	1	yes	characteristic
214	C₄	#1559	1	1	yes	characteristic
215	C₂	#8590	1	1	yes	characteristic
216	C₂	#8622	1	1	yes	characteristic
217	C₂	#8634	1	1	yes	characteristic
218	C₂	#8659	1	1	yes	characteristic
219	C₂	#8661	1	1	yes	characteristic
220	C₂	#8675	1	1	yes	characteristic
221	C₂	#8681	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×C₄⋊Q₈

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#8785

C₂

yes

maximal, characteristic

#7653

C₂

yes

maximal

#7040

C₂

yes

maximal

C₂×C₈×Q₈

C₄

yes

—

C₄×C₄⋊Q₈

C₂²

yes

characteristic

#1984

C₄

yes

—

#1939

—

#1939

—

#1842

C₂²

yes

characteristic

#1831

C₂²

yes

—

#1831

C₂²

yes

—

C₈×C₄⋊C₄

—

C₈×C₄⋊C₄

—

C₄²×C₈

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈

Q₈

yes

characteristic

C₂×C₄×Q₈

C₂×C₄

yes

—

C₈×Q₈

—

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

characteristic

#391

D₄

yes

—

#389

—

#389

—

#384

Q₈

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂³

yes

—

—

—

—

—

—

—

—

C₂²×Q₈

C₄⋊C₄

yes

—

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

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

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

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

#108

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

105

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

106

C₂²×C₄

#117

yes

characteristic

107

C₂²×C₄

#121

yes

characteristic

108

C₂²×C₄

#121

yes

characteristic

109

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

110

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

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

#100

yes

—

130

C₂×C₈

#103

yes

—

131

C₂×C₈

#103

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂×SD₁₆

yes

characteristic

147

C₄²

C₂×Q₁₆

yes

characteristic

148

C₄²

#135

yes

—

149

C₄²

AGL(1,ℤ/8)

yes

characteristic

150

C₄²

#137

yes

characteristic

151

C₄²

#137

yes

—

152

C₂³

#380

yes

characteristic

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

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

C₂×#100

yes

characteristic

175

C₂×C₄

#413

yes

characteristic

176

C₂×C₄

C₂×#103

yes

characteristic

177

C₂×C₄

#416

yes

characteristic

178

C₂×C₄

#417

yes

characteristic

179

C₂×C₄

#419

yes

characteristic

180

C₂×C₄

C₂×#108

yes

—

181

C₂×C₄

#460

yes

characteristic

182

C₂×C₄

#462

yes

characteristic

183

C₂×C₄

#464

yes

—

184

C₂×C₄

#475

yes

characteristic

185

C₂×C₄

#477

yes

characteristic

186

C₂×C₄

#479

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₂²

#1800

yes

characteristic

195

C₂²

#1803

yes

characteristic

196

C₂²

#1805

yes

characteristic

197

C₂²

#1810

yes

characteristic

198

C₂²

#1812

yes

characteristic

199

C₂²

#1814

yes

characteristic

200

C₂²

#1814

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

#1542

yes

characteristic

212

C₄

#1548

yes

characteristic

213

C₄

#1551

yes

characteristic

214

C₄

#1559

yes

characteristic

215

C₂

#8590

yes

characteristic

216

C₂

#8622

yes

characteristic

217

C₂

#8634

yes

characteristic

218

C₂

#8659

yes

characteristic

219

C₂

#8661

yes

characteristic

220

C₂

#8675

yes

characteristic

221

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
C₈×C₄⋊Q₈	2
#8807	2
#8785	2
#7653	2
#7040	2

subgroup

index

C₈×C₄⋊Q₈

Subgroups — #99072

Conjugacy classes of subgroups

Maximal subgroups