Group #99058 — subgroups

Subgroups — #99058

Order 512.

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

normal subgroups only

This group has 352 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	#8784	C₂	1	1	yes	maximal, characteristic
3	#8784	C₂	1	1	yes	maximal, characteristic
4	#8587	C₂	1	2	yes	maximal
5	#7050	C₂	1	2	yes	maximal
6	C₂×#446	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	2	yes	—
12	#1842	C₂²	1	2	yes	—
13	#1842	C₂²	1	1	yes	characteristic
14	#1770	—	2	1	no	—
15	#1770	—	2	1	no	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#333	—	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	C₂×C₄	1	2	yes	—
25	#446	—	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×#97	—	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	Q₈	1	2	yes	—
44	#389	—	2	1	no	—
45	#389	—	2	1	no	—
46	#384	D₄	1	1	yes	characteristic
47	C₄×C₄⋊C₄	—	2	1	no	—
48	C₄×C₄⋊C₄	—	2	1	no	—
49	C₄³	C₂³	1	1	yes	characteristic
50	#336	—	2	2	no	—
51	#336	—	2	2	no	—
52	#333	—	4	4	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	2	no	—
62	C₄⋊Q₈	—	2	2	no	—
63	C₄×Q₈	—	4	2	no	—
64	C₄×Q₈	—	4	2	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	2	1	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	C₄⋊C₄	1	2	yes	—
71	C₂×C₄²	—	2	2	no	—
72	C₂×C₄²	C₂²×C₄	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×D₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₄²	#41	1	1	yes	characteristic
78	C₂×C₄²	#41	1	1	yes	characteristic
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	4	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	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	—	8	2	no	—
92	C₄⋊C₈	—	8	2	no	—
93	#97	—	4	4	no	—
94	C₄×C₈	—	2	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	—	4	1	no	—
100	#95	—	2	2	no	—
101	#95	—	2	2	no	—
102	C₂×Q₈	—	2	4	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	4	2	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	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	4	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	—	8	2	no	—
133	C₂×C₈	—	8	2	no	—
134	C₂×C₈	#100	1	2	yes	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	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	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	2	no	—
151	C₄²	#135	1	2	yes	—
152	C₄²	#135	1	2	yes	—
153	C₄²	AGL(1,ℤ/8)	1	2	yes	—
154	C₄²	#137	1	2	yes	—
155	C₄²	#137	1	2	yes	—
156	C₄²	#137	1	2	yes	—
157	C₂³	#380	1	1	yes	characteristic
158	Q₈	—	8	4	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	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	4	2	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	#416	1	1	yes	characteristic
177	C₂×C₄	#416	1	1	yes	characteristic
178	C₂×C₄	#419	1	1	yes	characteristic
179	C₂×C₄	#419	1	1	yes	characteristic
180	C₂×C₄	#428	1	2	yes	—
181	C₂×C₄	#464	1	2	yes	—
182	C₂×C₄	#464	1	2	yes	—
183	C₂×C₄	#468	1	2	yes	—
184	C₂×C₄	#469	1	1	yes	characteristic
185	C₂×C₄	#470	1	1	yes	characteristic
186	C₂×C₄	#471	1	2	yes	—
187	C₂×C₄	#472	1	1	yes	characteristic
188	C₂×C₄	#473	1	1	yes	characteristic
189	C₂×C₄	#479	1	2	yes	—
190	C₂×C₄	#479	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	4	1	no	—
193	C₈	—	4	1	no	—
194	C₈	—	4	1	no	—
195	C₈	—	8	2	no	—
196	C₈	—	16	2	no	—
197	C₈	—	16	2	no	—
198	C₂²	#1800	1	1	yes	characteristic
199	C₂²	#1805	1	1	yes	characteristic
200	C₂²	#1805	1	1	yes	characteristic
201	C₂²	#1809	1	1	yes	characteristic
202	C₂²	#1813	1	1	yes	characteristic
203	C₂²	#1815	1	1	yes	characteristic
204	C₂²	#1816	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	4	1	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	#1543	1	1	yes	characteristic
212	C₄	#1554	1	1	yes	characteristic
213	C₄	#1554	1	1	yes	characteristic
214	C₄	#1558	1	1	yes	characteristic
215	C₄	#1698	1	2	yes	—
216	C₄	#1700	1	2	yes	—
217	C₄	#1702	1	2	yes	—
218	C₄	#1704	1	2	yes	—
219	C₂	#8594	1	1	yes	characteristic
220	C₂	#8618	1	1	yes	characteristic
221	C₂	#8640	1	1	yes	characteristic
222	C₂	#8655	1	1	yes	characteristic
223	C₂	#8657	1	1	yes	characteristic
224	C₂	#8687	1	1	yes	characteristic
225	C₂	#8689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21712

C₂

yes

maximal, characteristic

#8784

C₂

yes

maximal, characteristic

#8784

C₂

yes

maximal, characteristic

#8587

C₂

yes

maximal

#7050

C₂

yes

maximal

C₂×#446

C₄

yes

—

C₄×C₄⋊Q₈

C₂²

yes

characteristic

#1984

C₄

yes

—

#1939

—

#1939

—

#1842

C₂²

yes

—

#1842

C₂²

yes

—

#1842

C₂²

yes

characteristic

#1770

—

#1770

—

C₄²×C₈

C₂²

yes

—

—

—

—

—

—

C₂×C₄⋊Q₈

D₄

yes

characteristic

C₂×C₄×Q₈

C₂×C₄

yes

—

#446

—

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×#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

characteristic

#391

Q₈

yes

—

#389

—

#389

—

#384

D₄

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

—

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

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

#95

—

101

#95

—

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

—

133

C₂×C₈

—

134

C₂×C₈

#100

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

#135

yes

—

152

C₄²

#135

yes

—

153

C₄²

AGL(1,ℤ/8)

yes

—

154

C₄²

#137

yes

—

155

C₄²

#137

yes

—

156

C₄²

#137

yes

—

157

C₂³

#380

yes

characteristic

158

Q₈

—

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

#416

yes

characteristic

177

C₂×C₄

#416

yes

characteristic

178

C₂×C₄

#419

yes

characteristic

179

C₂×C₄

#419

yes

characteristic

180

C₂×C₄

#428

yes

—

181

C₂×C₄

#464

yes

—

182

C₂×C₄

#464

yes

—

183

C₂×C₄

#468

yes

—

184

C₂×C₄

#469

yes

characteristic

185

C₂×C₄

#470

yes

characteristic

186

C₂×C₄

#471

yes

—

187

C₂×C₄

#472

yes

characteristic

188

C₂×C₄

#473

yes

characteristic

189

C₂×C₄

#479

yes

—

190

C₂×C₄

#479

yes

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

198

C₂²

#1800

yes

characteristic

199

C₂²

#1805

yes

characteristic

200

C₂²

#1805

yes

characteristic

201

C₂²

#1809

yes

characteristic

202

C₂²

#1813

yes

characteristic

203

C₂²

#1815

yes

characteristic

204

C₂²

#1816

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

#1543

yes

characteristic

212

C₄

#1554

yes

characteristic

213

C₄

#1554

yes

characteristic

214

C₄

#1558

yes

characteristic

215

C₄

#1698

yes

—

216

C₄

#1700

yes

—

217

C₄

#1702

yes

—

218

C₄

#1704

yes

—

219

C₂

#8594

yes

characteristic

220

C₂

#8618

yes

characteristic

221

C₂

#8640

yes

characteristic

222

C₂

#8655

yes

characteristic

223

C₂

#8657

yes

characteristic

224

C₂

#8687

yes

characteristic

225

C₂

#8689

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
#8784	2
#8587	2
#7050	2

subgroup

index

Subgroups — #99058

Conjugacy classes of subgroups

Maximal subgroups