Group #105233 — subgroups

Subgroups — #105233

Order 512.

305 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 96 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	#9874	C₂	1	1	yes	maximal, characteristic
2	#9874	C₂	1	1	yes	maximal, characteristic
3	#9643	C₂	1	1	yes	maximal, characteristic
4	#8810	C₂	1	1	yes	maximal, characteristic
5	#8809	C₂	1	1	yes	maximal, characteristic
6	#8787	C₂	1	1	yes	maximal, characteristic
7	#7014	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#1842	C₂²	1	1	yes	characteristic
10	#1842	C₂²	1	1	yes	characteristic
11	#1842	C₂²	1	1	yes	characteristic
12	#1744	C₂²	1	1	yes	characteristic
13	#1744	C₂²	1	1	yes	characteristic
14	#1744	C₂²	1	1	yes	characteristic
15	C₂×#335	—	2	2	no	—
16	C₂×#334	—	2	2	no	—
17	C₂×Q₈⋊C₈	—	2	2	no	—
18	#1279	—	2	2	no	—
19	#1279	—	2	2	no	—
20	#1277	—	2	2	no	—
21	C₂×C₄⋊Q₈	Q₈	1	1	yes	characteristic
22	C₂×C₄×Q₈	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	1	no	—
24	C₂×C₄⋊C₈	—	2	1	no	—
25	C₂×C₄⋊C₈	—	2	1	no	—
26	C₂×C₄⋊C₈	—	2	1	no	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	1	no	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄×C₈	D₄	1	2	yes	—
33	C₂×C₄×C₈	D₄	1	2	yes	—
34	C₂×C₄×C₈	D₄	1	2	yes	—
35	#391	—	2	1	no	—
36	#389	—	2	1	no	—
37	#389	—	2	1	no	—
38	#384	Q₈	1	1	yes	characteristic
39	C₄×C₄⋊C₄	—	2	1	no	—
40	C₄×C₄⋊C₄	—	2	1	no	—
41	C₄³	C₂³	1	1	yes	characteristic
42	#336	—	2	1	no	—
43	#336	—	2	1	no	—
44	#336	—	2	1	no	—
45	#336	—	2	1	no	—
46	#336	—	2	1	no	—
47	#336	—	2	1	no	—
48	#335	—	4	4	no	—
49	#334	—	4	4	no	—
50	Q₈⋊C₈	—	4	4	no	—
51	C₂²×Q₈	—	2	1	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	—	2	2	no	—
54	C₂²×C₈	—	2	2	no	—
55	C₂²×C₈	—	4	1	no	—
56	C₂²×C₈	—	4	1	no	—
57	C₂²×C₈	—	4	1	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	C₄×Q₈	—	4	2	no	—
60	C₄×Q₈	—	4	2	no	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	—	2	1	no	—
63	C₂×C₄⋊C₄	—	2	1	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	1	no	—
66	C₂×C₄⋊C₄	—	4	1	no	—
67	C₂×C₄²	—	4	1	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₄²	#41	1	1	yes	characteristic
73	C₂×C₄²	#41	1	1	yes	characteristic
74	C₂×C₄²	#41	1	1	yes	characteristic
75	C₄⋊C₈	—	4	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄⋊C₈	—	4	2	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	D₈	1	4	yes	—
89	C₄×C₈	SD₁₆	1	4	yes	—
90	#95	—	4	1	no	—
91	#95	—	4	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	4	1	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂²×C₄	#123	1	1	yes	characteristic
102	C₂²×C₄	#123	1	1	yes	characteristic
103	C₂²×C₄	#123	1	1	yes	characteristic
104	C₂²×C₄	#127	1	1	yes	characteristic
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	—	8	1	no	—
116	C₂×C₈	—	8	1	no	—
117	C₂×C₈	—	8	1	no	—
118	C₂×C₈	—	8	1	no	—
119	C₂×C₈	—	8	1	no	—
120	C₂×C₈	—	8	1	no	—
121	C₂×C₈	—	8	1	no	—
122	C₂×C₈	—	8	1	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	2	no	—
131	C₄²	—	4	1	no	—
132	C₄²	—	4	1	no	—
133	C₄²	—	4	1	no	—
134	C₄²	—	4	1	no	—
135	C₄²	—	8	1	no	—
136	C₄²	—	8	1	no	—
137	C₄²	C₂×D₈	1	2	yes	—
138	C₄²	C₂×SD₁₆	1	2	yes	—
139	C₄²	#135	1	2	yes	—
140	C₄²	#137	1	2	yes	—
141	C₄²	#137	1	2	yes	—
142	C₄²	#137	1	2	yes	—
143	C₂³	#399	1	1	yes	characteristic
144	Q₈	—	4	4	no	—
145	Q₈	—	8	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	#474	1	2	yes	—
165	C₂×C₄	#475	1	2	yes	—
166	C₂×C₄	#479	1	2	yes	—
167	C₂×C₄	#480	1	2	yes	—
168	C₂×C₄	#483	1	2	yes	—
169	C₂×C₄	#485	1	2	yes	—
170	C₂×C₄	#492	1	1	yes	characteristic
171	C₂×C₄	#493	1	1	yes	characteristic
172	C₂×C₄	#495	1	1	yes	characteristic
173	C₂×C₄	#497	1	1	yes	characteristic
174	C₂×C₄	#497	1	1	yes	characteristic
175	C₂×C₄	#497	1	1	yes	characteristic
176	C₈	—	4	4	no	—
177	C₈	—	4	4	no	—
178	C₈	—	8	2	no	—
179	C₈	—	16	1	no	—
180	C₈	—	16	1	no	—
181	C₈	—	16	1	no	—
182	C₂²	#2091	1	1	yes	characteristic
183	C₂²	#2093	1	1	yes	characteristic
184	C₂²	#2096	1	1	yes	characteristic
185	C₂²	#2097	1	1	yes	characteristic
186	C₂²	#2098	1	1	yes	characteristic
187	C₂²	#2098	1	1	yes	characteristic
188	C₂²	#2098	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	8	2	no	—
196	C₄	—	16	1	no	—
197	C₄	#1706	1	1	yes	characteristic
198	C₄	#1716	1	1	yes	characteristic
199	C₄	#1718	1	1	yes	characteristic
200	C₄	#1722	1	1	yes	characteristic
201	C₂	#10660	1	1	yes	characteristic
202	C₂	#10663	1	1	yes	characteristic
203	C₂	#10664	1	1	yes	characteristic
204	C₂	#10667	1	1	yes	characteristic
205	C₂	#10669	1	1	yes	characteristic
206	C₂	#10670	1	1	yes	characteristic
207	C₂	#10675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#9874

C₂

yes

maximal, characteristic

#9874

C₂

yes

maximal, characteristic

#9643

C₂

yes

maximal, characteristic

#8810

C₂

yes

maximal, characteristic

#8809

C₂

yes

maximal, characteristic

#8787

C₂

yes

maximal, characteristic

#7014

C₂

yes

maximal, characteristic

C₄×C₄⋊Q₈

C₂²

yes

characteristic

#1842

C₂²

yes

characteristic

#1842

C₂²

yes

characteristic

#1842

C₂²

yes

characteristic

#1744

C₂²

yes

characteristic

#1744

C₂²

yes

characteristic

#1744

C₂²

yes

characteristic

C₂×#335

—

C₂×#334

—

C₂×Q₈⋊C₈

—

#1279

—

#1279

—

#1277

—

C₂×C₄⋊Q₈

Q₈

yes

characteristic

C₂×C₄×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

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

—

—

—

Q₈

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂³

yes

—

—

—

—

—

—

—

—

Q₈⋊C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

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

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

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

100

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

101

C₂²×C₄

#123

yes

characteristic

102

C₂²×C₄

#123

yes

characteristic

103

C₂²×C₄

#123

yes

characteristic

104

C₂²×C₄

#127

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

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

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂×D₈

yes

—

138

C₄²

C₂×SD₁₆

yes

—

139

C₄²

#135

yes

—

140

C₄²

#137

yes

—

141

C₄²

#137

yes

—

142

C₄²

#137

yes

—

143

C₂³

#399

yes

characteristic

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

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

#474

yes

—

165

C₂×C₄

#475

yes

—

166

C₂×C₄

#479

yes

—

167

C₂×C₄

#480

yes

—

168

C₂×C₄

#483

yes

—

169

C₂×C₄

#485

yes

—

170

C₂×C₄

#492

yes

characteristic

171

C₂×C₄

#493

yes

characteristic

172

C₂×C₄

#495

yes

characteristic

173

C₂×C₄

#497

yes

characteristic

174

C₂×C₄

#497

yes

characteristic

175

C₂×C₄

#497

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₂²

#2091

yes

characteristic

183

C₂²

#2093

yes

characteristic

184

C₂²

#2096

yes

characteristic

185

C₂²

#2097

yes

characteristic

186

C₂²

#2098

yes

characteristic

187

C₂²

#2098

yes

characteristic

188

C₂²

#2098

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#1706

yes

characteristic

198

C₄

#1716

yes

characteristic

199

C₄

#1718

yes

characteristic

200

C₄

#1722

yes

characteristic

201

C₂

#10660

yes

characteristic

202

C₂

#10663

yes

characteristic

203

C₂

#10664

yes

characteristic

204

C₂

#10667

yes

characteristic

205

C₂

#10669

yes

characteristic

206

C₂

#10670

yes

characteristic

207

C₂

#10675

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#399
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator, Frattini, cosocle

C₄³

C₂³

C₂³

G ▷ C₄³ ▷ C₂³ ▷ C₁

C₁ ◁ C₂²×C₄ ◁ 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
#9874	2
#9643	2
#8810	2
#8809	2
#8787	2
#7014	2

subgroup

index

Subgroups — #105233

Conjugacy classes of subgroups

Maximal subgroups