Group #125052 — subgroups

Subgroups — #125052

Order 512.

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

normal subgroups only

This group has 330 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	C₄×#500	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	#8784	C₂	1	1	yes	maximal, characteristic
4	#7121	C₂	1	4	yes	maximal
5	C₂×#500	C₄	1	1	yes	characteristic
6	C₄×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₄⋊C₁₆	—	2	1	no	—
8	#1990	—	2	1	no	—
9	#1942	—	2	1	no	—
10	#1942	C₂²	1	2	yes	—
11	#1845	C₄	1	1	yes	characteristic
12	#1842	—	2	1	no	—
13	C₄×#107	—	2	1	no	—
14	C₄×#107	C₄	1	2	yes	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	#1374	—	2	1	no	—
17	#1374	C₂²	1	2	yes	—
18	#1277	—	2	1	no	—
19	#1277	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈	—	2	1	no	—
21	C₂×C₄×Q₈	—	4	1	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	#500	—	2	4	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	—	2	1	no	—
26	C₂×#107	C₂×C₄	1	2	yes	—
27	C₂×#107	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	—	4	1	no	—
30	C₂×C₄⋊C₈	—	4	1	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	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	C₂³	1	1	yes	characteristic
37	#391	—	4	1	no	—
38	#389	—	2	2	no	—
39	#389	—	4	1	no	—
40	#384	—	2	1	no	—
41	C₄×C₄⋊C₄	—	2	2	no	—
42	C₄×C₄⋊C₄	—	4	1	no	—
43	C₄³	D₄	1	1	yes	characteristic
44	C₄⋊C₁₆	—	4	2	no	—
45	#340	—	2	2	no	—
46	#340	—	2	2	no	—
47	C₂²×Q₈	—	4	1	no	—
48	C₂²×C₈	—	4	2	no	—
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₄⋊Q₈	—	2	4	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	C₄×Q₈	—	8	4	no	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	—	4	1	no	—
59	C₂×C₄⋊C₄	—	4	1	no	—
60	C₂×C₄⋊C₄	—	4	1	no	—
61	C₂×C₄⋊C₄	—	4	2	no	—
62	C₂×C₄²	—	4	2	no	—
63	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	D₈	1	1	yes	characteristic
66	C₂×C₄²	D₈	1	1	yes	characteristic
67	C₂×C₄²	SD₁₆	1	1	yes	characteristic
68	C₂×C₄²	SD₁₆	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
70	C₂×C₁₆	—	4	2	no	—
71	C₂×C₁₆	—	4	2	no	—
72	C₂×C₁₆	—	4	2	no	—
73	#107	—	2	4	no	—
74	#107	—	2	2	no	—
75	#107	—	4	2	no	—
76	#107	—	4	2	no	—
77	#107	M₁₆	1	4	yes	—
78	C₄⋊C₈	—	4	2	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	8	2	no	—
81	C₄⋊C₈	—	8	2	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	2	2	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	—	4	1	no	—
87	C₄×C₈	SD₁₆	1	2	yes	—
88	C₄×C₈	Q₁₆	1	2	yes	—
89	#95	—	4	2	no	—
90	#95	—	4	2	no	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	—	8	2	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	—	4	2	no	—
96	C₂²×C₄	#102	1	1	yes	characteristic
97	C₂²×C₄	#102	1	1	yes	characteristic
98	C₂²×C₄	#102	1	1	yes	characteristic
99	C₂²×C₄	#102	1	1	yes	characteristic
100	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
102	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	8	2	no	—
113	C₂×C₈	—	8	2	no	—
114	C₂×C₈	—	8	2	no	—
115	C₂×C₈	#103	1	2	yes	—
116	C₂×C₈	C₂×M₁₆	1	2	yes	—
117	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	8	2	no	—
125	C₄⋊C₄	—	8	2	no	—
126	C₄⋊C₄	—	8	4	no	—
127	C₄²	—	2	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	4	1	no	—
134	C₄²	—	8	4	no	—
135	C₄²	#101	1	2	yes	—
136	C₄²	#103	1	2	yes	—
137	C₄²	SD₃₂	1	2	yes	—
138	C₄²	Q₃₂	1	2	yes	—
139	C₁₆	—	8	2	no	—
140	C₂³	C₂×#102	1	1	yes	characteristic
141	Q₈	—	8	8	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	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	—	4	4	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	#332	1	2	yes	—
160	C₂×C₄	#332	1	2	yes	—
161	C₂×C₄	#358	1	2	yes	—
162	C₂×C₄	#358	1	2	yes	—
163	C₂×C₄	#360	1	2	yes	—
164	C₂×C₄	#361	1	2	yes	—
165	C₂×C₄	#407	1	1	yes	characteristic
166	C₂×C₄	#410	1	1	yes	characteristic
167	C₂×C₄	C₂×#101	1	1	yes	characteristic
168	C₂×C₄	C₂×#103	1	1	yes	characteristic
169	C₂×C₄	#421	1	1	yes	characteristic
170	C₂×C₄	#421	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
172	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	4	1	no	—
178	C₈	—	16	2	no	—
179	C₂²	#1486	1	1	yes	characteristic
180	C₂²	#1517	1	1	yes	characteristic
181	C₂²	C₂×#332	1	1	yes	characteristic
182	C₂²	C₂×#358	1	1	yes	characteristic
183	C₂²	#2143	1	1	yes	characteristic
184	C₂²	C₂×#360	1	1	yes	characteristic
185	C₂²	C₂×#361	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	1	no	—
192	C₄	—	8	4	no	—
193	C₄	—	16	2	no	—
194	C₄	#1365	1	4	yes	—
195	C₄	#1467	1	1	yes	characteristic
196	C₄	#1487	1	1	yes	characteristic
197	C₄	#1518	1	1	yes	characteristic
198	C₄	#1537	1	1	yes	characteristic
199	C₂	#7652	1	1	yes	characteristic
200	C₂	#11658	1	1	yes	characteristic
201	C₂	#11663	1	1	yes	characteristic
202	C₂	#11857	1	1	yes	characteristic
203	C₂	#11864	1	1	yes	characteristic
204	C₂	#11936	1	1	yes	characteristic
205	C₂	C₂×#1365	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#500

C₂

yes

maximal, characteristic

#12879

C₂

yes

maximal, characteristic

#8784

C₂

yes

maximal, characteristic

#7121

C₂

yes

maximal

C₂×#500

C₄

yes

characteristic

C₄×C₄⋊Q₈

—

C₂×C₄⋊C₁₆

—

#1990

—

#1942

—

#1942

C₂²

yes

—

#1845

C₄

yes

—

—

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1374

—

#1374

C₂²

yes

—

#1277

—

#1277

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂²×C₁₆

—

—

—

—

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

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

C₂³

yes

—

—

—

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

#340

—

#340

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

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

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

M₁₆

yes

—

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

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

100

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

101

C₂²×C₄

C₂×D₈

yes

characteristic

102

C₂²×C₄

C₂×SD₁₆

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

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

#103

yes

—

116

C₂×C₈

C₂×M₁₆

yes

—

117

C₂×C₈

C₂×SD₁₆

yes

characteristic

118

C₂×C₈

C₂×Q₁₆

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

#101

yes

—

136

C₄²

#103

yes

—

137

C₄²

SD₃₂

yes

—

138

C₄²

Q₃₂

yes

—

139

C₁₆

—

140

C₂³

C₂×#102

yes

characteristic

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

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

#332

yes

—

160

C₂×C₄

#332

yes

—

161

C₂×C₄

#358

yes

—

162

C₂×C₄

#358

yes

—

163

C₂×C₄

#360

yes

—

164

C₂×C₄

#361

yes

—

165

C₂×C₄

#407

yes

characteristic

166

C₂×C₄

#410

yes

characteristic

167

C₂×C₄

C₂×#101

yes

characteristic

168

C₂×C₄

C₂×#103

yes

characteristic

169

C₂×C₄

#421

yes

characteristic

170

C₂×C₄

#421

yes

characteristic

171

C₂×C₄

C₂×SD₃₂

yes

characteristic

172

C₂×C₄

C₂×Q₃₂

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₂²

#1486

yes

characteristic

180

C₂²

#1517

yes

characteristic

181

C₂²

C₂×#332

yes

characteristic

182

C₂²

C₂×#358

yes

characteristic

183

C₂²

#2143

yes

characteristic

184

C₂²

C₂×#360

yes

characteristic

185

C₂²

C₂×#361

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#1365

yes

—

195

C₄

#1467

yes

characteristic

196

C₄

#1487

yes

characteristic

197

C₄

#1518

yes

characteristic

198

C₄

#1537

yes

characteristic

199

C₂

#7652

yes

characteristic

200

C₂

#11658

yes

characteristic

201

C₂

#11663

yes

characteristic

202

C₂

#11857

yes

characteristic

203

C₂

#11864

yes

characteristic

204

C₂

#11936

yes

characteristic

205

C₂

C₂×#1365

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#102

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#500	2
#12879	2
#8784	2
#7121	2

subgroup

index

Subgroups — #125052

Conjugacy classes of subgroups

Maximal subgroups