Group #134593 — subgroups

Subgroups — #134593

Order 512.

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

normal subgroups only

This group has 300 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	#21732	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	#11936	C₂	1	2	yes	maximal
4	#8807	C₂	1	1	yes	maximal, characteristic
5	#7280	C₂	1	2	yes	maximal
6	#3160	C₂²	1	1	yes	characteristic
7	#2960	—	2	1	no	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	—	2	1	no	—
10	#2150	C₂²	1	2	yes	—
11	#1994	—	2	1	no	—
12	#1945	—	2	1	no	—
13	#1945	C₄	1	2	yes	—
14	#1848	C₂²	1	1	yes	characteristic
15	#1831	—	2	1	no	—
16	#1772	—	2	1	no	—
17	#1772	C₄	1	2	yes	—
18	C₄²×C₈	C₂²	1	1	yes	characteristic
19	C₂×#335	—	2	1	no	—
20	C₂×#334	C₂²	1	2	yes	—
21	#1363	—	2	4	no	—
22	#533	—	2	1	no	—
23	C₂×M₃₂	—	2	2	no	—
24	#500	Q₈	1	1	yes	characteristic
25	#500	Q₈	1	1	yes	characteristic
26	#499	—	2	1	no	—
27	#499	D₄	1	2	yes	—
28	#498	—	2	1	no	—
29	#446	—	4	1	no	—
30	#446	—	4	1	no	—
31	C₈×Q₈	—	4	1	no	—
32	C₈×Q₈	—	4	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#108	C₂×C₄	1	2	yes	—
35	#427	—	2	1	no	—
36	#427	C₂×C₄	1	2	yes	—
37	#424	—	2	2	no	—
38	#423	—	2	1	no	—
39	C₂×C₄⋊C₈	—	2	2	no	—
40	#405	—	2	2	no	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₄³	D₄	1	1	yes	characteristic
48	#364	—	4	2	no	—
49	#364	—	4	2	no	—
50	C₄⋊C₁₆	—	2	2	no	—
51	C₄⋊C₁₆	—	2	2	no	—
52	#339	—	2	2	no	—
53	#339	—	2	2	no	—
54	#336	—	4	2	no	—
55	#335	—	4	1	no	—
56	#335	—	4	1	no	—
57	#334	—	2	2	no	—
58	#334	—	2	2	no	—
59	C₂×M₁₆	—	2	2	no	—
60	C₂²×C₈	—	4	2	no	—
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₄⋊Q₈	—	2	1	no	—
66	C₄⋊Q₈	—	2	1	no	—
67	#125	—	2	2	no	—
68	C₂²⋊Q₈	—	4	1	no	—
69	C₂²⋊Q₈	—	4	1	no	—
70	C₄×Q₈	—	4	1	no	—
71	C₄×Q₈	—	4	1	no	—
72	#117	—	2	2	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	—	2	1	no	—
75	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	M₃₂	—	4	2	no	—
79	M₃₂	—	4	2	no	—
80	C₂×C₁₆	—	4	2	no	—
81	#108	—	4	2	no	—
82	#108	—	4	1	no	—
83	#108	—	4	1	no	—
84	#107	—	2	1	no	—
85	#107	—	2	1	no	—
86	#107	C₄⋊C₄	1	2	yes	—
87	#107	C₄⋊C₄	1	2	yes	—
88	#106	—	2	2	no	—
89	#106	—	2	2	no	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	1	no	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄⋊C₈	—	4	1	no	—
96	#97	—	4	2	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	2	no	—
102	C₄×C₈	D₈	1	1	yes	characteristic
103	C₄×C₈	D₈	1	1	yes	characteristic
104	C₄×C₈	SD₁₆	1	2	yes	—
105	C₄×C₈	SD₁₆	1	1	yes	characteristic
106	C₄×C₈	SD₁₆	1	1	yes	characteristic
107	C₄×C₈	Q₁₆	1	2	yes	—
108	C₄×C₈	C₂×D₄	1	1	yes	characteristic
109	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₄×C₈	#41	1	1	yes	characteristic
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	4	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂²×C₄	#137	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	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	—	8	4	no	—
127	C₂×C₈	—	8	2	no	—
128	C₂×C₈	#101	1	2	yes	—
129	C₂×C₈	#102	1	1	yes	characteristic
130	C₂×C₈	#102	1	1	yes	characteristic
131	C₂×C₈	#102	1	1	yes	characteristic
132	C₂×C₈	#102	1	1	yes	characteristic
133	C₂×C₈	#103	1	2	yes	—
134	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₈	#117	1	1	yes	characteristic
136	C₂×C₈	#121	1	1	yes	characteristic
137	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₈	#137	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₂²⋊C₄	—	4	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	2	no	—
155	C₄²	#103	1	2	yes	—
156	C₄²	#121	1	1	yes	characteristic
157	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
158	C₄²	C₂×D₈	1	1	yes	characteristic
159	C₄²	C₂×SD₁₆	1	1	yes	characteristic
160	C₁₆	—	8	2	no	—
161	C₂³	#417	1	1	yes	characteristic
162	Q₈	—	8	2	no	—
163	Q₈	—	8	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	#380	1	1	yes	characteristic
175	C₂×C₄	C₂×#101	1	1	yes	characteristic
176	C₂×C₄	#413	1	1	yes	characteristic
177	C₂×C₄	C₂×#102	1	1	yes	characteristic
178	C₂×C₄	C₂×#103	1	1	yes	characteristic
179	C₂×C₄	#417	1	1	yes	characteristic
180	C₂×C₄	#459	1	1	yes	characteristic
181	C₂×C₄	#461	1	1	yes	characteristic
182	C₂×C₄	#474	1	1	yes	characteristic
183	C₂×C₄	#476	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	8	2	no	—
189	C₈	—	16	2	no	—
190	C₈	#332	1	2	yes	—
191	C₈	#332	1	2	yes	—
192	C₈	#460	1	1	yes	characteristic
193	C₈	#462	1	1	yes	characteristic
194	C₈	#475	1	1	yes	characteristic
195	C₈	#477	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	#1550	1	1	yes	characteristic
199	C₂²	#1561	1	1	yes	characteristic
200	C₂²	#1802	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	8	2	no	—
207	C₄	—	8	2	no	—
208	C₄	C₂×#332	1	1	yes	characteristic
209	C₄	#1547	1	1	yes	characteristic
210	C₄	#1801	1	1	yes	characteristic
211	C₄	#1803	1	1	yes	characteristic
212	C₄	#2231	1	2	yes	—
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	#8586	1	1	yes	characteristic
216	C₂	#12822	1	1	yes	characteristic
217	C₂	#12824	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21732

C₂

yes

maximal, characteristic

#12900

C₂

yes

maximal, characteristic

#11936

C₂

yes

maximal

#8807

C₂

yes

maximal, characteristic

#7280

C₂

yes

maximal

#3160

C₂²

yes

—

—

—

C₂²

yes

—

#1994

—

#1945

—

#1945

C₄

yes

—

#1848

C₂²

yes

—

—

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#334

C₂²

yes

—

#1363

—

#533

—

C₂×M₃₂

—

#500

Q₈

yes

characteristic

#500

Q₈

yes

characteristic

#499

—

#499

D₄

yes

—

#498

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

C₂×#108

—

C₂×#108

C₂×C₄

yes

—

#427

—

#427

C₂×C₄

yes

—

#424

—

#423

—

C₂×C₄⋊C₈

—

#405

—

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

characteristic

C₄³

D₄

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

—

—

C₂×M₁₆

—

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

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₄

yes

—

#107

C₄⋊C₄

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

D₈

yes

characteristic

103

C₄×C₈

D₈

yes

characteristic

104

C₄×C₈

SD₁₆

yes

—

105

C₄×C₈

SD₁₆

yes

characteristic

106

C₄×C₈

SD₁₆

yes

characteristic

107

C₄×C₈

Q₁₆

yes

—

108

C₄×C₈

C₂×D₄

yes

characteristic

109

C₄×C₈

C₂×Q₈

yes

characteristic

110

C₄×C₈

#41

yes

characteristic

111

C₄×C₈

#41

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

117

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

118

C₂²×C₄

#137

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

#101

yes

—

129

C₂×C₈

#102

yes

characteristic

130

C₂×C₈

#102

yes

characteristic

131

C₂×C₈

#102

yes

characteristic

132

C₂×C₈

#102

yes

characteristic

133

C₂×C₈

#103

yes

—

134

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

135

C₂×C₈

#117

yes

characteristic

136

C₂×C₈

#121

yes

characteristic

137

C₂×C₈

C₂²⋊Q₈

yes

characteristic

138

C₂×C₈

C₂×SD₁₆

yes

characteristic

139

C₂×C₈

C₂×Q₁₆

yes

characteristic

140

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

141

C₂×C₈

#137

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

#103

yes

—

156

C₄²

#121

yes

characteristic

157

C₄²

C₂²⋊Q₈

yes

characteristic

158

C₄²

C₂×D₈

yes

characteristic

159

C₄²

C₂×SD₁₆

yes

characteristic

160

C₁₆

—

161

C₂³

#417

yes

characteristic

162

Q₈

—

163

Q₈

—

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

#380

yes

characteristic

175

C₂×C₄

C₂×#101

yes

characteristic

176

C₂×C₄

#413

yes

characteristic

177

C₂×C₄

C₂×#102

yes

characteristic

178

C₂×C₄

C₂×#103

yes

characteristic

179

C₂×C₄

#417

yes

characteristic

180

C₂×C₄

#459

yes

characteristic

181

C₂×C₄

#461

yes

characteristic

182

C₂×C₄

#474

yes

characteristic

183

C₂×C₄

#476

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

#332

yes

—

191

C₈

#332

yes

—

192

C₈

#460

yes

characteristic

193

C₈

#462

yes

characteristic

194

C₈

#475

yes

characteristic

195

C₈

#477

yes

characteristic

196

C₂²

—

197

C₂²

—

198

C₂²

#1550

yes

characteristic

199

C₂²

#1561

yes

characteristic

200

C₂²

#1802

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

C₂×#332

yes

characteristic

209

C₄

#1547

yes

characteristic

210

C₄

#1801

yes

characteristic

211

C₄

#1803

yes

characteristic

212

C₄

#2231

yes

—

213

C₂

—

214

C₂

—

215

C₂

#8586

yes

characteristic

216

C₂

#12822

yes

characteristic

217

C₂

#12824

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#417
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1802

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#417

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#21732	2
#12900	2
#11936	2
#8807	2
#7280	2

subgroup

index

Subgroups — #134593

Conjugacy classes of subgroups

Maximal subgroups