Group #141059 — subgroups

Subgroups — #141059

Order 512.

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

normal subgroups only

This group has 382 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	#33242	C₂	1	1	yes	maximal, characteristic
2	#12653	C₂	1	1	yes	maximal, characteristic
3	C₂×#1364	C₂	1	2	yes	maximal
4	C₂×#1362	C₂	1	2	yes	maximal
5	#8806	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₂×#494	C₂²	1	1	yes	characteristic
8	C₂×#487	—	2	1	no	—
9	#2862	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#346	—	2	1	no	—
12	#1969	—	2	1	no	—
13	#1969	—	2	1	no	—
14	#1969	C₄	1	2	yes	—
15	#1969	C₄	1	2	yes	—
16	#1842	—	2	1	no	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#335	—	2	1	no	—
19	C₂×#335	C₂²	1	2	yes	—
20	#1374	—	2	1	no	—
21	#1364	—	2	4	no	—
22	#1362	—	2	4	no	—
23	C₂²×Q₁₆	—	2	1	no	—
24	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊Q₈	—	2	1	no	—
26	C₂×C₄⋊Q₈	—	2	1	no	—
27	C₂²×C₁₆	—	2	2	no	—
28	#500	—	2	2	no	—
29	#500	—	2	2	no	—
30	#494	—	2	2	no	—
31	#494	—	2	2	no	—
32	#487	—	4	2	no	—
33	#487	—	4	2	no	—
34	C₂×#107	—	2	1	no	—
35	C₂×#107	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×C₄⋊C₈	—	4	2	no	—
38	C₂×#103	—	2	2	no	—
39	C₂×#103	—	2	2	no	—
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	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	C₂³	1	1	yes	characteristic
46	#391	—	2	2	no	—
47	#391	—	2	2	no	—
48	#391	—	4	2	no	—
49	#382	—	4	2	no	—
50	C₄³	D₄	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	2	no	—
52	C₄⋊C₁₆	D₄	1	4	yes	—
53	#346	—	4	2	no	—
54	#335	—	2	2	no	—
55	#335	—	2	2	no	—
56	#335	—	4	2	no	—
57	C₂²×Q₈	—	2	2	no	—
58	C₂²×Q₈	—	4	1	no	—
59	C₂×Q₁₆	—	2	4	no	—
60	C₂×Q₁₆	—	4	2	no	—
61	C₂×Q₁₆	—	4	2	no	—
62	C₂×Q₁₆	—	4	2	no	—
63	C₂×Q₁₆	—	4	2	no	—
64	C₂×Q₁₆	—	4	1	no	—
65	C₂×Q₁₆	—	4	1	no	—
66	C₂²×C₈	—	4	2	no	—
67	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₄⋊Q₈	—	4	1	no	—
72	C₄⋊Q₈	—	4	1	no	—
73	C₄⋊Q₈	—	4	1	no	—
74	C₄⋊Q₈	—	4	1	no	—
75	C₄⋊Q₈	—	4	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	4	1	no	—
78	C₄⋊Q₈	—	4	1	no	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄⋊C₄	—	4	1	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄²	D₈	1	2	yes	—
86	C₂×C₄²	SD₁₆	1	2	yes	—
87	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₁₆	—	2	4	no	—
89	C₂×C₁₆	—	4	2	no	—
90	C₂×C₁₆	—	4	2	no	—
91	#107	—	2	4	no	—
92	#107	—	4	2	no	—
93	C₄⋊C₈	—	4	2	no	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄⋊C₈	—	8	4	no	—
96	#103	—	4	4	no	—
97	#103	—	4	4	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	D₈	1	2	yes	—
104	C₄×C₈	SD₁₆	1	2	yes	—
105	C₄×C₈	C₂×D₄	1	2	yes	—
106	C₄×C₈	#41	1	2	yes	—
107	#95	—	4	4	no	—
108	#95	—	4	4	no	—
109	C₂×Q₈	—	4	2	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	4	2	no	—
115	C₂×Q₈	—	8	1	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂×Q₈	—	8	2	no	—
118	C₂×Q₈	—	8	2	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#102	1	2	yes	—
122	C₂²×C₄	#102	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
125	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
126	Q₁₆	—	8	4	no	—
127	Q₁₆	—	8	2	no	—
128	Q₁₆	—	8	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	8	2	no	—
136	C₂×C₈	—	8	4	no	—
137	C₂×C₈	#102	1	2	yes	—
138	C₂×C₈	#102	1	2	yes	—
139	C₂×C₈	C₄×D₄	1	2	yes	—
140	C₂×C₈	#121	1	2	yes	—
141	C₂×C₈	#135	1	2	yes	—
142	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₈	#137	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	8	1	no	—
147	C₄⋊C₄	—	8	1	no	—
148	C₄⋊C₄	—	8	1	no	—
149	C₄⋊C₄	—	8	1	no	—
150	C₄⋊C₄	—	8	1	no	—
151	C₄⋊C₄	—	8	1	no	—
152	C₄²	—	2	4	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	#124	1	1	yes	characteristic
160	C₄²	#127	1	1	yes	characteristic
161	C₄²	C₂×D₈	1	1	yes	characteristic
162	C₄²	C₂×SD₁₆	1	1	yes	characteristic
163	C₁₆	—	4	4	no	—
164	C₂³	C₂×#102	1	1	yes	characteristic
165	Q₈	—	8	2	no	—
166	Q₈	—	8	2	no	—
167	Q₈	—	8	2	no	—
168	Q₈	—	8	2	no	—
169	Q₈	—	16	2	no	—
170	Q₈	—	16	1	no	—
171	Q₈	—	16	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	#362	1	4	yes	—
184	C₂×C₄	#390	1	1	yes	characteristic
185	C₂×C₄	#413	1	1	yes	characteristic
186	C₂×C₄	#413	1	1	yes	characteristic
187	C₂×C₄	C₂×#102	1	1	yes	characteristic
188	C₂×C₄	#416	1	1	yes	characteristic
189	C₂×C₄	#417	1	1	yes	characteristic
190	C₂×C₄	#486	1	1	yes	characteristic
191	C₂×C₄	#486	1	1	yes	characteristic
192	C₂×C₄	#492	1	1	yes	characteristic
193	C₂×C₄	#493	1	1	yes	characteristic
194	C₂×C₄	#509	1	2	yes	—
195	C₂×C₄	#510	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	2	2	no	—
199	C₈	—	16	2	no	—
200	C₈	#463	1	2	yes	—
201	C₈	#464	1	2	yes	—
202	C₂²	#1540	1	1	yes	characteristic
203	C₂²	#1547	1	1	yes	characteristic
204	C₂²	#1957	1	1	yes	characteristic
205	C₂²	#2139	1	2	yes	—
206	C₂²	C₂×#362	1	2	yes	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	4	no	—
210	C₄	—	4	2	no	—
211	C₄	—	16	2	no	—
212	C₄	—	16	2	no	—
213	C₄	#1540	1	1	yes	characteristic
214	C₄	#1547	1	1	yes	characteristic
215	C₄	#1967	1	1	yes	characteristic
216	C₄	#1983	1	1	yes	characteristic
217	C₄	#2243	1	2	yes	—
218	C₄	#2250	1	1	yes	characteristic
219	C₄	#2251	1	1	yes	characteristic
220	C₂	#9698	1	1	yes	characteristic
221	C₂	#11954	1	2	yes	—
222	C₂	#11988	1	2	yes	—
223	C₂	#13205	1	1	yes	characteristic
224	C₂	#13212	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33242

C₂

yes

maximal, characteristic

#12653

C₂

yes

maximal, characteristic

C₂×#1364

C₂

yes

maximal

C₂×#1362

C₂

yes

maximal

#8806

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

C₂×#494

C₂²

yes

characteristic

C₂×#487

—

#2862

—

C₂×C₄⋊C₁₆

C₂²

yes

—

—

—

—

C₄

yes

—

#1969

C₄

yes

—

#1842

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#335

C₂²

yes

—

#1374

—

#1364

—

#1362

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×C₄

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×#103

—

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

—

—

—

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

D₄

yes

—

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

—

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

D₈

yes

—

104

C₄×C₈

SD₁₆

yes

—

105

C₄×C₈

C₂×D₄

yes

—

106

C₄×C₈

#41

yes

—

107

#95

—

108

#95

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

#102

yes

—

122

C₂²×C₄

#102

yes

—

123

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

124

C₂²×C₄

C₂×D₈

yes

characteristic

125

C₂²×C₄

C₂×SD₁₆

yes

characteristic

126

Q₁₆

—

127

Q₁₆

—

128

Q₁₆

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

#102

yes

—

138

C₂×C₈

#102

yes

—

139

C₂×C₈

C₄×D₄

yes

—

140

C₂×C₈

#121

yes

—

141

C₂×C₈

#135

yes

—

142

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

143

C₂×C₈

#137

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

#124

yes

characteristic

160

C₄²

#127

yes

characteristic

161

C₄²

C₂×D₈

yes

characteristic

162

C₄²

C₂×SD₁₆

yes

characteristic

163

C₁₆

—

164

C₂³

C₂×#102

yes

characteristic

165

Q₈

—

166

Q₈

—

167

Q₈

—

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#362

yes

—

184

C₂×C₄

#390

yes

characteristic

185

C₂×C₄

#413

yes

characteristic

186

C₂×C₄

#413

yes

characteristic

187

C₂×C₄

C₂×#102

yes

characteristic

188

C₂×C₄

#416

yes

characteristic

189

C₂×C₄

#417

yes

characteristic

190

C₂×C₄

#486

yes

characteristic

191

C₂×C₄

#486

yes

characteristic

192

C₂×C₄

#492

yes

characteristic

193

C₂×C₄

#493

yes

characteristic

194

C₂×C₄

#509

yes

—

195

C₂×C₄

#510

yes

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

#463

yes

—

201

C₈

#464

yes

—

202

C₂²

#1540

yes

characteristic

203

C₂²

#1547

yes

characteristic

204

C₂²

#1957

yes

characteristic

205

C₂²

#2139

yes

—

206

C₂²

C₂×#362

yes

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

#1540

yes

characteristic

214

C₄

#1547

yes

characteristic

215

C₄

#1967

yes

characteristic

216

C₄

#1983

yes

characteristic

217

C₄

#2243

yes

—

218

C₄

#2250

yes

characteristic

219

C₄

#2251

yes

characteristic

220

C₂

#9698

yes

characteristic

221

C₂

#11954

yes

—

222

C₂

#11988

yes

—

223

C₂

#13205

yes

characteristic

224

C₂

#13212

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 ▷ C₂×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 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33242	2
#12653	2
C₂×#1364	2
C₂×#1362	2
#8806	2

subgroup

index

Subgroups — #141059

Conjugacy classes of subgroups

Maximal subgroups