Group #141010 — subgroups

Subgroups — #141010

Order 512.

366 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 138 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	#33313	C₂	1	1	yes	maximal, characteristic
2	#12653	C₂	1	1	yes	maximal, characteristic
3	C₂×#1365	C₂	1	2	yes	maximal
4	C₂×#1364	C₂	1	2	yes	maximal
5	#8806	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₂×#500	C₂²	1	1	yes	characteristic
8	C₂×#498	—	2	1	no	—
9	#2862	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#346	—	2	1	no	—
12	#1846	—	2	1	no	—
13	#1846	—	2	1	no	—
14	#1846	C₄	1	2	yes	—
15	#1846	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	#1365	—	2	4	no	—
22	#1364	—	2	4	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×C₄⋊Q₈	—	2	1	no	—
25	C₂²×C₁₆	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	—	2	2	no	—
28	#500	—	2	2	no	—
29	#500	—	2	2	no	—
30	#498	—	4	2	no	—
31	#498	—	4	2	no	—
32	C₂×#107	—	2	1	no	—
33	C₂×#107	—	2	1	no	—
34	C₂×#107	C₂×C₄	1	2	yes	—
35	C₂×#107	C₂×C₄	1	2	yes	—
36	C₂×#106	—	2	2	no	—
37	C₂×#106	—	2	2	no	—
38	C₂×C₄⋊C₈	—	2	2	no	—
39	C₂×C₄⋊C₈	—	4	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	—	4	2	no	—
47	#391	—	4	2	no	—
48	#382	—	2	2	no	—
49	#382	—	2	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	—	4	1	no	—
58	C₂²×Q₈	—	4	1	no	—
59	C₂²×C₈	—	4	2	no	—
60	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	1	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₄⋊Q₈	—	4	1	no	—
68	C₄⋊Q₈	—	4	1	no	—
69	C₄⋊Q₈	—	4	1	no	—
70	C₄⋊Q₈	—	4	1	no	—
71	C₄⋊Q₈	—	4	1	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	—	4	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄²	D₈	1	2	yes	—
79	C₂×C₄²	SD₁₆	1	2	yes	—
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	—	2	4	no	—
82	C₂×C₁₆	—	4	2	no	—
83	C₂×C₁₆	—	4	2	no	—
84	#107	—	2	4	no	—
85	#107	—	2	4	no	—
86	#107	—	4	2	no	—
87	#107	—	4	2	no	—
88	#106	—	4	4	no	—
89	#106	—	4	4	no	—
90	C₄⋊C₈	—	4	2	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	8	4	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	D₈	1	2	yes	—
97	C₄×C₈	SD₁₆	1	2	yes	—
98	C₄×C₈	SD₁₆	1	2	yes	—
99	C₄×C₈	Q₁₆	1	2	yes	—
100	C₄×C₈	C₂×D₄	1	2	yes	—
101	C₄×C₈	#41	1	2	yes	—
102	#95	—	4	4	no	—
103	#95	—	4	4	no	—
104	C₂×Q₈	—	8	4	no	—
105	C₂×Q₈	—	8	1	no	—
106	C₂×Q₈	—	8	1	no	—
107	C₂×Q₈	—	8	1	no	—
108	C₂×Q₈	—	8	1	no	—
109	C₂×Q₈	—	8	2	no	—
110	C₂×Q₈	—	8	2	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	—	4	2	no	—
113	C₂²×C₄	#102	1	2	yes	—
114	C₂²×C₄	#102	1	2	yes	—
115	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
117	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
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	—	8	2	no	—
123	C₂×C₈	—	8	4	no	—
124	C₂×C₈	#101	1	2	yes	—
125	C₂×C₈	#102	1	2	yes	—
126	C₂×C₈	#102	1	2	yes	—
127	C₂×C₈	#103	1	2	yes	—
128	C₂×C₈	C₄×D₄	1	2	yes	—
129	C₂×C₈	#121	1	2	yes	—
130	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₈	#137	1	2	yes	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	8	1	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	1	no	—
140	C₄⋊C₄	—	8	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#101	1	2	yes	—
147	C₄²	#103	1	2	yes	—
148	C₄²	#124	1	1	yes	characteristic
149	C₄²	#127	1	1	yes	characteristic
150	C₄²	C₂×D₈	1	1	yes	characteristic
151	C₄²	C₂×SD₁₆	1	1	yes	characteristic
152	C₁₆	—	4	4	no	—
153	C₂³	C₂×#102	1	1	yes	characteristic
154	Q₈	—	16	2	no	—
155	Q₈	—	16	2	no	—
156	Q₈	—	16	2	no	—
157	Q₈	—	16	1	no	—
158	Q₈	—	16	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	#332	1	4	yes	—
170	C₂×C₄	#390	1	1	yes	characteristic
171	C₂×C₄	C₂×#101	1	1	yes	characteristic
172	C₂×C₄	C₂×#101	1	1	yes	characteristic
173	C₂×C₄	C₂×#102	1	1	yes	characteristic
174	C₂×C₄	C₂×#103	1	1	yes	characteristic
175	C₂×C₄	#419	1	1	yes	characteristic
176	C₂×C₄	#486	1	1	yes	characteristic
177	C₂×C₄	#486	1	1	yes	characteristic
178	C₂×C₄	#492	1	1	yes	characteristic
179	C₂×C₄	#493	1	1	yes	characteristic
180	C₂×C₄	#509	1	2	yes	—
181	C₂×C₄	#510	1	2	yes	—
182	C₈	—	2	2	no	—
183	C₈	—	2	2	no	—
184	C₈	—	16	2	no	—
185	C₈	#332	1	4	yes	—
186	C₈	#461	1	2	yes	—
187	C₈	#462	1	2	yes	—
188	C₂²	C₂×#332	1	1	yes	characteristic
189	C₂²	#1546	1	1	yes	characteristic
190	C₂²	#1957	1	1	yes	characteristic
191	C₂²	#2139	1	2	yes	—
192	C₂²	#2148	1	2	yes	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	—	16	2	no	—
198	C₄	—	16	2	no	—
199	C₄	C₂×#332	1	1	yes	characteristic
200	C₄	#1546	1	1	yes	characteristic
201	C₄	#1966	1	1	yes	characteristic
202	C₄	#1981	1	1	yes	characteristic
203	C₄	#2243	1	2	yes	—
204	C₄	#2250	1	1	yes	characteristic
205	C₄	#2251	1	1	yes	characteristic
206	C₂	#9692	1	1	yes	characteristic
207	C₂	#11942	1	2	yes	—
208	C₂	#11976	1	2	yes	—
209	C₂	#13205	1	1	yes	characteristic
210	C₂	#13214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33313

C₂

yes

maximal, characteristic

#12653

C₂

yes

maximal, characteristic

C₂×#1365

C₂

yes

maximal

C₂×#1364

C₂

yes

maximal

#8806

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

C₂×#500

C₂²

yes

characteristic

C₂×#498

—

#2862

—

C₂×C₄⋊C₁₆

C₂²

yes

—

—

—

—

C₄

yes

—

#1846

C₄

yes

—

#1842

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#335

C₂²

yes

—

#1374

—

#1365

—

#1364

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

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

—

—

—

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

D₄

yes

—

—

—

—

—

C₂²×Q₈

—

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

—

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

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

100

C₄×C₈

C₂×D₄

yes

—

101

C₄×C₈

#41

yes

—

102

#95

—

103

#95

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

#102

yes

—

114

C₂²×C₄

#102

yes

—

115

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

116

C₂²×C₄

C₂×D₈

yes

characteristic

117

C₂²×C₄

C₂×SD₁₆

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#101

yes

—

125

C₂×C₈

#102

yes

—

126

C₂×C₈

#102

yes

—

127

C₂×C₈

#103

yes

—

128

C₂×C₈

C₄×D₄

yes

—

129

C₂×C₈

#121

yes

—

130

C₂×C₈

C₂×SD₁₆

yes

characteristic

131

C₂×C₈

C₂×Q₁₆

yes

characteristic

132

C₂×C₈

#137

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#101

yes

—

147

C₄²

#103

yes

—

148

C₄²

#124

yes

characteristic

149

C₄²

#127

yes

characteristic

150

C₄²

C₂×D₈

yes

characteristic

151

C₄²

C₂×SD₁₆

yes

characteristic

152

C₁₆

—

153

C₂³

C₂×#102

yes

characteristic

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

Q₈

—

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

#332

yes

—

170

C₂×C₄

#390

yes

characteristic

171

C₂×C₄

C₂×#101

yes

characteristic

172

C₂×C₄

C₂×#101

yes

characteristic

173

C₂×C₄

C₂×#102

yes

characteristic

174

C₂×C₄

C₂×#103

yes

characteristic

175

C₂×C₄

#419

yes

characteristic

176

C₂×C₄

#486

yes

characteristic

177

C₂×C₄

#486

yes

characteristic

178

C₂×C₄

#492

yes

characteristic

179

C₂×C₄

#493

yes

characteristic

180

C₂×C₄

#509

yes

—

181

C₂×C₄

#510

yes

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

#332

yes

—

186

C₈

#461

yes

—

187

C₈

#462

yes

—

188

C₂²

C₂×#332

yes

characteristic

189

C₂²

#1546

yes

characteristic

190

C₂²

#1957

yes

characteristic

191

C₂²

#2139

yes

—

192

C₂²

#2148

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

C₂×#332

yes

characteristic

200

C₄

#1546

yes

characteristic

201

C₄

#1966

yes

characteristic

202

C₄

#1981

yes

characteristic

203

C₄

#2243

yes

—

204

C₄

#2250

yes

characteristic

205

C₄

#2251

yes

characteristic

206

C₂

#9692

yes

characteristic

207

C₂

#11942

yes

—

208

C₂

#11976

yes

—

209

C₂

#13205

yes

characteristic

210

C₂

#13214

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
#33313	2
#12653	2
C₂×#1365	2
C₂×#1364	2
#8806	2

subgroup

index

Subgroups — #141010

Conjugacy classes of subgroups

Maximal subgroups