Group #140577 — subgroups

Subgroups — #140577

Order 512.

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

normal subgroups only

This group has 424 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	C₄×#494	C₂	1	1	yes	maximal, characteristic
2	#12591	C₂	1	1	yes	maximal, characteristic
3	C₂×#1334	C₂	1	4	yes	maximal
4	#8805	C₂	1	1	yes	maximal, characteristic
5	C₂×#494	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₁₆	—	2	1	no	—
7	C₂×C₄×Q₁₆	C₂²	1	2	yes	—
8	C₄×C₄⋊Q₈	—	2	1	no	—
9	C₂×#346	C₂²	1	2	yes	—
10	#1969	—	2	1	no	—
11	#1969	C₄	1	2	yes	—
12	#1956	—	2	1	no	—
13	#1845	C₄	1	1	yes	characteristic
14	#1842	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#334	—	2	1	no	—
17	C₂×#334	C₂²	1	2	yes	—
18	#1374	—	2	1	no	—
19	#1374	—	2	1	no	—
20	#1334	—	2	8	no	—
21	C₂²×Q₁₆	—	2	1	no	—
22	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×C₄×Q₈	—	2	2	no	—
25	C₂²×C₁₆	—	2	2	no	—
26	#494	—	2	4	no	—
27	C₄×Q₁₆	—	2	4	no	—
28	C₄×Q₁₆	—	2	4	no	—
29	C₄×Q₁₆	—	4	2	no	—
30	C₄×Q₁₆	—	4	2	no	—
31	C₂×#107	—	2	1	no	—
32	C₂×#107	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	—	4	2	no	—
35	C₂×#103	—	2	2	no	—
36	C₂×#103	—	2	2	no	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#391	—	2	2	no	—
44	#389	—	4	2	no	—
45	#384	—	2	1	no	—
46	C₄×C₄⋊C₄	—	4	2	no	—
47	C₄³	D₄	1	1	yes	characteristic
48	#346	—	2	2	no	—
49	#346	D₄	1	4	yes	—
50	#334	—	2	4	no	—
51	#334	—	4	2	no	—
52	C₂²×Q₈	—	2	2	no	—
53	C₂×Q₁₆	—	2	4	no	—
54	C₂×Q₁₆	—	2	4	no	—
55	C₂×Q₁₆	—	2	2	no	—
56	C₂×Q₁₆	—	2	4	no	—
57	C₂×Q₁₆	—	4	2	no	—
58	C₂×Q₁₆	—	4	2	no	—
59	C₂×Q₁₆	M₁₆	1	4	yes	—
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	4	no	—
66	C₄⋊Q₈	—	4	2	no	—
67	C₄×Q₈	—	4	2	no	—
68	C₄×Q₈	—	4	2	no	—
69	C₄×Q₈	—	4	2	no	—
70	C₄×Q₈	—	4	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	4	2	no	—
73	C₂×C₄⋊C₄	—	4	2	no	—
74	C₂×C₄⋊C₄	—	4	2	no	—
75	C₂×C₄²	—	4	2	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	—	4	2	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄⋊C₈	—	8	4	no	—
89	#103	—	4	4	no	—
90	#103	—	4	4	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	4	1	no	—
94	C₄×C₈	—	4	1	no	—
95	C₄×C₈	D₈	1	2	yes	—
96	C₄×C₈	SD₁₆	1	2	yes	—
97	C₄×C₈	C₂×D₄	1	2	yes	—
98	C₄×C₈	#41	1	2	yes	—
99	#95	—	4	4	no	—
100	C₂×Q₈	—	2	4	no	—
101	C₂×Q₈	—	4	2	no	—
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	4	2	no	—
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	—	4	2	no	—
108	C₂²×C₄	#102	1	2	yes	—
109	C₂²×C₄	#102	1	2	yes	—
110	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	4	8	no	—
115	Q₁₆	—	4	4	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
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	4	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	#102	1	2	yes	—
125	C₂×C₈	#102	1	2	yes	—
126	C₂×C₈	C₄×D₄	1	2	yes	—
127	C₂×C₈	#121	1	2	yes	—
128	C₂×C₈	C₂×M₁₆	1	2	yes	—
129	C₂×C₈	#131	1	2	yes	—
130	C₄⋊C₄	—	4	8	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	8	4	no	—
134	C₄⋊C₄	—	8	2	no	—
135	C₄⋊C₄	—	8	2	no	—
136	C₄²	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	8	2	no	—
143	C₄²	—	8	2	no	—
144	C₄²	#124	1	1	yes	characteristic
145	C₄²	#127	1	1	yes	characteristic
146	C₄²	C₂×D₈	1	1	yes	characteristic
147	C₄²	C₂×SD₁₆	1	1	yes	characteristic
148	C₁₆	—	4	4	no	—
149	C₂³	C₂×#102	1	1	yes	characteristic
150	Q₈	—	4	4	no	—
151	Q₈	—	4	4	no	—
152	Q₈	—	8	2	no	—
153	Q₈	—	8	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	#360	1	4	yes	—
166	C₂×C₄	#362	1	4	yes	—
167	C₂×C₄	#390	1	1	yes	characteristic
168	C₂×C₄	#407	1	1	yes	characteristic
169	C₂×C₄	#408	1	1	yes	characteristic
170	C₂×C₄	C₂×#102	1	1	yes	characteristic
171	C₂×C₄	#421	1	1	yes	characteristic
172	C₂×C₄	#421	1	1	yes	characteristic
173	C₂×C₄	#486	1	1	yes	characteristic
174	C₂×C₄	#486	1	1	yes	characteristic
175	C₂×C₄	#492	1	1	yes	characteristic
176	C₂×C₄	#493	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₈	—	16	2	no	—
181	C₈	#435	1	4	yes	—
182	C₂²	#1480	1	1	yes	characteristic
183	C₂²	#1486	1	1	yes	characteristic
184	C₂²	#1957	1	1	yes	characteristic
185	C₂²	C₂×#360	1	2	yes	—
186	C₂²	C₂×#362	1	2	yes	—
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	8	4	no	—
194	C₄	—	16	2	no	—
195	C₄	#1480	1	1	yes	characteristic
196	C₄	#1486	1	1	yes	characteristic
197	C₄	#1955	1	1	yes	characteristic
198	C₄	#1963	1	1	yes	characteristic
199	C₂	#9644	1	1	yes	characteristic
200	C₂	#11655	1	2	yes	—
201	C₂	#11664	1	2	yes	—
202	C₂	#13210	1	1	yes	characteristic
203	C₂	#13212	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#494

C₂

yes

maximal, characteristic

#12591

C₂

yes

maximal, characteristic

C₂×#1334

C₂

yes

maximal

#8805

C₂

yes

maximal, characteristic

C₂×#494

C₄

yes

characteristic

C₂×C₄×Q₁₆

—

C₂×C₄×Q₁₆

C₂²

yes

—

C₄×C₄⋊Q₈

—

C₂×#346

C₂²

yes

—

#1969

—

#1969

C₄

yes

—

#1956

—

#1845

C₄

yes

characteristic

#1842

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

—

C₂×#334

C₂²

yes

—

#1374

—

#1374

—

#1334

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×C₄

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂²×C₁₆

—

#494

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#107

—

C₂×#107

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

—

—

—

C₄×C₄⋊C₄

—

C₄³

D₄

yes

characteristic

#346

—

#346

D₄

yes

—

#334

—

#334

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

M₁₆

yes

—

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

—

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

—

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

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

#102

yes

—

109

C₂²×C₄

#102

yes

—

110

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

111

C₂²×C₄

C₂×D₈

yes

characteristic

112

C₂²×C₄

C₂×SD₁₆

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

115

Q₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#102

yes

—

125

C₂×C₈

#102

yes

—

126

C₂×C₈

C₄×D₄

yes

—

127

C₂×C₈

#121

yes

—

128

C₂×C₈

C₂×M₁₆

yes

—

129

C₂×C₈

#131

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

#124

yes

characteristic

145

C₄²

#127

yes

characteristic

146

C₄²

C₂×D₈

yes

characteristic

147

C₄²

C₂×SD₁₆

yes

characteristic

148

C₁₆

—

149

C₂³

C₂×#102

yes

characteristic

150

Q₈

—

151

Q₈

—

152

Q₈

—

153

Q₈

—

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

—

165

C₂×C₄

#360

yes

—

166

C₂×C₄

#362

yes

—

167

C₂×C₄

#390

yes

characteristic

168

C₂×C₄

#407

yes

characteristic

169

C₂×C₄

#408

yes

characteristic

170

C₂×C₄

C₂×#102

yes

characteristic

171

C₂×C₄

#421

yes

characteristic

172

C₂×C₄

#421

yes

characteristic

173

C₂×C₄

#486

yes

characteristic

174

C₂×C₄

#486

yes

characteristic

175

C₂×C₄

#492

yes

characteristic

176

C₂×C₄

#493

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

#435

yes

—

182

C₂²

#1480

yes

characteristic

183

C₂²

#1486

yes

characteristic

184

C₂²

#1957

yes

characteristic

185

C₂²

C₂×#360

yes

—

186

C₂²

C₂×#362

yes

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

#1480

yes

characteristic

196

C₄

#1486

yes

characteristic

197

C₄

#1955

yes

characteristic

198

C₄

#1963

yes

characteristic

199

C₂

#9644

yes

characteristic

200

C₂

#11655

yes

—

201

C₂

#11664

yes

—

202

C₂

#13210

yes

characteristic

203

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 ▷ 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 ▷ 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
C₄×#494	2
#12591	2
C₂×#1334	2
#8805	2

subgroup

index

Subgroups — #140577

Conjugacy classes of subgroups

Maximal subgroups