Group #134609 — subgroups

Subgroups — #134609

Order 512.

357 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 100 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	#21698	C₂	1	1	yes	maximal, characteristic
2	#12901	C₂	1	1	yes	maximal, characteristic
3	#12881	C₂	1	1	yes	maximal, characteristic
4	C₂×#1358	C₂	1	2	yes	maximal
5	#7271	C₂	1	2	yes	maximal
6	C₂×#494	C₂²	1	1	yes	characteristic
7	C₂×#441	—	2	1	no	—
8	C₂×C₄×Q₁₆	—	2	1	no	—
9	#2327	C₄	1	2	yes	—
10	C₂×#364	—	2	1	no	—
11	C₂×#364	C₂²	1	2	yes	—
12	C₂×#364	C₂²	1	2	yes	—
13	#2150	—	2	1	no	—
14	#1971	—	2	1	no	—
15	#1969	—	2	1	no	—
16	#1960	C₄	1	2	yes	—
17	#1846	C₂²	1	1	yes	characteristic
18	#1726	C₂²	1	1	yes	characteristic
19	#1377	—	2	1	no	—
20	#1375	—	2	1	no	—
21	#1358	—	2	4	no	—
22	C₂²×Q₁₆	—	2	1	no	—
23	C₂²×Q₁₆	—	2	1	no	—
24	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	C₂×M₃₂	—	2	2	no	—
28	#494	—	2	1	no	—
29	#494	—	2	1	no	—
30	#494	D₄	1	2	yes	—
31	#494	Q₈	1	2	yes	—
32	#441	—	4	4	no	—
33	C₄×Q₁₆	—	4	2	no	—
34	C₄×Q₁₆	—	4	2	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	C₂×#103	—	2	2	no	—
38	C₂×#103	—	2	2	no	—
39	C₂×#97	C₂×C₄	1	1	yes	characteristic
40	C₂×#97	D₄	1	1	yes	characteristic
41	C₂×#97	D₄	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	C₂³	1	1	yes	characteristic
45	#391	—	2	2	no	—
46	#387	—	4	4	no	—
47	#382	C₂×C₄	1	2	yes	—
48	#376	C₂×C₄	1	1	yes	characteristic
49	#364	—	2	2	no	—
50	#364	—	2	2	no	—
51	#364	—	2	2	no	—
52	#364	—	2	2	no	—
53	#364	—	4	2	no	—
54	C₄⋊C₁₆	—	4	2	no	—
55	C₂²×Q₈	—	2	2	no	—
56	C₂×Q₁₆	—	2	2	no	—
57	C₂×Q₁₆	—	2	2	no	—
58	C₂×Q₁₆	—	2	2	no	—
59	C₂×Q₁₆	—	2	2	no	—
60	C₂×Q₁₆	—	4	1	no	—
61	C₂×Q₁₆	—	4	1	no	—
62	C₂×Q₁₆	—	4	2	no	—
63	C₂×Q₁₆	—	4	1	no	—
64	C₂×Q₁₆	—	4	2	no	—
65	C₂×Q₁₆	—	4	1	no	—
66	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	—	4	2	no	—
72	C₄⋊Q₈	C₄⋊C₄	1	4	yes	—
73	C₄×Q₈	—	4	4	no	—
74	C₄×Q₈	—	4	4	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	4	2	no	—
77	C₂×C₄⋊C₄	—	4	2	no	—
78	C₂×C₄²	—	4	2	no	—
79	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
82	M₃₂	—	4	2	no	—
83	M₃₂	—	4	2	no	—
84	M₃₂	—	4	2	no	—
85	M₃₂	—	4	2	no	—
86	C₂×C₁₆	—	4	2	no	—
87	C₂×C₁₆	—	4	2	no	—
88	#107	—	4	1	no	—
89	#107	—	4	1	no	—
90	#106	—	4	2	no	—
91	#103	—	4	4	no	—
92	#103	—	4	4	no	—
93	#97	—	2	2	no	—
94	#97	—	2	2	no	—
95	#97	—	4	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	D₈	1	2	yes	—
99	C₄×C₈	SD₁₆	1	2	yes	—
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	#95	—	2	2	no	—
105	#95	—	4	2	no	—
106	#95	—	4	4	no	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	—	4	2	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	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	#100	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	Q₁₆	—	4	2	no	—
120	Q₁₆	—	4	2	no	—
121	Q₁₆	—	4	4	no	—
122	Q₁₆	—	4	2	no	—
123	Q₁₆	—	4	2	no	—
124	Q₁₆	—	4	4	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	#102	1	2	yes	—
131	C₂×C₈	#121	1	1	yes	characteristic
132	C₂×C₈	#121	1	1	yes	characteristic
133	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₈	C₂×D₈	1	1	yes	characteristic
136	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₈	#135	1	2	yes	—
138	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₈	#137	1	1	yes	characteristic
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	8	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	8	4	no	—
151	C₄²	#102	1	2	yes	—
152	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₄²	#117	1	1	yes	characteristic
154	C₁₆	—	8	2	no	—
155	C₁₆	—	8	2	no	—
156	C₂³	C₂×#100	1	1	yes	characteristic
157	Q₈	—	4	4	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	Q₈	—	8	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	#380	1	1	yes	characteristic
174	C₂×C₄	#413	1	1	yes	characteristic
175	C₂×C₄	C₂×#102	1	1	yes	characteristic
176	C₂×C₄	#416	1	1	yes	characteristic
177	C₂×C₄	#417	1	1	yes	characteristic
178	C₂×C₄	#418	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	4	1	no	—
181	C₈	#459	1	1	yes	characteristic
182	C₈	#461	1	1	yes	characteristic
183	C₈	#463	1	2	yes	—
184	C₈	#474	1	1	yes	characteristic
185	C₈	#476	1	1	yes	characteristic
186	C₈	#478	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#1545	1	1	yes	characteristic
189	C₂²	#1551	1	1	yes	characteristic
190	C₂²	#1800	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	8	4	no	—
196	C₄	—	16	2	no	—
197	C₄	#1540	1	1	yes	characteristic
198	C₄	#1557	1	1	yes	characteristic
199	C₄	#1802	1	1	yes	characteristic
200	C₄	#1804	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#8588	1	1	yes	characteristic
204	C₂	#11990	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21698

C₂

yes

maximal, characteristic

#12901

C₂

yes

maximal, characteristic

#12881

C₂

yes

maximal, characteristic

C₂×#1358

C₂

yes

maximal

#7271

C₂

yes

maximal

C₂×#494

C₂²

yes

characteristic

C₂×#441

—

C₂×C₄×Q₁₆

—

#2327

C₄

yes

—

C₂×#364

—

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

—

—

—

C₄

yes

—

#1846

C₂²

yes

characteristic

#1726

C₂²

yes

—

—

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄×Q₈

—

C₂×M₃₂

—

C₂×M₃₂

—

#494

—

#494

—

#494

D₄

yes

—

#494

Q₈

yes

—

#441

—

C₄×Q₁₆

—

C₄×Q₁₆

—

—

—

—

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

C₂×C₄

yes

—

#376

C₂×C₄

yes

—

—

—

—

—

C₄⋊C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄⋊C₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

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

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

100

C₄×C₈

C₂×D₄

yes

characteristic

101

C₄×C₈

C₂×Q₈

yes

characteristic

102

C₄×C₈

#41

yes

characteristic

103

C₄×C₈

#41

yes

characteristic

104

#95

—

105

#95

—

106

#95

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

#100

yes

—

118

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

119

Q₁₆

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#102

yes

—

131

C₂×C₈

#121

yes

characteristic

132

C₂×C₈

#121

yes

characteristic

133

C₂×C₈

C₂²⋊Q₈

yes

characteristic

134

C₂×C₈

C₂²⋊Q₈

yes

characteristic

135

C₂×C₈

C₂×D₈

yes

characteristic

136

C₂×C₈

C₂×SD₁₆

yes

characteristic

137

C₂×C₈

#135

yes

—

138

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

139

C₂×C₈

AGL(1,ℤ/8)

yes

—

140

C₂×C₈

#137

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

#102

yes

—

152

C₄²

C₂×C₄⋊C₄

yes

characteristic

153

C₄²

#117

yes

characteristic

154

C₁₆

—

155

C₁₆

—

156

C₂³

C₂×#100

yes

characteristic

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

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

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#380

yes

characteristic

174

C₂×C₄

#413

yes

characteristic

175

C₂×C₄

C₂×#102

yes

characteristic

176

C₂×C₄

#416

yes

characteristic

177

C₂×C₄

#417

yes

characteristic

178

C₂×C₄

#418

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

#459

yes

characteristic

182

C₈

#461

yes

characteristic

183

C₈

#463

yes

—

184

C₈

#474

yes

characteristic

185

C₈

#476

yes

characteristic

186

C₈

#478

yes

—

187

C₂²

—

188

C₂²

#1545

yes

characteristic

189

C₂²

#1551

yes

characteristic

190

C₂²

#1800

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#1540

yes

characteristic

198

C₄

#1557

yes

characteristic

199

C₄

#1802

yes

characteristic

200

C₄

#1804

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#8588

yes

characteristic

204

C₂

#11990

yes

—

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1551

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ 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
#21698	2
#12901	2
#12881	2
C₂×#1358	2
#7271	2

subgroup

index

Subgroups — #134609

Conjugacy classes of subgroups

Maximal subgroups