Group #102115 — subgroups

Subgroups — #102115

Order 512.

474 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 150 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	#9870	C₂	1	2	yes	maximal
3	#9870	C₂	1	2	yes	maximal
4	#8784	C₂	1	1	yes	maximal, characteristic
5	#8131	C₂	1	1	yes	maximal, characteristic
6	C₂×#494	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₁₆	—	2	2	no	—
8	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
9	#1969	—	2	2	no	—
10	#1956	C₄	1	2	yes	—
11	#1845	C₄	1	1	yes	characteristic
12	#1842	C₂²	1	2	yes	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	#1279	—	2	4	no	—
17	#1279	—	2	4	no	—
18	#1277	—	2	2	no	—
19	#1273	—	2	2	no	—
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×Q₈	—	2	2	no	—
23	#494	—	2	2	no	—
24	#494	C₈	1	4	yes	complemented
25	C₄×Q₁₆	—	4	8	no	—
26	C₂×#107	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×#103	—	2	4	no	—
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	—	2	1	no	—
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	2	yes	—
38	C₂×C₄×C₈	D₄	1	2	yes	—
39	C₂×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	D₄	1	2	yes	—
41	#391	—	2	2	no	—
42	#389	—	2	2	no	—
43	#389	—	2	2	no	—
44	#384	C₂×C₄	1	2	yes	—
45	C₄×C₄⋊C₄	—	2	2	no	—
46	C₄×C₄⋊C₄	—	2	2	no	—
47	C₄³	C₂³	1	1	yes	characteristic
48	C₂²×Q₈	—	2	2	no	—
49	C₂×Q₁₆	—	2	8	no	—
50	C₂×Q₁₆	—	4	4	no	—
51	C₂×Q₁₆	—	4	4	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	—	2	2	no	—
54	C₂²×C₈	—	4	2	no	—
55	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
57	C₄⋊Q₈	—	2	2	no	—
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	—	2	2	no	—
60	C₄⋊Q₈	C₂×C₈	1	4	yes	—
61	C₄×Q₈	—	4	8	no	—
62	C₂×C₄⋊C₄	—	2	2	no	—
63	C₂×C₄⋊C₄	—	2	2	no	—
64	C₂×C₄⋊C₄	—	2	2	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄²	—	4	2	no	—
69	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	#41	1	1	yes	characteristic
75	C₂×C₄²	#41	1	1	yes	characteristic
76	#107	—	4	4	no	—
77	C₄⋊C₈	—	4	4	no	—
78	C₄⋊C₈	—	4	4	no	—
79	C₄⋊C₈	—	4	4	no	—
80	#103	—	4	8	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	—	4	2	no	—
88	C₄×C₈	—	4	2	no	—
89	C₄×C₈	—	4	2	no	—
90	C₄×C₈	—	4	2	no	—
91	C₄×C₈	C₂×C₈	1	2	yes	—
92	C₄×C₈	M₁₆	1	2	yes	—
93	C₄×C₈	Q₁₆	1	4	yes	complemented
94	#95	—	4	2	no	—
95	#95	—	4	2	no	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	—	4	4	no	—
99	C₂×Q₈	—	4	4	no	—
100	C₂²×C₄	—	4	2	no	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂²×C₄	#121	1	1	yes	characteristic
106	C₂²×C₄	#121	1	1	yes	characteristic
107	C₂²×C₄	#124	1	1	yes	characteristic
108	C₂²×C₄	#127	1	1	yes	characteristic
109	Q₁₆	—	4	16	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	8	2	no	—
122	C₂×C₈	—	8	2	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	C₂²⋊C₈	1	4	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₄⋊C₄	—	4	4	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₄⋊C₄	—	8	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	8	4	no	—
137	C₄²	C₂²×C₈	1	1	yes	characteristic
138	C₄²	C₂×M₁₆	1	1	yes	characteristic
139	C₄²	#131	1	1	yes	characteristic
140	C₄²	#131	1	1	yes	characteristic
141	C₄²	C₂×Q₁₆	1	2	yes	—
142	C₄²	#135	1	2	yes	—
143	C₄²	AGL(1,ℤ/8)	1	4	yes	—
144	C₄²	#137	1	4	yes	—
145	C₂³	#390	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₄	#408	1	1	yes	characteristic
162	C₂×C₄	#418	1	2	yes	—
163	C₂×C₄	#421	1	2	yes	—
164	C₂×C₄	C₈×D₄	1	1	yes	characteristic
165	C₂×C₄	C₈×D₄	1	1	yes	characteristic
166	C₂×C₄	#436	1	1	yes	characteristic
167	C₂×C₄	#436	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₁₆	1	2	yes	—
169	C₂×C₄	#441	1	2	yes	—
170	C₂×C₄	#443	1	2	yes	—
171	C₂×C₄	#444	1	2	yes	—
172	C₂×C₄	#467	1	2	yes	—
173	C₂×C₄	#470	1	2	yes	—
174	C₂×C₄	#490	1	2	yes	—
175	C₂×C₄	#494	1	1	yes	characteristic
176	C₂×C₄	#495	1	1	yes	characteristic
177	C₂×C₄	#496	1	2	yes	—
178	C₂×C₄	#497	1	2	yes	—
179	C₈	—	2	4	no	—
180	C₈	—	4	4	no	—
181	C₈	—	4	4	no	—
182	C₈	—	4	2	no	—
183	C₈	—	16	2	no	—
184	C₂²	#1954	1	1	yes	characteristic
185	C₂²	#1961	1	1	yes	characteristic
186	C₂²	#1963	1	1	yes	characteristic
187	C₂²	#1969	1	1	yes	characteristic
188	C₂²	#1971	1	1	yes	characteristic
189	C₂²	#1974	1	1	yes	characteristic
190	C₂²	#1975	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	8	4	no	—
195	C₄	—	16	2	no	—
196	C₄	#1478	1	2	yes	—
197	C₄	#1496	1	2	yes	—
198	C₄	C₈×Q₁₆	1	2	yes	—
199	C₄	#1579	1	2	yes	—
200	C₄	#1591	1	2	yes	—
201	C₄	#1593	1	2	yes	—
202	C₄	#1720	1	2	yes	—
203	C₄	#1723	1	2	yes	—
204	C₂	#9642	1	1	yes	characteristic
205	C₂	#9656	1	1	yes	characteristic
206	C₂	#9658	1	1	yes	characteristic
207	C₂	#9835	1	1	yes	characteristic
208	C₂	#9848	1	1	yes	characteristic
209	C₂	#9853	1	1	yes	characteristic
210	C₂	#9860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#494

C₂

yes

maximal, characteristic

#9870

C₂

yes

maximal

#9870

C₂

yes

maximal

#8784

C₂

yes

maximal, characteristic

#8131

C₂

yes

maximal, characteristic

C₂×#494

C₄

yes

characteristic

C₂×C₄×Q₁₆

—

C₄×C₄⋊Q₈

C₂²

yes

—

#1969

—

#1956

C₄

yes

—

#1845

C₄

yes

characteristic

#1842

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄×Q₈

—

#494

—

#494

C₈

yes

complemented

C₄×Q₁₆

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

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

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂³

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₈

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

C₂×C₄²

yes

C₂×C₄²

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

Q₁₆

yes

complemented

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

103

C₂²×C₄

C₄×D₄

yes

characteristic

104

C₂²×C₄

C₄×D₄

yes

characteristic

105

C₂²×C₄

#121

yes

characteristic

106

C₂²×C₄

#121

yes

characteristic

107

C₂²×C₄

#124

yes

characteristic

108

C₂²×C₄

#127

yes

characteristic

109

Q₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

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

C₂²⋊C₈

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²×C₈

yes

characteristic

138

C₄²

C₂×M₁₆

yes

characteristic

139

C₄²

#131

yes

characteristic

140

C₄²

#131

yes

characteristic

141

C₄²

C₂×Q₁₆

yes

—

142

C₄²

#135

yes

—

143

C₄²

AGL(1,ℤ/8)

yes

—

144

C₄²

#137

yes

—

145

C₂³

#390

yes

characteristic

146

Q₈

—

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

161

C₂×C₄

#408

yes

characteristic

162

C₂×C₄

#418

yes

—

163

C₂×C₄

#421

yes

—

164

C₂×C₄

C₈×D₄

yes

characteristic

165

C₂×C₄

C₈×D₄

yes

characteristic

166

C₂×C₄

#436

yes

characteristic

167

C₂×C₄

#436

yes

characteristic

168

C₂×C₄

C₄×Q₁₆

yes

—

169

C₂×C₄

#441

yes

—

170

C₂×C₄

#443

yes

—

171

C₂×C₄

#444

yes

—

172

C₂×C₄

#467

yes

—

173

C₂×C₄

#470

yes

—

174

C₂×C₄

#490

yes

—

175

C₂×C₄

#494

yes

characteristic

176

C₂×C₄

#495

yes

characteristic

177

C₂×C₄

#496

yes

—

178

C₂×C₄

#497

yes

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₂²

#1954

yes

characteristic

185

C₂²

#1961

yes

characteristic

186

C₂²

#1963

yes

characteristic

187

C₂²

#1969

yes

characteristic

188

C₂²

#1971

yes

characteristic

189

C₂²

#1974

yes

characteristic

190

C₂²

#1975

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#1478

yes

—

197

C₄

#1496

yes

—

198

C₄

C₈×Q₁₆

yes

—

199

C₄

#1579

yes

—

200

C₄

#1591

yes

—

201

C₄

#1593

yes

—

202

C₄

#1720

yes

—

203

C₄

#1723

yes

—

204

C₂

#9642

yes

characteristic

205

C₂

#9656

yes

characteristic

206

C₂

#9658

yes

characteristic

207

C₂

#9835

yes

characteristic

208

C₂

#9848

yes

characteristic

209

C₂

#9853

yes

characteristic

210

C₂

#9860

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
24. #494	C₈	semidirect
93. C₄×C₈	Q₁₆	semidirect

H ≅ G/N

type

24. #494

C₈

semidirect

93. C₄×C₈

Q₁₆

semidirect

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

subgroup	index
C₄×#494	2
#9870	2
#8784	2
#8131	2

subgroup

index

Subgroups — #102115

Conjugacy classes of subgroups

Complements

Maximal subgroups