Group #102140 — subgroups

Subgroups — #102140

Order 512.

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

normal subgroups only

This group has 450 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×#500	C₂	1	1	yes	maximal, characteristic
2	#9870	C₂	1	2	yes	maximal
3	#9637	C₂	1	2	yes	maximal
4	#8805	C₂	1	1	yes	maximal, characteristic
5	#8131	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₄	1	1	yes	characteristic
7	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
8	#1990	C₄	1	2	yes	—
9	#1942	—	2	2	no	—
10	#1845	C₄	1	1	yes	characteristic
11	#1842	C₂²	1	2	yes	—
12	C₄×#107	—	2	2	no	—
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	C₂×#334	—	2	2	no	—
17	C₂×Q₈⋊C₈	—	2	4	no	—
18	#1279	—	2	4	no	—
19	#1273	—	2	2	no	—
20	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×Q₈	—	2	2	no	—
22	#500	—	2	4	no	—
23	C₂×#107	—	2	2	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×C₄⋊C₈	—	2	2	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×C₄×C₈	—	2	1	no	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	2	yes	—
36	C₂×C₄×C₈	D₄	1	2	yes	—
37	C₂×C₄×C₈	D₄	1	2	yes	—
38	#391	—	2	2	no	—
39	#389	—	2	2	no	—
40	#389	—	2	2	no	—
41	#384	C₂×C₄	1	2	yes	—
42	C₄×C₄⋊C₄	—	2	2	no	—
43	C₄×C₄⋊C₄	—	2	2	no	—
44	C₄³	C₂³	1	1	yes	characteristic
45	#340	—	2	4	no	—
46	#334	—	4	4	no	—
47	Q₈⋊C₈	—	4	8	no	—
48	C₂²×Q₈	—	2	2	no	—
49	C₂²×C₈	—	2	2	no	—
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	—	4	2	no	—
52	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
54	C₄⋊Q₈	—	2	2	no	—
55	C₄⋊Q₈	—	2	2	no	—
56	C₄⋊Q₈	—	2	2	no	—
57	C₄⋊Q₈	M₁₆	1	4	yes	—
58	C₄×Q₈	—	4	4	no	—
59	C₄×Q₈	—	4	4	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	2	2	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	—	4	2	no	—
66	C₂×C₄²	—	4	2	no	—
67	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₄	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	#41	1	1	yes	characteristic
73	C₂×C₄²	#41	1	1	yes	characteristic
74	#107	—	2	8	no	—
75	#107	—	4	4	no	—
76	C₄⋊C₈	—	4	4	no	—
77	C₄⋊C₈	—	4	4	no	—
78	C₄⋊C₈	—	4	4	no	—
79	C₄×C₈	—	2	4	no	—
80	C₄×C₈	—	2	4	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	—	4	2	no	—
86	C₄×C₈	—	4	2	no	—
87	C₄×C₈	—	4	2	no	—
88	C₄×C₈	SD₁₆	1	4	yes	—
89	C₄×C₈	Q₁₆	1	4	yes	—
90	#95	—	4	2	no	—
91	#95	—	4	2	no	—
92	C₂×Q₈	—	2	4	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	4	4	no	—
95	C₂×Q₈	—	4	4	no	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
100	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
101	C₂²×C₄	#121	1	1	yes	characteristic
102	C₂²×C₄	#121	1	1	yes	characteristic
103	C₂²×C₄	#124	1	1	yes	characteristic
104	C₂²×C₄	#127	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	8	2	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	#103	1	4	yes	—
119	C₂×C₈	#103	1	4	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	4	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	8	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	8	2	no	—
132	C₄²	—	8	2	no	—
133	C₄²	C₂×M₁₆	1	2	yes	—
134	C₄²	#131	1	2	yes	—
135	C₄²	C₂×SD₁₆	1	2	yes	—
136	C₄²	C₂×Q₁₆	1	2	yes	—
137	C₄²	AGL(1,ℤ/8)	1	4	yes	—
138	C₄²	#137	1	4	yes	—
139	C₂³	#390	1	1	yes	characteristic
140	Q₈	—	4	8	no	—
141	Q₈	—	8	4	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	#407	1	1	yes	characteristic
155	C₂×C₄	#408	1	1	yes	characteristic
156	C₂×C₄	C₂×#103	1	2	yes	—
157	C₂×C₄	#421	1	2	yes	—
158	C₂×C₄	#435	1	2	yes	—
159	C₂×C₄	#435	1	2	yes	—
160	C₂×C₄	#441	1	2	yes	—
161	C₂×C₄	#442	1	2	yes	—
162	C₂×C₄	#443	1	2	yes	—
163	C₂×C₄	#443	1	2	yes	—
164	C₂×C₄	#469	1	2	yes	—
165	C₂×C₄	#470	1	2	yes	—
166	C₂×C₄	#487	1	2	yes	—
167	C₂×C₄	#492	1	1	yes	characteristic
168	C₂×C₄	#494	1	1	yes	characteristic
169	C₂×C₄	#496	1	2	yes	—
170	C₂×C₄	#497	1	2	yes	—
171	C₈	—	2	4	no	—
172	C₈	—	2	4	no	—
173	C₈	—	4	4	no	—
174	C₈	—	4	4	no	—
175	C₈	—	16	2	no	—
176	C₂²	#1955	1	1	yes	characteristic
177	C₂²	#1956	1	1	yes	characteristic
178	C₂²	#1963	1	1	yes	characteristic
179	C₂²	#1971	1	1	yes	characteristic
180	C₂²	#1972	1	1	yes	characteristic
181	C₂²	#1974	1	1	yes	characteristic
182	C₂²	#1974	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	8	4	no	—
187	C₄	—	16	2	no	—
188	C₄	#1481	1	2	yes	—
189	C₄	#1487	1	2	yes	—
190	C₄	#1586	1	4	yes	—
191	C₄	#1587	1	4	yes	—
192	C₄	#1714	1	2	yes	—
193	C₄	#1723	1	2	yes	—
194	C₂	#9643	1	1	yes	characteristic
195	C₂	#9664	1	1	yes	characteristic
196	C₂	#9665	1	1	yes	characteristic
197	C₂	#9836	1	1	yes	characteristic
198	C₂	#9852	1	1	yes	characteristic
199	C₂	#9853	1	1	yes	characteristic
200	C₂	#9855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#500

C₂

yes

maximal, characteristic

#9870

C₂

yes

maximal

#9637

C₂

yes

maximal

#8805

C₂

yes

maximal, characteristic

#8131

C₂

yes

maximal, characteristic

C₂×#500

C₄

yes

characteristic

C₄×C₄⋊Q₈

C₂²

yes

—

#1990

C₄

yes

—

#1942

—

#1845

C₄

yes

characteristic

#1842

C₂²

yes

—

C₄×#107

—

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

—

C₂×Q₈⋊C₈

—

#1279

—

#1273

—

C₂×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄×Q₈

—

#500

—

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

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

#340

—

#334

—

Q₈⋊C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

M₁₆

yes

—

C₄×Q₈

—

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

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄×D₄

yes

characteristic

100

C₂²×C₄

C₄×D₄

yes

characteristic

101

C₂²×C₄

#121

yes

characteristic

102

C₂²×C₄

#121

yes

characteristic

103

C₂²×C₄

#124

yes

characteristic

104

C₂²×C₄

#127

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

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

#103

yes

—

119

C₂×C₈

#103

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂×M₁₆

yes

—

134

C₄²

#131

yes

—

135

C₄²

C₂×SD₁₆

yes

—

136

C₄²

C₂×Q₁₆

yes

—

137

C₄²

AGL(1,ℤ/8)

yes

—

138

C₄²

#137

yes

—

139

C₂³

#390

yes

characteristic

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#407

yes

characteristic

155

C₂×C₄

#408

yes

characteristic

156

C₂×C₄

C₂×#103

yes

—

157

C₂×C₄

#421

yes

—

158

C₂×C₄

#435

yes

—

159

C₂×C₄

#435

yes

—

160

C₂×C₄

#441

yes

—

161

C₂×C₄

#442

yes

—

162

C₂×C₄

#443

yes

—

163

C₂×C₄

#443

yes

—

164

C₂×C₄

#469

yes

—

165

C₂×C₄

#470

yes

—

166

C₂×C₄

#487

yes

—

167

C₂×C₄

#492

yes

characteristic

168

C₂×C₄

#494

yes

characteristic

169

C₂×C₄

#496

yes

—

170

C₂×C₄

#497

yes

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₂²

#1955

yes

characteristic

177

C₂²

#1956

yes

characteristic

178

C₂²

#1963

yes

characteristic

179

C₂²

#1971

yes

characteristic

180

C₂²

#1972

yes

characteristic

181

C₂²

#1974

yes

characteristic

182

C₂²

#1974

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

#1481

yes

—

189

C₄

#1487

yes

—

190

C₄

#1586

yes

—

191

C₄

#1587

yes

—

192

C₄

#1714

yes

—

193

C₄

#1723

yes

—

194

C₂

#9643

yes

characteristic

195

C₂

#9664

yes

characteristic

196

C₂

#9665

yes

characteristic

197

C₂

#9836

yes

characteristic

198

C₂

#9852

yes

characteristic

199

C₂

#9853

yes

characteristic

200

C₂

#9855

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

socle, prime radical

C₂³

#390

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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

subgroup	index
C₄×#500	2
#9870	2
#9637	2
#8805	2
#8131	2

subgroup

index

Subgroups — #102140

Conjugacy classes of subgroups

Maximal subgroups