Group #142138 — subgroups

Subgroups — #142138

Order 512.

324 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 110 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	#33311	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	#8273	C₂	1	1	yes	maximal, characteristic
4	#7275	C₂	1	4	yes	maximal
5	C₂×#499	—	2	1	no	—
6	C₂×#499	C₄	1	2	yes	—
7	#2859	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	#1846	—	2	1	no	—
10	#1846	C₂²	1	2	yes	—
11	#1845	C₄	1	1	yes	characteristic
12	#1845	C₄	1	1	yes	characteristic
13	#1844	—	2	1	no	—
14	#1767	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#334	C₂²	1	2	yes	—
17	#1374	—	2	1	no	—
18	#1374	—	2	1	no	—
19	#1277	—	2	1	no	—
20	C₂×#125	—	2	2	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	#499	—	4	4	no	—
23	#499	—	4	4	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	—	2	1	no	—
26	C₂×#107	C₂×C₄	1	2	yes	—
27	C₂×#107	C₂×C₄	1	2	yes	—
28	C₂×#106	—	2	2	no	—
29	C₂×#106	—	2	2	no	—
30	C₂×C₄⋊C₈	—	2	2	no	—
31	C₂×#97	—	4	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	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#389	—	4	1	no	—
39	#389	—	4	1	no	—
40	#389	—	4	1	no	—
41	#389	—	4	1	no	—
42	#384	—	2	1	no	—
43	#384	—	2	1	no	—
44	#382	—	2	2	no	—
45	C₄³	D₄	1	1	yes	characteristic
46	C₄⋊C₁₆	—	2	4	no	—
47	#336	—	4	1	no	—
48	#334	—	2	2	no	—
49	#334	Q₈	1	4	yes	—
50	C₂²×C₈	—	4	2	no	—
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	#125	—	4	4	no	—
56	#125	—	4	4	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	1	no	—
62	C₂×C₄⋊C₄	—	4	1	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	D₈	1	1	yes	characteristic
68	C₂×C₄²	D₈	1	1	yes	characteristic
69	C₂×C₄²	SD₁₆	1	1	yes	characteristic
70	C₂×C₄²	SD₁₆	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₁₆	—	4	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₂×C₁₆	—	4	2	no	—
75	#107	—	4	2	no	—
76	#107	—	4	2	no	—
77	#107	—	4	2	no	—
78	#107	—	4	2	no	—
79	#106	—	4	4	no	—
80	#106	—	4	4	no	—
81	C₄⋊C₈	—	2	4	no	—
82	C₄⋊C₈	—	4	2	no	—
83	#97	—	8	2	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	C₂×Q₈	1	2	yes	—
90	C₄×C₈	#41	1	2	yes	—
91	#95	—	4	2	no	—
92	#95	—	4	2	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	#102	1	1	yes	characteristic
96	C₂²×C₄	#102	1	1	yes	characteristic
97	C₂²×C₄	#102	1	1	yes	characteristic
98	C₂²×C₄	#102	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
101	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	4	4	no	—
109	C₂×C₈	—	8	2	no	—
110	C₂×C₈	—	8	2	no	—
111	C₂×C₈	#100	1	2	yes	—
112	C₂×C₈	#101	1	2	yes	—
113	C₂×C₈	C₄×Q₈	1	2	yes	—
114	C₂×C₈	#124	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₈	#137	1	2	yes	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	8	2	no	—
123	C₄⋊C₄	—	8	2	no	—
124	C₄⋊C₄	—	8	4	no	—
125	C₄⋊C₄	—	8	4	no	—
126	C₄⋊C₄	—	8	2	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	2	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	4	1	no	—
135	C₄²	—	4	1	no	—
136	C₄²	C₂²⋊Q₈	1	2	yes	—
137	C₄²	#137	1	2	yes	—
138	C₁₆	—	8	2	no	—
139	C₂³	C₂×#102	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	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	—	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	—	8	2	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	#359	1	2	yes	—
161	C₂×C₄	#360	1	2	yes	—
162	C₂×C₄	#391	1	1	yes	characteristic
163	C₂×C₄	C₂×#100	1	1	yes	characteristic
164	C₂×C₄	C₂×#101	1	1	yes	characteristic
165	C₂×C₄	#418	1	1	yes	characteristic
166	C₂×C₄	#419	1	1	yes	characteristic
167	C₂×C₄	#419	1	1	yes	characteristic
168	C₂×C₄	#474	1	2	yes	—
169	C₂×C₄	#475	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	—	8	4	no	—
174	C₈	#489	1	2	yes	—
175	C₈	#501	1	2	yes	—
176	C₂²	#1545	1	1	yes	characteristic
177	C₂²	#1546	1	1	yes	characteristic
178	C₂²	#1988	1	1	yes	characteristic
179	C₂²	C₂×#359	1	1	yes	characteristic
180	C₂²	#2143	1	1	yes	characteristic
181	C₂²	C₂×#360	1	1	yes	characteristic
182	C₂²	#2145	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	16	2	no	—
190	C₄	—	16	2	no	—
191	C₄	#1556	1	1	yes	characteristic
192	C₄	#1558	1	1	yes	characteristic
193	C₄	#1993	1	1	yes	characteristic
194	C₄	#1997	1	1	yes	characteristic
195	C₂	#9905	1	1	yes	characteristic
196	C₂	#11958	1	1	yes	characteristic
197	C₂	#11960	1	1	yes	characteristic
198	C₂	#11967	1	1	yes	characteristic
199	C₂	#11969	1	1	yes	characteristic
200	C₂	#13274	1	1	yes	characteristic
201	C₂	#13278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#12879

C₂

yes

maximal, characteristic

#8273

C₂

yes

maximal, characteristic

#7275

C₂

yes

maximal

C₂×#499

—

C₂×#499

C₄

yes

—

#2859

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1846

—

#1846

C₂²

yes

—

#1845

C₄

yes

characteristic

#1845

C₄

yes

characteristic

#1844

—

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

C₂²

yes

—

—

—

—

—

C₂²×C₁₆

—

—

—

—

—

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×#97

—

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

—

#336

—

#334

—

#334

Q₈

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

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

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

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂²×C₄

C₂×D₈

yes

characteristic

101

C₂²×C₄

C₂×SD₁₆

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#100

yes

—

112

C₂×C₈

#101

yes

—

113

C₂×C₈

C₄×Q₈

yes

—

114

C₂×C₈

#124

yes

characteristic

115

C₂×C₈

C₄⋊Q₈

yes

characteristic

116

C₂×C₈

AGL(1,ℤ/8)

yes

—

117

C₂×C₈

#137

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

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

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₂²⋊Q₈

yes

—

137

C₄²

#137

yes

—

138

C₁₆

—

139

C₂³

C₂×#102

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

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

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#359

yes

—

161

C₂×C₄

#360

yes

—

162

C₂×C₄

#391

yes

characteristic

163

C₂×C₄

C₂×#100

yes

characteristic

164

C₂×C₄

C₂×#101

yes

characteristic

165

C₂×C₄

#418

yes

characteristic

166

C₂×C₄

#419

yes

characteristic

167

C₂×C₄

#419

yes

characteristic

168

C₂×C₄

#474

yes

—

169

C₂×C₄

#475

yes

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

#489

yes

—

175

C₈

#501

yes

—

176

C₂²

#1545

yes

characteristic

177

C₂²

#1546

yes

characteristic

178

C₂²

#1988

yes

characteristic

179

C₂²

C₂×#359

yes

characteristic

180

C₂²

#2143

yes

characteristic

181

C₂²

C₂×#360

yes

characteristic

182

C₂²

#2145

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#1556

yes

characteristic

192

C₄

#1558

yes

characteristic

193

C₄

#1993

yes

characteristic

194

C₄

#1997

yes

characteristic

195

C₂

#9905

yes

characteristic

196

C₂

#11958

yes

characteristic

197

C₂

#11960

yes

characteristic

198

C₂

#11967

yes

characteristic

199

C₂

#11969

yes

characteristic

200

C₂

#13274

yes

characteristic

201

C₂

#13278

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
#33311	2
#12879	2
#8273	2
#7275	2

subgroup

index

Subgroups — #142138

Conjugacy classes of subgroups

Maximal subgroups