Group #124228 — subgroups

Subgroups — #124228

Order 512.

336 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 130 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	#21712	C₂	1	1	yes	maximal, characteristic
2	#12653	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#7118	C₂	1	4	yes	maximal
5	C₂×#446	—	2	1	no	—
6	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₁₆	—	2	1	no	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×#346	—	2	1	no	—
10	C₂×C₄×C₁₆	—	2	1	no	—
11	#1984	—	2	1	no	—
12	#1939	—	2	1	no	—
13	#1939	C₂²	1	2	yes	—
14	#1842	C₄	1	1	yes	characteristic
15	#1770	—	2	1	no	—
16	#1770	C₄	1	2	yes	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1374	C₂²	1	2	yes	—
19	#1374	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
21	C₂×C₄×Q₈	—	2	1	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	C₂²×C₁₆	—	2	2	no	—
24	#446	—	4	2	no	—
25	#446	—	4	2	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×#97	—	2	2	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	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	C₂³	1	1	yes	characteristic
37	#391	—	2	1	no	—
38	#389	—	2	1	no	—
39	#389	C₈	1	2	yes	—
40	#384	C₈	1	1	yes	characteristic
41	C₄×C₄⋊C₄	—	2	1	no	—
42	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₄³	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₁₆	—	2	4	no	—
45	C₄⋊C₁₆	—	2	4	no	—
46	#346	—	4	2	no	—
47	C₄×C₁₆	—	4	2	no	—
48	#336	—	2	2	no	—
49	#336	—	2	2	no	—
50	C₂²×Q₈	—	2	1	no	—
51	C₂²×C₈	—	2	2	no	—
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	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₄⋊Q₈	—	4	2	no	—
57	C₄×Q₈	—	4	2	no	—
58	C₄×Q₈	—	4	2	no	—
59	C₂×C₄⋊C₄	—	2	1	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	—	2	1	no	—
63	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
64	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
65	C₂×C₄²	—	2	2	no	—
66	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄²	M₁₆	1	1	yes	characteristic
71	C₂×C₄²	M₁₆	1	1	yes	characteristic
72	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₁₆	—	2	4	no	—
74	C₂×C₁₆	—	2	4	no	—
75	C₂×C₁₆	—	4	2	no	—
76	C₂×C₁₆	—	4	2	no	—
77	C₂×C₁₆	—	4	2	no	—
78	C₂×C₁₆	—	4	2	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	#97	—	4	4	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	4	1	no	—
89	C₄×C₈	D₈	1	2	yes	—
90	C₄×C₈	SD₁₆	1	2	yes	—
91	C₄×C₈	SD₁₆	1	2	yes	—
92	C₄×C₈	Q₁₆	1	2	yes	—
93	#95	—	2	2	no	—
94	#95	—	2	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	—	4	2	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
104	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
106	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	#101	1	2	yes	—
118	C₂×C₈	#102	1	2	yes	—
119	C₂×C₈	#103	1	2	yes	—
120	C₂×C₈	C₂×D₈	1	1	yes	characteristic
121	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	4	1	no	—
136	C₄²	—	4	2	no	—
137	C₄²	—	4	2	no	—
138	C₄²	#102	1	2	yes	—
139	C₄²	#103	1	2	yes	—
140	C₄²	#104	1	2	yes	—
141	C₄²	#104	1	2	yes	—
142	C₁₆	—	4	4	no	—
143	C₁₆	—	4	4	no	—
144	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	Q₈	—	8	4	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	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	D₄⋊C₈	1	2	yes	—
164	C₂×C₄	D₄⋊C₈	1	2	yes	—
165	C₂×C₄	Q₈⋊C₈	1	2	yes	—
166	C₂×C₄	Q₈⋊C₈	1	2	yes	—
167	C₂×C₄	#349	1	2	yes	—
168	C₂×C₄	#350	1	2	yes	—
169	C₂×C₄	#410	1	1	yes	characteristic
170	C₂×C₄	C₂×#101	1	1	yes	characteristic
171	C₂×C₄	C₂×#102	1	1	yes	characteristic
172	C₂×C₄	C₂×#103	1	1	yes	characteristic
173	C₂×C₄	C₂×#104	1	1	yes	characteristic
174	C₂×C₄	C₂×#104	1	1	yes	characteristic
175	C₂×C₄	#504	1	2	yes	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	4	1	no	—
181	C₈	—	8	2	no	—
182	C₈	#332	1	4	yes	—
183	C₂²	#1461	1	1	yes	characteristic
184	C₂²	C₂×D₄⋊C₈	1	1	yes	characteristic
185	C₂²	C₂×Q₈⋊C₈	1	1	yes	characteristic
186	C₂²	#2113	1	1	yes	characteristic
187	C₂²	C₂×#349	1	1	yes	characteristic
188	C₂²	C₂×#350	1	1	yes	characteristic
189	C₂²	#2117	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	8	2	no	—
196	C₄	—	8	2	no	—
197	C₄	#1330	1	4	yes	—
198	C₄	#1502	1	1	yes	characteristic
199	C₄	#1503	1	1	yes	characteristic
200	C₄	#1526	1	1	yes	characteristic
201	C₄	C₂×#332	1	1	yes	characteristic
202	C₂	#7570	1	1	yes	characteristic
203	C₂	#11495	1	1	yes	characteristic
204	C₂	#11498	1	1	yes	characteristic
205	C₂	#11582	1	1	yes	characteristic
206	C₂	C₂×#1330	1	1	yes	characteristic
207	C₂	#11608	1	1	yes	characteristic
208	C₂	#11611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21712

C₂

yes

maximal, characteristic

#12653

C₂

yes

maximal, characteristic

C₄×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#7118

C₂

yes

maximal

C₂×#446

—

C₄×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

—

C₂×#346

—

C₂×C₄×C₁₆

—

#1984

—

#1939

—

#1939

C₂²

yes

—

#1842

C₄

yes

characteristic

#1770

—

#1770

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1374

C₂²

yes

—

#1374

C₂²

yes

—

C₂×C₄⋊Q₈

C₈

yes

characteristic

C₂×C₄×Q₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#446

—

#446

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×#97

—

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

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

C₈

yes

—

#384

C₈

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂×C₄

yes

—

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

#336

—

#336

—

C₂²×Q₈

—

C₂²×C₈

—

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

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

M₁₆

yes

characteristic

C₂×C₄²

M₁₆

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂²⋊C₈

yes

characteristic

101

C₂²×C₄

C₂²⋊C₈

yes

characteristic

102

C₂²×C₄

C₂²⋊C₈

yes

characteristic

103

C₂²×C₄

C₂²⋊C₈

yes

characteristic

104

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

105

C₂²×C₄

C₂²×C₈

yes

characteristic

106

C₂²×C₄

C₂×M₁₆

yes

characteristic

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

#101

yes

—

118

C₂×C₈

#102

yes

—

119

C₂×C₈

#103

yes

—

120

C₂×C₈

C₂×D₈

yes

characteristic

121

C₂×C₈

C₂×SD₁₆

yes

characteristic

122

C₂×C₈

C₂×SD₁₆

yes

characteristic

123

C₂×C₈

C₂×Q₁₆

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

#102

yes

—

139

C₄²

#103

yes

—

140

C₄²

#104

yes

—

141

C₄²

#104

yes

—

142

C₁₆

—

143

C₁₆

—

144

C₂³

C₂×C₂²⋊C₈

yes

characteristic

145

Q₈

—

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

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

D₄⋊C₈

yes

—

164

C₂×C₄

D₄⋊C₈

yes

—

165

C₂×C₄

Q₈⋊C₈

yes

—

166

C₂×C₄

Q₈⋊C₈

yes

—

167

C₂×C₄

#349

yes

—

168

C₂×C₄

#350

yes

—

169

C₂×C₄

#410

yes

characteristic

170

C₂×C₄

C₂×#101

yes

characteristic

171

C₂×C₄

C₂×#102

yes

characteristic

172

C₂×C₄

C₂×#103

yes

characteristic

173

C₂×C₄

C₂×#104

yes

characteristic

174

C₂×C₄

C₂×#104

yes

characteristic

175

C₂×C₄

#504

yes

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

#332

yes

—

183

C₂²

#1461

yes

characteristic

184

C₂²

C₂×D₄⋊C₈

yes

characteristic

185

C₂²

C₂×Q₈⋊C₈

yes

characteristic

186

C₂²

#2113

yes

characteristic

187

C₂²

C₂×#349

yes

characteristic

188

C₂²

C₂×#350

yes

characteristic

189

C₂²

#2117

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#1330

yes

—

198

C₄

#1502

yes

characteristic

199

C₄

#1503

yes

characteristic

200

C₄

#1526

yes

characteristic

201

C₄

C₂×#332

yes

characteristic

202

C₂

#7570

yes

characteristic

203

C₂

#11495

yes

characteristic

204

C₂

#11498

yes

characteristic

205

C₂

#11582

yes

characteristic

206

C₂

C₂×#1330

yes

characteristic

207

C₂

#11608

yes

characteristic

208

C₂

#11611

yes

characteristic

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

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×C₂²⋊C₈

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#21712	2
#12653	2
C₄×C₄⋊C₁₆	2
#7118	2

subgroup

index

#21712

#12653

C₄×C₄⋊C₁₆

#7118

Subgroups — #124228

Conjugacy classes of subgroups

Maximal subgroups