Group #102365 — subgroups

Subgroups — #102365

Order 512.

354 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 114 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	#33290	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#7663	C₂	1	2	yes	maximal
5	#7662	C₂	1	2	yes	maximal
6	#2862	C₂²	1	1	yes	characteristic
7	#2859	C₂²	1	1	yes	characteristic
8	#1991	—	2	1	no	—
9	#1991	—	2	1	no	—
10	#1990	C₄	1	2	yes	—
11	#1990	C₄	1	2	yes	—
12	#1835	—	2	1	no	—
13	#1835	—	2	1	no	—
14	#1831	C₂²	1	2	yes	—
15	C₄×C₄⋊C₈	C₂²	1	2	yes	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#335	—	2	1	no	—
18	C₂×#334	—	2	1	no	—
19	C₂×C₈²	—	2	1	no	—
20	#1273	—	2	1	no	—
21	#1272	—	2	4	no	—
22	#1271	—	2	4	no	—
23	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	C₂×#125	—	2	1	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	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	#391	—	2	2	no	—
35	#391	—	2	2	no	—
36	#389	—	2	2	no	—
37	#389	—	2	2	no	—
38	#384	C₂×C₄	1	2	yes	—
39	#382	—	2	1	no	—
40	#382	—	2	1	no	—
41	#382	—	2	1	no	—
42	C₄³	C₂³	1	1	yes	characteristic
43	#340	—	2	2	no	—
44	#340	—	2	2	no	—
45	#340	—	2	2	no	—
46	#340	—	2	2	no	—
47	#336	—	2	2	no	—
48	#336	—	2	2	no	—
49	#336	D₄	1	4	yes	—
50	#335	—	4	2	no	—
51	#334	—	4	2	no	—
52	C₈²	—	4	2	no	—
53	C₂²×Q₈	—	2	2	no	—
54	C₂²×C₈	—	2	4	no	—
55	C₂²×C₈	—	4	2	no	—
56	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₄⋊Q₈	—	4	2	no	—
61	C₄⋊Q₈	—	4	2	no	—
62	#125	—	4	2	no	—
63	#125	—	4	1	no	—
64	#125	—	4	1	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	4	2	no	—
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄²	C₂×D₄	1	2	yes	—
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄²	#41	1	2	yes	—
76	C₄⋊C₈	—	2	8	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄⋊C₈	—	4	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	—	4	2	no	—
85	C₄×C₈	—	4	2	no	—
86	C₄×C₈	SD₁₆	1	2	yes	—
87	C₄×C₈	SD₁₆	1	2	yes	—
88	C₄×C₈	Q₁₆	1	2	yes	—
89	C₄×C₈	Q₁₆	1	2	yes	—
90	#95	—	4	2	no	—
91	#95	—	4	1	no	—
92	#95	—	4	1	no	—
93	#95	—	4	1	no	—
94	#95	—	4	1	no	—
95	C₂×Q₈	—	4	4	no	—
96	C₂×Q₈	—	8	2	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₄	C₄×D₄	1	2	yes	—
102	C₂²×C₄	#121	1	2	yes	—
103	C₂²×C₄	#124	1	1	yes	characteristic
104	C₂²×C₄	#127	1	1	yes	characteristic
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
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	8	no	—
114	C₂×C₈	—	4	4	no	—
115	C₂×C₈	—	4	4	no	—
116	C₂×C₈	—	8	4	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	#103	1	2	yes	—
119	C₂×C₈	#103	1	2	yes	—
120	C₂×C₈	#103	1	2	yes	—
121	C₂×C₈	#103	1	2	yes	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₄⋊C₄	—	8	2	no	—
129	C₄⋊C₄	—	8	2	no	—
130	C₄⋊C₄	—	8	2	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	4	no	—
134	C₄²	—	4	2	no	—
135	C₄²	C₂×SD₁₆	1	1	yes	characteristic
136	C₄²	C₂×SD₁₆	1	1	yes	characteristic
137	C₄²	C₂×Q₁₆	1	1	yes	characteristic
138	C₄²	C₂×Q₁₆	1	1	yes	characteristic
139	C₄²	#135	1	2	yes	—
140	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₄²	#137	1	1	yes	characteristic
142	C₂³	#390	1	1	yes	characteristic
143	Q₈	—	16	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	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	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	#413	1	1	yes	characteristic
157	C₂×C₄	#413	1	1	yes	characteristic
158	C₂×C₄	C₂×#103	1	1	yes	characteristic
159	C₂×C₄	C₂×#103	1	1	yes	characteristic
160	C₂×C₄	#416	1	1	yes	characteristic
161	C₂×C₄	#417	1	1	yes	characteristic
162	C₂×C₄	#443	1	4	yes	—
163	C₂×C₄	#463	1	2	yes	—
164	C₂×C₄	#464	1	2	yes	—
165	C₂×C₄	#487	1	1	yes	characteristic
166	C₂×C₄	#487	1	1	yes	characteristic
167	C₂×C₄	#492	1	1	yes	characteristic
168	C₂×C₄	#494	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	—	4	8	no	—
174	C₈	—	16	2	no	—
175	C₂²	#1956	1	1	yes	characteristic
176	C₂²	#1967	1	1	yes	characteristic
177	C₂²	#1974	1	2	yes	—
178	C₂²	#1976	1	2	yes	—
179	C₂²	#1983	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	4	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	2	no	—
188	C₄	—	16	2	no	—
189	C₄	#1542	1	1	yes	characteristic
190	C₄	#1542	1	1	yes	characteristic
191	C₄	#1550	1	1	yes	characteristic
192	C₄	#1550	1	1	yes	characteristic
193	C₂	#9697	1	1	yes	characteristic
194	C₂	#9751	1	2	yes	—
195	C₂	#9755	1	1	yes	characteristic
196	C₂	#9789	1	2	yes	—
197	C₂	#9855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33290

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

C₈×C₄⋊C₈

C₂

yes

maximal, characteristic

#7663

C₂

yes

maximal

#7662

C₂

yes

maximal

#2862

C₂²

yes

characteristic

#2859

C₂²

yes

—

—

C₄

yes

—

#1990

C₄

yes

—

#1835

—

#1835

—

#1831

C₂²

yes

—

C₄×C₄⋊C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#334

—

C₂×C₈²

—

#1273

—

#1272

—

#1271

—

C₂×C₄⋊Q₈

C₂×C₄

yes

—

C₂×#125

—

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

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#382

—

#382

—

#382

—

C₄³

C₂³

yes

—

—

—

—

—

—

D₄

yes

—

#335

—

#334

—

C₈²

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

#125

—

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

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

#41

yes

—

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

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

101

C₂²×C₄

C₄×D₄

yes

—

102

C₂²×C₄

#121

yes

—

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

C₂×C₈

#103

yes

—

121

C₂×C₈

#103

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂×SD₁₆

yes

characteristic

136

C₄²

C₂×SD₁₆

yes

characteristic

137

C₄²

C₂×Q₁₆

yes

characteristic

138

C₄²

C₂×Q₁₆

yes

characteristic

139

C₄²

#135

yes

—

140

C₄²

AGL(1,ℤ/8)

yes

characteristic

141

C₄²

#137

yes

characteristic

142

C₂³

#390

yes

characteristic

143

Q₈

—

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

#413

yes

characteristic

157

C₂×C₄

#413

yes

characteristic

158

C₂×C₄

C₂×#103

yes

characteristic

159

C₂×C₄

C₂×#103

yes

characteristic

160

C₂×C₄

#416

yes

characteristic

161

C₂×C₄

#417

yes

characteristic

162

C₂×C₄

#443

yes

—

163

C₂×C₄

#463

yes

—

164

C₂×C₄

#464

yes

—

165

C₂×C₄

#487

yes

characteristic

166

C₂×C₄

#487

yes

characteristic

167

C₂×C₄

#492

yes

characteristic

168

C₂×C₄

#494

yes

characteristic

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₂²

#1956

yes

characteristic

176

C₂²

#1967

yes

characteristic

177

C₂²

#1974

yes

—

178

C₂²

#1976

yes

—

179

C₂²

#1983

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#1542

yes

characteristic

190

C₄

#1542

yes

characteristic

191

C₄

#1550

yes

characteristic

192

C₄

#1550

yes

characteristic

193

C₂

#9697

yes

characteristic

194

C₂

#9751

yes

—

195

C₂

#9755

yes

characteristic

196

C₂

#9789

yes

—

197

C₂

#9855

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#390

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ 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
#33290	2
#8807	2
C₈×C₄⋊C₈	2
#7663	2
#7662	2

subgroup

index

C₈×C₄⋊C₈

Subgroups — #102365

Conjugacy classes of subgroups

Maximal subgroups