Group #102255 — subgroups

Subgroups — #102255

Order 512.

438 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 146 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	#33312	C₂	1	1	yes	maximal, characteristic
2	#9898	C₂	1	2	yes	maximal
3	#9694	C₂	1	2	yes	maximal
4	#8784	C₂	1	1	yes	maximal, characteristic
5	#8261	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	1	no	—
7	C₂×#500	—	2	1	no	—
8	#2868	C₂²	1	2	yes	—
9	#1845	—	2	2	no	—
10	#1845	C₄	1	4	yes	—
11	#1842	C₂²	1	2	yes	—
12	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#332	—	2	4	no	—
16	C₂×#322	—	2	1	no	—
17	#1296	—	2	4	no	—
18	#1277	—	2	2	no	—
19	#1273	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×C₄⋊Q₈	—	2	2	no	—
23	#500	—	4	2	no	—
24	#500	—	4	2	no	—
25	#500	—	4	2	no	—
26	#500	—	4	2	no	—
27	C₂×#107	—	2	4	no	—
28	C₂×#107	—	2	4	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄⋊C₈	—	2	2	no	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄⋊C₈	D₄	1	2	yes	—
33	C₂×C₄⋊C₈	D₄	1	2	yes	—
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	—	2	1	no	—
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₈	D₄	1	2	yes	—
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	#384	—	2	2	no	—
41	#384	—	2	2	no	—
42	#384	—	2	2	no	—
43	#384	C₂×C₄	1	4	yes	—
44	C₄³	C₂³	1	1	yes	characteristic
45	#336	—	2	1	no	—
46	#336	—	2	1	no	—
47	#332	—	4	8	no	—
48	#322	—	4	2	no	—
49	C₂²×Q₈	—	4	2	no	—
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	—	2	2	no	—
52	C₂²×C₈	—	4	2	no	—
53	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
55	C₄⋊Q₈	—	4	4	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	4	4	no	—
59	C₄⋊Q₈	—	4	2	no	—
60	C₄⋊Q₈	—	4	2	no	—
61	C₄⋊Q₈	—	8	4	no	—
62	C₂×C₄⋊C₄	—	2	4	no	—
63	C₂×C₄⋊C₄	—	2	4	no	—
64	C₂×C₄⋊C₄	—	2	4	no	—
65	C₂×C₄⋊C₄	—	4	2	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄²	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄²	#41	1	1	yes	characteristic
74	C₂×C₄²	#41	1	1	yes	characteristic
75	#107	—	4	8	no	—
76	#107	—	4	8	no	—
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	2	4	no	—
79	C₄⋊C₈	—	4	4	no	—
80	C₄⋊C₈	—	4	4	no	—
81	C₄⋊C₈	—	4	4	no	—
82	C₄⋊C₈	D₈	1	4	yes	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	—	4	1	no	—
88	C₄×C₈	—	4	1	no	—
89	C₄×C₈	—	4	1	no	—
90	C₄×C₈	—	4	1	no	—
91	C₄×C₈	D₈	1	2	yes	—
92	C₄×C₈	SD₁₆	1	2	yes	—
93	C₂×Q₈	—	8	2	no	—
94	C₂×Q₈	—	8	2	no	—
95	C₂×Q₈	—	8	4	no	—
96	C₂×Q₈	—	8	4	no	—
97	C₂²×C₄	—	4	4	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	—	4	2	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	#101	1	4	yes	—
119	C₂×C₈	#102	1	4	yes	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	8	no	—
123	C₄⋊C₄	—	8	4	no	—
124	C₄⋊C₄	—	8	4	no	—
125	C₄⋊C₄	—	8	2	no	—
126	C₄⋊C₄	—	8	2	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₄⋊C₄	—	8	2	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	C₂×D₈	1	1	yes	characteristic
133	C₄²	C₂×D₈	1	2	yes	—
134	C₄²	C₂×SD₁₆	1	1	yes	characteristic
135	C₄²	#135	1	2	yes	—
136	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₄²	#137	1	4	yes	—
139	C₄²	#137	1	4	yes	—
140	C₂³	#390	1	1	yes	characteristic
141	Q₈	—	16	4	no	—
142	Q₈	—	16	4	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	—	8	4	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	8	4	no	—
152	C₂×C₄	C₂×#101	1	2	yes	—
153	C₂×C₄	C₂×#102	1	1	yes	characteristic
154	C₂×C₄	#418	1	1	yes	characteristic
155	C₂×C₄	#419	1	2	yes	—
156	C₂×C₄	C₄×D₈	1	2	yes	—
157	C₂×C₄	#441	1	2	yes	—
158	C₂×C₄	#459	1	1	yes	characteristic
159	C₂×C₄	#459	1	1	yes	characteristic
160	C₂×C₄	#460	1	1	yes	characteristic
161	C₂×C₄	#460	1	1	yes	characteristic
162	C₂×C₄	#466	1	2	yes	—
163	C₂×C₄	#470	1	2	yes	—
164	C₂×C₄	#472	1	2	yes	—
165	C₂×C₄	#473	1	2	yes	—
166	C₂×C₄	#491	1	2	yes	—
167	C₂×C₄	#493	1	1	yes	characteristic
168	C₂×C₄	#495	1	1	yes	characteristic
169	C₂×C₄	#497	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	—	8	2	no	—
175	C₈	—	16	2	no	—
176	C₂²	#1960	1	1	yes	characteristic
177	C₂²	#1966	1	1	yes	characteristic
178	C₂²	#1970	1	1	yes	characteristic
179	C₂²	#1971	1	1	yes	characteristic
180	C₂²	#1978	1	1	yes	characteristic
181	C₂²	#1979	1	1	yes	characteristic
182	C₂²	#1982	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	16	4	no	—
187	C₄	#1546	1	2	yes	—
188	C₄	#1556	1	2	yes	—
189	C₄	#1667	1	2	yes	—
190	C₄	#1668	1	2	yes	—
191	C₄	#1689	1	2	yes	—
192	C₄	#1690	1	2	yes	—
193	C₄	#1718	1	2	yes	—
194	C₄	#1722	1	2	yes	—
195	C₂	#9705	1	1	yes	characteristic
196	C₂	#9734	1	1	yes	characteristic
197	C₂	#9742	1	1	yes	characteristic
198	C₂	#9778	1	1	yes	characteristic
199	C₂	#9799	1	1	yes	characteristic
200	C₂	#9810	1	1	yes	characteristic
201	C₂	#9831	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal, characteristic

#9898

C₂

yes

maximal

#9694

C₂

yes

maximal

#8784

C₂

yes

maximal, characteristic

#8261

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

—

#2868

C₂²

yes

—

#1845

—

#1845

C₄

yes

—

#1842

C₂²

yes

—

C₄×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₄⋊C₈

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

—

C₂×C₄⋊C₈

D₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

—

—

C₂×C₄

yes

—

C₄³

C₂³

yes

—

—

—

—

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

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

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

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

D₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

#101

yes

—

119

C₂×C₈

#102

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

C₄⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂×D₈

yes

characteristic

133

C₄²

C₂×D₈

yes

—

134

C₄²

C₂×SD₁₆

yes

characteristic

135

C₄²

#135

yes

—

136

C₄²

AGL(1,ℤ/8)

yes

characteristic

137

C₄²

AGL(1,ℤ/8)

yes

characteristic

138

C₄²

#137

yes

—

139

C₄²

#137

yes

—

140

C₂³

#390

yes

characteristic

141

Q₈

—

142

Q₈

—

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

C₂×#101

yes

—

153

C₂×C₄

C₂×#102

yes

characteristic

154

C₂×C₄

#418

yes

characteristic

155

C₂×C₄

#419

yes

—

156

C₂×C₄

C₄×D₈

yes

—

157

C₂×C₄

#441

yes

—

158

C₂×C₄

#459

yes

characteristic

159

C₂×C₄

#459

yes

characteristic

160

C₂×C₄

#460

yes

characteristic

161

C₂×C₄

#460

yes

characteristic

162

C₂×C₄

#466

yes

—

163

C₂×C₄

#470

yes

—

164

C₂×C₄

#472

yes

—

165

C₂×C₄

#473

yes

—

166

C₂×C₄

#491

yes

—

167

C₂×C₄

#493

yes

characteristic

168

C₂×C₄

#495

yes

characteristic

169

C₂×C₄

#497

yes

—

170

C₂×C₄

#497

yes

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₂²

#1960

yes

characteristic

177

C₂²

#1966

yes

characteristic

178

C₂²

#1970

yes

characteristic

179

C₂²

#1971

yes

characteristic

180

C₂²

#1978

yes

characteristic

181

C₂²

#1979

yes

characteristic

182

C₂²

#1982

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#1546

yes

—

188

C₄

#1556

yes

—

189

C₄

#1667

yes

—

190

C₄

#1668

yes

—

191

C₄

#1689

yes

—

192

C₄

#1690

yes

—

193

C₄

#1718

yes

—

194

C₄

#1722

yes

—

195

C₂

#9705

yes

characteristic

196

C₂

#9734

yes

characteristic

197

C₂

#9742

yes

characteristic

198

C₂

#9778

yes

characteristic

199

C₂

#9799

yes

characteristic

200

C₂

#9810

yes

characteristic

201

C₂

#9831

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
#33312	2
#9898	2
#9694	2
#8784	2
#8261	2

subgroup

index

Subgroups — #102255

Conjugacy classes of subgroups

Maximal subgroups