Group #102252 — subgroups

Subgroups — #102252

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	#9896	C₂	1	2	yes	maximal
3	#9691	C₂	1	2	yes	maximal
4	#8784	C₂	1	1	yes	maximal, characteristic
5	#8261	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₄	1	4	yes	—
7	#2868	C₂²	1	2	yes	—
8	#1845	—	2	1	no	—
9	#1845	—	2	2	no	—
10	#1845	—	2	1	no	—
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×#333	—	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	C₂×C₄	1	4	yes	—
23	#500	—	4	8	no	—
24	C₂×#107	—	2	4	no	—
25	C₂×#107	—	2	4	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	D₄	1	2	yes	—
30	C₂×C₄⋊C₈	D₄	1	2	yes	—
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	—	2	1	no	—
33	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	2	yes	—
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	#384	—	2	2	no	—
38	#384	—	2	2	no	—
39	#384	—	2	2	no	—
40	#384	—	2	2	no	—
41	C₄³	C₂³	1	1	yes	characteristic
42	#336	—	2	1	no	—
43	#336	—	2	1	no	—
44	#333	—	4	8	no	—
45	#322	—	4	2	no	—
46	C₂²×Q₈	—	2	4	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	—	4	2	no	—
50	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
52	C₄⋊Q₈	—	4	8	no	—
53	C₄⋊Q₈	—	4	4	no	—
54	C₄⋊Q₈	—	4	4	no	—
55	C₄⋊Q₈	—	8	4	no	—
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	—	2	4	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	2	no	—
62	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄²	#41	1	1	yes	characteristic
68	C₂×C₄²	#41	1	1	yes	characteristic
69	#107	—	4	8	no	—
70	#107	—	4	8	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	—	2	4	no	—
73	C₄⋊C₈	—	4	4	no	—
74	C₄⋊C₈	—	4	4	no	—
75	C₄⋊C₈	—	4	4	no	—
76	C₄⋊C₈	D₈	1	4	yes	—
77	C₄×C₈	—	2	4	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	4	no	—
81	C₄×C₈	—	4	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	SD₁₆	1	2	yes	—
86	C₄×C₈	Q₁₆	1	2	yes	—
87	C₂×Q₈	—	4	8	no	—
88	C₂×Q₈	—	8	4	no	—
89	C₂×Q₈	—	8	4	no	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₄	#121	1	1	yes	characteristic
95	C₂²×C₄	#121	1	1	yes	characteristic
96	C₂²×C₄	#124	1	1	yes	characteristic
97	C₂²×C₄	#127	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	8	2	no	—
109	C₂×C₈	—	8	2	no	—
110	C₂×C₈	—	8	2	no	—
111	C₂×C₈	#100	1	4	yes	—
112	C₂×C₈	#103	1	4	yes	—
113	C₄⋊C₄	—	4	8	no	—
114	C₄⋊C₄	—	8	4	no	—
115	C₄⋊C₄	—	8	4	no	—
116	C₄⋊C₄	—	8	4	no	—
117	C₄⋊C₄	—	8	4	no	—
118	C₄⋊C₄	—	8	2	no	—
119	C₄⋊C₄	—	8	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	C₂×D₈	1	2	yes	—
124	C₄²	C₂×SD₁₆	1	1	yes	characteristic
125	C₄²	C₂×Q₁₆	1	1	yes	characteristic
126	C₄²	#135	1	2	yes	—
127	C₄²	#137	1	1	yes	characteristic
128	C₄²	#137	1	1	yes	characteristic
129	C₄²	#137	1	4	yes	—
130	C₄²	#137	1	4	yes	—
131	C₂³	#390	1	1	yes	characteristic
132	Q₈	—	16	8	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	4	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	C₂×#100	1	2	yes	—
143	C₂×C₄	C₂×#103	1	1	yes	characteristic
144	C₂×C₄	#419	1	2	yes	—
145	C₂×C₄	#419	1	1	yes	characteristic
146	C₂×C₄	C₄×D₈	1	2	yes	—
147	C₂×C₄	#441	1	2	yes	—
148	C₂×C₄	#461	1	1	yes	characteristic
149	C₂×C₄	#461	1	1	yes	characteristic
150	C₂×C₄	#462	1	1	yes	characteristic
151	C₂×C₄	#462	1	1	yes	characteristic
152	C₂×C₄	#466	1	2	yes	—
153	C₂×C₄	#470	1	2	yes	—
154	C₂×C₄	#472	1	2	yes	—
155	C₂×C₄	#473	1	2	yes	—
156	C₂×C₄	#491	1	2	yes	—
157	C₂×C₄	#493	1	1	yes	characteristic
158	C₂×C₄	#495	1	1	yes	characteristic
159	C₂×C₄	#497	1	2	yes	—
160	C₂×C₄	#497	1	2	yes	—
161	C₈	—	2	4	no	—
162	C₈	—	2	4	no	—
163	C₈	—	4	4	no	—
164	C₈	—	8	2	no	—
165	C₈	—	16	2	no	—
166	C₂²	#1960	1	1	yes	characteristic
167	C₂²	#1965	1	1	yes	characteristic
168	C₂²	#1970	1	1	yes	characteristic
169	C₂²	#1971	1	1	yes	characteristic
170	C₂²	#1978	1	1	yes	characteristic
171	C₂²	#1979	1	1	yes	characteristic
172	C₂²	#1981	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	16	4	no	—
177	C₄	#1548	1	2	yes	—
178	C₄	#1558	1	2	yes	—
179	C₄	#1673	1	2	yes	—
180	C₄	#1674	1	2	yes	—
181	C₄	#1695	1	2	yes	—
182	C₄	#1696	1	2	yes	—
183	C₄	#1718	1	2	yes	—
184	C₄	#1722	1	2	yes	—
185	C₂	#9702	1	1	yes	characteristic
186	C₂	#9731	1	1	yes	characteristic
187	C₂	#9739	1	1	yes	characteristic
188	C₂	#9778	1	1	yes	characteristic
189	C₂	#9802	1	1	yes	characteristic
190	C₂	#9807	1	1	yes	characteristic
191	C₂	#9831	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal, characteristic

#9896

C₂

yes

maximal

#9691

C₂

yes

maximal

#8784

C₂

yes

maximal, characteristic

#8261

C₂

yes

maximal, characteristic

C₂×#500

C₄

yes

—

#2868

C₂²

yes

—

—

—

—

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

yes

—

#500

—

C₂×#107

—

C₂×#107

—

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

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

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

—

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

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₄×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

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

#103

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₂×D₈

yes

—

124

C₄²

C₂×SD₁₆

yes

characteristic

125

C₄²

C₂×Q₁₆

yes

characteristic

126

C₄²

#135

yes

—

127

C₄²

#137

yes

characteristic

128

C₄²

#137

yes

characteristic

129

C₄²

#137

yes

—

130

C₄²

#137

yes

—

131

C₂³

#390

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×#100

yes

—

143

C₂×C₄

C₂×#103

yes

characteristic

144

C₂×C₄

#419

yes

—

145

C₂×C₄

#419

yes

characteristic

146

C₂×C₄

C₄×D₈

yes

—

147

C₂×C₄

#441

yes

—

148

C₂×C₄

#461

yes

characteristic

149

C₂×C₄

#461

yes

characteristic

150

C₂×C₄

#462

yes

characteristic

151

C₂×C₄

#462

yes

characteristic

152

C₂×C₄

#466

yes

—

153

C₂×C₄

#470

yes

—

154

C₂×C₄

#472

yes

—

155

C₂×C₄

#473

yes

—

156

C₂×C₄

#491

yes

—

157

C₂×C₄

#493

yes

characteristic

158

C₂×C₄

#495

yes

characteristic

159

C₂×C₄

#497

yes

—

160

C₂×C₄

#497

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₂²

#1960

yes

characteristic

167

C₂²

#1965

yes

characteristic

168

C₂²

#1970

yes

characteristic

169

C₂²

#1971

yes

characteristic

170

C₂²

#1978

yes

characteristic

171

C₂²

#1979

yes

characteristic

172

C₂²

#1981

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#1548

yes

—

178

C₄

#1558

yes

—

179

C₄

#1673

yes

—

180

C₄

#1674

yes

—

181

C₄

#1695

yes

—

182

C₄

#1696

yes

—

183

C₄

#1718

yes

—

184

C₄

#1722

yes

—

185

C₂

#9702

yes

characteristic

186

C₂

#9731

yes

characteristic

187

C₂

#9739

yes

characteristic

188

C₂

#9778

yes

characteristic

189

C₂

#9802

yes

characteristic

190

C₂

#9807

yes

characteristic

191

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
#9896	2
#9691	2
#8784	2
#8261	2

subgroup

index

Subgroups — #102252

Conjugacy classes of subgroups

Maximal subgroups