Group #102382 — subgroups

Subgroups — #102382

Order 512.

374 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 134 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	#8807	C₂	1	1	yes	maximal, characteristic
3	#8260	C₂	1	1	yes	maximal, characteristic
4	#7663	C₂	1	4	yes	maximal
5	C₂×#499	—	2	1	no	—
6	C₂×#499	—	2	1	no	—
7	#2859	C₂²	1	2	yes	—
8	#1846	—	2	1	no	—
9	#1846	—	2	1	no	—
10	#1845	C₄	1	2	yes	—
11	#1844	C₄	1	2	yes	—
12	#1831	C₂²	1	2	yes	—
13	C₄×C₄⋊C₈	C₂²	1	2	yes	—
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#335	—	2	1	no	—
16	C₂×#334	—	2	1	no	—
17	C₂×#322	—	2	1	no	—
18	#1273	—	2	1	no	—
19	#1271	—	2	8	no	—
20	C₂×#125	—	2	2	no	—
21	#499	—	4	4	no	—
22	#499	—	4	2	no	—
23	#499	—	4	2	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×#107	—	2	2	no	—
26	C₂×#106	—	2	2	no	—
27	C₂×#106	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄×C₈	—	2	2	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
37	#389	—	2	4	no	—
38	#389	—	2	4	no	—
39	#384	C₂×C₄	1	4	yes	—
40	#382	—	2	2	no	—
41	C₄³	C₂³	1	1	yes	characteristic
42	#336	—	2	4	no	—
43	#336	D₄	1	4	yes	—
44	#335	—	4	2	no	—
45	#334	—	4	2	no	—
46	#322	—	4	2	no	—
47	C₂²×C₈	—	2	4	no	—
48	C₂²×C₈	—	4	2	no	—
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	#125	—	4	2	no	—
54	#125	—	4	2	no	—
55	#125	—	4	4	no	—
56	C₂×C₄⋊C₄	—	2	4	no	—
57	C₂×C₄⋊C₄	—	2	4	no	—
58	C₂×C₄⋊C₄	—	2	4	no	—
59	C₂×C₄⋊C₄	—	4	4	no	—
60	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂×D₄	1	2	yes	—
62	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄²	#41	1	2	yes	—
65	#107	—	4	4	no	—
66	#107	—	4	4	no	—
67	#106	—	4	4	no	—
68	#106	—	4	4	no	—
69	C₄⋊C₈	—	4	4	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	4	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	4	2	no	—
76	C₄×C₈	—	4	2	no	—
77	C₄×C₈	D₈	1	2	yes	—
78	C₄×C₈	SD₁₆	1	2	yes	—
79	C₄×C₈	SD₁₆	1	2	yes	—
80	C₄×C₈	SD₁₆	1	2	yes	—
81	C₄×C₈	Q₁₆	1	2	yes	—
82	C₄×C₈	Q₁₆	1	2	yes	—
83	#95	—	4	2	no	—
84	#95	—	4	2	no	—
85	C₂²×C₄	—	4	4	no	—
86	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₄	1	2	yes	—
88	C₂²×C₄	#121	1	2	yes	—
89	C₂²×C₄	#124	1	1	yes	characteristic
90	C₂²×C₄	#127	1	1	yes	characteristic
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	4	4	no	—
96	C₂×C₈	—	4	4	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	8	4	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	#101	1	2	yes	—
101	C₂×C₈	#101	1	2	yes	—
102	C₂×C₈	#102	1	2	yes	—
103	C₂×C₈	#102	1	2	yes	—
104	C₂×C₈	#103	1	2	yes	—
105	C₂×C₈	#103	1	2	yes	—
106	C₂×C₈	#103	1	2	yes	—
107	C₂×C₈	#103	1	2	yes	—
108	C₄⋊C₄	—	4	4	no	—
109	C₄⋊C₄	—	4	4	no	—
110	C₄⋊C₄	—	4	8	no	—
111	C₄⋊C₄	—	8	4	no	—
112	C₄⋊C₄	—	8	4	no	—
113	C₄⋊C₄	—	8	4	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	4	no	—
117	C₄²	—	4	2	no	—
118	C₄²	C₂×D₈	1	1	yes	characteristic
119	C₄²	C₂×SD₁₆	1	1	yes	characteristic
120	C₄²	C₂×SD₁₆	1	1	yes	characteristic
121	C₄²	C₂×SD₁₆	1	1	yes	characteristic
122	C₄²	C₂×Q₁₆	1	1	yes	characteristic
123	C₄²	C₂×Q₁₆	1	1	yes	characteristic
124	C₄²	AGL(1,ℤ/8)	1	2	yes	—
125	C₂³	#390	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	—	8	4	no	—
135	C₂×C₄	C₂×#101	1	1	yes	characteristic
136	C₂×C₄	C₂×#101	1	1	yes	characteristic
137	C₂×C₄	C₂×#102	1	1	yes	characteristic
138	C₂×C₄	C₂×#103	1	1	yes	characteristic
139	C₂×C₄	C₂×#103	1	1	yes	characteristic
140	C₂×C₄	#418	1	1	yes	characteristic
141	C₂×C₄	#444	1	4	yes	—
142	C₂×C₄	#459	1	2	yes	—
143	C₂×C₄	#460	1	2	yes	—
144	C₂×C₄	#487	1	1	yes	characteristic
145	C₂×C₄	#487	1	1	yes	characteristic
146	C₂×C₄	#492	1	1	yes	characteristic
147	C₂×C₄	#494	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	8	4	no	—
151	C₈	—	16	2	no	—
152	C₈	#332	1	4	yes	—
153	C₈	#332	1	4	yes	—
154	C₂²	#1956	1	1	yes	characteristic
155	C₂²	#1966	1	1	yes	characteristic
156	C₂²	#1975	1	2	yes	—
157	C₂²	#1976	1	2	yes	—
158	C₂²	#1982	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	16	4	no	—
167	C₄	C₂×#332	1	1	yes	characteristic
168	C₄	C₂×#332	1	1	yes	characteristic
169	C₄	#1549	1	1	yes	characteristic
170	C₄	#1549	1	1	yes	characteristic
171	C₂	#9694	1	1	yes	characteristic
172	C₂	#9753	1	2	yes	—
173	C₂	#9755	1	1	yes	characteristic
174	C₂	#9792	1	2	yes	—
175	C₂	#9861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#8260

C₂

yes

maximal, characteristic

#7663

C₂

yes

—

—

C₂²

yes

—

#1846

—

#1846

—

#1845

C₄

yes

—

#1844

C₄

yes

—

#1831

C₂²

yes

—

C₄×C₄⋊C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

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

—

C₄³

C₂³

yes

characteristic

#336

—

#336

D₄

yes

—

#335

—

#334

—

#322

—

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

—

—

—

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

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄×D₄

yes

—

C₂²×C₄

#121

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

#101

yes

—

101

C₂×C₈

#101

yes

—

102

C₂×C₈

#102

yes

—

103

C₂×C₈

#102

yes

—

104

C₂×C₈

#103

yes

—

105

C₂×C₈

#103

yes

—

106

C₂×C₈

#103

yes

—

107

C₂×C₈

#103

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×D₈

yes

characteristic

119

C₄²

C₂×SD₁₆

yes

characteristic

120

C₄²

C₂×SD₁₆

yes

characteristic

121

C₄²

C₂×SD₁₆

yes

characteristic

122

C₄²

C₂×Q₁₆

yes

characteristic

123

C₄²

C₂×Q₁₆

yes

characteristic

124

C₄²

AGL(1,ℤ/8)

yes

—

125

C₂³

#390

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×#101

yes

characteristic

136

C₂×C₄

C₂×#101

yes

characteristic

137

C₂×C₄

C₂×#102

yes

characteristic

138

C₂×C₄

C₂×#103

yes

characteristic

139

C₂×C₄

C₂×#103

yes

characteristic

140

C₂×C₄

#418

yes

characteristic

141

C₂×C₄

#444

yes

—

142

C₂×C₄

#459

yes

—

143

C₂×C₄

#460

yes

—

144

C₂×C₄

#487

yes

characteristic

145

C₂×C₄

#487

yes

characteristic

146

C₂×C₄

#492

yes

characteristic

147

C₂×C₄

#494

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

#332

yes

—

153

C₈

#332

yes

—

154

C₂²

#1956

yes

characteristic

155

C₂²

#1966

yes

characteristic

156

C₂²

#1975

yes

—

157

C₂²

#1976

yes

—

158

C₂²

#1982

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

C₂×#332

yes

characteristic

168

C₄

C₂×#332

yes

characteristic

169

C₄

#1549

yes

characteristic

170

C₄

#1549

yes

characteristic

171

C₂

#9694

yes

characteristic

172

C₂

#9753

yes

—

173

C₂

#9755

yes

characteristic

174

C₂

#9792

yes

—

175

C₂

#9861

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
#33311	2
#8807	2
#8260	2
#7663	2

subgroup

index

Subgroups — #102382

Conjugacy classes of subgroups

Maximal subgroups