Group #142102 — subgroups

Subgroups — #142102

Order 512.

342 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 138 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	#12653	C₂	1	1	yes	maximal, characteristic
3	#8806	C₂	1	1	yes	maximal, characteristic
4	#7274	C₂	1	4	yes	maximal
5	C₂×#499	—	2	1	no	—
6	C₂×#499	C₄	1	2	yes	—
7	#2859	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#346	—	2	1	no	—
10	#1846	—	2	1	no	—
11	#1846	C₄	1	2	yes	—
12	#1845	C₂²	1	2	yes	—
13	#1844	—	2	1	no	—
14	#1842	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#335	—	2	1	no	—
17	C₂×#335	C₂²	1	2	yes	—
18	#1374	—	2	1	no	—
19	C₂×#125	—	2	2	no	—
20	C₂²×C₁₆	—	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	C₂×C₄	1	4	yes	—
26	C₂×#106	—	2	2	no	—
27	C₂×#106	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	2	no	—
29	C₂×C₄⋊C₈	—	4	2	no	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
36	#389	—	4	2	no	—
37	#389	—	4	2	no	—
38	#384	—	2	2	no	—
39	#382	—	2	2	no	—
40	C₄³	D₄	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	Q₈	1	4	yes	—
43	#346	—	4	2	no	—
44	#335	—	2	4	no	—
45	#335	—	4	2	no	—
46	C₂²×C₈	—	4	2	no	—
47	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	#125	—	4	4	no	—
52	#125	—	4	4	no	—
53	C₂×C₄⋊C₄	—	2	4	no	—
54	C₂×C₄⋊C₄	—	4	4	no	—
55	C₂×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄²	D₈	1	2	yes	—
60	C₂×C₄²	SD₁₆	1	2	yes	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	4	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₂×C₁₆	—	4	2	no	—
65	#107	—	4	2	no	—
66	#107	—	4	2	no	—
67	#107	—	4	4	no	—
68	#106	—	4	4	no	—
69	#106	—	4	4	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	4	2	no	—
72	C₄⋊C₈	—	8	4	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	D₈	1	2	yes	—
77	C₄×C₈	SD₁₆	1	2	yes	—
78	C₄×C₈	SD₁₆	1	2	yes	—
79	C₄×C₈	Q₁₆	1	2	yes	—
80	C₄×C₈	C₂×Q₈	1	2	yes	—
81	C₄×C₈	#41	1	2	yes	—
82	#95	—	4	4	no	—
83	C₂²×C₄	—	4	4	no	—
84	C₂²×C₄	#102	1	2	yes	—
85	C₂²×C₄	#102	1	2	yes	—
86	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	8	2	no	—
94	C₂×C₈	—	8	4	no	—
95	C₂×C₈	#101	1	2	yes	—
96	C₂×C₈	#102	1	2	yes	—
97	C₂×C₈	#103	1	2	yes	—
98	C₂×C₈	#103	1	2	yes	—
99	C₂×C₈	C₄×Q₈	1	2	yes	—
100	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₈	C₂×D₈	1	1	yes	characteristic
102	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₈	#137	1	2	yes	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	8	4	no	—
107	C₄⋊C₄	—	8	8	no	—
108	C₄⋊C₄	—	8	2	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	4	2	no	—
115	C₄²	#100	1	2	yes	—
116	C₄²	#102	1	2	yes	—
117	C₄²	#124	1	1	yes	characteristic
118	C₄²	C₄⋊Q₈	1	1	yes	characteristic
119	C₄²	C₂×SD₁₆	1	1	yes	characteristic
120	C₄²	C₂×Q₁₆	1	1	yes	characteristic
121	C₁₆	—	4	4	no	—
122	C₂³	C₂×#102	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	#331	1	4	yes	—
133	C₂×C₄	#391	1	1	yes	characteristic
134	C₂×C₄	C₂×#100	1	1	yes	characteristic
135	C₂×C₄	C₂×#101	1	1	yes	characteristic
136	C₂×C₄	C₂×#102	1	1	yes	characteristic
137	C₂×C₄	C₂×#103	1	1	yes	characteristic
138	C₂×C₄	#419	1	1	yes	characteristic
139	C₂×C₄	#486	1	1	yes	characteristic
140	C₂×C₄	#487	1	1	yes	characteristic
141	C₂×C₄	#498	1	1	yes	characteristic
142	C₂×C₄	#500	1	1	yes	characteristic
143	C₂×C₄	#509	1	2	yes	—
144	C₂×C₄	#510	1	2	yes	—
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	16	2	no	—
148	C₈	#332	1	4	yes	—
149	C₈	#474	1	2	yes	—
150	C₈	#475	1	2	yes	—
151	C₂²	C₂×#331	1	1	yes	characteristic
152	C₂²	#1546	1	1	yes	characteristic
153	C₂²	#1990	1	1	yes	characteristic
154	C₂²	#2139	1	2	yes	—
155	C₂²	#2148	1	2	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	16	4	no	—
161	C₄	C₂×#332	1	1	yes	characteristic
162	C₄	#1548	1	1	yes	characteristic
163	C₄	#1988	1	1	yes	characteristic
164	C₄	#1997	1	1	yes	characteristic
165	C₄	#2244	1	1	yes	characteristic
166	C₄	#2245	1	1	yes	characteristic
167	C₄	#2255	1	2	yes	—
168	C₂	#9898	1	1	yes	characteristic
169	C₂	#11930	1	2	yes	—
170	C₂	#11976	1	2	yes	—
171	C₂	#13284	1	1	yes	characteristic
172	C₂	#13289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#12653

C₂

yes

maximal, characteristic

#8806

C₂

yes

maximal, characteristic

#7274

C₂

yes

maximal

C₂×#499

—

C₂×#499

C₄

yes

—

#2859

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×#346

—

#1846

—

#1846

C₄

yes

—

#1845

C₂²

yes

—

#1844

—

#1842

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#335

C₂²

yes

—

#1374

—

C₂×#125

—

C₂²×C₁₆

—

—

—

—

—

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

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

C₂³

yes

—

—

—

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

Q₈

yes

—

#346

—

#335

—

#335

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

—

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

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

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#102

yes

—

C₂²×C₄

#102

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#101

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

C₄×Q₈

yes

—

100

C₂×C₈

C₂²⋊Q₈

yes

—

101

C₂×C₈

C₂×D₈

yes

characteristic

102

C₂×C₈

C₂×SD₁₆

yes

characteristic

103

C₂×C₈

#137

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

#100

yes

—

116

C₄²

#102

yes

—

117

C₄²

#124

yes

characteristic

118

C₄²

C₄⋊Q₈

yes

characteristic

119

C₄²

C₂×SD₁₆

yes

characteristic

120

C₄²

C₂×Q₁₆

yes

characteristic

121

C₁₆

—

122

C₂³

C₂×#102

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#331

yes

—

133

C₂×C₄

#391

yes

characteristic

134

C₂×C₄

C₂×#100

yes

characteristic

135

C₂×C₄

C₂×#101

yes

characteristic

136

C₂×C₄

C₂×#102

yes

characteristic

137

C₂×C₄

C₂×#103

yes

characteristic

138

C₂×C₄

#419

yes

characteristic

139

C₂×C₄

#486

yes

characteristic

140

C₂×C₄

#487

yes

characteristic

141

C₂×C₄

#498

yes

characteristic

142

C₂×C₄

#500

yes

characteristic

143

C₂×C₄

#509

yes

—

144

C₂×C₄

#510

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#332

yes

—

149

C₈

#474

yes

—

150

C₈

#475

yes

—

151

C₂²

C₂×#331

yes

characteristic

152

C₂²

#1546

yes

characteristic

153

C₂²

#1990

yes

characteristic

154

C₂²

#2139

yes

—

155

C₂²

#2148

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂×#332

yes

characteristic

162

C₄

#1548

yes

characteristic

163

C₄

#1988

yes

characteristic

164

C₄

#1997

yes

characteristic

165

C₄

#2244

yes

characteristic

166

C₄

#2245

yes

characteristic

167

C₄

#2255

yes

—

168

C₂

#9898

yes

characteristic

169

C₂

#11930

yes

—

170

C₂

#11976

yes

—

171

C₂

#13284

yes

characteristic

172

C₂

#13289

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#102

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33311	2
#12653	2
#8806	2
#7274	2

subgroup

index

Subgroups — #142102

Conjugacy classes of subgroups

Maximal subgroups