Group #141012 — subgroups

Subgroups — #141012

Order 512.

392 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 142 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	#12653	C₂	1	1	yes	maximal, characteristic
3	C₂×#1365	C₂	1	4	yes	maximal
4	#8807	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	1	no	—
6	C₂×#500	C₂²	1	2	yes	—
7	#2868	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#346	—	2	1	no	—
10	#1845	—	2	1	no	—
11	#1845	—	2	1	no	—
12	#1845	C₄	1	2	yes	—
13	#1845	C₄	1	2	yes	—
14	#1831	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#335	C₂²	1	2	yes	—
17	C₂×#334	—	2	1	no	—
18	#1374	—	2	1	no	—
19	#1365	—	2	8	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	4	2	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	#500	—	2	4	no	—
24	#500	—	2	4	no	—
25	#500	—	4	2	no	—
26	#500	—	4	2	no	—
27	C₂×#107	—	2	2	no	—
28	C₂×#107	—	2	2	no	—
29	C₂×#107	—	2	2	no	—
30	C₂×#107	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	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	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#384	—	2	2	no	—
39	#384	—	2	2	no	—
40	#384	—	4	2	no	—
41	C₄³	D₄	1	1	yes	characteristic
42	C₄⋊C₁₆	—	2	2	no	—
43	C₄⋊C₁₆	D₄	1	4	yes	—
44	#346	—	4	2	no	—
45	#336	—	4	2	no	—
46	#335	—	2	4	no	—
47	#334	—	4	2	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂²×C₈	—	4	2	no	—
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	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	8	8	no	—
59	C₂×C₄⋊C₄	—	2	4	no	—
60	C₂×C₄⋊C₄	—	4	4	no	—
61	C₂×C₄⋊C₄	—	4	4	no	—
62	C₂×C₄⋊C₄	—	4	2	no	—
63	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	D₈	1	2	yes	—
66	C₂×C₄²	SD₁₆	1	2	yes	—
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₁₆	—	2	4	no	—
69	C₂×C₁₆	—	4	2	no	—
70	C₂×C₁₆	—	4	2	no	—
71	#107	—	2	8	no	—
72	#107	—	4	4	no	—
73	#107	—	4	4	no	—
74	#107	—	4	4	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	D₈	1	2	yes	—
81	C₄×C₈	SD₁₆	1	2	yes	—
82	C₄×C₈	SD₁₆	1	2	yes	—
83	C₄×C₈	Q₁₆	1	2	yes	—
84	C₄×C₈	C₂×D₄	1	2	yes	—
85	C₄×C₈	#41	1	2	yes	—
86	C₂×Q₈	—	8	8	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂²×C₄	—	4	4	no	—
90	C₂²×C₄	#102	1	2	yes	—
91	C₂²×C₄	#102	1	2	yes	—
92	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
94	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	—	8	4	no	—
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	C₄×D₄	1	2	yes	—
106	C₂×C₈	#121	1	2	yes	—
107	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₈	#137	1	2	yes	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	8	8	no	—
113	C₄⋊C₄	—	8	8	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄²	—	2	4	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	4	no	—
120	C₄²	#101	1	2	yes	—
121	C₄²	#103	1	2	yes	—
122	C₄²	#124	1	1	yes	characteristic
123	C₄²	#127	1	1	yes	characteristic
124	C₄²	C₂×D₈	1	1	yes	characteristic
125	C₄²	C₂×SD₁₆	1	1	yes	characteristic
126	C₁₆	—	4	4	no	—
127	C₂³	C₂×#102	1	1	yes	characteristic
128	Q₈	—	16	4	no	—
129	Q₈	—	16	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	8	8	no	—
136	C₂×C₄	—	8	4	no	—
137	C₂×C₄	#332	1	4	yes	—
138	C₂×C₄	#361	1	4	yes	—
139	C₂×C₄	#390	1	1	yes	characteristic
140	C₂×C₄	C₂×#101	1	1	yes	characteristic
141	C₂×C₄	C₂×#101	1	1	yes	characteristic
142	C₂×C₄	C₂×#102	1	1	yes	characteristic
143	C₂×C₄	C₂×#103	1	1	yes	characteristic
144	C₂×C₄	#419	1	1	yes	characteristic
145	C₂×C₄	#486	1	1	yes	characteristic
146	C₂×C₄	#486	1	1	yes	characteristic
147	C₂×C₄	#492	1	1	yes	characteristic
148	C₂×C₄	#493	1	1	yes	characteristic
149	C₂×C₄	#510	1	4	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	16	2	no	—
153	C₈	#332	1	4	yes	—
154	C₈	#461	1	2	yes	—
155	C₈	#462	1	2	yes	—
156	C₂²	C₂×#332	1	1	yes	characteristic
157	C₂²	#1546	1	1	yes	characteristic
158	C₂²	#1957	1	1	yes	characteristic
159	C₂²	#2141	1	2	yes	—
160	C₂²	C₂×#361	1	2	yes	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	4	no	—
165	C₄	—	16	4	no	—
166	C₄	C₂×#332	1	1	yes	characteristic
167	C₄	#1546	1	1	yes	characteristic
168	C₄	#1966	1	1	yes	characteristic
169	C₄	#1981	1	1	yes	characteristic
170	C₄	#2245	1	2	yes	—
171	C₄	#2251	1	2	yes	—
172	C₂	#9692	1	1	yes	characteristic
173	C₂	#11941	1	2	yes	—
174	C₂	#11975	1	2	yes	—
175	C₂	#13207	1	1	yes	characteristic
176	C₂	#13213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal, characteristic

#12653

C₂

yes

maximal, characteristic

C₂×#1365

C₂

yes

maximal

#8807

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

#2868

—

C₂×C₄⋊C₁₆

C₂²

yes

—

—

—

—

C₄

yes

—

#1845

C₄

yes

—

#1831

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

C₂×#334

—

#1374

—

#1365

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

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

D₄

yes

—

—

—

—

—

C₂²×Q₈

—

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

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

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

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

100

C₂×C₈

—

101

C₂×C₈

#101

yes

—

102

C₂×C₈

#102

yes

—

103

C₂×C₈

#102

yes

—

104

C₂×C₈

#103

yes

—

105

C₂×C₈

C₄×D₄

yes

—

106

C₂×C₈

#121

yes

—

107

C₂×C₈

C₂×SD₁₆

yes

characteristic

108

C₂×C₈

C₂×Q₁₆

yes

characteristic

109

C₂×C₈

#137

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

#101

yes

—

121

C₄²

#103

yes

—

122

C₄²

#124

yes

characteristic

123

C₄²

#127

yes

characteristic

124

C₄²

C₂×D₈

yes

characteristic

125

C₄²

C₂×SD₁₆

yes

characteristic

126

C₁₆

—

127

C₂³

C₂×#102

yes

characteristic

128

Q₈

—

129

Q₈

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#332

yes

—

138

C₂×C₄

#361

yes

—

139

C₂×C₄

#390

yes

characteristic

140

C₂×C₄

C₂×#101

yes

characteristic

141

C₂×C₄

C₂×#101

yes

characteristic

142

C₂×C₄

C₂×#102

yes

characteristic

143

C₂×C₄

C₂×#103

yes

characteristic

144

C₂×C₄

#419

yes

characteristic

145

C₂×C₄

#486

yes

characteristic

146

C₂×C₄

#486

yes

characteristic

147

C₂×C₄

#492

yes

characteristic

148

C₂×C₄

#493

yes

characteristic

149

C₂×C₄

#510

yes

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

#332

yes

—

154

C₈

#461

yes

—

155

C₈

#462

yes

—

156

C₂²

C₂×#332

yes

characteristic

157

C₂²

#1546

yes

characteristic

158

C₂²

#1957

yes

characteristic

159

C₂²

#2141

yes

—

160

C₂²

C₂×#361

yes

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

C₂×#332

yes

characteristic

167

C₄

#1546

yes

characteristic

168

C₄

#1966

yes

characteristic

169

C₄

#1981

yes

characteristic

170

C₄

#2245

yes

—

171

C₄

#2251

yes

—

172

C₂

#9692

yes

characteristic

173

C₂

#11941

yes

—

174

C₂

#11975

yes

—

175

C₂

#13207

yes

characteristic

176

C₂

#13213

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
#33312	2
#12653	2
C₂×#1365	2
#8807	2

subgroup

index

Subgroups — #141012

Conjugacy classes of subgroups

Maximal subgroups