Group #140339 — subgroups

Subgroups — #140339

Order 512.

264 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 126 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	#13159	C₂	1	2	yes	maximal
2	C₂×#1370	C₂	1	1	yes	maximal, characteristic
3	C₂×#1311	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	—	2	2	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×#107	C₄	1	1	yes	characteristic
11	C₈×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#335	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1370	—	2	2	no	—
16	#1311	—	2	2	no	—
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂×#107	C₈	1	2	yes	—
23	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	—	2	2	no	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄⋊C₁₆	—	2	2	no	—
31	C₄⋊C₁₆	—	4	4	no	—
32	C₄×C₁₆	—	2	1	no	—
33	C₄×C₁₆	—	2	1	no	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	Q₈	1	2	yes	—
36	#340	C₈	1	1	yes	characteristic
37	#340	C₈	1	1	yes	characteristic
38	#336	—	2	2	no	—
39	#335	—	2	2	no	—
40	C₈²	D₄	1	2	yes	—
41	C₈²	Q₈	1	2	yes	—
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₈	#41	1	1	yes	characteristic
48	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
49	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
50	C₂×C₄²	—	2	2	no	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	4	2	no	—
60	C₂×C₁₆	—	4	2	no	—
61	C₂×C₁₆	—	4	2	no	—
62	C₂×C₁₆	D₈	1	2	yes	—
63	C₂×C₁₆	Q₁₆	1	2	yes	—
64	#107	—	2	4	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	4	2	no	—
69	C₄×C₈	—	4	2	no	—
70	C₄×C₈	C₄⋊C₄	1	2	yes	—
71	C₄×C₈	C₄⋊C₄	1	2	yes	—
72	C₄×C₈	M₁₆	1	2	yes	—
73	C₄×C₈	C₂×D₄	1	1	yes	characteristic
74	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₄×C₈	#41	1	1	yes	characteristic
77	#95	—	2	2	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
81	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
82	C₂²×C₄	#131	1	1	yes	characteristic
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	C₄⋊C₈	1	2	yes	—
93	C₂×C₈	C₄⋊C₈	1	2	yes	—
94	C₂×C₈	#121	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₈	#125	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₈	C₂×D₈	1	1	yes	characteristic
99	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₈	#135	1	1	yes	characteristic
101	C₂×C₈	#135	1	1	yes	characteristic
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	M₃₂	1	4	yes	—
106	C₄²	—	4	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₄²	C₄×Q₈	1	1	yes	characteristic
110	C₄²	C₂×M₁₆	1	1	yes	characteristic
111	C₄²	#131	1	1	yes	characteristic
112	C₁₆	—	2	4	no	—
113	C₁₆	—	2	2	no	—
114	C₁₆	—	2	2	no	—
115	C₁₆	—	8	2	no	—
116	C₂³	C₈×D₄	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	#389	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂×C₄	#424	1	1	yes	characteristic
127	C₂×C₄	#436	1	1	yes	characteristic
128	C₂×C₄	C₄×D₈	1	1	yes	characteristic
129	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
130	C₂×C₄	#443	1	1	yes	characteristic
131	C₂×C₄	#443	1	1	yes	characteristic
132	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
133	C₂×C₄	#446	1	1	yes	characteristic
134	C₂×C₄	C₂×M₃₂	1	2	yes	—
135	C₂×C₄	#504	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	—	8	2	no	—
142	C₈	#466	1	1	yes	characteristic
143	C₈	#467	1	1	yes	characteristic
144	C₈	#499	1	1	yes	characteristic
145	C₈	#500	1	1	yes	characteristic
146	C₂²	C₈×D₈	1	1	yes	characteristic
147	C₂²	C₈×Q₁₆	1	1	yes	characteristic
148	C₂²	#1939	1	1	yes	characteristic
149	C₂²	#2168	1	2	yes	—
150	C₂²	#2171	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	4	no	—
154	C₄	—	8	2	no	—
155	C₄	#1589	1	1	yes	characteristic
156	C₄	#1591	1	1	yes	characteristic
157	C₄	#1942	1	1	yes	characteristic
158	C₄	#1945	1	1	yes	characteristic
159	C₄	#2150	1	1	yes	characteristic
160	C₄	#2151	1	1	yes	characteristic
161	C₄	#2183	1	2	yes	—
162	C₂	#9594	1	1	yes	characteristic
163	C₂	#12038	1	2	yes	—
164	C₂	#12053	1	2	yes	—
165	C₂	#13160	1	1	yes	characteristic
166	C₂	#13162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13159

C₂

yes

maximal

C₂×#1370

C₂

yes

maximal, characteristic

C₂×#1311

C₂

yes

maximal, characteristic

C₈×#107

C₂

yes

maximal, characteristic

#7167

C₂

yes

maximal

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#107

C₄

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

C₂×#335

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#107

C₈

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

Q₈

yes

—

#340

C₈

yes

characteristic

#340

C₈

yes

characteristic

#336

—

#335

—

C₈²

D₄

yes

—

C₈²

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

Q₁₆

yes

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

#131

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

100

C₂×C₈

#135

yes

characteristic

101

C₂×C₈

#135

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

M₃₂

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₂×C₄⋊C₄

yes

characteristic

109

C₄²

C₄×Q₈

yes

characteristic

110

C₄²

C₂×M₁₆

yes

characteristic

111

C₄²

#131

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

C₈×D₄

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#389

yes

characteristic

125

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

126

C₂×C₄

#424

yes

characteristic

127

C₂×C₄

#436

yes

characteristic

128

C₂×C₄

C₄×D₈

yes

characteristic

129

C₂×C₄

C₄×Q₁₆

yes

characteristic

130

C₂×C₄

#443

yes

characteristic

131

C₂×C₄

#443

yes

characteristic

132

C₂×C₄

C₈×Q₈

yes

characteristic

133

C₂×C₄

#446

yes

characteristic

134

C₂×C₄

C₂×M₃₂

yes

—

135

C₂×C₄

#504

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

#466

yes

characteristic

143

C₈

#467

yes

characteristic

144

C₈

#499

yes

characteristic

145

C₈

#500

yes

characteristic

146

C₂²

C₈×D₈

yes

characteristic

147

C₂²

C₈×Q₁₆

yes

characteristic

148

C₂²

#1939

yes

characteristic

149

C₂²

#2168

yes

—

150

C₂²

#2171

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#1589

yes

characteristic

156

C₄

#1591

yes

characteristic

157

C₄

#1942

yes

characteristic

158

C₄

#1945

yes

characteristic

159

C₄

#2150

yes

characteristic

160

C₄

#2151

yes

characteristic

161

C₄

#2183

yes

—

162

C₂

#9594

yes

characteristic

163

C₂

#12038

yes

—

164

C₂

#12053

yes

—

165

C₂

#13160

yes

characteristic

166

C₂

#13162

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂×D₄

commutator

C₂²×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₈×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
#13159	2
C₂×#1370	2
C₂×#1311	2
C₈×#107	2
#7167	2

subgroup

index

Subgroups — #140339

Conjugacy classes of subgroups

Maximal subgroups