Group #140340 — subgroups

Subgroups — #140340

Order 512.

247 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 110 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	#13163	C₂	1	2	yes	maximal
2	#12004	C₂	1	1	yes	maximal, characteristic
3	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	1	yes	maximal, characteristic
5	#7169	C₂	1	2	yes	maximal
6	C₂×#364	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	C₄×M₃₂	C₂²	1	1	yes	characteristic
9	C₄×M₃₂	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	#1376	—	2	2	no	—
15	#1370	—	2	1	no	—
16	#1370	—	2	1	no	—
17	#1311	—	2	1	no	—
18	C₈×C₁₆	—	2	1	no	—
19	#1279	C₄	1	2	yes	—
20	C₂×M₃₂	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	D₄	1	1	yes	characteristic
23	C₂×M₃₂	D₄	1	1	yes	characteristic
24	C₂×#107	—	2	1	no	—
25	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	—	2	2	no	—
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	#364	—	4	4	no	—
32	C₄⋊C₁₆	—	2	2	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	#346	—	2	1	no	—
35	#346	Q₈	1	2	yes	—
36	C₄×C₁₆	—	2	1	no	—
37	C₄×C₁₆	—	2	1	no	—
38	#340	—	2	1	no	—
39	#336	—	2	2	no	—
40	#335	C₈	1	2	yes	—
41	#335	C₈	1	2	yes	—
42	C₈²	D₄	1	1	yes	characteristic
43	C₈²	D₄	1	1	yes	characteristic
44	C₈²	Q₈	1	1	yes	characteristic
45	C₈²	Q₈	1	1	yes	characteristic
46	C₂²×C₈	—	2	2	no	—
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	M₁₆	1	1	yes	characteristic
49	C₂²×C₈	M₁₆	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	#41	1	1	yes	characteristic
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄²	—	2	2	no	—
54	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
55	M₃₂	—	2	2	no	—
56	M₃₂	—	2	2	no	—
57	M₃₂	—	2	1	no	—
58	M₃₂	—	2	1	no	—
59	M₃₂	—	4	2	no	—
60	M₃₂	—	4	2	no	—
61	M₃₂	D₈	1	2	yes	—
62	M₃₂	Q₁₆	1	2	yes	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	—	2	1	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	4	2	no	—
67	#107	—	2	4	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	C₂×C₈	1	4	yes	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	4	2	no	—
72	C₄×C₈	—	4	2	no	—
73	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₈	C₂×C₈	1	1	yes	characteristic
78	C₄×C₈	C₂×C₈	1	1	yes	characteristic
79	C₄×C₈	C₂×D₄	1	1	yes	characteristic
80	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	#95	—	2	2	no	—
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
87	C₂²×C₄	#131	1	1	yes	characteristic
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₈	#121	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₈	#125	1	1	yes	characteristic
105	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₈	C₂×D₈	1	1	yes	characteristic
107	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₈	#135	1	1	yes	characteristic
109	C₂×C₈	#135	1	1	yes	characteristic
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	—	4	2	no	—
113	C₄²	—	4	2	no	—
114	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₄²	C₄×Q₈	1	1	yes	characteristic
116	C₄²	C₂²×C₈	1	1	yes	characteristic
117	C₄²	#131	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	8	2	no	—
122	C₂³	#436	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	#389	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₄	#424	1	1	yes	characteristic
132	C₂×C₄	C₈×D₄	1	1	yes	characteristic
133	C₂×C₄	C₄×D₈	1	1	yes	characteristic
134	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
135	C₂×C₄	#443	1	1	yes	characteristic
136	C₂×C₄	#443	1	1	yes	characteristic
137	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
138	C₂×C₄	#446	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	1	no	—
146	C₈	—	4	4	no	—
147	C₈	#364	1	2	yes	complemented
148	C₈	#466	1	1	yes	characteristic
149	C₈	#467	1	1	yes	characteristic
150	C₈	#499	1	1	yes	characteristic
151	C₈	#500	1	1	yes	characteristic
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#1589	1	1	yes	characteristic
155	C₂²	#1591	1	1	yes	characteristic
156	C₂²	#1939	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	8	4	no	—
161	C₄	C₈×D₈	1	1	yes	characteristic
162	C₄	C₈×Q₁₆	1	1	yes	characteristic
163	C₄	#1942	1	1	yes	characteristic
164	C₄	#1945	1	1	yes	characteristic
165	C₄	C₂×#364	1	1	yes	characteristic
166	C₄	#2153	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#9594	1	1	yes	characteristic
170	C₂	#13163	1	1	yes	characteristic
171	C₂	#13164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13163

C₂

yes

maximal

#12004

C₂

yes

maximal, characteristic

C₈×M₃₂

C₂

yes

maximal, characteristic

C₈×#107

C₂

yes

maximal, characteristic

#7169

C₂

yes

maximal

C₂×#364

—

#2150

C₂²

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₄×#107

C₄

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

C₂×#335

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

—

C₈×C₁₆

—

#1279

C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

characteristic

C₂×M₃₂

D₄

yes

characteristic

C₂×#107

—

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

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

#336

—

#335

C₈

yes

—

#335

C₈

yes

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₈

yes

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂×M₁₆

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

—

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

100

C₂×C₈

C₄⋊C₈

yes

characteristic

101

C₂×C₈

C₄⋊C₈

yes

characteristic

102

C₂×C₈

#121

yes

characteristic

103

C₂×C₈

C₂²⋊Q₈

yes

characteristic

104

C₂×C₈

#125

yes

characteristic

105

C₂×C₈

C₄⋊Q₈

yes

characteristic

106

C₂×C₈

C₂×D₈

yes

characteristic

107

C₂×C₈

C₂×Q₁₆

yes

characteristic

108

C₂×C₈

#135

yes

characteristic

109

C₂×C₈

#135

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×C₄⋊C₄

yes

characteristic

115

C₄²

C₄×Q₈

yes

characteristic

116

C₄²

C₂²×C₈

yes

characteristic

117

C₄²

#131

yes

characteristic

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₂³

#436

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#389

yes

characteristic

130

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

131

C₂×C₄

#424

yes

characteristic

132

C₂×C₄

C₈×D₄

yes

characteristic

133

C₂×C₄

C₄×D₈

yes

characteristic

134

C₂×C₄

C₄×Q₁₆

yes

characteristic

135

C₂×C₄

#443

yes

characteristic

136

C₂×C₄

#443

yes

characteristic

137

C₂×C₄

C₈×Q₈

yes

characteristic

138

C₂×C₄

#446

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

#364

yes

complemented

148

C₈

#466

yes

characteristic

149

C₈

#467

yes

characteristic

150

C₈

#499

yes

characteristic

151

C₈

#500

yes

characteristic

152

C₂²

—

153

C₂²

—

154

C₂²

#1589

yes

characteristic

155

C₂²

#1591

yes

characteristic

156

C₂²

#1939

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₈×D₈

yes

characteristic

162

C₄

C₈×Q₁₆

yes

characteristic

163

C₄

#1942

yes

characteristic

164

C₄

#1945

yes

characteristic

165

C₄

C₂×#364

yes

characteristic

166

C₄

#2153

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#9594

yes

characteristic

170

C₂

#13163

yes

characteristic

171

C₂

#13164

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1939
prime radical	C₂³	#436
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#121

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
147. C₈	#364	semidirect

H ≅ G/N

type

147. C₈

#364

semidirect

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

subgroup	index
#13163	2
#12004	2
C₈×M₃₂	2
C₈×#107	2
#7169	2

subgroup

index

C₈×M₃₂

Subgroups — #140340

Conjugacy classes of subgroups

Complements

Maximal subgroups