Group #146131 — subgroups

Subgroups — #146131

Order 512.

172 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 54 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	#12034	C₂	1	1	yes	maximal, characteristic
2	#11638	C₂	1	1	yes	maximal, characteristic
3	#8679	C₂	1	1	yes	maximal, characteristic
4	#8635	C₂	1	1	yes	maximal, characteristic
5	#7289	C₂	1	1	yes	maximal, characteristic
6	#7218	C₂	1	1	yes	maximal, characteristic
7	#7218	C₂	1	1	yes	maximal, characteristic
8	#2960	C₂²	1	1	yes	characteristic
9	#2162	C₂²	1	1	yes	characteristic
10	#2162	C₂²	1	1	yes	characteristic
11	#1835	C₂²	1	1	yes	characteristic
12	#1568	C₂²	1	1	yes	characteristic
13	#1568	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1537	—	2	1	no	—
16	#1518	—	2	1	no	—
17	#1372	—	2	1	no	—
18	#1371	—	2	1	no	—
19	Q₈⋊C₁₆	—	2	1	no	—
20	Q₈⋊C₁₆	—	2	1	no	—
21	#1307	—	2	1	no	—
22	#1286	—	2	1	no	—
23	#533	D₄	1	1	yes	characteristic
24	#446	—	2	1	no	—
25	#446	—	2	1	no	—
26	C₈×Q₈	—	2	1	no	—
27	C₈×Q₈	—	2	1	no	—
28	#432	—	2	1	no	—
29	#432	—	2	1	no	—
30	#431	—	2	1	no	—
31	#424	—	2	1	no	—
32	#423	D₄	1	1	yes	characteristic
33	#405	—	2	1	no	—
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	#382	—	2	1	no	—
36	C₄⋊C₁₆	—	2	1	no	—
37	C₄⋊C₁₆	—	2	1	no	—
38	C₂²⋊C₁₆	—	2	1	no	—
39	C₂²⋊C₁₆	—	2	1	no	—
40	C₄×C₁₆	—	2	1	no	—
41	C₄×C₁₆	D₄	1	1	yes	characteristic
42	C₄×C₁₆	D₄	1	1	yes	characteristic
43	#340	—	2	1	no	—
44	#340	—	2	1	no	—
45	#335	—	2	1	no	—
46	#335	—	2	1	no	—
47	#335	—	2	1	no	—
48	#334	—	2	1	no	—
49	#334	—	2	1	no	—
50	#334	—	2	1	no	—
51	#330	—	4	2	no	—
52	#328	—	4	2	no	—
53	C₂×M₁₆	—	2	1	no	—
54	C₂²×C₈	#41	1	1	yes	characteristic
55	C₂²×C₈	#41	1	1	yes	characteristic
56	C₄⋊Q₈	Q₁₆	1	1	yes	characteristic
57	C₄⋊Q₈	Q₁₆	1	1	yes	characteristic
58	#125	—	2	1	no	—
59	C₂²⋊Q₈	—	2	1	no	—
60	C₂²⋊Q₈	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	#117	—	2	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	—	2	1	no	—
68	C₂×C₁₆	—	4	1	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	4	1	no	—
72	C₄⋊C₈	—	4	1	no	—
73	C₄⋊C₈	—	4	1	no	—
74	C₄⋊C₈	—	4	1	no	—
75	C₄⋊C₈	—	4	1	no	—
76	C₂²⋊C₈	—	4	1	no	—
77	C₂²⋊C₈	—	4	1	no	—
78	C₂²⋊C₈	—	4	1	no	—
79	#97	—	4	1	no	—
80	#97	—	4	1	no	—
81	#97	—	4	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄×C₈	#41	1	1	yes	characteristic
86	C₄×C₈	#41	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	#95	—	4	1	no	—
89	#95	—	4	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂²×C₄	—	8	1	no	—
93	C₂²×C₄	#123	1	1	yes	characteristic
94	C₂²×C₄	#135	1	1	yes	characteristic
95	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
96	M₁₆	—	8	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	8	1	no	—
101	C₂×C₈	—	8	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	#123	1	1	yes	characteristic
104	C₂×C₈	#124	1	1	yes	characteristic
105	C₂×C₈	#126	1	1	yes	characteristic
106	C₂×C₈	#126	1	1	yes	characteristic
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	8	1	no	—
114	C₄⋊C₄	—	8	1	no	—
115	C₂²⋊C₄	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	#121	1	1	yes	characteristic
118	C₄²	#123	1	1	yes	characteristic
119	C₄²	C₂×Q₁₆	1	1	yes	characteristic
120	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₁₆	—	4	1	no	—
122	C₁₆	—	4	1	no	—
123	C₁₆	—	8	1	no	—
124	C₂³	#482	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	16	1	no	—
134	C₂×C₄	—	16	1	no	—
135	C₂×C₄	—	16	1	no	—
136	C₂×C₄	#397	1	1	yes	characteristic
137	C₂×C₄	#467	1	1	yes	characteristic
138	C₂×C₄	#469	1	1	yes	characteristic
139	C₂×C₄	#482	1	1	yes	characteristic
140	C₂×C₄	#484	1	1	yes	characteristic
141	C₂×C₄	#484	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	1	no	—
147	C₈	—	16	1	no	—
148	C₈	—	16	1	no	—
149	C₈	—	16	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	#2076	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	8	1	no	—
160	C₄	—	8	1	no	—
161	C₄	—	32	1	no	—
162	C₄	#2080	1	1	yes	characteristic
163	C₄	#2082	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#10640	1	1	yes	characteristic
166	C₂	#13499	1	1	yes	characteristic
167	C₂	#13500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12034

C₂

yes

maximal, characteristic

#11638

C₂

yes

maximal, characteristic

#8679

C₂

yes

maximal, characteristic

#8635

C₂

yes

maximal, characteristic

#7289

C₂

yes

maximal, characteristic

#7218

C₂

yes

maximal, characteristic

#7218

C₂

yes

maximal, characteristic

#2960

C₂²

yes

characteristic

#2162

C₂²

yes

characteristic

#2162

C₂²

yes

characteristic

#1835

C₂²

yes

characteristic

#1568

C₂²

yes

characteristic

#1568

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

Q₈⋊C₁₆

—

Q₈⋊C₁₆

—

#1307

—

#1286

—

#533

D₄

yes

characteristic

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

—

—

—

—

D₄

yes

characteristic

#405

—

C₂×C₄×C₈

C₂³

yes

characteristic

#382

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

yes

C₂²×C₈

yes

C₄⋊Q₈

Q₁₆

yes

characteristic

C₄⋊Q₈

Q₁₆

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

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

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

#123

yes

characteristic

104

C₂×C₈

#124

yes

characteristic

105

C₂×C₈

#126

yes

characteristic

106

C₂×C₈

#126

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

#121

yes

characteristic

118

C₄²

#123

yes

characteristic

119

C₄²

C₂×Q₁₆

yes

characteristic

120

C₄²

AGL(1,ℤ/8)

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#482

yes

characteristic

125

Q₈

—

126

Q₈

—

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

—

136

C₂×C₄

#397

yes

characteristic

137

C₂×C₄

#467

yes

characteristic

138

C₂×C₄

#469

yes

characteristic

139

C₂×C₄

#482

yes

characteristic

140

C₂×C₄

#484

yes

characteristic

141

C₂×C₄

#484

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#2076

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#2080

yes

characteristic

163

C₄

#2082

yes

characteristic

164

C₂

—

165

C₂

#10640

yes

characteristic

166

C₂

#13499

yes

characteristic

167

C₂

#13500

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#482
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2076

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#482

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ 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
#12034	2
#11638	2
#8679	2
#8635	2
#7289	2
#7218	2

subgroup

index

Subgroups — #146131

Conjugacy classes of subgroups

Maximal subgroups