Group #146129 — subgroups

Subgroups — #146129

Order 512.

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

normal subgroups only

This group has 179 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	#11639	C₂	1	1	yes	maximal, characteristic
3	#8661	C₂	1	1	yes	maximal, characteristic
4	#8621	C₂	1	1	yes	maximal, characteristic
5	#7293	C₂	1	1	yes	maximal, characteristic
6	#7224	C₂	1	1	yes	maximal, characteristic
7	#7220	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
9	#2163	C₂²	1	1	yes	characteristic
10	#2162	C₂²	1	1	yes	characteristic
11	#1837	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	#1487	—	2	1	no	—
16	#1467	—	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	#1301	—	2	1	no	—
22	#1290	—	2	1	no	—
23	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
24	C₈×Q₈	—	2	1	no	—
25	C₈×Q₈	—	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	#423	—	2	1	no	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
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	#346	—	2	1	no	—
41	C₄×C₁₆	D₄	1	1	yes	characteristic
42	C₄×C₁₆	D₄	1	1	yes	characteristic
43	#341	—	2	1	no	—
44	#341	—	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	Q₈⋊C₈	—	4	2	no	—
52	#324	—	4	2	no	—
53	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
54	C₂×M₁₆	—	4	1	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	#41	1	1	yes	characteristic
57	C₂²×C₈	#41	1	1	yes	characteristic
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×Q₈	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	C₂×C₄⋊C₄	—	2	1	no	—
63	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
64	C₂×C₄²	—	2	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	C₄⋊C₈	—	4	1	no	—
77	C₄⋊C₈	—	4	1	no	—
78	C₂²⋊C₈	—	4	1	no	—
79	C₂²⋊C₈	—	4	1	no	—
80	C₂²⋊C₈	—	4	1	no	—
81	#97	—	4	1	no	—
82	#97	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	#123	1	1	yes	characteristic
96	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
97	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
98	M₁₆	—	8	1	no	—
99	M₁₆	—	8	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	8	1	no	—
104	C₂×C₈	—	8	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	—	8	1	no	—
107	C₂×C₈	—	8	1	no	—
108	C₂×C₈	#123	1	1	yes	characteristic
109	C₂×C₈	#124	1	1	yes	characteristic
110	C₂×C₈	#126	1	1	yes	characteristic
111	C₂×C₈	#126	1	1	yes	characteristic
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	#121	1	1	yes	characteristic
119	C₄²	#123	1	1	yes	characteristic
120	C₄²	#135	1	1	yes	characteristic
121	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₁₆	—	4	1	no	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	8	1	no	—
125	C₂³	#484	1	1	yes	characteristic
126	Q₈	—	2	2	no	—
127	Q₈	—	2	2	no	—
128	Q₈	—	4	1	no	—
129	Q₈	—	4	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	#397	1	1	yes	characteristic
140	C₂×C₄	#467	1	1	yes	characteristic
141	C₂×C₄	#469	1	1	yes	characteristic
142	C₂×C₄	#482	1	1	yes	characteristic
143	C₂×C₄	#482	1	1	yes	characteristic
144	C₂×C₄	#484	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	8	1	no	—
150	C₈	—	16	1	no	—
151	C₈	—	16	1	no	—
152	C₈	—	16	1	no	—
153	C₈	—	16	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#2076	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	8	1	no	—
165	C₄	#2080	1	1	yes	characteristic
166	C₄	#2082	1	1	yes	characteristic
167	C₂	—	4	1	no	—
168	C₂	#10640	1	1	yes	characteristic
169	C₂	#13498	1	1	yes	characteristic
170	C₂	#13500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12034

C₂

yes

maximal, characteristic

#11639

C₂

yes

maximal, characteristic

#8661

C₂

yes

maximal, characteristic

#8621

C₂

yes

maximal, characteristic

#7293

C₂

yes

maximal, characteristic

#7224

C₂

yes

maximal, characteristic

#7220

C₂

yes

maximal, characteristic

C₂×C₈×Q₈

C₂²

yes

characteristic

#2163

C₂²

yes

characteristic

#2162

C₂²

yes

characteristic

#1837

C₂²

yes

characteristic

#1568

C₂²

yes

characteristic

#1568

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

Q₈⋊C₁₆

—

Q₈⋊C₁₆

—

#1301

—

#1290

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

Q₈⋊C₈

—

#324

—

C₂²×Q₈

Q₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

yes

C₂²×C₈

yes

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₁₆

yes

characteristic

C₂×C₄²

—

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

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#123

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

#123

yes

characteristic

109

C₂×C₈

#124

yes

characteristic

110

C₂×C₈

#126

yes

characteristic

111

C₂×C₈

#126

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

#121

yes

characteristic

119

C₄²

#123

yes

characteristic

120

C₄²

#135

yes

characteristic

121

C₄²

AGL(1,ℤ/8)

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

#484

yes

characteristic

126

Q₈

—

127

Q₈

—

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

—

138

C₂×C₄

—

139

C₂×C₄

#397

yes

characteristic

140

C₂×C₄

#467

yes

characteristic

141

C₂×C₄

#469

yes

characteristic

142

C₂×C₄

#482

yes

characteristic

143

C₂×C₄

#482

yes

characteristic

144

C₂×C₄

#484

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#2076

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#2080

yes

characteristic

166

C₄

#2082

yes

characteristic

167

C₂

—

168

C₂

#10640

yes

characteristic

169

C₂

#13498

yes

characteristic

170

C₂

#13500

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#484
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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

#484

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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
#11639	2
#8661	2
#8621	2
#7293	2
#7224	2
#7220	2

subgroup

index

Subgroups — #146129

Conjugacy classes of subgroups

Maximal subgroups