Group #145581 — subgroups

Subgroups — #145581

Order 512.

262 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 78 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	#33728	C₂	1	1	yes	maximal, characteristic
2	#13160	C₂	1	1	yes	maximal, characteristic
3	#11939	C₂	1	2	yes	maximal
4	#9749	C₂	1	2	yes	maximal
5	#7157	C₂	1	1	yes	maximal, characteristic
6	#3160	C₂²	1	2	yes	—
7	#2255	—	2	2	no	—
8	#2255	—	2	2	no	—
9	#2158	—	2	1	no	—
10	#2158	—	2	1	no	—
11	C₂×C₄⋊C₁₆	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#346	C₂²	1	1	yes	characteristic
14	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	#1564	C₂²	1	2	yes	—
16	#1503	—	2	2	no	—
17	#1481	—	2	2	no	—
18	#1374	—	2	1	no	—
19	#1365	—	2	2	no	—
20	#1364	—	2	2	no	—
21	#533	—	2	2	no	—
22	C₂²×C₁₆	—	2	1	no	—
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	D₄	1	1	yes	characteristic
25	#500	D₄	1	2	yes	—
26	#500	D₄	1	2	yes	—
27	#499	—	2	2	no	—
28	#499	—	2	2	no	—
29	#498	—	2	2	no	—
30	#431	—	2	2	no	—
31	#427	—	2	2	no	—
32	#427	—	2	2	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×C₄⋊C₄	D₄	1	1	yes	characteristic
37	#367	—	2	2	no	—
38	#367	—	2	2	no	—
39	#366	—	2	1	no	—
40	#366	—	2	1	no	—
41	#366	—	2	1	no	—
42	#366	—	2	1	no	—
43	C₄⋊C₁₆	—	2	2	no	—
44	C₄⋊C₁₆	—	4	2	no	—
45	#346	—	2	2	no	—
46	#336	—	2	1	no	—
47	#335	—	2	2	no	—
48	#334	—	2	2	no	—
49	#328	—	4	2	no	—
50	#328	—	4	2	no	—
51	Q₈⋊C₈	—	4	4	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	#41	1	1	yes	characteristic
55	C₄⋊Q₈	—	2	2	no	—
56	C₄⋊Q₈	—	2	2	no	—
57	#125	—	4	2	no	—
58	C₂²⋊Q₈	—	4	2	no	—
59	C₂²⋊Q₈	—	4	2	no	—
60	C₄×Q₈	—	4	2	no	—
61	C₄×Q₈	—	4	2	no	—
62	#117	—	4	2	no	—
63	C₂×C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	4	1	no	—
71	C₂×C₁₆	—	4	1	no	—
72	C₂×C₁₆	—	4	1	no	—
73	C₂×C₁₆	SD₁₆	1	2	yes	—
74	C₂×C₁₆	Q₁₆	1	2	yes	—
75	#107	—	2	2	no	—
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	2	2	no	—
79	#106	—	4	2	no	—
80	#106	—	4	2	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₂²⋊C₈	—	4	2	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	—	4	2	no	—
87	C₄×C₈	C₂×D₄	1	2	yes	—
88	C₄×C₈	#41	1	2	yes	—
89	#95	D₈	1	1	yes	characteristic
90	#95	D₈	1	1	yes	characteristic
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	#121	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
97	C₂²×C₄	#135	1	1	yes	characteristic
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	8	2	no	—
104	C₂×C₈	#120	1	1	yes	characteristic
105	C₂×C₈	#123	1	2	yes	—
106	C₂×C₈	#124	1	1	yes	characteristic
107	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₈	#137	1	1	yes	characteristic
110	C₂×C₈	#137	1	1	yes	characteristic
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₂²⋊C₄	—	8	2	no	—
119	C₄²	—	4	1	no	—
120	C₄²	—	4	1	no	—
121	C₄²	—	8	2	no	—
122	C₄²	#121	1	2	yes	—
123	C₄²	AGL(1,ℤ/8)	1	2	yes	—
124	C₁₆	—	2	2	no	—
125	C₁₆	—	2	2	no	—
126	C₁₆	—	8	1	no	—
127	C₂³	#466	1	1	yes	characteristic
128	Q₈	—	8	4	no	—
129	Q₈	—	16	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	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	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	#394	1	1	yes	characteristic
141	C₂×C₄	#459	1	2	yes	—
142	C₂×C₄	#461	1	1	yes	characteristic
143	C₂×C₄	#462	1	1	yes	characteristic
144	C₂×C₄	#463	1	2	yes	—
145	C₂×C₄	#469	1	1	yes	characteristic
146	C₂×C₄	#473	1	1	yes	characteristic
147	C₂×C₄	#473	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	8	1	no	—
150	C₈	—	16	2	no	—
151	C₈	#448	1	1	yes	characteristic
152	C₈	#451	1	1	yes	characteristic
153	C₈	#487	1	1	yes	characteristic
154	C₈	#491	1	1	yes	characteristic
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#1673	1	1	yes	characteristic
158	C₂²	#1674	1	1	yes	characteristic
159	C₂²	#2054	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	2	no	—
163	C₄	—	8	1	no	—
164	C₄	—	16	2	no	—
165	C₄	—	16	2	no	—
166	C₄	#1693	1	1	yes	characteristic
167	C₄	#1694	1	1	yes	characteristic
168	C₄	#2043	1	1	yes	characteristic
169	C₄	#2047	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	#10545	1	1	yes	characteristic
173	C₂	#13444	1	1	yes	characteristic
174	C₂	#13447	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33728

C₂

yes

maximal, characteristic

#13160

C₂

yes

maximal, characteristic

#11939

C₂

yes

maximal

#9749

C₂

yes

maximal

#7157

C₂

yes

maximal, characteristic

#3160

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

—

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

#500

D₄

yes

—

#500

D₄

yes

—

—

—

—

—

—

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

D₄

yes

—

—

—

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

—

Q₈⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

D₈

yes

characteristic

#95

D₈

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

#120

yes

characteristic

105

C₂×C₈

#123

yes

—

106

C₂×C₈

#124

yes

characteristic

107

C₂×C₈

C₂×SD₁₆

yes

characteristic

108

C₂×C₈

C₂×Q₁₆

yes

characteristic

109

C₂×C₈

#137

yes

characteristic

110

C₂×C₈

#137

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

#121

yes

—

123

C₄²

AGL(1,ℤ/8)

yes

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

—

127

C₂³

#466

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

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#394

yes

characteristic

141

C₂×C₄

#459

yes

—

142

C₂×C₄

#461

yes

characteristic

143

C₂×C₄

#462

yes

characteristic

144

C₂×C₄

#463

yes

—

145

C₂×C₄

#469

yes

characteristic

146

C₂×C₄

#473

yes

characteristic

147

C₂×C₄

#473

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

#448

yes

characteristic

152

C₈

#451

yes

characteristic

153

C₈

#487

yes

characteristic

154

C₈

#491

yes

characteristic

155

C₂²

—

156

C₂²

—

157

C₂²

#1673

yes

characteristic

158

C₂²

#1674

yes

characteristic

159

C₂²

#2054

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#1693

yes

characteristic

167

C₄

#1694

yes

characteristic

168

C₄

#2043

yes

characteristic

169

C₄

#2047

yes

characteristic

170

C₂

—

171

C₂

—

172

C₂

#10545

yes

characteristic

173

C₂

#13444

yes

characteristic

174

C₂

#13447

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2054
prime radical	C₂³	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#469

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ 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
#33728	2
#13160	2
#11939	2
#9749	2
#7157	2

subgroup

index

Subgroups — #145581

Conjugacy classes of subgroups

Maximal subgroups