Group #145977 — subgroups

Subgroups — #145977

Order 512.

198 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 74 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	#13289	C₂	1	1	yes	maximal, characteristic
2	#13273	C₂	1	1	yes	maximal, characteristic
3	#13188	C₂	1	1	yes	maximal, characteristic
4	#13164	C₂	1	1	yes	maximal, characteristic
5	#12011	C₂	1	1	yes	maximal, characteristic
6	C₂×#1366	C₂	1	1	yes	maximal, characteristic
7	#7270	C₂	1	1	yes	maximal, characteristic
8	C₂×#499	C₂²	1	1	yes	characteristic
9	#2155	—	2	1	no	—
10	#2153	—	2	1	no	—
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	C₂×#364	C₂²	1	1	yes	characteristic
14	C₄×M₃₂	C₂²	1	1	yes	characteristic
15	#1848	C₂²	1	1	yes	characteristic
16	#1772	C₂²	1	1	yes	characteristic
17	#1388	—	2	1	no	—
18	#1387	—	2	1	no	—
19	#1387	—	2	1	no	—
20	#1386	—	2	1	no	—
21	#1383	—	2	1	no	—
22	#1378	—	2	1	no	—
23	#1373	—	2	2	no	—
24	#1366	—	2	2	no	—
25	C₂×#125	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	D₄	1	1	yes	characteristic
30	C₂×M₃₂	D₄	1	1	yes	characteristic
31	#499	—	2	1	no	—
32	#499	—	2	1	no	—
33	#499	—	2	1	no	—
34	#499	Q₈	1	2	yes	—
35	C₂×#108	—	2	1	no	—
36	C₂×#108	—	2	1	no	—
37	C₂×#108	D₄	1	1	yes	characteristic
38	#427	D₄	1	1	yes	characteristic
39	C₂×#107	—	2	1	no	—
40	C₂×#106	—	2	1	no	—
41	#423	—	2	1	no	—
42	#405	—	2	1	no	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#365	—	4	2	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	—	2	1	no	—
48	#364	—	2	1	no	—
49	#364	—	2	1	no	—
50	#364	—	4	2	no	—
51	#364	D₄	1	2	yes	—
52	#346	—	2	1	no	—
53	C₄×C₁₆	—	2	1	no	—
54	#339	—	2	1	no	—
55	C₂×M₁₆	—	2	1	no	—
56	C₂×M₁₆	—	4	1	no	—
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	#125	—	2	2	no	—
60	#125	—	4	1	no	—
61	#125	—	8	1	no	—
62	#117	—	2	1	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	8	1	no	—
65	C₂×C₄²	#41	1	1	yes	characteristic
66	M₃₂	—	2	1	no	—
67	M₃₂	—	2	2	no	—
68	M₃₂	—	2	1	no	—
69	M₃₂	—	2	1	no	—
70	M₃₂	—	4	1	no	—
71	M₃₂	—	4	1	no	—
72	M₃₂	—	4	1	no	—
73	M₃₂	—	4	1	no	—
74	M₃₂	—	4	1	no	—
75	M₃₂	SD₁₆	1	2	yes	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	4	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	#108	—	2	1	no	—
82	#108	—	4	1	no	—
83	#108	—	4	1	no	—
84	#108	—	4	1	no	—
85	#108	—	4	1	no	—
86	#108	Q₁₆	1	2	yes	—
87	#107	—	2	2	no	—
88	#107	—	2	1	no	—
89	#107	—	4	1	no	—
90	#106	—	2	2	no	—
91	#106	—	2	1	no	—
92	#106	—	4	1	no	—
93	C₄⋊C₈	—	8	1	no	—
94	#97	—	4	1	no	—
95	C₄×C₈	—	4	1	no	—
96	C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₄×C₈	#41	1	1	yes	characteristic
99	C₄×C₈	#41	1	1	yes	characteristic
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	8	1	no	—
102	C₂²×C₄	#121	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	1	no	—
105	M₁₆	—	8	1	no	—
106	M₁₆	—	8	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	8	1	no	—
109	C₂×C₈	#121	1	1	yes	characteristic
110	C₂×C₈	#121	1	1	yes	characteristic
111	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₈	#135	1	1	yes	characteristic
116	C₂×C₈	#135	1	1	yes	characteristic
117	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₈	#137	1	1	yes	characteristic
119	C₂×C₈	#137	1	1	yes	characteristic
120	C₂×C₈	#137	1	1	yes	characteristic
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	8	2	no	—
124	C₄⋊C₄	—	8	2	no	—
125	C₄⋊C₄	—	8	1	no	—
126	C₂²⋊C₄	—	4	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	4	1	no	—
129	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
130	C₄²	#126	1	1	yes	characteristic
131	C₁₆	—	2	2	no	—
132	C₁₆	—	4	2	no	—
133	C₁₆	—	4	1	no	—
134	C₁₆	—	8	1	no	—
135	C₁₆	—	8	1	no	—
136	C₂³	#471	1	1	yes	characteristic
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	16	2	no	—
143	C₂×C₄	#396	1	1	yes	characteristic
144	C₂×C₄	#461	1	1	yes	characteristic
145	C₂×C₄	#464	1	1	yes	characteristic
146	C₂×C₄	#467	1	1	yes	characteristic
147	C₂×C₄	#468	1	1	yes	characteristic
148	C₂×C₄	#472	1	1	yes	characteristic
149	C₈	—	4	2	no	—
150	C₈	—	16	1	no	—
151	C₈	#462	1	1	yes	characteristic
152	C₈	#463	1	1	yes	characteristic
153	C₈	#465	1	1	yes	characteristic
154	C₈	#470	1	1	yes	characteristic
155	C₈	#475	1	1	yes	characteristic
156	C₈	#477	1	1	yes	characteristic
157	C₈	#479	1	1	yes	characteristic
158	C₈	#479	1	1	yes	characteristic
159	C₂²	—	4	1	no	—
160	C₂²	#1684	1	1	yes	characteristic
161	C₂²	#1691	1	1	yes	characteristic
162	C₂²	#2063	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	16	1	no	—
166	C₄	—	16	2	no	—
167	C₄	#1675	1	1	yes	characteristic
168	C₄	#1698	1	1	yes	characteristic
169	C₄	#2066	1	1	yes	characteristic
170	C₄	#2068	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#10601	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13289

C₂

yes

maximal, characteristic

#13273

C₂

yes

maximal, characteristic

#13188

C₂

yes

maximal, characteristic

#13164

C₂

yes

maximal, characteristic

#12011

C₂

yes

maximal, characteristic

C₂×#1366

C₂

yes

maximal, characteristic

#7270

C₂

yes

maximal, characteristic

C₂×#499

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#364

C₂²

yes

characteristic

C₂×#364

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1772

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

characteristic

C₂×M₃₂

D₄

yes

—

—

—

Q₈

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

D₄

yes

characteristic

#427

D₄

yes

—

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

#346

—

C₄×C₁₆

—

#339

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

Q₁₆

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

#121

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

#121

yes

characteristic

110

C₂×C₈

#121

yes

characteristic

111

C₂×C₈

C₂²⋊Q₈

yes

characteristic

112

C₂×C₈

C₂²⋊Q₈

yes

characteristic

113

C₂×C₈

C₂×SD₁₆

yes

characteristic

114

C₂×C₈

C₂×Q₁₆

yes

characteristic

115

C₂×C₈

#135

yes

characteristic

116

C₂×C₈

#135

yes

characteristic

117

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₈

#137

yes

characteristic

119

C₂×C₈

#137

yes

characteristic

120

C₂×C₈

#137

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

C₂²⋊Q₈

yes

characteristic

130

C₄²

#126

yes

characteristic

131

C₁₆

—

132

C₁₆

—

133

C₁₆

—

134

C₁₆

—

135

C₁₆

—

136

C₂³

#471

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#396

yes

characteristic

144

C₂×C₄

#461

yes

characteristic

145

C₂×C₄

#464

yes

characteristic

146

C₂×C₄

#467

yes

characteristic

147

C₂×C₄

#468

yes

characteristic

148

C₂×C₄

#472

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

#462

yes

characteristic

152

C₈

#463

yes

characteristic

153

C₈

#465

yes

characteristic

154

C₈

#470

yes

characteristic

155

C₈

#475

yes

characteristic

156

C₈

#477

yes

characteristic

157

C₈

#479

yes

characteristic

158

C₈

#479

yes

characteristic

159

C₂²

—

160

C₂²

#1684

yes

characteristic

161

C₂²

#1691

yes

characteristic

162

C₂²

#2063

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#1675

yes

characteristic

168

C₄

#1698

yes

characteristic

169

C₄

#2066

yes

characteristic

170

C₄

#2068

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#10601

yes

characteristic

	subgroup	G/H
center, socle	C₂	#10601
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#10601

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#471

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
#13289	2
#13273	2
#13188	2
#13164	2
#12011	2
C₂×#1366	2
#7270	2

subgroup

index

Subgroups — #145977

Conjugacy classes of subgroups

Maximal subgroups