Group #146180 — subgroups

Subgroups — #146180

Order 512.

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

normal subgroups only

This group has 187 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	#13184	C₂	1	1	yes	maximal, characteristic
2	#13179	C₂	1	1	yes	maximal, characteristic
3	#9607	C₂	1	1	yes	maximal, characteristic
4	#8793	C₂	1	1	yes	maximal, characteristic
5	#7280	C₂	1	1	yes	maximal, characteristic
6	#7275	C₂	1	1	yes	maximal, characteristic
7	#7153	C₂	1	1	yes	maximal, characteristic
8	C₂×#368	—	2	1	no	—
9	C₂×#365	—	2	1	no	—
10	#2150	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	#1848	C₂²	1	1	yes	characteristic
13	#1846	C₂²	1	1	yes	characteristic
14	C₄×#108	C₂²	1	1	yes	characteristic
15	#1565	—	2	1	no	—
16	C₂×#334	C₂²	1	1	yes	characteristic
17	#1450	C₂²	1	1	yes	characteristic
18	#1383	—	2	1	no	—
19	#1381	—	2	1	no	—
20	#1286	—	2	1	no	—
21	#1286	—	2	1	no	—
22	#1283	—	2	1	no	—
23	C₂×M₃₂	—	2	1	no	—
24	C₂²×C₁₆	—	2	1	no	—
25	#431	D₄	1	1	yes	characteristic
26	#431	D₄	1	1	yes	characteristic
27	C₂×#108	—	2	1	no	—
28	C₂×#108	—	2	1	no	—
29	C₂×#108	D₄	1	1	yes	characteristic
30	C₂×#108	D₄	1	1	yes	characteristic
31	C₂×#107	—	2	1	no	—
32	C₂×#106	—	2	1	no	—
33	#423	—	2	1	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₄×M₁₆	—	2	1	no	—
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#382	—	2	1	no	—
38	#368	—	4	2	no	—
39	#365	—	4	2	no	—
40	C₄⋊C₁₆	—	2	1	no	—
41	C₄⋊C₁₆	—	2	1	no	—
42	C₄⋊C₁₆	Q₈	1	2	yes	—
43	C₄⋊C₁₆	Q₈	1	2	yes	—
44	#341	—	2	1	no	—
45	#335	—	4	1	no	—
46	#334	—	2	1	no	—
47	#334	—	2	1	no	—
48	#334	—	4	1	no	—
49	#322	—	2	1	no	—
50	C₈²	—	2	1	no	—
51	C₂×M₁₆	—	2	1	no	—
52	C₂×M₁₆	—	2	1	no	—
53	C₂×M₁₆	—	4	1	no	—
54	C₂²×C₈	—	4	1	no	—
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₈	#41	1	1	yes	characteristic
57	C₂×C₄⋊C₄	—	4	1	no	—
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	M₃₂	—	4	1	no	—
60	M₃₂	—	4	1	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	4	1	no	—
64	C₂×C₁₆	—	4	1	no	—
65	#108	—	2	1	no	—
66	#108	—	2	1	no	—
67	#108	—	2	1	no	—
68	#108	—	4	1	no	—
69	#108	—	4	1	no	—
70	#108	—	8	1	no	—
71	#108	SD₁₆	1	2	yes	—
72	#107	—	8	1	no	—
73	#106	—	4	1	no	—
74	#106	—	4	1	no	—
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	—	4	1	no	—
78	C₄⋊C₈	—	4	1	no	—
79	C₄⋊C₈	—	4	1	no	—
80	C₄⋊C₈	—	4	1	no	—
81	C₂²⋊C₈	—	2	1	no	—
82	C₂²⋊C₈	—	2	1	no	—
83	#97	—	4	1	no	—
84	C₄×C₈	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	4	1	no	—
87	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	#95	—	4	1	no	—
92	#95	—	4	1	no	—
93	C₂²×C₄	—	8	1	no	—
94	C₂²×C₄	#121	1	1	yes	characteristic
95	C₂²×C₄	#135	1	1	yes	characteristic
96	C₂²×C₄	#137	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	—	4	1	no	—
99	M₁₆	—	4	1	no	—
100	M₁₆	—	4	1	no	—
101	M₁₆	—	8	1	no	—
102	M₁₆	—	8	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	8	1	no	—
110	C₂×C₈	—	8	1	no	—
111	C₂×C₈	—	16	1	no	—
112	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₈	#125	1	1	yes	characteristic
115	C₂×C₈	#125	1	1	yes	characteristic
116	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₈	#135	1	1	yes	characteristic
118	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₈	#137	1	1	yes	characteristic
120	C₄⋊C₄	—	8	1	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄²	#123	1	1	yes	characteristic
123	C₄²	C₄⋊Q₈	1	1	yes	characteristic
124	C₄²	#135	1	1	yes	characteristic
125	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₁₆	—	4	2	no	—
127	C₁₆	—	4	2	no	—
128	C₂³	#464	1	1	yes	characteristic
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	16	1	no	—
133	C₂×C₄	—	16	1	no	—
134	C₂×C₄	#398	1	1	yes	characteristic
135	C₂×C₄	#463	1	1	yes	characteristic
136	C₂×C₄	#465	1	1	yes	characteristic
137	C₂×C₄	#468	1	1	yes	characteristic
138	C₂×C₄	#472	1	1	yes	characteristic
139	C₂×C₄	#473	1	1	yes	characteristic
140	C₂×C₄	#482	1	1	yes	characteristic
141	C₂×C₄	#485	1	1	yes	characteristic
142	C₂×C₄	#499	1	1	yes	characteristic
143	C₂×C₄	#501	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	4	2	no	—
147	C₈	—	8	1	no	—
148	C₈	—	8	1	no	—
149	C₈	—	8	1	no	—
150	C₈	—	16	1	no	—
151	C₈	—	16	1	no	—
152	C₈	#475	1	1	yes	characteristic
153	C₈	#476	1	1	yes	characteristic
154	C₈	#478	1	1	yes	characteristic
155	C₈	#479	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#1680	1	1	yes	characteristic
159	C₂²	#1698	1	1	yes	characteristic
160	C₂²	#2085	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	32	1	no	—
166	C₄	#1677	1	1	yes	characteristic
167	C₄	#1699	1	1	yes	characteristic
168	C₄	#2088	1	1	yes	characteristic
169	C₄	#2089	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	#10657	1	1	yes	characteristic
173	C₂	#13512	1	1	yes	characteristic
174	C₂	#13513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13184

C₂

yes

maximal, characteristic

#13179

C₂

yes

maximal, characteristic

#9607

C₂

yes

maximal, characteristic

#8793

C₂

yes

maximal, characteristic

#7280

C₂

yes

maximal, characteristic

#7275

C₂

yes

maximal, characteristic

#7153

C₂

yes

maximal, characteristic

C₂×#368

—

C₂×#365

—

#2150

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

C₄×#108

C₂²

yes

characteristic

#1565

—

C₂×#334

C₂²

yes

characteristic

#1450

C₂²

yes

—

—

—

—

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

D₄

yes

characteristic

#431

D₄

yes

—

—

D₄

yes

characteristic

C₂×#108

D₄

yes

—

—

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

Q₈

yes

—

C₄⋊C₁₆

Q₈

yes

—

—

—

—

—

—

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

SD₁₆

yes

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²⋊Q₈

yes

characteristic

113

C₂×C₈

C₂²⋊Q₈

yes

characteristic

114

C₂×C₈

#125

yes

characteristic

115

C₂×C₈

#125

yes

characteristic

116

C₂×C₈

C₂×SD₁₆

yes

characteristic

117

C₂×C₈

#135

yes

characteristic

118

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

119

C₂×C₈

#137

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

#123

yes

characteristic

123

C₄²

C₄⋊Q₈

yes

characteristic

124

C₄²

#135

yes

characteristic

125

C₄²

AGL(1,ℤ/8)

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₂³

#464

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#398

yes

characteristic

135

C₂×C₄

#463

yes

characteristic

136

C₂×C₄

#465

yes

characteristic

137

C₂×C₄

#468

yes

characteristic

138

C₂×C₄

#472

yes

characteristic

139

C₂×C₄

#473

yes

characteristic

140

C₂×C₄

#482

yes

characteristic

141

C₂×C₄

#485

yes

characteristic

142

C₂×C₄

#499

yes

characteristic

143

C₂×C₄

#501

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

#475

yes

characteristic

153

C₈

#476

yes

characteristic

154

C₈

#478

yes

characteristic

155

C₈

#479

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

#1680

yes

characteristic

159

C₂²

#1698

yes

characteristic

160

C₂²

#2085

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#1677

yes

characteristic

167

C₄

#1699

yes

characteristic

168

C₄

#2088

yes

characteristic

169

C₄

#2089

yes

characteristic

170

C₂

—

171

C₂

—

172

C₂

#10657

yes

characteristic

173

C₂

#13512

yes

characteristic

174

C₂

#13513

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2085
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#464
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2085

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#464

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
#13184	2
#13179	2
#9607	2
#8793	2
#7280	2
#7275	2
#7153	2

subgroup

index

Subgroups — #146180

Conjugacy classes of subgroups

Maximal subgroups