Group #140404 — subgroups

Subgroups — #140404

Order 512.

248 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 110 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	#12653	C₂	1	1	yes	maximal, characteristic
2	C₂×#1368	C₂	1	2	yes	maximal
3	#8807	C₂	1	1	yes	maximal, characteristic
4	#8273	C₂	1	1	yes	maximal, characteristic
5	#7156	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
7	C₂×#346	C₂²	1	2	yes	—
8	#1831	—	2	1	no	—
9	#1767	—	2	1	no	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	—	2	1	no	—
12	C₂×#334	C₂²	1	2	yes	—
13	C₂×#334	C₂²	1	1	yes	characteristic
14	C₂×#334	C₂²	1	1	yes	characteristic
15	#1374	C₄	1	1	yes	characteristic
16	#1374	C₄	1	1	yes	characteristic
17	#1368	—	2	4	no	—
18	#1277	—	2	1	no	—
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₈	—	2	1	no	—
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×#97	—	4	1	no	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	C₄³	D₄	1	1	yes	characteristic
32	C₄⋊C₁₆	—	2	4	no	—
33	#346	—	2	2	no	—
34	#346	—	2	2	no	—
35	#336	—	4	1	no	—
36	#336	—	4	1	no	—
37	#336	—	4	1	no	—
38	#335	—	4	2	no	—
39	#334	—	2	2	no	—
40	#334	—	2	1	no	—
41	#334	—	2	1	no	—
42	#334	—	2	2	no	—
43	#334	D₄	1	2	yes	—
44	#334	Q₈	1	2	yes	—
45	C₂²×C₈	—	4	1	no	—
46	C₂²×C₈	—	4	1	no	—
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	SD₁₆	1	1	yes	characteristic
55	C₂×C₄²	SD₁₆	1	1	yes	characteristic
56	C₂×C₄²	SD₁₆	1	1	yes	characteristic
57	C₂×C₄²	SD₁₆	1	1	yes	characteristic
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	M₁₆	1	4	yes	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	C₄⋊C₈	—	4	1	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄⋊C₈	—	4	1	no	—
71	#97	—	8	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	C₂×D₄	1	1	yes	characteristic
79	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₄×C₈	#41	1	1	yes	characteristic
81	C₄×C₈	#41	1	1	yes	characteristic
82	C₂²×C₄	#106	1	1	yes	characteristic
83	C₂²×C₄	#106	1	1	yes	characteristic
84	C₂²×C₄	#106	1	1	yes	characteristic
85	C₂²×C₄	#106	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
88	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	8	1	no	—
99	C₂×C₈	—	8	1	no	—
100	C₂×C₈	—	8	1	no	—
101	C₂×C₈	—	8	1	no	—
102	C₂×C₈	—	8	2	no	—
103	C₂×C₈	—	8	2	no	—
104	C₂×C₈	—	8	2	no	—
105	C₂×C₈	#108	1	2	yes	—
106	C₂×C₈	C₄×D₄	1	1	yes	characteristic
107	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
108	C₂×C₈	#125	1	1	yes	characteristic
109	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₈	C₂×M₁₆	1	2	yes	—
111	C₂×C₈	#131	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	—	4	1	no	—
119	C₄²	—	4	1	no	—
120	C₄²	#121	1	1	yes	characteristic
121	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
122	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₄²	#137	1	1	yes	characteristic
124	C₁₆	—	2	4	no	—
125	C₁₆	—	4	2	no	—
126	C₂³	C₂×#106	1	1	yes	characteristic
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	#365	1	2	yes	—
143	C₂×C₄	#389	1	1	yes	characteristic
144	C₂×C₄	#423	1	1	yes	characteristic
145	C₂×C₄	#424	1	1	yes	characteristic
146	C₂×C₄	#428	1	1	yes	characteristic
147	C₂×C₄	C₂×#108	1	1	yes	characteristic
148	C₂×C₄	#430	1	1	yes	characteristic
149	C₂×C₄	#460	1	1	yes	characteristic
150	C₂×C₄	#461	1	1	yes	characteristic
151	C₂×C₄	#475	1	1	yes	characteristic
152	C₂×C₄	#476	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	8	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	16	2	no	—
160	C₈	#435	1	2	yes	—
161	C₈	#446	1	2	yes	—
162	C₂²	#1566	1	1	yes	characteristic
163	C₂²	#1570	1	1	yes	characteristic
164	C₂²	#1948	1	1	yes	characteristic
165	C₂²	C₂×#365	1	1	yes	characteristic
166	C₂²	#2155	1	1	yes	characteristic
167	C₂²	#2155	1	1	yes	characteristic
168	C₂²	#2155	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	2	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	4	1	no	—
177	C₄	#1569	1	1	yes	characteristic
178	C₄	#1573	1	1	yes	characteristic
179	C₄	#1940	1	1	yes	characteristic
180	C₄	#1950	1	1	yes	characteristic
181	C₂	#9605	1	1	yes	characteristic
182	C₂	#12019	1	1	yes	characteristic
183	C₂	#12021	1	1	yes	characteristic
184	C₂	#12029	1	1	yes	characteristic
185	C₂	#12031	1	1	yes	characteristic
186	C₂	#13184	1	1	yes	characteristic
187	C₂	#13185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12653

C₂

yes

maximal, characteristic

C₂×#1368

C₂

yes

maximal

#8807

C₂

yes

maximal, characteristic

#8273

C₂

yes

maximal, characteristic

#7156

C₂

yes

maximal

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×#346

C₂²

yes

—

#1831

—

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×#334

C₂²

yes

—

C₂×#334

C₂²

yes

characteristic

C₂×#334

C₂²

yes

characteristic

#1374

C₄

yes

characteristic

#1374

C₄

yes

characteristic

#1368

—

#1277

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

—

—

—

—

—

—

—

—

—

—

D₄

yes

—

#334

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

#108

yes

—

106

C₂×C₈

C₄×D₄

yes

characteristic

107

C₂×C₈

C₄×Q₈

yes

characteristic

108

C₂×C₈

#125

yes

characteristic

109

C₂×C₈

C₄⋊Q₈

yes

characteristic

110

C₂×C₈

C₂×M₁₆

yes

—

111

C₂×C₈

#131

yes

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

#121

yes

characteristic

121

C₄²

C₂²⋊Q₈

yes

characteristic

122

C₄²

AGL(1,ℤ/8)

yes

characteristic

123

C₄²

#137

yes

characteristic

124

C₁₆

—

125

C₁₆

—

126

C₂³

C₂×#106

yes

characteristic

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

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#365

yes

—

143

C₂×C₄

#389

yes

characteristic

144

C₂×C₄

#423

yes

characteristic

145

C₂×C₄

#424

yes

characteristic

146

C₂×C₄

#428

yes

characteristic

147

C₂×C₄

C₂×#108

yes

characteristic

148

C₂×C₄

#430

yes

characteristic

149

C₂×C₄

#460

yes

characteristic

150

C₂×C₄

#461

yes

characteristic

151

C₂×C₄

#475

yes

characteristic

152

C₂×C₄

#476

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

#435

yes

—

161

C₈

#446

yes

—

162

C₂²

#1566

yes

characteristic

163

C₂²

#1570

yes

characteristic

164

C₂²

#1948

yes

characteristic

165

C₂²

C₂×#365

yes

characteristic

166

C₂²

#2155

yes

characteristic

167

C₂²

#2155

yes

characteristic

168

C₂²

#2155

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#1569

yes

characteristic

178

C₄

#1573

yes

characteristic

179

C₄

#1940

yes

characteristic

180

C₄

#1950

yes

characteristic

181

C₂

#9605

yes

characteristic

182

C₂

#12019

yes

characteristic

183

C₂

#12021

yes

characteristic

184

C₂

#12029

yes

characteristic

185

C₂

#12031

yes

characteristic

186

C₂

#13184

yes

characteristic

187

C₂

#13185

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#106

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×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
#12653	2
C₂×#1368	2
#8807	2
#8273	2
#7156	2

subgroup

index

Subgroups — #140404

Conjugacy classes of subgroups

Maximal subgroups