Group #140751 — subgroups

Subgroups — #140751

Order 512.

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

normal subgroups only

This group has 384 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	C₂×#2177	C₂	1	1	yes	maximal, characteristic
2	C₂×#1368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1333	C₂	1	2	yes	maximal, complemented
4	C₂×#1313	C₂	1	1	yes	maximal, characteristic, complemented
5	#9659	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×D₈	C₂²	1	2	yes	—
8	#2177	—	2	2	no	—
9	#2177	—	2	2	no	—
10	C₂×#367	C₄	1	1	yes	characteristic
11	C₂×#357	C₄	1	2	yes	—
12	C₂×#346	C₂²	1	1	yes	characteristic
13	C₂×#346	C₂²	1	1	yes	characteristic
14	#1954	—	2	2	no	—
15	C₂×#334	C₂²	1	2	yes	—
16	C₂×D₄⋊C₈	—	2	2	no	—
17	C₂×#322	C₂²	1	1	yes	characteristic
18	#1368	—	2	2	no	—
19	#1333	—	2	4	no	—
20	#1313	—	2	2	no	—
21	C₂²×D₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₄	—	2	2	no	—
23	C₂×D₁₆	—	2	2	no	—
24	C₂×D₁₆	—	2	2	no	—
25	C₂×D₁₆	C₈	1	4	yes	complemented
26	C₂²×C₁₆	—	2	1	no	—
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₄×D₈	—	2	2	no	—
30	C₄×D₈	—	2	2	no	—
31	C₄×D₈	—	2	2	no	—
32	C₄×D₈	—	2	2	no	—
33	C₂×#107	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×#102	—	2	2	no	—
36	C₂×C₂²⋊C₈	—	4	4	no	—
37	C₂×C₄×C₈	D₄	1	2	yes	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#367	—	2	2	no	—
40	#357	—	2	4	no	—
41	#346	—	2	1	no	—
42	#346	D₄	1	2	yes	—
43	#346	D₄	1	2	yes	—
44	#346	D₄	1	2	yes	—
45	#334	—	2	2	no	—
46	#334	—	2	2	no	—
47	D₄⋊C₈	—	4	4	no	—
48	#322	—	2	1	no	—
49	#322	—	2	1	no	—
50	C₂²×D₄	—	2	2	no	—
51	C₂³×C₄	—	4	2	no	—
52	C₂×D₈	—	2	2	no	—
53	C₂×D₈	—	2	2	no	—
54	C₂×D₈	—	2	2	no	—
55	C₂×D₈	—	2	2	no	—
56	C₂×D₈	—	2	2	no	—
57	C₂×D₈	C₂×C₈	1	4	yes	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	—	4	2	no	—
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₈	#41	1	1	yes	characteristic
62	C₄×D₄	—	4	2	no	—
63	C₄×D₄	—	4	2	no	—
64	C₄×D₄	—	4	2	no	—
65	C₄×D₄	—	4	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₂²⋊C₄	—	4	2	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	D₁₆	—	2	8	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	1	no	—
74	C₂×C₁₆	—	4	1	no	—
75	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
76	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
77	C₂×C₁₆	C₂×C₈	1	2	yes	—
78	C₂×C₁₆	M₁₆	1	2	yes	—
79	#107	—	2	2	no	—
80	#107	—	2	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	#102	—	4	2	no	—
84	#102	—	4	2	no	—
85	C₂²⋊C₈	—	8	8	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	C₂×D₄	1	1	yes	characteristic
89	C₄×C₈	C₂×D₄	1	1	yes	characteristic
90	C₄×C₈	C₂×D₄	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₂⁴	—	4	2	no	—
93	C₂×D₄	—	2	4	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	8	4	no	—
101	C₂²×C₄	—	8	4	no	—
102	C₂²×C₄	—	8	2	no	—
103	C₂²×C₄	—	8	2	no	—
104	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
106	D₈	—	2	4	no	—
107	D₈	—	2	4	no	—
108	D₈	—	4	2	no	—
109	D₈	—	4	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	4	4	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	8	4	no	—
115	C₂×C₈	—	8	2	no	—
116	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₈	C₄×D₄	1	1	yes	characteristic
118	C₂×C₈	#121	1	1	yes	characteristic
119	C₂×C₈	#124	1	1	yes	characteristic
120	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₈	#131	1	1	yes	characteristic
123	C₂×C₈	#131	1	1	yes	characteristic
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	—	8	2	no	—
127	C₂²⋊C₄	—	8	2	no	—
128	C₄²	#121	1	1	yes	characteristic
129	C₄²	#127	1	1	yes	characteristic
130	C₄²	AGL(1,ℤ/8)	1	2	yes	—
131	C₁₆	—	2	2	no	—
132	C₁₆	—	4	2	no	—
133	C₁₆	C₂²⋊C₈	1	4	yes	complemented
134	C₂³	—	4	4	no	—
135	C₂³	—	8	2	no	—
136	C₂³	—	8	4	no	—
137	C₂³	—	8	4	no	—
138	C₂³	—	8	2	no	—
139	C₂³	#442	1	1	yes	characteristic
140	D₄	—	4	4	no	—
141	D₄	—	4	4	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	16	4	no	—
148	C₂×C₄	—	16	2	no	—
149	C₂×C₄	—	16	2	no	—
150	C₂×C₄	#390	1	1	yes	characteristic
151	C₂×C₄	C₈×D₄	1	1	yes	characteristic
152	C₂×C₄	#436	1	1	yes	characteristic
153	C₂×C₄	#442	1	1	yes	characteristic
154	C₂×C₄	#444	1	1	yes	characteristic
155	C₂×C₄	#444	1	1	yes	characteristic
156	C₂×C₄	#469	1	2	yes	—
157	C₂×C₄	#496	1	2	yes	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	8	2	no	—
161	C₈	—	16	2	no	—
162	C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₈	#408	1	1	yes	characteristic
164	C₈	C₈×D₄	1	1	yes	characteristic
165	C₈	#436	1	1	yes	characteristic
166	C₂²	—	8	8	no	—
167	C₂²	—	8	4	no	—
168	C₂²	—	16	4	no	—
169	C₂²	—	16	2	no	—
170	C₂²	—	16	2	no	—
171	C₂²	#1580	1	1	yes	characteristic
172	C₂²	#1594	1	1	yes	characteristic
173	C₂²	#1972	1	1	yes	characteristic
174	C₂²	AGL(1,ℤ/16)	1	4	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	16	2	no	—
178	C₄	#1580	1	1	yes	characteristic
179	C₄	#1594	1	1	yes	characteristic
180	C₄	#1954	1	1	yes	characteristic
181	C₄	#1975	1	1	yes	characteristic
182	C₂	—	16	4	no	—
183	C₂	#9659	1	1	yes	characteristic
184	C₂	#12067	1	2	yes	—
185	C₂	#12089	1	2	yes	—
186	C₂	#13249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2177

C₂

yes

maximal, characteristic

C₂×#1368

C₂

yes

maximal, characteristic, complemented

C₂×#1333

C₂

yes

maximal, complemented

C₂×#1313

C₂

yes

maximal, characteristic, complemented

#9659

C₂

yes

maximal, complemented

C₂²×D₁₆

C₄

yes

characteristic

C₂×C₄×D₈

C₂²

yes

—

#2177

—

#2177

—

C₂×#367

C₄

yes

characteristic

C₂×#357

C₄

yes

—

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1954

—

C₂×#334

C₂²

yes

—

C₂×D₄⋊C₈

—

C₂×#322

C₂²

yes

—

—

—

C₂²×D₈

C₂×C₄

yes

—

C₂×C₄×D₄

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

C₈

yes

complemented

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₄×D₈

—

C₄×D₈

—

C₄×D₈

—

C₄×D₈

—

C₂×#107

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×#102

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

D₄

yes

—

#346

D₄

yes

—

#346

D₄

yes

—

#334

—

#334

—

D₄⋊C₈

—

#322

—

#322

—

C₂²×D₄

—

C₂³×C₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×C₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₄

yes

characteristic

105

C₂²×C₄

AGL(1,ℤ/8)

yes

—

106

D₈

—

107

D₈

—

108

D₈

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

117

C₂×C₈

C₄×D₄

yes

characteristic

118

C₂×C₈

#121

yes

characteristic

119

C₂×C₈

#124

yes

characteristic

120

C₂×C₈

C₂²×C₈

yes

characteristic

121

C₂×C₈

C₂×M₁₆

yes

characteristic

122

C₂×C₈

#131

yes

characteristic

123

C₂×C₈

#131

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

#121

yes

characteristic

129

C₄²

#127

yes

characteristic

130

C₄²

AGL(1,ℤ/8)

yes

—

131

C₁₆

—

132

C₁₆

—

133

C₁₆

C₂²⋊C₈

yes

complemented

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

#442

yes

characteristic

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#390

yes

characteristic

151

C₂×C₄

C₈×D₄

yes

characteristic

152

C₂×C₄

#436

yes

characteristic

153

C₂×C₄

#442

yes

characteristic

154

C₂×C₄

#444

yes

characteristic

155

C₂×C₄

#444

yes

characteristic

156

C₂×C₄

#469

yes

—

157

C₂×C₄

#496

yes

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

C₂×C₂²⋊C₈

yes

characteristic

163

C₈

#408

yes

characteristic

164

C₈

C₈×D₄

yes

characteristic

165

C₈

#436

yes

characteristic

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#1580

yes

characteristic

172

C₂²

#1594

yes

characteristic

173

C₂²

#1972

yes

characteristic

174

C₂²

AGL(1,ℤ/16)

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#1580

yes

characteristic

179

C₄

#1594

yes

characteristic

180

C₄

#1954

yes

characteristic

181

C₄

#1975

yes

characteristic

182

C₂

—

183

C₂

#9659

yes

characteristic

184

C₂

#12067

yes

—

185

C₂

#12089

yes

—

186

C₂

#13249

yes

—

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂³

#442

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂²×D₁₆

C₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×#1368	C₂	semidirect
3. C₂×#1333	C₂	semidirect
4. C₂×#1313	C₂	semidirect
5. #9659	C₂	semidirect
25. C₂×D₁₆	C₈	semidirect
133. C₁₆	C₂²⋊C₈	semidirect

H ≅ G/N

type

2. C₂×#1368

C₂

semidirect

3. C₂×#1333

C₂

semidirect

4. C₂×#1313

C₂

semidirect

5. #9659

C₂

semidirect

25. C₂×D₁₆

C₈

semidirect

133. C₁₆

C₂²⋊C₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#2177	2
C₂×#1368	2
C₂×#1333	2
C₂×#1313	2
#9659	2

subgroup

index

Subgroups — #140751

Conjugacy classes of subgroups

Complements

Maximal subgroups