Group #141239 — subgroups

Subgroups — #141239

Order 512.

370 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 122 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#13275	C₂	1	2	yes	maximal
3	C₂×#1370	C₂	1	1	yes	maximal, characteristic
4	C₂×#1364	C₂	1	2	yes	maximal
5	C₈×#107	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	1	no	—
7	C₂×#500	C₂²	1	2	yes	—
8	C₂×#494	—	2	1	no	—
9	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₄×#107	C₂²	1	1	yes	characteristic
11	C₈×C₄⋊C₄	—	2	1	no	—
12	#1713	—	2	2	no	—
13	#1713	C₄	1	4	yes	complemented
14	C₂×#335	C₂²	1	1	yes	characteristic
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1380	—	2	4	no	—
17	#1370	—	2	2	no	—
18	#1364	—	2	4	no	—
19	#1279	—	2	1	no	—
20	C₂²×Q₁₆	—	4	1	no	—
21	C₂²×Q₁₆	—	4	1	no	—
22	C₂×C₄⋊Q₈	—	2	2	no	—
23	C₂²×C₁₆	—	2	2	no	—
24	#500	—	2	4	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	—	2	2	no	—
28	#500	—	4	2	no	—
29	#500	C₂×C₄	1	4	yes	—
30	#494	—	2	4	no	—
31	#494	—	4	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	4	2	no	—
35	C₂×#107	D₄	1	2	yes	—
36	C₂×C₄⋊C₈	—	2	1	no	—
37	C₂×C₄×C₈	—	4	1	no	—
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	C₄×C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₁₆	—	2	2	no	—
43	C₄⋊C₁₆	—	2	2	no	—
44	#340	—	2	1	no	—
45	#336	—	4	1	no	—
46	#335	—	2	1	no	—
47	#335	D₄	1	2	yes	—
48	C₈²	C₂×C₄	1	2	yes	—
49	C₈²	D₄	1	2	yes	—
50	C₂²×Q₈	—	4	2	no	—
51	C₂×Q₁₆	—	4	4	no	—
52	C₂×Q₁₆	—	4	4	no	—
53	C₂×Q₁₆	—	8	2	no	—
54	C₂×Q₁₆	—	8	2	no	—
55	C₂×Q₁₆	—	8	2	no	—
56	C₂×Q₁₆	—	8	2	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	4	1	no	—
59	C₂²×C₈	D₈	1	1	yes	characteristic
60	C₂²×C₈	D₈	1	1	yes	characteristic
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	#41	1	1	yes	characteristic
63	C₄⋊Q₈	—	2	4	no	—
64	C₄⋊Q₈	—	4	2	no	—
65	C₄⋊Q₈	—	4	2	no	—
66	C₄⋊Q₈	—	4	2	no	—
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄²	—	4	1	no	—
71	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₁₆	—	4	2	no	—
73	C₂×C₁₆	—	4	4	no	—
74	#107	—	2	2	no	—
75	#107	—	2	4	no	—
76	#107	—	2	4	no	—
77	#107	—	4	8	no	—
78	#107	—	4	2	no	—
79	#107	—	4	2	no	—
80	#107	Q₁₆	1	4	yes	complemented
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	4	1	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	8	2	no	—
85	C₄×C₈	C₂²⋊C₄	1	2	yes	—
86	C₄×C₈	C₂²⋊C₄	1	2	yes	—
87	C₄×C₈	SD₁₆	1	2	yes	—
88	C₄×C₈	C₂²×C₄	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	#95	—	4	1	no	—
93	C₂×Q₈	—	4	4	no	—
94	C₂×Q₈	—	8	2	no	—
95	C₂×Q₈	—	8	2	no	—
96	C₂×Q₈	—	8	2	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂×Q₈	—	8	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	8	2	no	—
101	C₂²×C₄	#121	1	1	yes	characteristic
102	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
103	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
104	Q₁₆	—	8	8	no	—
105	Q₁₆	—	8	8	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	8	1	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	#102	1	2	yes	—
115	C₂×C₈	#102	1	2	yes	—
116	C₂×C₈	D₁₆	1	2	yes	—
117	C₂×C₈	SD₃₂	1	2	yes	—
118	C₂×C₈	C₄×D₄	1	1	yes	characteristic
119	C₂×C₈	C₄×D₄	1	1	yes	characteristic
120	C₂×C₈	#124	1	1	yes	characteristic
121	C₂×C₈	#127	1	1	yes	characteristic
122	C₂×C₈	C₂×Q₁₆	1	2	yes	—
123	C₂×C₈	#135	1	2	yes	—
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₄⋊C₄	—	8	2	no	—
129	C₄⋊C₄	—	8	2	no	—
130	C₄²	—	8	1	no	—
131	C₄²	—	8	1	no	—
132	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₄²	#121	1	1	yes	characteristic
134	C₄²	C₂×SD₁₆	1	1	yes	characteristic
135	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₁₆	—	8	2	no	—
137	C₂³	#459	1	1	yes	characteristic
138	Q₈	—	8	4	no	—
139	Q₈	—	8	4	no	—
140	Q₈	—	16	2	no	—
141	Q₈	—	16	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	4	no	—
147	C₂×C₄	—	16	2	no	—
148	C₂×C₄	—	16	2	no	—
149	C₂×C₄	#390	1	1	yes	characteristic
150	C₂×C₄	C₂×#102	1	1	yes	characteristic
151	C₂×C₄	#418	1	1	yes	characteristic
152	C₂×C₄	#459	1	1	yes	characteristic
153	C₂×C₄	#460	1	1	yes	characteristic
154	C₂×C₄	#460	1	1	yes	characteristic
155	C₂×C₄	#467	1	2	yes	—
156	C₂×C₄	#473	1	2	yes	—
157	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
158	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
159	C₂×C₄	#509	1	1	yes	characteristic
160	C₂×C₄	#509	1	1	yes	characteristic
161	C₈	—	2	4	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	#357	1	4	yes	—
167	C₈	C₄×Q₁₆	1	2	yes	—
168	C₈	#494	1	1	yes	characteristic
169	C₈	#495	1	1	yes	characteristic
170	C₂²	#1669	1	2	yes	—
171	C₂²	#1982	1	1	yes	characteristic
172	C₂²	#2206	1	1	yes	characteristic
173	C₂²	#2207	1	1	yes	characteristic
174	C₂²	#2214	1	2	yes	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	8	4	no	—
178	C₄	—	16	4	no	—
179	C₄	#1670	1	2	yes	—
180	C₄	#1969	1	1	yes	characteristic
181	C₄	#1979	1	1	yes	characteristic
182	C₄	C₂×#357	1	1	yes	characteristic
183	C₄	#2140	1	1	yes	characteristic
184	C₄	#2206	1	1	yes	characteristic
185	C₄	#2207	1	1	yes	characteristic
186	C₂	#9733	1	1	yes	characteristic
187	C₂	#12445	1	2	yes	—
188	C₂	#12446	1	2	yes	—
189	C₂	#13252	1	1	yes	characteristic
190	C₂	#13262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#13275

C₂

yes

maximal

C₂×#1370

C₂

yes

maximal, characteristic

C₂×#1364

C₂

yes

maximal

C₈×#107

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

C₂×#494

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₄×#107

C₂²

yes

characteristic

C₈×C₄⋊C₄

—

#1713

—

#1713

C₄

yes

complemented

C₂×#335

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

—

—

—

—

—

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

D₄

yes

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

Q₁₆

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

#121

yes

characteristic

102

C₂²×C₄

C₂×D₈

yes

characteristic

103

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

104

Q₁₆

—

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

#102

yes

—

115

C₂×C₈

#102

yes

—

116

C₂×C₈

D₁₆

yes

—

117

C₂×C₈

SD₃₂

yes

—

118

C₂×C₈

C₄×D₄

yes

characteristic

119

C₂×C₈

C₄×D₄

yes

characteristic

120

C₂×C₈

#124

yes

characteristic

121

C₂×C₈

#127

yes

characteristic

122

C₂×C₈

C₂×Q₁₆

yes

—

123

C₂×C₈

#135

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂×C₂²⋊C₄

yes

characteristic

133

C₄²

#121

yes

characteristic

134

C₄²

C₂×SD₁₆

yes

characteristic

135

C₄²

AGL(1,ℤ/8)

yes

characteristic

136

C₁₆

—

137

C₂³

#459

yes

characteristic

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

#390

yes

characteristic

150

C₂×C₄

C₂×#102

yes

characteristic

151

C₂×C₄

#418

yes

characteristic

152

C₂×C₄

#459

yes

characteristic

153

C₂×C₄

#460

yes

characteristic

154

C₂×C₄

#460

yes

characteristic

155

C₂×C₄

#467

yes

—

156

C₂×C₄

#473

yes

—

157

C₂×C₄

C₂×D₁₆

yes

characteristic

158

C₂×C₄

C₂×SD₃₂

yes

characteristic

159

C₂×C₄

#509

yes

characteristic

160

C₂×C₄

#509

yes

characteristic

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

#357

yes

—

167

C₈

C₄×Q₁₆

yes

—

168

C₈

#494

yes

characteristic

169

C₈

#495

yes

characteristic

170

C₂²

#1669

yes

—

171

C₂²

#1982

yes

characteristic

172

C₂²

#2206

yes

characteristic

173

C₂²

#2207

yes

characteristic

174

C₂²

#2214

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#1670

yes

—

180

C₄

#1969

yes

characteristic

181

C₄

#1979

yes

characteristic

182

C₄

C₂×#357

yes

characteristic

183

C₄

#2140

yes

characteristic

184

C₄

#2206

yes

characteristic

185

C₄

#2207

yes

characteristic

186

C₂

#9733

yes

characteristic

187

C₂

#12445

yes

—

188

C₂

#12446

yes

—

189

C₂

#13252

yes

characteristic

190

C₂

#13262

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#459

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #1713	C₄	semidirect
80. #107	Q₁₆	semidirect

H ≅ G/N

type

13. #1713

C₄

semidirect

80. #107

Q₁₆

semidirect

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

subgroup	index
C₂×#1713	2
#13275	2
C₂×#1370	2
C₂×#1364	2
C₈×#107	2

subgroup

index

Subgroups — #141239

Conjugacy classes of subgroups

Complements

Maximal subgroups