Group #141240 — subgroups

Subgroups — #141240

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	#13276	C₂	1	2	yes	maximal
3	C₂×#1370	C₂	1	1	yes	maximal, characteristic
4	C₂×#1365	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	#1365	—	2	4	no	—
19	#1279	—	2	1	no	—
20	C₂²×Q₁₆	—	4	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	#500	—	2	4	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	—	4	2	no	—
28	#500	C₂×C₄	1	4	yes	—
29	#494	—	2	4	no	—
30	#494	—	4	2	no	—
31	C₂×#107	—	2	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	4	1	no	—
34	C₂×#107	—	4	1	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	8	no	—
52	C₂×Q₁₆	—	8	8	no	—
53	C₂²×C₈	—	2	1	no	—
54	C₂²×C₈	—	4	1	no	—
55	C₂²×C₈	D₈	1	1	yes	characteristic
56	C₂²×C₈	D₈	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	#41	1	1	yes	characteristic
59	C₄⋊Q₈	—	2	4	no	—
60	C₄⋊Q₈	—	4	2	no	—
61	C₄⋊Q₈	—	4	2	no	—
62	C₄⋊Q₈	—	4	2	no	—
63	C₂×C₄⋊C₄	—	2	2	no	—
64	C₂×C₄⋊C₄	—	4	2	no	—
65	C₂×C₄⋊C₄	—	4	2	no	—
66	C₂×C₄²	—	4	1	no	—
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₁₆	—	4	2	no	—
69	C₂×C₁₆	—	4	4	no	—
70	#107	—	2	2	no	—
71	#107	—	2	4	no	—
72	#107	—	2	4	no	—
73	#107	—	4	4	no	—
74	#107	—	4	4	no	—
75	#107	—	4	2	no	—
76	#107	—	4	2	no	—
77	#107	Q₁₆	1	4	yes	complemented
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	4	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	8	2	no	—
82	C₄×C₈	C₂²⋊C₄	1	2	yes	—
83	C₄×C₈	C₂²⋊C₄	1	2	yes	—
84	C₄×C₈	SD₁₆	1	2	yes	—
85	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	C₂×D₄	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	#95	—	4	1	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	8	8	no	—
92	C₂×Q₈	—	8	2	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	8	2	no	—
95	C₂²×C₄	#121	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
97	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
98	Q₁₆	—	8	16	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	—	8	1	no	—
107	C₂×C₈	#102	1	2	yes	—
108	C₂×C₈	#102	1	2	yes	—
109	C₂×C₈	SD₃₂	1	2	yes	—
110	C₂×C₈	Q₃₂	1	2	yes	—
111	C₂×C₈	C₄×D₄	1	1	yes	characteristic
112	C₂×C₈	C₄×D₄	1	1	yes	characteristic
113	C₂×C₈	#124	1	1	yes	characteristic
114	C₂×C₈	#127	1	1	yes	characteristic
115	C₂×C₈	C₂×Q₁₆	1	2	yes	—
116	C₂×C₈	#135	1	2	yes	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	4	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	8	2	no	—
122	C₄⋊C₄	—	8	2	no	—
123	C₄²	—	8	1	no	—
124	C₄²	—	8	1	no	—
125	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₄²	#121	1	1	yes	characteristic
127	C₄²	C₂×SD₁₆	1	1	yes	characteristic
128	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₁₆	—	8	2	no	—
130	C₂³	#459	1	1	yes	characteristic
131	Q₈	—	8	8	no	—
132	Q₈	—	16	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	16	4	no	—
139	C₂×C₄	#390	1	1	yes	characteristic
140	C₂×C₄	C₂×#102	1	1	yes	characteristic
141	C₂×C₄	#418	1	1	yes	characteristic
142	C₂×C₄	#459	1	1	yes	characteristic
143	C₂×C₄	#460	1	1	yes	characteristic
144	C₂×C₄	#460	1	1	yes	characteristic
145	C₂×C₄	#467	1	2	yes	—
146	C₂×C₄	#473	1	2	yes	—
147	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
149	C₂×C₄	#510	1	1	yes	characteristic
150	C₂×C₄	#510	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	8	2	no	—
156	C₈	#358	1	4	yes	—
157	C₈	C₄×Q₁₆	1	2	yes	—
158	C₈	#494	1	1	yes	characteristic
159	C₈	#495	1	1	yes	characteristic
160	C₂²	#1669	1	2	yes	—
161	C₂²	#1982	1	1	yes	characteristic
162	C₂²	#2208	1	1	yes	characteristic
163	C₂²	#2209	1	1	yes	characteristic
164	C₂²	#2213	1	2	yes	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	8	4	no	—
168	C₄	—	16	4	no	—
169	C₄	#1670	1	2	yes	—
170	C₄	#1969	1	1	yes	characteristic
171	C₄	#1979	1	1	yes	characteristic
172	C₄	C₂×#358	1	1	yes	characteristic
173	C₄	#2141	1	1	yes	characteristic
174	C₄	#2208	1	1	yes	characteristic
175	C₄	#2209	1	1	yes	characteristic
176	C₂	#9733	1	1	yes	characteristic
177	C₂	#12447	1	2	yes	—
178	C₂	#12448	1	2	yes	—
179	C₂	#13253	1	1	yes	characteristic
180	C₂	#13261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#13276

C₂

yes

maximal

C₂×#1370

C₂

yes

maximal, characteristic

C₂×#1365

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

—

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

—

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

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

#102

yes

—

108

C₂×C₈

#102

yes

—

109

C₂×C₈

SD₃₂

yes

—

110

C₂×C₈

Q₃₂

yes

—

111

C₂×C₈

C₄×D₄

yes

characteristic

112

C₂×C₈

C₄×D₄

yes

characteristic

113

C₂×C₈

#124

yes

characteristic

114

C₂×C₈

#127

yes

characteristic

115

C₂×C₈

C₂×Q₁₆

yes

—

116

C₂×C₈

#135

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂×C₂²⋊C₄

yes

characteristic

126

C₄²

#121

yes

characteristic

127

C₄²

C₂×SD₁₆

yes

characteristic

128

C₄²

AGL(1,ℤ/8)

yes

characteristic

129

C₁₆

—

130

C₂³

#459

yes

characteristic

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#390

yes

characteristic

140

C₂×C₄

C₂×#102

yes

characteristic

141

C₂×C₄

#418

yes

characteristic

142

C₂×C₄

#459

yes

characteristic

143

C₂×C₄

#460

yes

characteristic

144

C₂×C₄

#460

yes

characteristic

145

C₂×C₄

#467

yes

—

146

C₂×C₄

#473

yes

—

147

C₂×C₄

C₂×SD₃₂

yes

characteristic

148

C₂×C₄

C₂×Q₃₂

yes

characteristic

149

C₂×C₄

#510

yes

characteristic

150

C₂×C₄

#510

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

#358

yes

—

157

C₈

C₄×Q₁₆

yes

—

158

C₈

#494

yes

characteristic

159

C₈

#495

yes

characteristic

160

C₂²

#1669

yes

—

161

C₂²

#1982

yes

characteristic

162

C₂²

#2208

yes

characteristic

163

C₂²

#2209

yes

characteristic

164

C₂²

#2213

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#1670

yes

—

170

C₄

#1969

yes

characteristic

171

C₄

#1979

yes

characteristic

172

C₄

C₂×#358

yes

characteristic

173

C₄

#2141

yes

characteristic

174

C₄

#2208

yes

characteristic

175

C₄

#2209

yes

characteristic

176

C₂

#9733

yes

characteristic

177

C₂

#12447

yes

—

178

C₂

#12448

yes

—

179

C₂

#13253

yes

characteristic

180

C₂

#13261

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
77. #107	Q₁₆	semidirect

H ≅ G/N

type

13. #1713

C₄

semidirect

77. #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
#13276	2
C₂×#1370	2
C₂×#1365	2
C₈×#107	2

subgroup

index

Subgroups — #141240

Conjugacy classes of subgroups

Complements

Maximal subgroups