Group #141776 — subgroups

Subgroups — #141776

Order 512.

400 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 150 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×#1707	C₂	1	1	yes	maximal, characteristic
2	#13282	C₂	1	2	yes	maximal
3	#13203	C₂	1	2	yes	maximal
4	C₂×#1312	C₂	1	1	yes	maximal, characteristic
5	#8818	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₂×#498	—	2	1	no	—
8	C₂×#494	C₂²	1	1	yes	characteristic
9	C₂×#487	—	2	1	no	—
10	C₂×#358	—	2	4	no	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	#1845	—	2	1	no	—
13	#1845	—	2	1	no	—
14	#1845	C₂²	1	2	yes	—
15	#1707	—	2	2	no	—
16	#1707	C₄	1	4	yes	complemented
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1380	—	2	4	no	—
19	#1312	—	2	2	no	—
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₄⋊Q₈	—	2	1	no	—
23	C₂²×C₁₆	D₄	1	4	yes	—
24	#500	—	2	1	no	—
25	#500	—	2	2	no	—
26	#500	C₂×C₄	1	2	yes	—
27	#498	—	2	4	no	—
28	#498	—	4	2	no	—
29	#494	—	2	1	no	—
30	#494	—	2	2	no	—
31	#494	C₂×C₄	1	2	yes	—
32	#487	—	2	4	no	—
33	#487	—	4	2	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#107	—	2	2	no	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	—	4	2	no	—
38	C₂×#107	—	4	2	no	—
39	C₂×#106	—	4	2	no	—
40	C₂×#103	—	4	2	no	—
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#384	—	2	2	no	—
45	#358	—	4	8	no	—
46	C₄×C₁₆	—	2	2	no	—
47	C₄×C₁₆	—	2	2	no	—
48	C₈²	C₂×C₄	1	2	yes	—
49	C₈²	D₄	1	2	yes	—
50	C₂²×Q₈	—	4	1	no	—
51	C₂²×Q₈	—	4	1	no	—
52	C₂×Q₁₆	—	2	4	no	—
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	4	4	no	—
55	C₂×Q₁₆	—	4	4	no	—
56	C₂²×C₈	D₈	1	2	yes	—
57	C₂²×C₈	D₈	1	2	yes	—
58	C₂²×C₈	C₂×D₄	1	2	yes	—
59	C₄⋊Q₈	—	2	2	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	4	1	no	—
62	C₄⋊Q₈	—	4	1	no	—
63	C₄⋊Q₈	—	4	1	no	—
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	1	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₂×C₄⋊C₄	—	4	1	no	—
68	C₂×C₄⋊C₄	—	4	1	no	—
69	C₂×C₄⋊C₄	—	4	1	no	—
70	C₂×C₄⋊C₄	—	4	1	no	—
71	C₂×C₄⋊C₄	—	4	2	no	—
72	C₂×C₄⋊C₄	—	4	2	no	—
73	C₂×C₄⋊C₄	—	4	2	no	—
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	—	2	4	no	—
78	#107	—	2	4	no	—
79	#107	—	4	2	no	—
80	#107	—	4	4	no	—
81	#107	—	4	4	no	—
82	#107	—	8	4	no	—
83	#107	—	8	4	no	—
84	#106	—	4	8	no	—
85	#103	—	4	8	no	—
86	C₄×C₈	C₂²⋊C₄	1	2	yes	—
87	C₄×C₈	C₂²⋊C₄	1	2	yes	—
88	C₄×C₈	SD₁₆	1	2	yes	—
89	C₄×C₈	SD₁₆	1	2	yes	—
90	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₄×C₈	C₂×D₄	1	1	yes	characteristic
92	C₄×C₈	#41	1	2	yes	—
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	8	1	no	—
96	C₂×Q₈	—	8	1	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂×Q₈	—	8	2	no	—
99	C₂×Q₈	—	8	2	no	—
100	C₂×Q₈	—	8	2	no	—
101	C₂²×C₄	—	8	1	no	—
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	—	8	2	no	—
104	C₂²×C₄	#127	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
107	Q₁₆	—	4	8	no	—
108	Q₁₆	—	8	4	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	#102	1	4	yes	—
112	C₂×C₈	#102	1	4	yes	—
113	C₂×C₈	SD₃₂	1	4	yes	—
114	C₂×C₈	Q₃₂	1	4	yes	—
115	C₂×C₈	C₄×D₄	1	2	yes	—
116	C₂×C₈	#121	1	2	yes	—
117	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
118	C₂×C₈	#137	1	4	yes	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	8	1	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₄⋊C₄	—	8	1	no	—
126	C₄⋊C₄	—	8	1	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₄⋊C₄	—	8	2	no	—
129	C₄⋊C₄	—	8	4	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₄²	#124	1	1	yes	characteristic
133	C₄²	C₂×SD₁₆	1	1	yes	characteristic
134	C₄²	C₂×SD₁₆	1	1	yes	characteristic
135	C₁₆	—	4	4	no	—
136	C₂³	#493	1	1	yes	characteristic
137	Q₈	—	8	4	no	—
138	Q₈	—	8	4	no	—
139	Q₈	—	16	2	no	—
140	Q₈	—	16	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	16	1	no	—
144	C₂×C₄	—	16	1	no	—
145	C₂×C₄	—	16	1	no	—
146	C₂×C₄	—	16	1	no	—
147	C₂×C₄	—	16	2	no	—
148	C₂×C₄	—	16	2	no	—
149	C₂×C₄	—	16	2	no	—
150	C₂×C₄	#390	1	1	yes	characteristic
151	C₂×C₄	C₂×#102	1	1	yes	characteristic
152	C₂×C₄	C₂×#102	1	1	yes	characteristic
153	C₂×C₄	#486	1	1	yes	characteristic
154	C₂×C₄	#486	1	1	yes	characteristic
155	C₂×C₄	#492	1	1	yes	characteristic
156	C₂×C₄	#496	1	2	yes	—
157	C₂×C₄	#497	1	2	yes	—
158	C₂×C₄	C₂×SD₃₂	1	2	yes	—
159	C₂×C₄	C₂×Q₃₂	1	2	yes	—
160	C₂×C₄	#509	1	2	yes	—
161	C₂×C₄	#510	1	2	yes	—
162	C₈	—	2	4	no	—
163	C₈	#358	1	8	yes	complemented
164	C₈	#440	1	4	yes	—
165	C₈	#468	1	4	yes	—
166	C₂²	#1721	1	1	yes	characteristic
167	C₂²	#1722	1	1	yes	characteristic
168	C₂²	#1957	1	1	yes	characteristic
169	C₂²	#2247	1	1	yes	characteristic
170	C₂²	#2248	1	1	yes	characteristic
171	C₂²	#2250	1	1	yes	characteristic
172	C₂²	#2251	1	1	yes	characteristic
173	C₄	—	16	2	no	—
174	C₄	—	16	2	no	—
175	C₄	—	32	2	no	—
176	C₄	#1714	1	2	yes	—
177	C₄	#1973	1	2	yes	—
178	C₄	C₂×#358	1	2	yes	—
179	C₄	#2139	1	2	yes	—
180	C₄	#2241	1	2	yes	—
181	C₄	#2243	1	2	yes	—
182	C₂	#9842	1	1	yes	characteristic
183	C₂	#12528	1	1	yes	characteristic
184	C₂	#12529	1	1	yes	characteristic
185	C₂	#12534	1	1	yes	characteristic
186	C₂	#12535	1	1	yes	characteristic
187	C₂	#13203	1	1	yes	characteristic
188	C₂	#13205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1707

C₂

yes

maximal, characteristic

#13282

C₂

yes

maximal

#13203

C₂

yes

maximal

C₂×#1312

C₂

yes

maximal, characteristic

#8818

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

C₂×#498

—

C₂×#494

C₂²

yes

characteristic

C₂×#487

—

C₂×#358

—

C₂×C₄×C₁₆

C₂²

yes

—

#1845

—

#1845

—

#1845

C₂²

yes

—

#1707

—

#1707

C₄

yes

complemented

C₂×C₈²

C₂²

yes

characteristic

#1380

—

#1312

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

#500

—

#500

—

#500

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

#127

yes

characteristic

105

C₂²×C₄

C₂×D₈

yes

characteristic

106

C₂²×C₄

C₂×D₈

yes

characteristic

107

Q₁₆

—

108

Q₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#102

yes

—

112

C₂×C₈

#102

yes

—

113

C₂×C₈

SD₃₂

yes

—

114

C₂×C₈

Q₃₂

yes

—

115

C₂×C₈

C₄×D₄

yes

—

116

C₂×C₈

#121

yes

—

117

C₂×C₈

AGL(1,ℤ/8)

yes

—

118

C₂×C₈

#137

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

C₂×C₂²⋊C₄

yes

characteristic

132

C₄²

#124

yes

characteristic

133

C₄²

C₂×SD₁₆

yes

characteristic

134

C₄²

C₂×SD₁₆

yes

characteristic

135

C₁₆

—

136

C₂³

#493

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

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

C₂×#102

yes

characteristic

152

C₂×C₄

C₂×#102

yes

characteristic

153

C₂×C₄

#486

yes

characteristic

154

C₂×C₄

#486

yes

characteristic

155

C₂×C₄

#492

yes

characteristic

156

C₂×C₄

#496

yes

—

157

C₂×C₄

#497

yes

—

158

C₂×C₄

C₂×SD₃₂

yes

—

159

C₂×C₄

C₂×Q₃₂

yes

—

160

C₂×C₄

#509

yes

—

161

C₂×C₄

#510

yes

—

162

C₈

—

163

C₈

#358

yes

complemented

164

C₈

#440

yes

—

165

C₈

#468

yes

—

166

C₂²

#1721

yes

characteristic

167

C₂²

#1722

yes

characteristic

168

C₂²

#1957

yes

characteristic

169

C₂²

#2247

yes

characteristic

170

C₂²

#2248

yes

characteristic

171

C₂²

#2250

yes

characteristic

172

C₂²

#2251

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#1714

yes

—

177

C₄

#1973

yes

—

178

C₄

C₂×#358

yes

—

179

C₄

#2139

yes

—

180

C₄

#2241

yes

—

181

C₄

#2243

yes

—

182

C₂

#9842

yes

characteristic

183

C₂

#12528

yes

characteristic

184

C₂

#12529

yes

characteristic

185

C₂

#12534

yes

characteristic

186

C₂

#12535

yes

characteristic

187

C₂

#13203

yes

characteristic

188

C₂

#13205

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#493
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

#493

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
16. #1707	C₄	semidirect
163. C₈	#358	semidirect

H ≅ G/N

type

16. #1707

C₄

semidirect

163. C₈

#358

semidirect

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

subgroup	index
C₂×#1707	2
#13282	2
#13203	2
C₂×#1312	2
#8818	2

subgroup

index

Subgroups — #141776

Conjugacy classes of subgroups

Complements

Maximal subgroups