Group #128743 — subgroups

Subgroups — #128743

Order 512.

301 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 112 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×#1593	C₂	1	1	yes	maximal, characteristic
2	C₄×#362	C₂	1	1	yes	maximal, characteristic
3	#11594	C₂	1	1	yes	maximal, characteristic
4	#11404	C₂	1	1	yes	maximal, characteristic
5	#8295	C₂	1	1	yes	maximal, characteristic
6	#7173	C₂	1	1	yes	maximal, characteristic
7	#7158	C₂	1	1	yes	maximal, characteristic
8	C₂×#446	—	2	1	no	—
9	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×#362	C₄	1	2	yes	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₄×#108	C₂²	1	1	yes	characteristic
14	#1593	—	2	2	no	—
15	#1593	C₄	1	4	yes	—
16	#1564	C₂²	1	1	yes	characteristic
17	C₂×#334	C₂²	1	1	yes	characteristic
18	C₂×Q₈⋊C₈	—	2	1	no	—
19	#1451	—	2	1	no	—
20	C₂×#322	C₂²	1	1	yes	characteristic
21	#1387	C₄	1	2	yes	—
22	#1332	C₄	1	4	yes	—
23	#1313	—	2	2	no	—
24	#1278	—	2	1	no	—
25	C₂²×Q₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Q₈	—	2	1	no	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂×M₃₂	C₂×C₄	1	2	yes	—
29	#446	—	2	4	no	—
30	#446	—	4	2	no	—
31	C₄×Q₁₆	—	2	1	no	—
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	C₂×C₄	1	2	yes	—
34	C₄×Q₁₆	C₂×C₄	1	2	yes	—
35	#431	—	2	1	no	—
36	C₂×#108	C₂×C₄	1	2	yes	—
37	C₂×#107	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	—	4	1	no	—
40	C₂×#103	—	2	1	no	—
41	C₄×M₁₆	—	2	1	no	—
42	C₂×#97	—	4	1	no	—
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	#362	—	2	4	no	—
47	#346	—	2	1	no	—
48	#346	—	2	1	no	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	C₂×C₄	1	2	yes	—
51	#341	—	2	1	no	—
52	#335	C₂×C₄	1	2	yes	—
53	#334	—	2	1	no	—
54	#334	—	2	1	no	—
55	#334	C₂×C₄	1	2	yes	—
56	Q₈⋊C₈	—	2	4	no	—
57	#322	—	4	2	no	—
58	#322	C₂×C₄	1	2	yes	—
59	#322	D₄	1	2	yes	—
60	C₂²×Q₈	—	2	1	no	—
61	C₂×Q₁₆	—	2	1	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₂×Q₁₆	—	2	2	no	—
64	C₂×Q₁₆	C₄²	1	2	yes	—
65	C₂×M₁₆	—	2	2	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	4	1	no	—
69	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₈	#41	1	1	yes	characteristic
71	C₄×Q₈	—	2	2	no	—
72	C₄×Q₈	—	2	2	no	—
73	C₄×Q₈	—	4	1	no	—
74	C₄×Q₈	—	4	1	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄²	—	4	1	no	—
78	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	M₃₂	—	4	2	no	—
81	M₃₂	M₁₆	1	4	yes	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	2	2	no	—
84	#108	—	2	2	no	—
85	#108	—	2	2	no	—
86	#107	—	2	1	no	—
87	#107	C₄²	1	2	yes	—
88	C₄⋊C₈	—	2	2	no	—
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	4	4	no	—
91	C₄⋊C₈	—	4	1	no	—
92	#103	—	2	2	no	—
93	#103	—	2	2	no	—
94	C₂²⋊C₈	—	4	1	no	—
95	#97	—	4	4	no	—
96	#97	—	4	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	C₂²⋊C₄	1	2	yes	—
102	C₄×C₈	C₂²⋊C₄	1	2	yes	—
103	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₄×C₈	C₂×D₄	1	1	yes	characteristic
106	C₄×C₈	#41	1	1	yes	characteristic
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	4	2	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂²×C₄	—	4	1	no	—
112	C₂²×C₄	—	4	1	no	—
113	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂²×C₄	#137	1	1	yes	characteristic
116	Q₁₆	—	2	2	no	—
117	Q₁₆	—	2	2	no	—
118	Q₁₆	—	4	2	no	—
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₈	#117	1	1	yes	characteristic
132	C₂×C₈	C₄×D₄	1	1	yes	characteristic
133	C₂×C₈	C₄×D₄	1	1	yes	characteristic
134	C₂×C₈	C₂×M₁₆	1	2	yes	—
135	C₂×C₈	#131	1	2	yes	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄²	—	4	2	no	—
141	C₄²	—	4	2	no	—
142	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₄²	C₄×D₄	1	1	yes	characteristic
144	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₁₆	—	2	4	no	—
147	C₁₆	—	4	2	no	—
148	C₂³	#440	1	1	yes	characteristic
149	Q₈	—	4	2	no	—
150	Q₈	—	4	2	no	—
151	Q₈	—	8	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#417	1	1	yes	characteristic
160	C₂×C₄	#418	1	1	yes	characteristic
161	C₂×C₄	#436	1	2	yes	—
162	C₂×C₄	#440	1	1	yes	characteristic
163	C₂×C₄	#442	1	1	yes	characteristic
164	C₂×C₄	#442	1	1	yes	characteristic
165	C₂×C₄	#444	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	4	2	no	—
169	C₈	—	4	2	no	—
170	C₈	—	8	2	no	—
171	C₈	—	8	2	no	—
172	C₈	—	8	2	no	—
173	C₈	C₄×M₁₆	1	1	yes	characteristic
174	C₈	#405	1	1	yes	characteristic
175	C₈	#435	1	2	yes	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#1592	1	2	yes	—
179	C₂²	#1762	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	8	2	no	—
184	C₄	—	8	2	no	—
185	C₄	#1594	1	2	yes	—
186	C₄	#1753	1	1	yes	characteristic
187	C₄	#1765	1	1	yes	characteristic
188	C₄	#2180	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	#8208	1	1	yes	characteristic
191	C₂	#12648	1	1	yes	characteristic
192	C₂	#12651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1593

C₂

yes

maximal, characteristic

C₄×#362

C₂

yes

maximal, characteristic

#11594

C₂

yes

maximal, characteristic

#11404

C₂

yes

maximal, characteristic

#8295

C₂

yes

maximal, characteristic

#7173

C₂

yes

maximal, characteristic

#7158

C₂

yes

maximal, characteristic

C₂×#446

—

C₂×C₄×Q₁₆

C₂²

yes

characteristic

C₂×#362

C₄

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₄×#108

C₂²

yes

characteristic

#1593

—

#1593

C₄

yes

—

#1564

C₂²

yes

characteristic

C₂×#334

C₂²

yes

characteristic

C₂×Q₈⋊C₈

—

#1451

—

C₂×#322

C₂²

yes

characteristic

#1387

C₄

yes

—

#1332

C₄

yes

—

#1313

—

#1278

—

C₂²×Q₁₆

C₂×C₄

yes

characteristic

C₂×C₄×Q₈

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

—

#446

—

#446

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂×C₄

yes

—

C₄×Q₁₆

C₂×C₄

yes

—

#431

—

C₂×#108

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×M₁₆

—

C₂×#97

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

#341

—

#335

C₂×C₄

yes

—

#334

—

#334

—

#334

C₂×C₄

yes

—

Q₈⋊C₈

—

#322

—

#322

C₂×C₄

yes

—

#322

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₄²

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

M₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

C₄²

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂²⋊C₄

yes

—

102

C₄×C₈

C₂²⋊C₄

yes

—

103

C₄×C₈

C₂²×C₄

yes

characteristic

104

C₄×C₈

C₂²×C₄

yes

characteristic

105

C₄×C₈

C₂×D₄

yes

characteristic

106

C₄×C₈

#41

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₄×D₄

yes

characteristic

114

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

115

C₂²×C₄

#137

yes

characteristic

116

Q₁₆

—

117

Q₁₆

—

118

Q₁₆

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₂×C₄²

yes

characteristic

131

C₂×C₈

#117

yes

characteristic

132

C₂×C₈

C₄×D₄

yes

characteristic

133

C₂×C₈

C₄×D₄

yes

characteristic

134

C₂×C₈

C₂×M₁₆

yes

—

135

C₂×C₈

#131

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

C₂×C₂²⋊C₄

yes

characteristic

143

C₄²

C₄×D₄

yes

characteristic

144

C₄²

AGL(1,ℤ/8)

yes

characteristic

145

C₄²

AGL(1,ℤ/8)

yes

characteristic

146

C₁₆

—

147

C₁₆

—

148

C₂³

#440

yes

characteristic

149

Q₈

—

150

Q₈

—

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₄×C₂²⋊C₄

yes

characteristic

159

C₂×C₄

#417

yes

characteristic

160

C₂×C₄

#418

yes

characteristic

161

C₂×C₄

#436

yes

—

162

C₂×C₄

#440

yes

characteristic

163

C₂×C₄

#442

yes

characteristic

164

C₂×C₄

#442

yes

characteristic

165

C₂×C₄

#444

yes

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

C₄×M₁₆

yes

characteristic

174

C₈

#405

yes

characteristic

175

C₈

#435

yes

—

176

C₂²

—

177

C₂²

—

178

C₂²

#1592

yes

—

179

C₂²

#1762

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#1594

yes

—

186

C₄

#1753

yes

characteristic

187

C₄

#1765

yes

characteristic

188

C₄

#2180

yes

—

189

C₂

—

190

C₂

#8208

yes

characteristic

191

C₂

#12648

yes

characteristic

192

C₂

#12651

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1762
prime radical	C₂³	#440
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#442

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ 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
C₂×#1593	2
C₄×#362	2
#11594	2
#11404	2
#8295	2
#7173	2
#7158	2

subgroup

index

Subgroups — #128743

Conjugacy classes of subgroups

Maximal subgroups