Group #144169 — subgroups

Subgroups — #144169

Order 512.

449 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2241	C₂	1	2	yes	maximal
2	C₂×#1713	C₂	1	1	yes	maximal, characteristic
3	#13203	C₂	1	2	yes	maximal
4	#9823	C₂	1	1	yes	maximal, characteristic
5	#7136	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	1	no	—
7	C₂×#500	—	2	1	no	—
8	C₂×#500	C₂²	1	1	yes	characteristic
9	C₂×#494	C₂²	1	2	yes	—
10	#2241	—	2	4	no	—
11	#2241	—	2	4	no	—
12	C₂×#358	—	2	4	no	—
13	C₂×#358	—	2	4	no	—
14	C₂×C₄×C₁₆	C₂²	1	2	yes	—
15	#1990	—	2	1	no	—
16	#1956	—	2	1	no	—
17	#1845	C₂²	1	1	yes	characteristic
18	#1713	—	2	4	no	—
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂²×Q₁₆	—	2	2	no	—
22	C₂×C₄⋊Q₈	—	2	2	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂²×C₁₆	D₄	1	4	yes	—
25	#500	—	2	4	no	—
26	#500	—	2	4	no	—
27	#500	—	2	2	no	—
28	#500	—	2	1	no	—
29	#500	—	4	2	no	—
30	#500	—	4	2	no	—
31	#500	D₄	1	2	yes	—
32	#494	—	2	2	no	—
33	#494	—	2	2	no	—
34	#494	—	2	2	no	—
35	#494	D₄	1	4	yes	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#107	—	4	2	no	—
39	C₂×#107	—	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	1	no	—
45	#358	—	2	16	no	—
46	#358	—	4	8	no	—
47	C₄×C₁₆	—	2	2	no	—
48	C₄×C₁₆	—	2	2	no	—
49	#340	—	4	2	no	—
50	C₈²	—	2	2	no	—
51	C₂²×Q₈	—	4	2	no	—
52	C₂²×Q₈	—	4	1	no	—
53	C₂×Q₁₆	—	2	4	no	—
54	C₂×Q₁₆	—	2	4	no	—
55	C₂×Q₁₆	—	4	2	no	—
56	C₂×Q₁₆	—	4	2	no	—
57	C₂×Q₁₆	—	4	4	no	—
58	C₂×Q₁₆	—	4	4	no	—
59	C₂×Q₁₆	—	4	4	no	—
60	C₂×Q₁₆	—	4	4	no	—
61	C₂²×C₈	D₈	1	2	yes	—
62	C₂²×C₈	D₈	1	2	yes	—
63	C₂²×C₈	C₂×D₄	1	2	yes	—
64	C₄⋊Q₈	—	2	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₄⋊Q₈	—	4	2	no	—
68	C₄⋊Q₈	—	4	1	no	—
69	C₄⋊Q₈	—	4	2	no	—
70	C₄⋊Q₈	—	4	2	no	—
71	C₄⋊Q₈	—	4	1	no	—
72	C₂×C₄⋊C₄	—	4	1	no	—
73	C₂×C₄⋊C₄	—	4	1	no	—
74	C₂×C₄⋊C₄	—	4	2	no	—
75	C₂×C₄⋊C₄	—	4	2	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄⋊C₄	—	4	1	no	—
78	C₂×C₄⋊C₄	—	4	1	no	—
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₁₆	—	2	4	no	—
81	C₂×C₁₆	—	2	4	no	—
82	C₂×C₁₆	—	2	4	no	—
83	#107	—	2	4	no	—
84	#107	—	4	8	no	—
85	#107	—	4	8	no	—
86	#107	—	4	2	no	—
87	#107	—	4	4	no	—
88	#103	—	8	4	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	C₂×D₄	1	2	yes	—
94	C₄×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄×C₈	#41	1	1	yes	characteristic
96	C₂×Q₈	—	4	4	no	—
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	—	8	1	no	—
99	C₂×Q₈	—	8	2	no	—
100	C₂×Q₈	—	8	2	no	—
101	C₂×Q₈	—	8	2	no	—
102	C₂×Q₈	—	8	2	no	—
103	C₂×Q₈	—	8	2	no	—
104	C₂×Q₈	—	8	2	no	—
105	C₂×Q₈	—	8	2	no	—
106	C₂²×C₄	—	8	2	no	—
107	C₂²×C₄	—	8	1	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	#127	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
112	Q₁₆	—	4	8	no	—
113	Q₁₆	—	4	8	no	—
114	Q₁₆	—	8	4	no	—
115	Q₁₆	—	8	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	SD₃₂	1	4	yes	—
121	C₂×C₈	Q₃₂	1	4	yes	—
122	C₂×C₈	#120	1	2	yes	—
123	C₂×C₈	#121	1	2	yes	—
124	C₂×C₈	#135	1	4	yes	—
125	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₄⋊C₄	—	4	4	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	1	no	—
131	C₄⋊C₄	—	8	2	no	—
132	C₄⋊C₄	—	8	2	no	—
133	C₄⋊C₄	—	8	1	no	—
134	C₄⋊C₄	—	8	2	no	—
135	C₄⋊C₄	—	8	2	no	—
136	C₄⋊C₄	—	8	2	no	—
137	C₄²	#120	1	1	yes	characteristic
138	C₄²	#121	1	1	yes	characteristic
139	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₁₆	—	4	4	no	—
142	C₂³	#493	1	1	yes	characteristic
143	Q₈	—	8	4	no	—
144	Q₈	—	8	4	no	—
145	Q₈	—	8	4	no	—
146	Q₈	—	16	2	no	—
147	Q₈	—	16	2	no	—
148	Q₈	—	16	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	16	2	no	—
152	C₂×C₄	—	16	2	no	—
153	C₂×C₄	—	16	1	no	—
154	C₂×C₄	—	16	1	no	—
155	C₂×C₄	—	16	1	no	—
156	C₂×C₄	—	16	1	no	—
157	C₂×C₄	—	16	1	no	—
158	C₂×C₄	#392	1	1	yes	characteristic
159	C₂×C₄	#447	1	1	yes	characteristic
160	C₂×C₄	#447	1	1	yes	characteristic
161	C₂×C₄	#459	1	1	yes	characteristic
162	C₂×C₄	#459	1	1	yes	characteristic
163	C₂×C₄	#495	1	2	yes	—
164	C₂×C₄	#496	1	1	yes	characteristic
165	C₂×C₄	#496	1	2	yes	—
166	C₂×C₄	C₂×SD₃₂	1	2	yes	—
167	C₂×C₄	C₂×Q₃₂	1	2	yes	—
168	C₂×C₄	#510	1	2	yes	—
169	C₂×C₄	#510	1	2	yes	—
170	C₈	—	2	4	no	—
171	C₈	—	2	4	no	—
172	C₈	#449	1	4	yes	—
173	C₈	#463	1	4	yes	—
174	C₂²	#1709	1	1	yes	characteristic
175	C₂²	#1721	1	1	yes	characteristic
176	C₂²	#2011	1	1	yes	characteristic
177	C₂²	#2247	1	1	yes	characteristic
178	C₂²	#2248	1	1	yes	characteristic
179	C₂²	#2251	1	1	yes	characteristic
180	C₂²	#2251	1	1	yes	characteristic
181	C₄	—	16	2	no	—
182	C₄	—	16	4	no	—
183	C₄	—	32	1	no	—
184	C₄	#1717	1	2	yes	—
185	C₄	#2012	1	2	yes	—
186	C₄	#2185	1	2	yes	—
187	C₄	#2188	1	2	yes	—
188	C₄	#2208	1	2	yes	—
189	C₄	#2209	1	2	yes	—
190	C₂	#10307	1	1	yes	characteristic
191	C₂	#12510	1	1	yes	characteristic
192	C₂	#12514	1	1	yes	characteristic
193	C₂	#12526	1	1	yes	characteristic
194	C₂	#12528	1	1	yes	characteristic
195	C₂	#13327	1	1	yes	characteristic
196	C₂	#13330	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2241

C₂

yes

maximal

C₂×#1713

C₂

yes

maximal, characteristic

#13203

C₂

yes

maximal

#9823

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

—

C₂×#500

C₂²

yes

characteristic

C₂×#494

C₂²

yes

—

—

—

—

—

C₂×C₄×C₁₆

C₂²

yes

—

#1990

—

#1956

—

#1845

C₂²

yes

characteristic

#1713

—

C₂×C₈²

C₂²

yes

characteristic

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

—

—

—

—

—

—

D₄

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

C₈²

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

#127

yes

characteristic

110

C₂²×C₄

C₂×D₈

yes

characteristic

111

C₂²×C₄

C₂×D₈

yes

characteristic

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

115

Q₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

SD₃₂

yes

—

121

C₂×C₈

Q₃₂

yes

—

122

C₂×C₈

#120

yes

—

123

C₂×C₈

#121

yes

—

124

C₂×C₈

#135

yes

—

125

C₂×C₈

AGL(1,ℤ/8)

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

#120

yes

characteristic

138

C₄²

#121

yes

characteristic

139

C₄²

AGL(1,ℤ/8)

yes

characteristic

140

C₄²

AGL(1,ℤ/8)

yes

characteristic

141

C₁₆

—

142

C₂³

#493

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

Q₈

—

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#392

yes

characteristic

159

C₂×C₄

#447

yes

characteristic

160

C₂×C₄

#447

yes

characteristic

161

C₂×C₄

#459

yes

characteristic

162

C₂×C₄

#459

yes

characteristic

163

C₂×C₄

#495

yes

—

164

C₂×C₄

#496

yes

characteristic

165

C₂×C₄

#496

yes

—

166

C₂×C₄

C₂×SD₃₂

yes

—

167

C₂×C₄

C₂×Q₃₂

yes

—

168

C₂×C₄

#510

yes

—

169

C₂×C₄

#510

yes

—

170

C₈

—

171

C₈

—

172

C₈

#449

yes

—

173

C₈

#463

yes

—

174

C₂²

#1709

yes

characteristic

175

C₂²

#1721

yes

characteristic

176

C₂²

#2011

yes

characteristic

177

C₂²

#2247

yes

characteristic

178

C₂²

#2248

yes

characteristic

179

C₂²

#2251

yes

characteristic

180

C₂²

#2251

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#1717

yes

—

185

C₄

#2012

yes

—

186

C₄

#2185

yes

—

187

C₄

#2188

yes

—

188

C₄

#2208

yes

—

189

C₄

#2209

yes

—

190

C₂

#10307

yes

characteristic

191

C₂

#12510

yes

characteristic

192

C₂

#12514

yes

characteristic

193

C₂

#12526

yes

characteristic

194

C₂

#12528

yes

characteristic

195

C₂

#13327

yes

characteristic

196

C₂

#13330

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×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, Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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×#2241	2
C₂×#1713	2
#13203	2
#9823	2
#7136	2

subgroup

index

Subgroups — #144169

Conjugacy classes of subgroups

Maximal subgroups