Group #507423 — subgroups

Subgroups — #507423

Order 512.

548 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 166 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	#33311	C₂	1	1	yes	maximal, characteristic, complemented
2	#21705	C₂	1	1	yes	maximal, characteristic
3	#13372	C₂	1	4	yes	maximal, complemented
4	#13211	C₂	1	2	yes	maximal
5	#12887	C₂	1	2	yes	maximal, complemented
6	#12823	C₂	1	4	yes	maximal, complemented
7	C₄×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#499	—	2	1	no	—
9	C₂×#499	C₂²	1	2	yes	—
10	#3147	C₂²	1	1	yes	characteristic
11	C₂×#443	—	2	1	no	—
12	C₂×#443	C₂²	1	2	yes	—
13	#2931	—	2	1	no	—
14	#2859	—	2	1	no	—
15	C₂×#364	C₂²	1	2	yes	—
16	C₂×#360	C₂²	1	8	yes	—
17	C₄×M₃₂	C₂²	1	2	yes	—
18	#1974	—	2	1	no	—
19	#1974	C₂²	1	2	yes	—
20	#1962	—	2	1	no	—
21	#1848	C₂²	1	1	yes	characteristic
22	#1846	—	2	1	no	—
23	#1846	C₂²	1	2	yes	—
24	#1845	—	2	1	no	—
25	#1844	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1383	C₂²	1	8	yes	—
28	#1376	C₂²	1	2	yes	—
29	C₂×#135	—	2	1	no	—
30	C₂×#135	C₂³	1	2	yes	—
31	C₂×#131	—	2	2	no	—
32	#532	—	2	1	no	—
33	C₂×#125	—	2	2	no	—
34	C₂×M₃₂	C₂³	1	4	yes	—
35	#499	—	4	4	no	—
36	#499	—	4	4	no	—
37	#495	—	2	1	no	—
38	#495	Q₈	1	2	yes	—
39	#494	Q₈	1	1	yes	characteristic
40	#493	Q₈	1	1	yes	characteristic
41	#492	—	2	1	no	—
42	#443	—	2	8	no	—
43	#443	—	4	4	no	—
44	#436	—	4	2	no	—
45	C₈×D₄	—	4	2	no	—
46	C₂×#108	—	2	1	no	—
47	C₂×#108	C₂³	1	2	yes	—
48	C₂×#107	—	2	2	no	—
49	C₂×#107	—	2	2	no	—
50	C₂×#106	—	2	2	no	—
51	C₂×#106	C₂³	1	4	yes	—
52	#423	—	2	1	no	—
53	C₂×#104	—	2	2	no	—
54	C₂×#104	—	2	2	no	—
55	#408	—	2	2	no	—
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
62	#389	—	4	4	no	—
63	#384	—	2	2	no	—
64	#382	—	2	2	no	—
65	C₄³	D₄	1	1	yes	characteristic
66	#364	—	2	4	no	—
67	#360	—	4	8	no	—
68	#346	—	2	2	no	—
69	C₄×C₁₆	—	2	2	no	—
70	C₂×#41	—	2	2	no	—
71	#135	—	2	4	no	—
72	#135	—	2	4	no	—
73	#135	—	4	2	no	—
74	#135	—	4	2	no	—
75	C₂×Q₁₆	—	2	1	no	—
76	C₂×Q₁₆	C₂×Q₈	1	2	yes	—
77	C₂×SD₁₆	—	2	2	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×D₈	—	2	1	no	—
80	C₂×D₈	C₂×Q₈	1	2	yes	—
81	#131	—	4	4	no	—
82	#131	—	4	4	no	—
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	—	4	2	no	—
85	C₂²×C₈	—	4	2	no	—
86	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₈	C₂⁴	1	1	yes	characteristic
90	C₄⋊Q₈	—	2	1	no	—
91	#127	—	2	1	no	—
92	#125	—	4	4	no	—
93	#125	—	4	4	no	—
94	#124	—	2	2	no	—
95	#121	—	4	2	no	—
96	C₄×D₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄⋊C₄	—	4	4	no	—
99	C₂×C₄⋊C₄	—	4	4	no	—
100	C₂×C₄²	—	2	2	no	—
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
104	M₃₂	—	2	4	no	—
105	M₃₂	—	2	4	no	—
106	C₂×C₁₆	—	2	4	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#107	—	4	4	no	—
110	#107	—	4	4	no	—
111	#106	—	2	8	no	—
112	#106	—	4	4	no	—
113	C₄⋊C₈	—	4	2	no	—
114	#104	—	4	4	no	—
115	#104	—	4	4	no	—
116	C₂²⋊C₈	—	4	4	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	D₈	1	4	yes	—
120	C₄×C₈	SD₁₆	1	2	yes	—
121	C₄×C₈	SD₁₆	1	2	yes	—
122	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
123	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
124	C₄×C₈	#41	1	1	yes	characteristic
125	C₄×C₈	#41	1	1	yes	characteristic
126	C₄×C₈	#41	1	4	yes	—
127	#95	—	4	4	no	—
128	#41	—	4	2	no	—
129	#41	—	4	2	no	—
130	#41	—	4	2	no	—
131	#41	—	4	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	—	4	2	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	4	4	no	—
140	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
142	Q₁₆	—	2	4	no	—
143	Q₁₆	—	4	2	no	—
144	SD₁₆	—	4	4	no	—
145	SD₁₆	—	4	4	no	—
146	D₈	—	2	4	no	—
147	D₈	—	4	2	no	—
148	M₁₆	—	4	2	no	—
149	M₁₆	—	4	2	no	—
150	M₁₆	—	8	4	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	8	4	no	—
155	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₈	#124	1	2	yes	—
158	C₂×C₈	C₂×D₈	1	2	yes	—
159	C₂×C₈	C₂×D₈	1	2	yes	—
160	C₂×C₈	C₂×SD₁₆	1	2	yes	—
161	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
162	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
163	C₂×C₈	#137	1	2	yes	—
164	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₈	C₂×#41	1	2	yes	—
166	C₂×C₈	C₂×#41	1	1	yes	characteristic
167	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	8	8	no	—
172	C₄⋊C₄	—	8	8	no	—
173	C₂²⋊C₄	—	4	4	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
178	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
179	C₄²	#124	1	2	yes	—
180	C₄²	C₂×D₈	1	2	yes	—
181	C₄²	C₂×SD₁₆	1	2	yes	—
182	C₁₆	—	4	4	no	—
183	C₂³	—	4	2	no	—
184	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	4	no	—
190	D₄	—	8	2	no	—
191	D₄	—	8	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	8	no	—
201	C₂×C₄	—	8	4	no	—
202	C₂×C₄	#476	1	2	yes	—
203	C₂×C₄	#476	1	2	yes	—
204	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
205	C₂×C₄	C₂×#124	1	1	yes	characteristic
206	C₂×C₄	#535	1	1	yes	characteristic
207	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
208	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
209	C₂×C₄	C₂×#137	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	8	2	no	—
212	C₈	#474	1	2	yes	—
213	C₈	#474	1	2	yes	—
214	C₈	#486	1	2	yes	—
215	C₈	#486	1	2	yes	—
216	C₈	#554	1	2	yes	—
217	C₈	#555	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	4	no	—
220	C₂²	—	8	2	no	—
221	C₂²	C₂×#476	1	1	yes	characteristic
222	C₂²	#3141	1	1	yes	characteristic
223	C₂²	#3205	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	8	2	no	—
229	C₄	—	16	4	no	—
230	C₄	#2563	1	1	yes	characteristic
231	C₄	C₂×#474	1	1	yes	characteristic
232	C₄	C₂×#486	1	1	yes	characteristic
233	C₄	#3206	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	16	2	no	—
237	C₂	#29139	1	1	yes	characteristic
238	C₂	#33637	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic, complemented

#21705

C₂

yes

maximal, characteristic

#13372

C₂

yes

maximal, complemented

#13211

C₂

yes

maximal

#12887

C₂

yes

maximal, complemented

#12823

C₂

yes

maximal, complemented

C₄×#364

C₂

yes

maximal, characteristic, complemented

C₂×#499

—

C₂×#499

C₂²

yes

—

#3147

C₂²

yes

characteristic

C₂×#443

—

C₂×#443

C₂²

yes

—

#2931

—

#2859

—

C₂×#364

C₂²

yes

—

C₂×#360

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1974

—

#1974

C₂²

yes

—

#1962

—

#1848

C₂²

yes

characteristic

#1846

—

#1846

C₂²

yes

—

#1845

—

#1844

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1383

C₂²

yes

—

#1376

C₂²

yes

—

C₂×#135

—

C₂×#135

C₂³

yes

—

C₂×#131

—

#532

—

C₂×#125

—

C₂×M₃₂

C₂³

yes

—

—

—

—

Q₈

yes

—

#494

Q₈

yes

characteristic

#493

Q₈

yes

—

—

—

—

C₈×D₄

—

C₂×#108

—

C₂×#108

C₂³

yes

—

—

—

—

C₂³

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄³

D₄

yes

—

—

—

C₄×C₁₆

—

C₂×#41

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×Q₈

yes

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

C₂×Q₈

yes

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

—

—

—

—

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₄

yes

characteristic

102

C₂×C₄²

C₂×D₄

yes

characteristic

103

C₂×C₄²

C₂×D₄

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

108

#108

—

109

#107

—

110

#107

—

111

#106

—

112

#106

—

113

C₄⋊C₈

—

114

#104

—

115

#104

—

116

C₂²⋊C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

D₈

yes

—

120

C₄×C₈

SD₁₆

yes

—

121

C₄×C₈

SD₁₆

yes

—

122

C₄×C₈

C₂×Q₈

yes

characteristic

123

C₄×C₈

C₂×Q₈

yes

characteristic

124

C₄×C₈

#41

yes

characteristic

125

C₄×C₈

#41

yes

characteristic

126

C₄×C₈

#41

yes

—

127

#95

—

128

#41

—

129

#41

—

130

#41

—

131

#41

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

AGL(1,ℤ/8)

yes

—

141

C₂²×C₄

C₂²×D₄

yes

characteristic

142

Q₁₆

—

143

Q₁₆

—

144

SD₁₆

—

145

SD₁₆

—

146

D₈

—

147

D₈

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂²⋊Q₈

yes

characteristic

156

C₂×C₈

C₂²⋊Q₈

yes

characteristic

157

C₂×C₈

#124

yes

—

158

C₂×C₈

C₂×D₈

yes

—

159

C₂×C₈

C₂×D₈

yes

—

160

C₂×C₈

C₂×SD₁₆

yes

—

161

C₂×C₈

C₂×SD₁₆

yes

characteristic

162

C₂×C₈

C₂×SD₁₆

yes

characteristic

163

C₂×C₈

#137

yes

—

164

C₂×C₈

C₂²×Q₈

yes

characteristic

165

C₂×C₈

C₂×#41

yes

—

166

C₂×C₈

C₂×#41

yes

characteristic

167

C₂×C₈

H(5,𝔽₂)

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₄²

C₂²⋊Q₈

yes

characteristic

178

C₄²

C₂²⋊Q₈

yes

characteristic

179

C₄²

#124

yes

—

180

C₄²

C₂×D₈

yes

—

181

C₄²

C₂×SD₁₆

yes

—

182

C₁₆

—

183

C₂³

—

184

C₂³

C₂×AGL(1,ℤ/8)

yes

characteristic

185

Q₈

—

186

Q₈

—

187

D₄

—

188

D₄

—

189

D₄

—

190

D₄

—

191

D₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

#476

yes

—

203

C₂×C₄

#476

yes

—

204

C₂×C₄

C₂×C₂²⋊Q₈

yes

characteristic

205

C₂×C₄

C₂×#124

yes

characteristic

206

C₂×C₄

#535

yes

characteristic

207

C₂×C₄

C₂²×D₈

yes

characteristic

208

C₂×C₄

C₂²×SD₁₆

yes

characteristic

209

C₂×C₄

C₂×#137

yes

characteristic

210

C₈

—

211

C₈

—

212

C₈

#474

yes

—

213

C₈

#474

yes

—

214

C₈

#486

yes

—

215

C₈

#486

yes

—

216

C₈

#554

yes

—

217

C₈

#555

yes

—

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

C₂×#476

yes

characteristic

222

C₂²

#3141

yes

characteristic

223

C₂²

#3205

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

#2563

yes

characteristic

231

C₄

C₂×#474

yes

characteristic

232

C₄

C₂×#486

yes

characteristic

233

C₄

#3206

yes

characteristic

234

C₂

—

235

C₂

—

236

C₂

—

237

C₂

#29139

yes

characteristic

238

C₂

#33637

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#476

commutator, Frattini, cosocle

C₂²×C₈

C₂⁴

prime radical

#3147

C₂²

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #33311	C₂	semidirect
3. #13372	C₂	semidirect
5. #12887	C₂	semidirect
6. #12823	C₂	semidirect
7. C₄×#364	C₂	semidirect

H ≅ G/N

type

1. #33311

C₂

semidirect

3. #13372

C₂

semidirect

5. #12887

C₂

semidirect

6. #12823

C₂

semidirect

7. C₄×#364

C₂

semidirect

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

subgroup	index
#33311	2
#21705	2
#13372	2
#13211	2
#12887	2
#12823	2
C₄×#364	2

subgroup

index

Subgroups — #507423

Conjugacy classes of subgroups

Complements

Maximal subgroups