Group #507426 — subgroups

Subgroups — #507426

Order 512.

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

normal subgroups only

This group has 556 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	#33313	C₂	1	1	yes	maximal, characteristic, complemented
2	#21705	C₂	1	1	yes	maximal, characteristic
3	#13371	C₂	1	4	yes	maximal, complemented
4	#13211	C₂	1	2	yes	maximal
5	#12887	C₂	1	2	yes	maximal, complemented
6	#12824	C₂	1	4	yes	maximal, complemented
7	C₄×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#500	—	2	1	no	—
9	C₂×#498	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	#2862	—	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	—	2	1	no	—
24	#1846	C₂²	1	2	yes	—
25	#1846	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×C₄⋊Q₈	—	2	2	no	—
34	C₂×M₃₂	C₂³	1	4	yes	—
35	#500	—	4	4	no	—
36	#498	—	4	4	no	—
37	#495	—	2	1	no	—
38	#495	Q₈	1	2	yes	complemented
39	#494	Q₈	1	1	yes	characteristic, complemented
40	#493	Q₈	1	1	yes	characteristic, complemented
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	#391	—	4	4	no	—
63	#382	—	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	C₂²×Q₈	—	4	2	no	—
72	#135	—	2	4	no	—
73	#135	—	2	4	no	—
74	#135	—	4	2	no	—
75	#135	—	4	2	no	—
76	C₂×Q₁₆	—	2	1	no	—
77	C₂×Q₁₆	C₂×Q₈	1	2	yes	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	2	2	no	—
80	C₂×D₈	—	2	1	no	—
81	C₂×D₈	C₂×Q₈	1	2	yes	—
82	#131	—	4	4	no	—
83	#131	—	4	4	no	—
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	4	2	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂⁴	1	1	yes	characteristic
91	C₄⋊Q₈	—	2	1	no	—
92	C₄⋊Q₈	—	4	4	no	—
93	C₄⋊Q₈	—	4	4	no	—
94	#127	—	2	1	no	—
95	#124	—	2	2	no	—
96	#121	—	4	2	no	—
97	C₄×D₄	—	4	2	no	—
98	C₂×C₄⋊C₄	—	2	4	no	—
99	C₂×C₄⋊C₄	—	4	2	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	#95	—	4	4	no	—
129	#41	—	4	2	no	—
130	#41	—	4	2	no	—
131	#41	—	4	2	no	—
132	#41	—	4	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	8	4	no	—
135	C₂×Q₈	—	8	8	no	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
144	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
145	Q₁₆	—	2	4	no	—
146	Q₁₆	—	4	2	no	—
147	SD₁₆	—	4	4	no	—
148	SD₁₆	—	4	4	no	—
149	D₈	—	2	4	no	—
150	D₈	—	4	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	8	4	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	8	4	no	—
158	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
160	C₂×C₈	#124	1	2	yes	—
161	C₂×C₈	C₂×D₈	1	2	yes	—
162	C₂×C₈	C₂×D₈	1	2	yes	—
163	C₂×C₈	C₂×SD₁₆	1	2	yes	—
164	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
165	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₈	#137	1	2	yes	—
167	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₈	C₂×#41	1	2	yes	—
169	C₂×C₈	C₂×#41	1	1	yes	characteristic
170	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	8	4	no	—
175	C₂²⋊C₄	—	4	4	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	4	4	no	—
179	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
180	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
181	C₄²	#124	1	2	yes	—
182	C₄²	C₂×D₈	1	2	yes	—
183	C₄²	C₂×SD₁₆	1	2	yes	—
184	C₁₆	—	4	4	no	—
185	C₂³	—	4	2	no	—
186	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	Q₈	—	4	2	no	—
188	Q₈	—	4	2	no	—
189	Q₈	—	16	8	no	—
190	D₄	—	4	2	no	—
191	D₄	—	4	2	no	—
192	D₄	—	8	4	no	—
193	D₄	—	8	2	no	—
194	D₄	—	8	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	8	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	#476	1	2	yes	—
206	C₂×C₄	#476	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
208	C₂×C₄	C₂×#124	1	1	yes	characteristic
209	C₂×C₄	#535	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
211	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
212	C₂×C₄	C₂×#137	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	8	2	no	—
215	C₈	#474	1	2	yes	—
216	C₈	#474	1	2	yes	—
217	C₈	#486	1	2	yes	—
218	C₈	#486	1	2	yes	—
219	C₈	#554	1	2	yes	—
220	C₈	#555	1	2	yes	—
221	C₂²	—	2	2	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	2	no	—
224	C₂²	C₂×#476	1	1	yes	characteristic
225	C₂²	#3141	1	1	yes	characteristic
226	C₂²	#3205	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	8	2	no	—
232	C₄	—	16	4	no	—
233	C₄	#2563	1	1	yes	characteristic
234	C₄	C₂×#474	1	1	yes	characteristic
235	C₄	C₂×#486	1	1	yes	characteristic
236	C₄	#3206	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	1	no	—
239	C₂	—	16	2	no	—
240	C₂	#29139	1	1	yes	characteristic
241	C₂	#33637	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33313

C₂

yes

maximal, characteristic, complemented

#21705

C₂

yes

maximal, characteristic

#13371

C₂

yes

maximal, complemented

#13211

C₂

yes

maximal

#12887

C₂

yes

maximal, complemented

#12824

C₂

yes

maximal, complemented

C₄×#364

C₂

yes

maximal, characteristic, complemented

C₂×#500

—

C₂×#498

C₂²

yes

—

#3147

C₂²

yes

characteristic

C₂×#443

—

C₂×#443

C₂²

yes

—

#2931

—

#2862

—

C₂×#364

C₂²

yes

—

C₂×#360

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1974

—

#1974

C₂²

yes

—

#1962

—

#1848

C₂²

yes

—

—

C₂²

yes

—

#1846

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1383

C₂²

yes

—

#1376

C₂²

yes

—

C₂×#135

—

C₂×#135

C₂³

yes

—

C₂×#131

—

#532

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂³

yes

—

—

—

—

Q₈

yes

complemented

#494

Q₈

yes

characteristic, complemented

#493

Q₈

yes

characteristic, complemented

—

—

—

—

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

—

C₂²×Q₈

—

—

—

—

—

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⋊Q₈

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

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

#95

—

129

#41

—

130

#41

—

131

#41

—

132

#41

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

AGL(1,ℤ/8)

yes

—

144

C₂²×C₄

C₂²×D₄

yes

characteristic

145

Q₁₆

—

146

Q₁₆

—

147

SD₁₆

—

148

SD₁₆

—

149

D₈

—

150

D₈

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

C₂²⋊Q₈

yes

characteristic

159

C₂×C₈

C₂²⋊Q₈

yes

characteristic

160

C₂×C₈

#124

yes

—

161

C₂×C₈

C₂×D₈

yes

—

162

C₂×C₈

C₂×D₈

yes

—

163

C₂×C₈

C₂×SD₁₆

yes

—

164

C₂×C₈

C₂×SD₁₆

yes

characteristic

165

C₂×C₈

C₂×SD₁₆

yes

characteristic

166

C₂×C₈

#137

yes

—

167

C₂×C₈

C₂²×Q₈

yes

characteristic

168

C₂×C₈

C₂×#41

yes

—

169

C₂×C₈

C₂×#41

yes

characteristic

170

C₂×C₈

H(5,𝔽₂)

yes

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂²⋊Q₈

yes

characteristic

180

C₄²

C₂²⋊Q₈

yes

characteristic

181

C₄²

#124

yes

—

182

C₄²

C₂×D₈

yes

—

183

C₄²

C₂×SD₁₆

yes

—

184

C₁₆

—

185

C₂³

—

186

C₂³

C₂×AGL(1,ℤ/8)

yes

characteristic

187

Q₈

—

188

Q₈

—

189

Q₈

—

190

D₄

—

191

D₄

—

192

D₄

—

193

D₄

—

194

D₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

#476

yes

—

206

C₂×C₄

#476

yes

—

207

C₂×C₄

C₂×C₂²⋊Q₈

yes

characteristic

208

C₂×C₄

C₂×#124

yes

characteristic

209

C₂×C₄

#535

yes

characteristic

210

C₂×C₄

C₂²×D₈

yes

characteristic

211

C₂×C₄

C₂²×SD₁₆

yes

characteristic

212

C₂×C₄

C₂×#137

yes

characteristic

213

C₈

—

214

C₈

—

215

C₈

#474

yes

—

216

C₈

#474

yes

—

217

C₈

#486

yes

—

218

C₈

#486

yes

—

219

C₈

#554

yes

—

220

C₈

#555

yes

—

221

C₂²

—

222

C₂²

—

223

C₂²

—

224

C₂²

C₂×#476

yes

characteristic

225

C₂²

#3141

yes

characteristic

226

C₂²

#3205

yes

characteristic

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

#2563

yes

characteristic

234

C₄

C₂×#474

yes

characteristic

235

C₄

C₂×#486

yes

characteristic

236

C₄

#3206

yes

characteristic

237

C₂

—

238

C₂

—

239

C₂

—

240

C₂

#29139

yes

characteristic

241

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. #33313	C₂	semidirect
3. #13371	C₂	semidirect
5. #12887	C₂	semidirect
6. #12824	C₂	semidirect
7. C₄×#364	C₂	semidirect
38. #495	Q₈	semidirect
39. #494	Q₈	semidirect
40. #493	Q₈	semidirect

H ≅ G/N

type

1. #33313

C₂

semidirect

3. #13371

C₂

semidirect

5. #12887

C₂

semidirect

6. #12824

C₂

semidirect

7. C₄×#364

C₂

semidirect

38. #495

Q₈

semidirect

39. #494

Q₈

semidirect

40. #493

Q₈

semidirect

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

subgroup	index
#33313	2
#21705	2
#13371	2
#13211	2
#12887	2
#12824	2
C₄×#364	2

subgroup

index

Subgroups — #507426

Conjugacy classes of subgroups

Complements

Maximal subgroups