Group #498573 — subgroups

Subgroups — #498573

Order 512.

642 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 194 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×#2247	C₂	1	1	yes	maximal, characteristic
2	#33458	C₂	1	2	yes	maximal
3	#23200	C₂	1	2	yes	maximal
4	#13262	C₂	1	2	yes	maximal
5	#13244	C₂	1	2	yes	maximal
6	#13020	C₂	1	4	yes	maximal
7	#12670	C₂	1	1	yes	maximal, characteristic
8	#12664	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₃₂	C₂²	1	2	yes	—
10	C₂²×Q₃₂	C₂²	1	2	yes	—
11	C₂×#494	C₂²	1	2	yes	—
12	#2950	—	2	2	no	—
13	#2950	C₂²	1	4	yes	—
14	#2247	C₄	1	8	yes	complemented
15	#2176	—	2	4	no	—
16	C₄×Q₃₂	—	2	4	no	—
17	C₂×#368	C₂²	1	1	yes	characteristic
18	#2158	C₂²	1	1	yes	characteristic
19	C₂×#364	C₂²	1	1	yes	characteristic
20	C₂×#362	C₂²	1	4	yes	—
21	#2141	C₂²	1	4	yes	—
22	#2108	C₂²	1	2	yes	—
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	#1979	—	2	2	no	—
25	#1979	C₂²	1	4	yes	—
26	#1886	—	2	4	no	—
27	#1772	C₂²	1	2	yes	—
28	#1772	C₂²	1	1	yes	characteristic
29	#1382	C₂²	1	2	yes	—
30	#1379	C₂²	1	2	yes	—
31	C₂²×Q₁₆	—	2	2	no	—
32	C₂²×Q₁₆	C₂³	1	4	yes	—
33	C₂×C₄⋊Q₈	—	2	2	no	—
34	#519	—	2	4	no	—
35	C₂×Q₃₂	—	2	2	no	—
36	C₂×Q₃₂	—	2	2	no	—
37	C₂×Q₃₂	—	2	4	no	—
38	C₂×Q₃₂	—	4	2	no	—
39	C₂×Q₃₂	—	4	2	no	—
40	C₂×Q₃₂	C₂×C₄	1	8	yes	—
41	C₂×Q₃₂	C₂×C₄	1	8	yes	—
42	C₂×M₃₂	C₂³	1	2	yes	—
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	#494	—	2	4	no	—
46	#494	C₂×C₄	1	8	yes	—
47	#441	—	4	4	no	—
48	#441	—	4	4	no	—
49	C₄×Q₁₆	—	2	8	no	—
50	C₄×Q₁₆	—	4	4	no	—
51	C₂×#108	D₄	1	1	yes	characteristic
52	C₂×#108	C₂³	1	2	yes	—
53	#427	D₄	1	1	yes	characteristic
54	#427	C₂³	1	2	yes	—
55	#419	—	2	4	no	—
56	#419	—	2	4	no	—
57	C₂×#101	—	2	4	no	—
58	#405	—	2	2	no	—
59	#405	C₂³	1	2	yes	—
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	#368	—	2	2	no	—
62	#367	D₄	1	2	yes	—
63	#366	D₄	1	2	yes	—
64	#364	—	2	2	no	—
65	#362	—	4	4	no	—
66	#358	—	2	8	no	—
67	#346	—	2	2	no	—
68	C₄×C₁₆	—	2	2	no	—
69	C₄×C₁₆	C₂×C₄	1	4	yes	—
70	#339	—	2	2	no	—
71	#339	D₄	1	2	yes	—
72	C₂²×Q₈	—	2	4	no	—
73	C₂×Q₁₆	—	2	8	no	—
74	C₂×Q₁₆	—	2	4	no	—
75	C₂×Q₁₆	—	2	4	no	—
76	C₂×Q₁₆	—	2	4	no	—
77	C₂×Q₁₆	—	2	4	no	—
78	C₂×Q₁₆	—	2	4	no	—
79	C₂×Q₁₆	—	4	2	no	—
80	C₂×Q₁₆	—	4	2	no	—
81	C₂×Q₁₆	—	4	4	no	—
82	C₂×Q₁₆	C₂²×C₄	1	8	yes	—
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	C₂×D₄	1	2	yes	—
85	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂⁴	1	1	yes	characteristic
87	C₄⋊Q₈	—	2	8	no	—
88	C₄⋊Q₈	—	8	2	no	—
89	C₄×Q₈	—	4	8	no	—
90	C₄×Q₈	—	4	8	no	—
91	#117	—	2	2	no	—
92	#117	—	8	4	no	—
93	#117	C₂×D₄	1	2	yes	—
94	C₂×C₄⋊C₄	—	8	2	no	—
95	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
96	Q₃₂	—	2	8	no	—
97	Q₃₂	—	2	8	no	—
98	Q₃₂	—	4	8	no	—
99	M₃₂	—	2	2	no	—
100	M₃₂	—	2	2	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	C₂²×C₄	1	2	yes	—
105	C₂×C₁₆	C₂²×C₄	1	2	yes	—
106	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
107	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
108	C₂×C₁₆	#41	1	2	yes	—
109	#108	—	2	2	no	—
110	#108	—	2	2	no	—
111	#108	—	2	2	no	—
112	#107	D₈	1	2	yes	—
113	#107	C₂×D₄	1	4	yes	—
114	#106	—	2	2	no	—
115	#106	D₈	1	2	yes	—
116	#103	—	4	8	no	—
117	#103	—	4	8	no	—
118	#101	—	8	4	no	—
119	#97	—	2	2	no	—
120	#97	—	4	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	4	2	no	—
123	C₄×C₈	C₂²×C₄	1	2	yes	—
124	C₄×C₈	#41	1	2	yes	—
125	C₂×Q₈	—	2	8	no	—
126	C₂×Q₈	—	4	4	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂×Q₈	—	4	4	no	—
130	C₂×Q₈	—	4	4	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	4	4	no	—
133	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
134	Q₁₆	—	2	8	no	—
135	Q₁₆	—	2	8	no	—
136	Q₁₆	—	4	4	no	—
137	Q₁₆	—	4	4	no	—
138	Q₁₆	—	4	8	no	—
139	Q₁₆	—	4	8	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	—	2	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	C₄×D₄	1	2	yes	—
147	C₂×C₈	C₂×D₈	1	1	yes	characteristic
148	C₂×C₈	C₂×D₈	1	1	yes	characteristic
149	C₂×C₈	#135	1	2	yes	—
150	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₈	C₂×#41	1	1	yes	characteristic
153	C₂×C₈	C₂×#41	1	1	yes	characteristic
154	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	8	8	no	—
158	C₄⋊C₄	—	8	8	no	—
159	C₄⋊C₄	C₂×D₈	1	4	yes	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	8	8	no	—
166	C₄²	C₄×D₄	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	—	4	2	no	—
169	C₁₆	C₄×D₄	1	2	yes	—
170	C₁₆	C₄×D₄	1	2	yes	—
171	C₂³	#576	1	1	yes	characteristic
172	Q₈	—	4	8	no	—
173	Q₈	—	4	8	no	—
174	Q₈	—	8	4	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	—	8	4	no	—
183	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂×C₄	D₄²	1	1	yes	characteristic
185	C₂×C₄	#550	1	1	yes	characteristic
186	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
187	C₂×C₄	C₂×#135	1	1	yes	characteristic
188	C₂×C₄	#576	1	1	yes	characteristic
189	C₈	—	4	2	no	—
190	C₈	—	8	2	no	—
191	C₈	C₄×D₈	1	2	yes	—
192	C₈	C₄×D₈	1	2	yes	—
193	C₈	C₂×C₄×D₄	1	1	yes	characteristic
194	C₈	#518	1	1	yes	characteristic
195	C₈	#550	1	1	yes	characteristic
196	C₈	#559	1	1	yes	characteristic
197	C₂²	—	2	2	no	—
198	C₂²	#2948	1	1	yes	characteristic
199	C₂²	D₄×D₈	1	1	yes	characteristic
200	C₂²	#3396	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	—	4	4	no	—
203	C₄	—	8	4	no	—
204	C₄	—	8	8	no	—
205	C₄	#2366	1	1	yes	characteristic
206	C₄	C₂×C₄×D₈	1	1	yes	characteristic
207	C₄	#3287	1	1	yes	characteristic
208	C₄	#3391	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	#23185	1	1	yes	characteristic
212	C₂	#33860	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2247

C₂

yes

maximal, characteristic

#33458

C₂

yes

maximal

#23200

C₂

yes

maximal

#13262

C₂

yes

maximal

#13244

C₂

yes

maximal

#13020

C₂

yes

maximal

#12670

C₂

yes

maximal, characteristic

#12664

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

—

C₂²×Q₃₂

C₂²

yes

—

C₂×#494

C₂²

yes

—

#2950

—

#2950

C₂²

yes

—

#2247

C₄

yes

complemented

#2176

—

C₄×Q₃₂

—

C₂×#368

C₂²

yes

characteristic

#2158

C₂²

yes

characteristic

C₂×#364

C₂²

yes

characteristic

C₂×#362

C₂²

yes

—

#2141

C₂²

yes

—

#2108

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1979

—

#1979

C₂²

yes

—

#1886

—

#1772

C₂²

yes

—

#1772

C₂²

yes

characteristic

#1382

C₂²

yes

—

#1379

C₂²

yes

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂³

yes

—

C₂×C₄⋊Q₈

—

#519

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×C₄

yes

—

C₂×Q₃₂

C₂×C₄

yes

—

C₂×M₃₂

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

#494

—

#494

C₂×C₄

yes

—

#441

—

#441

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#108

D₄

yes

characteristic

C₂×#108

C₂³

yes

—

#427

D₄

yes

characteristic

#427

C₂³

yes

—

—

—

—

—

C₂³

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#368

—

#367

D₄

yes

—

#366

D₄

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

#339

—

#339

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂²×C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

#117

C₂×D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

C₂²×C₄

yes

—

105

C₂×C₁₆

C₂²×C₄

yes

—

106

C₂×C₁₆

C₂×D₄

yes

characteristic

107

C₂×C₁₆

C₂×D₄

yes

characteristic

108

C₂×C₁₆

#41

yes

—

109

#108

—

110

#108

—

111

#108

—

112

#107

D₈

yes

—

113

#107

C₂×D₄

yes

—

114

#106

—

115

#106

D₈

yes

—

116

#103

—

117

#103

—

118

#101

—

119

#97

—

120

#97

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

C₂²×C₄

yes

—

124

C₄×C₈

#41

yes

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²×D₄

yes

characteristic

134

Q₁₆

—

135

Q₁₆

—

136

Q₁₆

—

137

Q₁₆

—

138

Q₁₆

—

139

Q₁₆

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

C₄×D₄

yes

—

147

C₂×C₈

C₂×D₈

yes

characteristic

148

C₂×C₈

C₂×D₈

yes

characteristic

149

C₂×C₈

#135

yes

—

150

C₂×C₈

C₂³×C₄

yes

characteristic

151

C₂×C₈

C₂²×D₄

yes

characteristic

152

C₂×C₈

C₂×#41

yes

characteristic

153

C₂×C₈

C₂×#41

yes

characteristic

154

C₂×C₈

H(5,𝔽₂)

yes

characteristic

155

C₂×C₈

H(5,𝔽₂)

yes

characteristic

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

C₂×D₈

yes

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

C₄×D₄

yes

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

C₄×D₄

yes

—

170

C₁₆

C₄×D₄

yes

—

171

C₂³

#576

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

C₂×C₄×D₄

yes

characteristic

184

C₂×C₄

D₄²

yes

characteristic

185

C₂×C₄

#550

yes

characteristic

186

C₂×C₄

C₂²×D₈

yes

characteristic

187

C₂×C₄

C₂×#135

yes

characteristic

188

C₂×C₄

#576

yes

characteristic

189

C₈

—

190

C₈

—

191

C₈

C₄×D₈

yes

—

192

C₈

C₄×D₈

yes

—

193

C₈

C₂×C₄×D₄

yes

characteristic

194

C₈

#518

yes

characteristic

195

C₈

#550

yes

characteristic

196

C₈

#559

yes

characteristic

197

C₂²

—

198

C₂²

#2948

yes

characteristic

199

C₂²

D₄×D₈

yes

characteristic

200

C₂²

#3396

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#2366

yes

characteristic

206

C₄

C₂×C₄×D₈

yes

characteristic

207

C₄

#3287

yes

characteristic

208

C₄

#3391

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

#23185

yes

characteristic

212

C₂

#33860

yes

—

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#576
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

D₄×D₈

commutator

C₂×C₈

C₂³×C₄

Frattini, cosocle

C₂²×C₈

C₂⁴

prime radical

C₂³

#576

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #2247	C₄	semidirect

H ≅ G/N

type

14. #2247

C₄

semidirect

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

subgroup	index
C₂×#2247	2
#33458	2
#23200	2
#13262	2
#13244	2
#13020	2
#12670	2
#12664	2

subgroup

index

Subgroups — #498573

Conjugacy classes of subgroups

Complements

Maximal subgroups