Group #507493 — subgroups

Subgroups — #507493

Order 512.

470 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 182 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
2	#21732	C₂	1	1	yes	maximal, characteristic
3	#13457	C₂	1	4	yes	maximal
4	#13288	C₂	1	2	yes	maximal
5	#13187	C₂	1	4	yes	maximal
6	#12904	C₂	1	2	yes	maximal
7	C₄×#364	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	C₂×#500	—	2	1	no	—
10	C₂×#498	C₂²	1	2	yes	—
11	#2960	—	2	1	no	—
12	#2862	—	2	1	no	—
13	C₂×#365	C₂²	1	8	yes	—
14	C₂×#364	C₂²	1	2	yes	—
15	C₄×M₃₂	C₂²	1	2	yes	—
16	#1994	—	2	1	no	—
17	#1945	—	2	1	no	—
18	#1945	C₂²	1	2	yes	—
19	#1848	C₂²	1	1	yes	characteristic
20	#1846	—	2	1	no	—
21	#1846	—	2	1	no	—
22	#1846	C₂²	1	2	yes	—
23	#1846	C₂²	1	2	yes	—
24	#1772	—	2	1	no	—
25	#1772	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1386	C₂²	1	8	yes	—
28	#1376	C₂²	1	2	yes	—
29	#533	—	2	1	no	—
30	C₂×C₄⋊Q₈	—	2	2	no	—
31	C₂×M₃₂	C₂³	1	4	yes	—
32	#500	—	4	4	no	—
33	#500	Q₈	1	2	yes	complemented
34	#499	—	2	1	no	—
35	#499	Q₈	1	2	yes	complemented
36	#498	—	2	1	no	—
37	#498	—	4	4	no	—
38	#446	—	4	2	no	—
39	C₈×Q₈	—	4	2	no	—
40	C₂×#108	—	2	1	no	—
41	C₂×#108	C₂³	1	2	yes	—
42	#427	—	2	1	no	—
43	#427	C₂³	1	2	yes	—
44	C₂×#107	—	2	2	no	—
45	C₂×#107	—	2	2	no	—
46	C₂×#106	—	2	2	no	—
47	C₂×#106	C₂³	1	4	yes	—
48	#424	—	2	2	no	—
49	#423	—	2	1	no	—
50	#405	—	2	2	no	—
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	#391	—	4	4	no	—
58	#382	—	2	2	no	—
59	#382	—	2	2	no	—
60	C₄³	D₄	1	1	yes	characteristic
61	#365	—	4	8	no	—
62	#364	Q₈	1	8	yes	complemented
63	#346	—	2	2	no	—
64	C₄×C₁₆	—	2	2	no	—
65	#339	—	2	2	no	—
66	#339	—	2	2	no	—
67	C₂²×Q₈	—	4	2	no	—
68	C₂×M₁₆	—	2	2	no	—
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂⁴	1	1	yes	characteristic
73	C₄⋊Q₈	—	2	2	no	—
74	C₄⋊Q₈	—	4	4	no	—
75	C₄⋊Q₈	—	4	4	no	—
76	#125	—	2	2	no	—
77	C₂²⋊Q₈	—	4	2	no	—
78	C₄×Q₈	—	4	2	no	—
79	#117	—	2	2	no	—
80	C₂×C₄⋊C₄	—	2	4	no	—
81	C₂×C₄⋊C₄	—	4	2	no	—
82	C₂×C₄²	—	2	2	no	—
83	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
86	M₃₂	—	2	4	no	—
87	M₃₂	C₂×Q₈	1	8	yes	—
88	C₂×C₁₆	—	2	4	no	—
89	#108	—	2	4	no	—
90	#108	—	4	2	no	—
91	#107	—	2	2	no	—
92	#107	—	4	4	no	—
93	#107	—	4	4	no	—
94	#107	C₂×Q₈	1	4	yes	—
95	#106	—	2	2	no	—
96	#106	—	2	2	no	—
97	#106	—	2	8	no	—
98	#106	—	4	4	no	—
99	C₄⋊C₈	—	4	4	no	—
100	C₄⋊C₈	—	4	2	no	—
101	#97	—	4	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	4	2	no	—
105	C₄×C₈	SD₁₆	1	2	yes	—
106	C₄×C₈	SD₁₆	1	2	yes	—
107	C₄×C₈	SD₁₆	1	4	yes	—
108	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₄×C₈	C₂×Q₈	1	4	yes	—
111	C₄×C₈	#41	1	1	yes	characteristic
112	C₄×C₈	#41	1	1	yes	characteristic
113	#95	—	4	4	no	—
114	#95	—	4	4	no	—
115	C₂×Q₈	—	4	2	no	—
116	C₂×Q₈	—	8	4	no	—
117	C₂×Q₈	—	8	8	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
122	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Q₈	1	2	yes	—
131	C₂×C₈	C₂×SD₁₆	1	2	yes	—
132	C₂×C₈	C₂×SD₁₆	1	2	yes	—
133	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₈	C₂×SD₁₆	1	2	yes	—
136	C₂×C₈	#137	1	2	yes	—
137	C₂×C₈	C₂²×Q₈	1	2	yes	—
138	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₈	C₂×#41	1	1	yes	characteristic
140	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	8	4	no	—
147	C₂²⋊C₄	—	4	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	4	4	no	—
151	C₄²	—	4	2	no	—
152	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
153	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
154	C₄²	C₄⋊Q₈	1	2	yes	—
155	C₄²	C₂×SD₁₆	1	2	yes	—
156	C₄²	C₂×SD₁₆	1	2	yes	—
157	C₁₆	—	2	8	no	—
158	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	16	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	8	8	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	#476	1	2	yes	—
170	C₂×C₄	#476	1	2	yes	—
171	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
173	C₂×C₄	#535	1	1	yes	characteristic
174	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
175	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂×#137	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	8	2	no	—
179	C₈	#475	1	2	yes	—
180	C₈	#475	1	2	yes	—
181	C₈	#498	1	2	yes	—
182	C₈	#498	1	2	yes	—
183	C₈	#554	1	2	yes	—
184	C₈	Q₈²	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#476	1	1	yes	characteristic
187	C₂²	#3165	1	1	yes	characteristic
188	C₂²	#3208	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	—	8	4	no	—
194	C₄	—	16	4	no	—
195	C₄	#2567	1	1	yes	characteristic
196	C₄	C₂×#475	1	1	yes	characteristic
197	C₄	C₂×#498	1	1	yes	characteristic
198	C₄	#3206	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#29206	1	1	yes	characteristic
202	C₂	#33637	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33313

C₂

yes

maximal, characteristic

#21732

C₂

yes

maximal, characteristic

#13457

C₂

yes

maximal

#13288

C₂

yes

maximal

#13187

C₂

yes

maximal

#12904

C₂

yes

maximal

C₄×#364

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

C₂×#500

—

C₂×#498

C₂²

yes

—

#2960

—

#2862

—

C₂×#365

C₂²

yes

—

C₂×#364

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1994

—

#1945

—

#1945

C₂²

yes

—

#1848

C₂²

yes

—

—

C₂²

yes

—

#1846

C₂²

yes

—

#1772

—

#1772

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1386

C₂²

yes

—

#1376

C₂²

yes

—

#533

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

C₂³

yes

—

#500

—

#500

Q₈

yes

complemented

#499

—

#499

Q₈

yes

—

—

—

C₈×Q₈

—

C₂×#108

—

C₂×#108

C₂³

yes

—

#427

—

#427

C₂³

yes

—

—

—

—

C₂³

yes

—

#424

—

#423

—

#405

—

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

characteristic

#365

—

#364

Q₈

yes

complemented

#346

—

C₄×C₁₆

—

#339

—

#339

—

C₂²×Q₈

—

C₂×M₁₆

—

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

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

C₂×Q₈

yes

—

C₂×C₁₆

—

—

—

—

—

—

C₂×Q₈

yes

—

—

—

—

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#97

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

SD₁₆

yes

—

106

C₄×C₈

SD₁₆

yes

—

107

C₄×C₈

SD₁₆

yes

—

108

C₄×C₈

C₂×Q₈

yes

characteristic

109

C₄×C₈

C₂×Q₈

yes

characteristic

110

C₄×C₈

C₂×Q₈

yes

—

111

C₄×C₈

#41

yes

characteristic

112

C₄×C₈

#41

yes

characteristic

113

#95

—

114

#95

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

AGL(1,ℤ/8)

yes

—

122

C₂²×C₄

C₂²×D₄

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₂²⋊Q₈

yes

characteristic

129

C₂×C₈

C₂²⋊Q₈

yes

characteristic

130

C₂×C₈

C₄⋊Q₈

yes

—

131

C₂×C₈

C₂×SD₁₆

yes

—

132

C₂×C₈

C₂×SD₁₆

yes

—

133

C₂×C₈

C₂×SD₁₆

yes

characteristic

134

C₂×C₈

C₂×SD₁₆

yes

characteristic

135

C₂×C₈

C₂×SD₁₆

yes

—

136

C₂×C₈

#137

yes

—

137

C₂×C₈

C₂²×Q₈

yes

—

138

C₂×C₈

C₂²×Q₈

yes

characteristic

139

C₂×C₈

C₂×#41

yes

characteristic

140

C₂×C₈

H(5,𝔽₂)

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

C₂²⋊Q₈

yes

characteristic

153

C₄²

C₂²⋊Q₈

yes

characteristic

154

C₄²

C₄⋊Q₈

yes

—

155

C₄²

C₂×SD₁₆

yes

—

156

C₄²

C₂×SD₁₆

yes

—

157

C₁₆

—

158

C₂³

C₂×AGL(1,ℤ/8)

yes

characteristic

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

#476

yes

—

170

C₂×C₄

#476

yes

—

171

C₂×C₄

C₂×C₂²⋊Q₈

yes

characteristic

172

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

173

C₂×C₄

#535

yes

characteristic

174

C₂×C₄

C₂²×SD₁₆

yes

characteristic

175

C₂×C₄

C₂²×SD₁₆

yes

characteristic

176

C₂×C₄

C₂×#137

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

#475

yes

—

180

C₈

#475

yes

—

181

C₈

#498

yes

—

182

C₈

#498

yes

—

183

C₈

#554

yes

—

184

C₈

Q₈²

yes

—

185

C₂²

—

186

C₂²

C₂×#476

yes

characteristic

187

C₂²

#3165

yes

characteristic

188

C₂²

#3208

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

#2567

yes

characteristic

196

C₄

C₂×#475

yes

characteristic

197

C₄

C₂×#498

yes

characteristic

198

C₄

#3206

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#29206

yes

characteristic

202

C₂

#33637

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
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

C₂³

C₂×AGL(1,ℤ/8)

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
33. #500	Q₈	semidirect
35. #499	Q₈	semidirect
62. #364	Q₈	semidirect

H ≅ G/N

type

33. #500

Q₈

semidirect

35. #499

Q₈

semidirect

62. #364

Q₈

semidirect

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

subgroup	index
#33313	2
#21732	2
#13457	2
#13288	2
#13187	2
#12904	2
C₄×#364	2

subgroup

index

Subgroups — #507493

Conjugacy classes of subgroups

Complements

Maximal subgroups