Group #497102 — subgroups

Subgroups — #497102

Order 512.

444 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 190 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	#33726	C₂	1	1	yes	maximal, characteristic
2	#21732	C₂	1	2	yes	maximal
3	#13285	C₂	1	2	yes	maximal
4	#13113	C₂	1	8	yes	maximal
5	#12891	C₂	1	1	yes	maximal, characteristic
6	#12590	C₂	1	1	yes	maximal, characteristic
7	#3160	C₂²	1	2	yes	—
8	#2960	—	2	2	no	—
9	#2254	C₄	1	2	yes	complemented
10	#2253	C₄	1	4	yes	complemented
11	#2252	C₄	1	1	yes	characteristic, complemented
12	#2252	C₄	1	1	yes	characteristic, complemented
13	#2158	C₂²	1	4	yes	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1994	—	2	2	no	—
17	#1945	—	2	2	no	—
18	#1945	C₂²	1	4	yes	—
19	#1848	C₂²	1	2	yes	—
20	#1772	—	2	2	no	—
21	#1772	C₂²	1	4	yes	—
22	C₄²×C₈	C₂²	1	1	yes	characteristic
23	#1382	C₂²	1	8	yes	—
24	#1381	C₂²	1	4	yes	—
25	#1374	C₂²	1	4	yes	—
26	#533	—	2	2	no	—
27	C₂²×C₁₆	C₂³	1	2	yes	—
28	C₂²×C₁₆	C₂³	1	2	yes	—
29	#500	C₂×C₄	1	2	yes	—
30	#500	C₂×C₄	1	2	yes	—
31	#499	—	2	2	no	—
32	#499	C₂×C₄	1	4	yes	—
33	#498	—	2	2	no	—
34	#446	—	4	2	no	—
35	#446	—	4	2	no	—
36	C₈×Q₈	—	4	2	no	—
37	C₈×Q₈	—	4	2	no	—
38	C₂×#108	—	2	2	no	—
39	C₂×#108	C₂³	1	4	yes	—
40	#427	—	2	2	no	—
41	#427	C₂³	1	4	yes	—
42	#424	—	2	4	no	—
43	#423	—	2	2	no	—
44	#405	—	2	4	no	—
45	C₂×C₄×C₈	D₄	1	2	yes	—
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	2	yes	—
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₄³	D₄	1	1	yes	characteristic
50	#367	C₂×C₄	1	8	yes	—
51	#366	C₂×C₄	1	4	yes	—
52	#366	C₂×C₄	1	4	yes	—
53	C₄×C₁₆	—	2	1	no	—
54	C₄×C₁₆	—	2	1	no	—
55	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
56	C₄×C₁₆	C₂×C₄	1	2	yes	—
57	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
58	#339	—	2	4	no	—
59	#339	—	2	4	no	—
60	C₂×M₁₆	—	2	4	no	—
61	C₂²×C₈	C₂×D₄	1	2	yes	—
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂⁴	1	1	yes	characteristic
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#125	—	2	4	no	—
67	C₂²⋊Q₈	—	4	2	no	—
68	C₂²⋊Q₈	—	4	2	no	—
69	C₄×Q₈	—	4	2	no	—
70	C₄×Q₈	—	4	2	no	—
71	#117	—	2	4	no	—
72	C₂×C₄²	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	C₂×D₄	1	2	yes	—
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	C₂²×C₄	1	2	yes	—
80	C₂×C₁₆	C₂²×C₄	1	2	yes	—
81	#108	—	2	8	no	—
82	#108	—	4	4	no	—
83	#107	—	2	2	no	—
84	#107	—	2	2	no	—
85	#107	C₂²×C₄	1	4	yes	—
86	#107	C₂²×C₄	1	4	yes	—
87	#106	—	2	4	no	—
88	#106	—	2	4	no	—
89	C₄⋊C₈	—	4	4	no	—
90	C₄⋊C₈	—	4	4	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	#97	—	4	4	no	—
94	C₄×C₈	—	2	4	no	—
95	C₄×C₈	—	2	4	no	—
96	C₄×C₈	—	4	4	no	—
97	C₄×C₈	D₈	1	1	yes	characteristic
98	C₄×C₈	D₈	1	2	yes	—
99	C₄×C₈	D₈	1	1	yes	characteristic
100	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	4	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	#135	1	1	yes	characteristic
108	C₂²×C₄	#135	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
110	M₁₆	—	4	4	no	—
111	M₁₆	—	4	4	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	4	no	—
115	C₂×C₈	C₄×D₄	1	1	yes	characteristic
116	C₂×C₈	C₄×D₄	1	1	yes	characteristic
117	C₂×C₈	C₂×D₈	1	2	yes	—
118	C₂×C₈	C₂×D₈	1	2	yes	—
119	C₂×C₈	#137	1	4	yes	—
120	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₈	C₂×#41	1	1	yes	characteristic
122	C₂×C₈	2₋¹⁺⁴	1	4	yes	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₂²⋊C₄	—	4	4	no	—
130	C₄²	—	2	4	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	4	2	no	—
134	C₄²	—	4	4	no	—
135	C₄²	C₄×D₄	1	1	yes	characteristic
136	C₄²	C₄×D₄	1	1	yes	characteristic
137	C₄²	C₂×D₈	1	1	yes	characteristic
138	C₄²	C₂×D₈	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	4	2	no	—
142	C₂³	C₂×#135	1	1	yes	characteristic
143	Q₈	—	8	4	no	—
144	Q₈	—	8	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	C₄×D₈	1	1	yes	characteristic
153	C₂×C₄	C₄×D₈	1	1	yes	characteristic
154	C₂×C₄	C₄×D₈	1	1	yes	characteristic
155	C₂×C₄	C₄×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂×C₄	#536	1	2	yes	—
158	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
159	C₂×C₄	C₂×#137	1	2	yes	—
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	8	4	no	—
163	C₈	#441	1	2	yes	—
164	C₈	#441	1	2	yes	—
165	C₈	#519	1	2	yes	—
166	C₈	#541	1	2	yes	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
170	C₂²	#3231	1	2	yes	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	4	no	—
174	C₄	—	8	4	no	—
175	C₄	—	8	4	no	—
176	C₄	#2178	1	2	yes	—
177	C₄	#2179	1	2	yes	—
178	C₄	#2327	1	1	yes	characteristic
179	C₄	C₂×#441	1	1	yes	characteristic
180	C₄	#3226	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	#22247	1	1	yes	characteristic
183	C₂	C₂×#2178	1	1	yes	characteristic
184	C₂	C₂×#2179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33726

C₂

yes

maximal, characteristic

#21732

C₂

yes

maximal

#13285

C₂

yes

maximal

#13113

C₂

yes

maximal

#12891

C₂

yes

maximal, characteristic

#12590

C₂

yes

maximal, characteristic

#3160

C₂²

yes

—

#2960

—

#2254

C₄

yes

complemented

#2253

C₄

yes

complemented

#2252

C₄

yes

characteristic, complemented

#2252

C₄

yes

characteristic, complemented

#2158

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

C₂²

yes

—

#1848

C₂²

yes

—

#1772

—

#1772

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1382

C₂²

yes

—

#1381

C₂²

yes

—

#1374

C₂²

yes

—

#533

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

#500

C₂×C₄

yes

—

#500

C₂×C₄

yes

—

#499

—

#499

C₂×C₄

yes

—

#498

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

C₂×#108

—

C₂×#108

C₂³

yes

—

#427

—

#427

C₂³

yes

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

#367

C₂×C₄

yes

—

#366

C₂×C₄

yes

—

#366

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

characteristic

#339

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

—

—

—

—

C₂²×C₄

yes

—

#107

C₂²×C₄

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

D₈

yes

characteristic

100

C₄×C₈

C₂²×C₄

yes

characteristic

101

C₄×C₈

C₂²×C₄

yes

characteristic

102

C₄×C₈

#41

yes

characteristic

103

C₄×C₈

#41

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

#135

yes

characteristic

108

C₂²×C₄

#135

yes

characteristic

109

C₂²×C₄

C₂²×D₄

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₄×D₄

yes

characteristic

116

C₂×C₈

C₄×D₄

yes

characteristic

117

C₂×C₈

C₂×D₈

yes

—

118

C₂×C₈

C₂×D₈

yes

—

119

C₂×C₈

#137

yes

—

120

C₂×C₈

C₂³×C₄

yes

characteristic

121

C₂×C₈

C₂×#41

yes

characteristic

122

C₂×C₈

2₋¹⁺⁴

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₄×D₄

yes

characteristic

136

C₄²

C₄×D₄

yes

characteristic

137

C₄²

C₂×D₈

yes

characteristic

138

C₄²

C₂×D₈

yes

characteristic

139

C₁₆

—

140

C₁₆

—

141

C₁₆

—

142

C₂³

C₂×#135

yes

characteristic

143

Q₈

—

144

Q₈

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₄×D₈

yes

characteristic

153

C₂×C₄

C₄×D₈

yes

characteristic

154

C₂×C₄

C₄×D₈

yes

characteristic

155

C₂×C₄

C₄×D₈

yes

characteristic

156

C₂×C₄

C₂×C₄×D₄

yes

characteristic

157

C₂×C₄

#536

yes

—

158

C₂×C₄

C₂²×D₈

yes

characteristic

159

C₂×C₄

C₂×#137

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

#441

yes

—

164

C₈

#441

yes

—

165

C₈

#519

yes

—

166

C₈

#541

yes

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂×C₄×D₈

yes

characteristic

170

C₂²

#3231

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#2178

yes

—

177

C₄

#2179

yes

—

178

C₄

#2327

yes

characteristic

179

C₄

C₂×#441

yes

characteristic

180

C₄

#3226

yes

—

181

C₂

—

182

C₂

#22247

yes

characteristic

183

C₂

C₂×#2178

yes

characteristic

184

C₂

C₂×#2179

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₈

commutator

C₂×C₈

C₂³×C₄

Frattini, cosocle

C₂²×C₈

C₂⁴

socle

C₂²

C₂×C₄×D₈

prime radical

C₂³

C₂×#135

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
9. #2254	C₄	semidirect
10. #2253	C₄	semidirect
11. #2252	C₄	semidirect
12. #2252	C₄	semidirect

H ≅ G/N

type

9. #2254

C₄

semidirect

10. #2253

C₄

semidirect

11. #2252

C₄

semidirect

12. #2252

C₄

semidirect

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

subgroup	index
#33726	2
#21732	2
#13285	2
#13113	2
#12891	2
#12590	2

subgroup

index

Subgroups — #497102

Conjugacy classes of subgroups

Complements

Maximal subgroups