Group #496165 — subgroups

Subgroups — #496165

Order 512.

484 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 234 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	#33728	C₂	1	1	yes	maximal, characteristic
2	#21732	C₂	1	2	yes	maximal
3	#13286	C₂	1	2	yes	maximal
4	#13169	C₂	1	4	yes	maximal
5	#12910	C₂	1	1	yes	maximal, characteristic
6	#12671	C₂	1	4	yes	maximal
7	C₄×#346	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	2	yes	—
9	#2960	—	2	2	no	—
10	#2255	C₄	1	8	yes	complemented
11	C₂×#368	C₂²	1	4	yes	—
12	#2158	C₂²	1	4	yes	—
13	C₂×#346	C₂²	1	1	yes	characteristic
14	C₂×#346	C₂²	1	1	yes	characteristic
15	C₂×#346	C₂²	1	2	yes	—
16	C₂×C₄×C₁₆	C₂²	1	2	yes	—
17	#1994	—	2	2	no	—
18	#1945	—	2	2	no	—
19	#1945	C₂²	1	4	yes	—
20	#1848	C₂²	1	2	yes	—
21	#1772	—	2	2	no	—
22	#1772	C₂²	1	4	yes	—
23	C₄²×C₈	C₂²	1	1	yes	characteristic
24	#1382	C₂²	1	8	yes	—
25	#533	—	2	2	no	—
26	C₂²×C₁₆	C₂³	1	2	yes	—
27	C₂²×C₁₆	C₂³	1	2	yes	—
28	#500	C₂×C₄	1	4	yes	—
29	#499	—	2	2	no	—
30	#499	C₂×C₄	1	4	yes	—
31	#498	—	2	2	no	—
32	#446	—	4	4	no	—
33	C₈×Q₈	—	4	4	no	—
34	C₂×#108	—	2	2	no	—
35	C₂×#108	C₂³	1	4	yes	—
36	#427	—	2	2	no	—
37	#427	C₂³	1	4	yes	—
38	#424	—	2	4	no	—
39	#423	—	2	2	no	—
40	#405	—	2	4	no	—
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₄³	D₄	1	1	yes	characteristic
48	#368	—	2	8	no	—
49	#367	C₂×C₄	1	8	yes	—
50	#366	C₂×C₄	1	8	yes	—
51	#346	—	2	2	no	—
52	#346	C₂×C₄	1	4	yes	—
53	#346	Q₈	1	8	yes	complemented
54	C₄×C₁₆	—	2	4	no	—
55	#339	—	2	4	no	—
56	#339	—	2	4	no	—
57	C₂×M₁₆	—	2	4	no	—
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	C₂⁴	1	1	yes	characteristic
62	C₄⋊Q₈	—	2	4	no	—
63	#125	—	2	4	no	—
64	C₂²⋊Q₈	—	4	4	no	—
65	C₄×Q₈	—	4	4	no	—
66	#117	—	2	4	no	—
67	C₂×C₄²	—	2	2	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	C₂²×C₄	1	4	yes	—
74	C₂×C₁₆	C₂×Q₈	1	4	yes	—
75	C₂×C₁₆	C₂×Q₈	1	4	yes	—
76	C₂×C₁₆	#41	1	4	yes	—
77	#108	—	2	8	no	—
78	#108	—	4	4	no	—
79	#107	—	2	4	no	—
80	#107	C₂²×C₄	1	8	yes	—
81	#106	—	2	4	no	—
82	#106	—	2	4	no	—
83	C₄⋊C₈	—	4	8	no	—
84	C₄⋊C₈	—	4	4	no	—
85	#97	—	4	4	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	4	4	no	—
89	C₄×C₈	D₈	1	4	yes	—
90	C₄×C₈	Q₁₆	1	4	yes	—
91	C₄×C₈	C₂²×C₄	1	2	yes	—
92	C₄×C₈	C₂×Q₈	1	4	yes	—
93	C₄×C₈	#41	1	2	yes	—
94	C₂×Q₈	—	4	4	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	#135	1	2	yes	—
98	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
99	M₁₆	—	4	4	no	—
100	M₁₆	—	4	4	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	4	4	no	—
104	C₂×C₈	C₄×D₄	1	2	yes	—
105	C₂×C₈	C₄⋊Q₈	1	2	yes	—
106	C₂×C₈	C₂×D₈	1	2	yes	—
107	C₂×C₈	C₂×D₈	1	2	yes	—
108	C₂×C₈	C₂×Q₁₆	1	2	yes	—
109	C₂×C₈	C₂×Q₁₆	1	2	yes	—
110	C₂×C₈	#135	1	2	yes	—
111	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₈	C₂²×Q₈	1	2	yes	—
113	C₂×C₈	C₂×#41	1	2	yes	—
114	C₂×C₈	C₂×#41	1	1	yes	characteristic
115	C₄⋊C₄	—	2	8	no	—
116	C₄⋊C₄	—	4	8	no	—
117	C₄⋊C₄	—	4	4	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	4	4	no	—
122	C₄²	—	4	4	no	—
123	C₄²	C₄×D₄	1	2	yes	—
124	C₄²	C₄⋊Q₈	1	2	yes	—
125	C₄²	C₂×D₈	1	2	yes	—
126	C₄²	C₂×Q₁₆	1	2	yes	—
127	C₁₆	—	2	4	no	—
128	C₁₆	C₄×Q₈	1	8	yes	—
129	C₂³	C₂×#135	1	1	yes	characteristic
130	Q₈	—	8	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	C₄×D₈	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂×C₄	#532	1	1	yes	characteristic
141	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
142	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
143	C₂×C₄	C₂×#135	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	8	4	no	—
146	C₈	C₄×Q₁₆	1	4	yes	—
147	C₈	#500	1	4	yes	—
148	C₈	C₂×C₄×Q₈	1	2	yes	—
149	C₈	#533	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
152	C₂²	#3160	1	1	yes	characteristic
153	C₂²	#3170	1	1	yes	characteristic
154	C₄	—	2	4	no	—
155	C₄	—	2	4	no	—
156	C₄	—	4	2	no	—
157	C₄	—	8	8	no	—
158	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
159	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
160	C₄	C₂×#500	1	1	yes	characteristic
161	C₄	#3169	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	C₄×#500	1	1	yes	characteristic
165	C₂	#33463	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33728

C₂

yes

maximal, characteristic

#21732

C₂

yes

maximal

#13286

C₂

yes

maximal

#13169

C₂

yes

maximal

#12910

C₂

yes

maximal, characteristic

#12671

C₂

yes

maximal

C₄×#346

C₂

yes

maximal, characteristic

#3160

C₂²

yes

—

#2960

—

#2255

C₄

yes

complemented

C₂×#368

C₂²

yes

—

#2158

C₂²

yes

—

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

#1994

—

#1945

—

#1945

C₂²

yes

—

#1848

C₂²

yes

—

#1772

—

#1772

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1382

C₂²

yes

—

#533

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

#500

C₂×C₄

yes

—

#499

—

#499

C₂×C₄

yes

—

#498

—

#446

—

C₈×Q₈

—

C₂×#108

—

C₂×#108

C₂³

yes

—

#427

—

#427

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

characteristic

C₄³

D₄

yes

characteristic

#368

—

#367

C₂×C₄

yes

—

#366

C₂×C₄

yes

—

#346

—

#346

C₂×C₄

yes

—

#346

Q₈

yes

complemented

C₄×C₁₆

—

#339

—

#339

—

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

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

—

—

—

C₂²×C₄

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#135

yes

—

C₂²×C₄

C₂²×D₄

yes

characteristic

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄×D₄

yes

—

105

C₂×C₈

C₄⋊Q₈

yes

—

106

C₂×C₈

C₂×D₈

yes

—

107

C₂×C₈

C₂×D₈

yes

—

108

C₂×C₈

C₂×Q₁₆

yes

—

109

C₂×C₈

C₂×Q₁₆

yes

—

110

C₂×C₈

#135

yes

—

111

C₂×C₈

C₂³×C₄

yes

characteristic

112

C₂×C₈

C₂²×Q₈

yes

—

113

C₂×C₈

C₂×#41

yes

—

114

C₂×C₈

C₂×#41

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₄×D₄

yes

—

124

C₄²

C₄⋊Q₈

yes

—

125

C₄²

C₂×D₈

yes

—

126

C₄²

C₂×Q₁₆

yes

—

127

C₁₆

—

128

C₁₆

C₄×Q₈

yes

—

129

C₂³

C₂×#135

yes

characteristic

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄×D₈

yes

—

138

C₂×C₄

C₂×C₄×D₄

yes

characteristic

139

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

140

C₂×C₄

#532

yes

characteristic

141

C₂×C₄

C₂²×D₈

yes

characteristic

142

C₂×C₄

C₂²×Q₁₆

yes

characteristic

143

C₂×C₄

C₂×#135

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

C₄×Q₁₆

yes

—

147

C₈

#500

yes

—

148

C₈

C₂×C₄×Q₈

yes

—

149

C₈

#533

yes

—

150

C₂²

—

151

C₂²

C₂×C₄×D₈

yes

characteristic

152

C₂²

#3160

yes

characteristic

153

C₂²

#3170

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₄×C₄⋊Q₈

yes

characteristic

159

C₄

C₂×C₄×Q₁₆

yes

characteristic

160

C₄

C₂×#500

yes

characteristic

161

C₄

#3169

yes

characteristic

162

C₂

—

163

C₂

—

164

C₂

C₄×#500

yes

characteristic

165

C₂

#33463

yes

—

	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
10. #2255	C₄	semidirect
53. #346	Q₈	semidirect

H ≅ G/N

type

10. #2255

C₄

semidirect

53. #346

Q₈

semidirect

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

subgroup	index
#33728	2
#21732	2
#13286	2
#13169	2
#12910	2
#12671	2
C₄×#346	2

subgroup

index

Subgroups — #496165

Conjugacy classes of subgroups

Complements

Maximal subgroups