Group #512942 — subgroups

Subgroups — #512942

Order 512.

420 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₂²×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#2258	C₂	1	1	yes	maximal, characteristic
3	#33608	C₂	1	2	yes	maximal
4	#13698	C₂	1	4	yes	maximal, complemented
5	#13695	C₂	1	4	yes	maximal, complemented
6	#13633	C₂	1	1	yes	maximal, characteristic, complemented
7	#13618	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₃₂	C₂²	1	2	yes	—
9	C₂×#504	C₂²	1	1	yes	characteristic
10	#3059	—	2	2	no	—
11	C₂×Q₆₄	—	2	4	no	—
12	C₂×Q₆₄	C₂²	1	2	yes	—
13	C₂×Q₆₄	C₂²	1	2	yes	—
14	#2258	—	2	4	no	—
15	C₂×M₆₄	C₂²	1	1	yes	characteristic
16	C₂×M₆₄	C₂²	1	2	yes	—
17	C₂²×C₃₂	C₂²	1	2	yes	—
18	C₂²×C₃₂	C₂²	1	1	yes	characteristic
19	#2219	—	2	4	no	—
20	#2216	C₂²	1	8	yes	—
21	#2158	C₂²	1	2	yes	—
22	#2141	—	2	2	no	—
23	#1424	C₂²	1	2	yes	—
24	#1423	C₂²	1	2	yes	—
25	#1416	C₂²	1	8	yes	—
26	C₂²×Q₁₆	—	2	2	no	—
27	C₂×#131	D₄	1	1	yes	characteristic
28	#536	—	4	2	no	—
29	C₂×Q₃₂	—	2	4	no	—
30	C₂×Q₃₂	—	2	2	no	—
31	C₂×Q₃₂	—	2	4	no	—
32	C₂×Q₃₂	C₂³	1	4	yes	—
33	#504	—	2	2	no	—
34	#504	—	2	2	no	—
35	C₂×M₃₂	D₄	1	1	yes	characteristic
36	C₂×M₃₂	D₄	1	2	yes	—
37	C₂²×C₁₆	C₂³	1	2	yes	—
38	C₂²×C₁₆	C₂³	1	1	yes	characteristic
39	#470	—	4	4	no	—
40	#467	—	2	8	no	—
41	#427	—	2	2	no	—
42	#419	—	4	2	no	—
43	Q₆₄	—	2	4	no	—
44	Q₆₄	—	2	4	no	—
45	M₆₄	—	2	2	no	—
46	M₆₄	—	2	4	no	—
47	C₂×C₃₂	—	2	4	no	—
48	C₂×C₃₂	D₄	1	4	yes	—
49	C₂×C₃₂	C₂³	1	2	yes	—
50	C₂×C₃₂	C₂³	1	1	yes	characteristic
51	C₂×C₃₂	C₂³	1	1	yes	characteristic
52	#367	—	2	2	no	—
53	#366	C₂³	1	4	yes	—
54	#358	—	2	8	no	—
55	C₂×#41	D₈	1	1	yes	characteristic
56	C₂²×Q₈	—	4	2	no	—
57	C₂×Q₁₆	—	2	4	no	—
58	C₂×Q₁₆	—	4	4	no	—
59	C₂×Q₁₆	—	4	2	no	—
60	C₂×Q₁₆	—	4	4	no	—
61	#131	—	2	2	no	—
62	#131	—	2	2	no	—
63	C₂×M₁₆	D₈	1	1	yes	characteristic
64	C₂×M₁₆	D₈	1	2	yes	—
65	C₂²×C₈	C₂×D₄	1	2	yes	—
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₄⋊Q₈	—	8	2	no	—
68	#124	—	8	2	no	—
69	#123	—	8	4	no	—
70	C₂²⋊Q₈	—	4	8	no	—
71	#117	—	4	2	no	—
72	Q₃₂	—	2	4	no	—
73	Q₃₂	—	2	4	no	—
74	Q₃₂	—	4	4	no	—
75	M₃₂	—	2	2	no	—
76	M₃₂	—	2	4	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	C₂×D₄	1	2	yes	—
80	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	C₂×D₄	1	2	yes	—
82	C₂×C₁₆	#41	1	2	yes	—
83	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
84	#107	—	2	4	no	—
85	#106	—	4	2	no	—
86	#103	—	4	8	no	—
87	C₃₂	—	2	2	no	—
88	C₃₂	C₂×D₄	1	2	yes	—
89	C₃₂	C₂×D₄	1	2	yes	—
90	#41	—	2	2	no	—
91	#41	—	2	2	no	—
92	C₂×Q₈	—	4	4	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂×Q₈	—	8	4	no	—
95	C₂×Q₈	—	8	4	no	—
96	C₂×Q₈	D₁₆	1	1	yes	characteristic
97	C₂×D₄	D₁₆	1	2	yes	—
98	C₂×D₄	D₁₆	1	1	yes	characteristic
99	C₂²×C₄	—	8	2	no	—
100	C₂²×C₄	C₂×D₈	1	2	yes	—
101	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
102	Q₁₆	—	4	4	no	—
103	Q₁₆	—	4	4	no	—
104	Q₁₆	—	8	4	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	4	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	#121	1	2	yes	—
110	C₂×C₈	C₂×D₈	1	2	yes	—
111	C₂×C₈	C₂×D₈	1	1	yes	characteristic
112	C₂×C₈	#135	1	2	yes	—
113	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	8	8	no	—
116	C₂²⋊C₄	—	8	2	no	—
117	C₂²⋊C₄	—	8	8	no	—
118	C₄²	—	8	2	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	#121	1	2	yes	—
121	C₁₆	C₂²×D₄	1	1	yes	characteristic
122	C₁₆	C₂×#41	1	1	yes	characteristic
123	C₂³	C₂×D₁₆	1	2	yes	—
124	C₂³	C₂×D₁₆	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	8	4	no	—
127	Q₈	—	8	4	no	—
128	Q₈	—	16	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	—	8	4	no	—
135	C₂×C₄	—	16	4	no	—
136	C₂×C₄	#466	1	2	yes	—
137	C₂×C₄	C₂×D₁₆	1	2	yes	—
138	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
139	C₂×C₄	#508	1	2	yes	—
140	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	#466	1	2	yes	—
143	C₈	C₂×#121	1	1	yes	characteristic
144	C₈	C₂×#135	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	2	4	no	—
148	C₂²	#2215	1	2	yes	—
149	C₂²	C₂²×D₁₆	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	16	4	no	—
152	C₄	—	16	4	no	—
153	C₄	#2215	1	2	yes	—
154	C₄	C₂×#466	1	1	yes	characteristic
155	C₄	C₂×#508	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	—	4	2	no	—
158	C₂	C₂×#2215	1	1	yes	characteristic
159	C₂	#33984	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₆₄

C₂

yes

maximal, characteristic, complemented

C₂×#2258

C₂

yes

maximal, characteristic

#33608

C₂

yes

maximal

#13698

C₂

yes

maximal, complemented

#13695

C₂

yes

maximal, complemented

#13633

C₂

yes

maximal, characteristic, complemented

#13618

C₂

yes

maximal, complemented

C₂²×Q₃₂

C₂²

yes

—

C₂×#504

C₂²

yes

characteristic

#3059

—

C₂×Q₆₄

—

C₂×Q₆₄

C₂²

yes

—

C₂×Q₆₄

C₂²

yes

—

#2258

—

C₂×M₆₄

C₂²

yes

characteristic

C₂×M₆₄

C₂²

yes

—

C₂²×C₃₂

C₂²

yes

—

C₂²×C₃₂

C₂²

yes

characteristic

#2219

—

#2216

C₂²

yes

—

#2158

C₂²

yes

—

#2141

—

#1424

C₂²

yes

—

#1423

C₂²

yes

—

#1416

C₂²

yes

—

C₂²×Q₁₆

—

C₂×#131

D₄

yes

characteristic

#536

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂³

yes

—

#504

—

#504

—

C₂×M₃₂

D₄

yes

characteristic

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

—

—

—

Q₆₄

—

Q₆₄

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

D₄

yes

—

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

characteristic

#367

—

#366

C₂³

yes

—

#358

—

C₂×#41

D₈

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

#131

—

#131

—

C₂×M₁₆

D₈

yes

characteristic

C₂×M₁₆

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

C₃₂

—

C₃₂

C₂×D₄

yes

—

C₃₂

C₂×D₄

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₆

yes

characteristic

C₂×D₄

D₁₆

yes

—

C₂×D₄

D₁₆

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

C₂×D₈

yes

—

101

C₂²×C₄

C₂×D₈

yes

characteristic

102

Q₁₆

—

103

Q₁₆

—

104

Q₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

#121

yes

—

110

C₂×C₈

C₂×D₈

yes

—

111

C₂×C₈

C₂×D₈

yes

characteristic

112

C₂×C₈

#135

yes

—

113

C₂×C₈

C₂²×D₄

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

#121

yes

—

121

C₁₆

C₂²×D₄

yes

characteristic

122

C₁₆

C₂×#41

yes

characteristic

123

C₂³

C₂×D₁₆

yes

—

124

C₂³

C₂×D₁₆

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#466

yes

—

137

C₂×C₄

C₂×D₁₆

yes

—

138

C₂×C₄

C₂×D₁₆

yes

characteristic

139

C₂×C₄

#508

yes

—

140

C₂×C₄

C₂²×D₈

yes

characteristic

141

C₈

—

142

C₈

#466

yes

—

143

C₈

C₂×#121

yes

characteristic

144

C₈

C₂×#135

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#2215

yes

—

149

C₂²

C₂²×D₁₆

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#2215

yes

—

154

C₄

C₂×#466

yes

characteristic

155

C₄

C₂×#508

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

C₂×#2215

yes

characteristic

159

C₂

#33984

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂⁴

prime radical

C₂×#41

D₈

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂²×Q₆₄	C₂	semidirect
4. #13698	C₂	semidirect
5. #13695	C₂	semidirect
6. #13633	C₂	semidirect
7. #13618	C₂	semidirect

H ≅ G/N

type

1. C₂²×Q₆₄

C₂

semidirect

4. #13698

C₂

semidirect

5. #13695

C₂

semidirect

6. #13633

C₂

semidirect

7. #13618

C₂

semidirect

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

subgroup	index
C₂²×Q₆₄	2
C₂×#2258	2
#33608	2
#13698	2
#13695	2
#13633	2
#13618	2

subgroup

index

C₂²×Q₆₄

Subgroups — #512942

Conjugacy classes of subgroups

Complements

Maximal subgroups