Group #512557 — subgroups

Subgroups — #512557

Order 512.

756 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 352 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	#62624	C₂	1	1	yes	maximal, characteristic
2	#33974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#2258	C₂	1	1	yes	maximal, characteristic, complemented
4	#13558	C₂	1	6	yes	maximal, complemented
5	C₂×#1401	C₂	1	6	yes	maximal, complemented
6	#3573	C₄	1	1	yes	characteristic
7	#3407	—	2	4	no	—
8	#3407	C₄	1	6	yes	—
9	#3407	C₄	1	1	yes	characteristic
10	C₂×#504	—	2	4	no	—
11	C₂×#504	C₂²	1	6	yes	—
12	C₂×#504	C₂²	1	1	yes	characteristic
13	#2258	—	2	4	no	—
14	#2258	—	2	4	no	—
15	C₂×M₆₄	C₂²	1	6	yes	complemented
16	C₂×M₆₄	C₂²	1	3	yes	complemented
17	C₂²×C₃₂	C₂²	1	3	yes	complemented
18	#1401	C₂²	1	4	yes	complemented
19	#1401	C₂²	1	12	yes	complemented
20	#585	C₈	1	1	yes	characteristic
21	#568	—	2	4	no	—
22	#568	C₈	1	6	yes	—
23	#568	C₈	1	1	yes	characteristic
24	C₂×#131	—	2	4	no	—
25	C₂×#131	C₂×C₄	1	6	yes	—
26	C₂×#131	C₂×C₄	1	1	yes	characteristic
27	#504	—	2	4	no	—
28	#504	—	2	12	no	—
29	#504	—	2	6	no	—
30	#504	—	2	6	no	—
31	#504	C₂×C₄	1	12	yes	—
32	#504	D₄	1	4	yes	—
33	#504	D₄	1	4	yes	—
34	#504	C₂³	1	4	yes	—
35	C₂×M₃₂	—	2	12	no	—
36	C₂×M₃₂	—	2	6	no	—
37	C₂×M₃₂	C₂×C₄	1	6	yes	—
38	C₂×M₃₂	C₂×C₄	1	3	yes	—
39	C₂²×C₁₆	—	2	6	no	—
40	C₂²×C₁₆	C₂³	1	3	yes	—
41	M₆₄	—	2	6	no	—
42	M₆₄	—	2	6	no	—
43	M₆₄	C₂³	1	4	yes	—
44	C₂×C₃₂	—	2	6	no	—
45	C₂×C₃₂	—	2	6	no	—
46	C₂×C₃₂	C₂³	1	1	yes	characteristic
47	C₂×C₃₂	C₂³	1	3	yes	—
48	2₋¹⁺⁴	—	2	1	no	—
49	2₋¹⁺⁴	C₁₆	1	4	yes	—
50	H(5,𝔽₂)	—	2	3	no	—
51	H(5,𝔽₂)	C₁₆	1	4	yes	—
52	C₂×#41	—	2	4	no	—
53	C₂×#41	C₂×C₈	1	6	yes	—
54	C₂×#41	C₂×C₈	1	1	yes	characteristic
55	#131	—	2	4	no	—
56	#131	—	2	12	no	—
57	#131	—	2	6	no	—
58	#131	—	2	6	no	—
59	#131	C₂²⋊C₄	1	4	yes	—
60	#131	C₂×C₈	1	12	yes	—
61	#131	M₁₆	1	4	yes	—
62	#131	C₂²×C₄	1	4	yes	—
63	C₂×M₁₆	—	2	12	no	—
64	C₂×M₁₆	—	2	6	no	—
65	C₂×M₁₆	C₂×C₈	1	6	yes	—
66	C₂×M₁₆	C₂×C₈	1	3	yes	—
67	C₂²×C₈	—	2	6	no	—
68	C₂²×C₈	C₂²×C₄	1	3	yes	—
69	M₃₂	—	2	6	no	—
70	M₃₂	—	2	12	no	—
71	M₃₂	—	2	4	no	—
72	M₃₂	C₂²⋊C₄	1	12	yes	—
73	M₃₂	C₂²×C₄	1	4	yes	—
74	C₂×C₁₆	—	2	6	no	—
75	C₂×C₁₆	—	2	12	no	—
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	C₂²×C₄	1	3	yes	—
78	C₂×C₁₆	C₂×D₄	1	6	yes	—
79	C₂×C₁₆	C₂×D₄	1	6	yes	—
80	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
81	C₃₂	—	2	1	no	—
82	C₃₂	—	2	3	no	—
83	C₃₂	—	2	4	no	—
84	#41	—	2	4	no	—
85	#41	—	2	12	no	—
86	#41	—	2	6	no	—
87	#41	—	2	6	no	—
88	#41	C₂²⋊C₈	1	4	yes	—
89	#41	C₂×C₁₆	1	12	yes	—
90	#41	C₂×C₁₆	1	4	yes	—
91	#41	C₂²×C₈	1	4	yes	—
92	C₂×Q₈	—	2	4	no	—
93	C₂×Q₈	C₂×C₁₆	1	6	yes	—
94	C₂×Q₈	M₃₂	1	1	yes	characteristic
95	C₂×D₄	—	2	6	no	—
96	C₂×D₄	—	2	12	no	—
97	C₂×D₄	C₂×C₁₆	1	6	yes	—
98	C₂×D₄	M₃₂	1	3	yes	—
99	C₂²×C₄	—	2	6	no	—
100	C₂²×C₄	C₂²×C₈	1	3	yes	—
101	M₁₆	—	2	6	no	—
102	M₁₆	—	2	4	no	—
103	M₁₆	—	2	12	no	—
104	M₁₆	C₂²⋊C₈	1	12	yes	—
105	M₁₆	C₂²×C₈	1	4	yes	—
106	C₂×C₈	—	2	6	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	12	no	—
109	C₂×C₈	C₂×C₂²⋊C₄	1	6	yes	—
110	C₂×C₈	C₂²×C₈	1	3	yes	—
111	C₂×C₈	C₂×M₁₆	1	6	yes	—
112	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
113	C₁₆	—	2	4	no	—
114	C₁₆	C₂×C₂²⋊C₄	1	6	yes	—
115	C₁₆	C₂²×D₄	1	1	yes	characteristic
116	C₁₆	C₂²×D₄	1	1	yes	characteristic
117	C₂³	—	2	6	no	—
118	C₂³	C₂×M₃₂	1	3	yes	—
119	Q₈	—	2	6	no	—
120	Q₈	C₂²⋊C₁₆	1	4	yes	—
121	Q₈	C₂²×C₁₆	1	4	yes	—
122	D₄	—	2	12	no	—
123	D₄	—	2	4	no	—
124	D₄	—	2	6	no	—
125	D₄	C₂²⋊C₁₆	1	12	yes	—
126	D₄	C₂²×C₁₆	1	4	yes	—
127	C₂×C₄	—	2	6	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	12	no	—
130	C₂×C₄	C₂×C₂²⋊C₈	1	6	yes	—
131	C₂×C₄	C₂²×C₁₆	1	6	yes	—
132	C₂×C₄	C₂×M₃₂	1	3	yes	—
133	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
134	C₈	—	2	4	no	—
135	C₈	C₂×C₂²⋊C₈	1	6	yes	—
136	C₈	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₈	C₂²×M₁₆	1	1	yes	characteristic
138	C₂²	—	2	4	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	8	6	no	—
141	C₂²	C₂×C₂²⋊C₁₆	1	6	yes	—
142	C₂²	C₂²×M₃₂	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	C₂×C₂²⋊C₁₆	1	6	yes	—
145	C₄	C₂²×C₂²⋊C₈	1	1	yes	characteristic
146	C₄	C₂³×C₁₆	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	6	no	—
149	C₂	—	4	4	no	—
150	C₂	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#62624

C₂

yes

maximal, characteristic

#33974

C₂

yes

maximal, characteristic, complemented

C₂×#2258

C₂

yes

maximal, characteristic, complemented

#13558

C₂

yes

maximal, complemented

C₂×#1401

C₂

yes

maximal, complemented

#3573

C₄

yes

characteristic

#3407

—

#3407

C₄

yes

—

#3407

C₄

yes

characteristic

C₂×#504

—

C₂×#504

C₂²

yes

—

C₂×#504

C₂²

yes

characteristic

#2258

—

#2258

—

C₂×M₆₄

C₂²

yes

complemented

C₂×M₆₄

C₂²

yes

complemented

C₂²×C₃₂

C₂²

yes

complemented

#1401

C₂²

yes

complemented

#1401

C₂²

yes

complemented

#585

C₈

yes

characteristic

#568

—

#568

C₈

yes

—

#568

C₈

yes

characteristic

C₂×#131

—

C₂×#131

C₂×C₄

yes

—

C₂×#131

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

#504

D₄

yes

—

#504

D₄

yes

—

#504

C₂³

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂³

yes

—

M₆₄

—

M₆₄

—

M₆₄

C₂³

yes

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

2₋¹⁺⁴

—

2₋¹⁺⁴

C₁₆

yes

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₁₆

yes

—

C₂×#41

—

C₂×#41

C₂×C₈

yes

—

C₂×#41

C₂×C₈

yes

—

—

—

—

C₂²⋊C₄

yes

—

#131

C₂×C₈

yes

—

#131

M₁₆

yes

—

#131

C₂²×C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₈

yes

—

C₂×M₁₆

C₂×C₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂²⋊C₄

yes

—

M₃₂

C₂²×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

—

—

—

—

C₂²⋊C₈

yes

—

#41

C₂×C₁₆

yes

—

#41

C₂×C₁₆

yes

—

#41

C₂²×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×C₁₆

yes

—

C₂×Q₈

M₃₂

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₁₆

yes

—

C₂×D₄

M₃₂

yes

—

C₂²×C₄

—

100

C₂²×C₄

C₂²×C₈

yes

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

C₂²⋊C₈

yes

—

105

M₁₆

C₂²×C₈

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×C₂²⋊C₄

yes

—

110

C₂×C₈

C₂²×C₈

yes

—

111

C₂×C₈

C₂×M₁₆

yes

—

112

C₂×C₈

C₂³×C₄

yes

characteristic

113

C₁₆

—

114

C₁₆

C₂×C₂²⋊C₄

yes

—

115

C₁₆

C₂²×D₄

yes

characteristic

116

C₁₆

C₂²×D₄

yes

characteristic

117

C₂³

—

118

C₂³

C₂×M₃₂

yes

—

119

Q₈

—

120

Q₈

C₂²⋊C₁₆

yes

—

121

Q₈

C₂²×C₁₆

yes

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

C₂²⋊C₁₆

yes

—

126

D₄

C₂²×C₁₆

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×C₂²⋊C₈

yes

—

131

C₂×C₄

C₂²×C₁₆

yes

—

132

C₂×C₄

C₂×M₃₂

yes

—

133

C₂×C₄

C₂³×C₈

yes

characteristic

134

C₈

—

135

C₈

C₂×C₂²⋊C₈

yes

—

136

C₈

C₂²×C₂²⋊C₄

yes

characteristic

137

C₈

C₂²×M₁₆

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊C₁₆

yes

—

142

C₂²

C₂²×M₃₂

yes

characteristic

143

C₄

—

144

C₄

C₂×C₂²⋊C₁₆

yes

—

145

C₄

C₂²×C₂²⋊C₈

yes

characteristic

146

C₄

C₂³×C₁₆

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

C₂²×C₂²⋊C₁₆

yes

characteristic

	subgroup	G/H
center	C₁₆	C₂²×D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂²×C₂²⋊C₁₆
prime radical	#585	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₁₆

C₂²×D₄

commutator

C₄

C₂³×C₁₆

Frattini, cosocle

C₂×C₁₆

C₂⁴

socle

C₂

C₂²×C₂²⋊C₁₆

C₈

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₁₆ ◁ C₂×#504 ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #33974	C₂	semidirect
3. C₂×#2258	C₂	semidirect
4. #13558	C₂	semidirect
5. C₂×#1401	C₂	semidirect
15. C₂×M₆₄	C₂²	semidirect
16. C₂×M₆₄	C₂²	semidirect
17. C₂²×C₃₂	C₂²	semidirect
18. #1401	C₂²	semidirect
19. #1401	C₂²	semidirect

H ≅ G/N

type

2. #33974

C₂

semidirect

3. C₂×#2258

C₂

semidirect

4. #13558

C₂

semidirect

5. C₂×#1401

C₂

semidirect

15. C₂×M₆₄

C₂²

semidirect

16. C₂×M₆₄

C₂²

semidirect

17. C₂²×C₃₂

C₂²

semidirect

18. #1401

C₂²

semidirect

19. #1401

C₂²

semidirect

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

subgroup	index
#62624	2
#33974	2
C₂×#2258	2
#13558	2
C₂×#1401	2

subgroup

index

Subgroups — #512557

Conjugacy classes of subgroups

Complements

Maximal subgroups