Group #512804 — subgroups

Subgroups — #512804

Order 512.

359 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 230 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	#33434	C₂	1	1	yes	maximal, characteristic
2	#13645	C₂	1	8	yes	maximal, complemented
3	#13640	C₂	1	2	yes	maximal, complemented
4	#13554	C₂	1	4	yes	maximal, complemented
5	C₂×#504	C₂²	1	2	yes	—
6	#2931	C₄	1	1	yes	characteristic
7	C₂×M₆₄	C₂²	1	4	yes	—
8	C₂²×C₃₂	C₂²	1	4	yes	—
9	#2169	C₄	1	4	yes	—
10	D₄×C₁₆	C₂²	1	4	yes	—
11	#2150	C₄	1	1	yes	characteristic
12	#2113	C₄	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄⋊C₃₂	C₂²	1	4	yes	—
15	C₂²⋊C₃₂	C₂²	1	4	yes	—
16	C₂²⋊C₃₂	C₂²	1	8	yes	—
17	#1397	C₂²	1	4	yes	—
18	C₂×#131	C₂×C₄	1	2	yes	—
19	#532	C₈	1	1	yes	characteristic
20	#504	—	4	2	no	—
21	#504	—	4	2	no	—
22	C₂×M₃₂	C₂×C₄	1	2	yes	—
23	C₂×M₃₂	C₂×C₄	1	4	yes	—
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂³	1	1	yes	characteristic
26	C₂²×C₁₆	C₂³	1	4	yes	—
27	#436	C₈	1	4	yes	—
28	C₈×D₄	C₂×C₄	1	4	yes	—
29	#423	C₈	1	1	yes	characteristic
30	#408	C₈	1	2	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	M₆₄	—	4	4	no	—
33	C₂×C₃₂	—	4	4	no	—
34	C₂×C₃₂	C₂³	1	8	yes	—
35	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
36	C₂²⋊C₁₆	C₂×C₄	1	8	yes	—
37	C₄×C₁₆	C₂×C₄	1	2	yes	—
38	C₄×C₁₆	C₂³	1	2	yes	—
39	C₂×#41	C₂×C₈	1	2	yes	—
40	#131	—	4	2	no	—
41	#131	—	4	2	no	—
42	C₂×M₁₆	C₂×C₈	1	2	yes	—
43	C₂×M₁₆	C₂×C₈	1	4	yes	—
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₈	C₂²×C₄	1	4	yes	—
47	C₄⋊Q₈	C₁₆	1	1	yes	characteristic
48	#127	C₁₆	1	1	yes	characteristic
49	#124	C₁₆	1	2	yes	—
50	#121	C₁₆	1	4	yes	—
51	C₄×D₄	C₂×C₈	1	4	yes	—
52	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
53	M₃₂	—	4	2	no	—
54	M₃₂	—	4	4	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	—	2	4	no	—
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	—	4	1	no	—
59	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₁₆	C₂²×C₄	1	2	yes	—
61	C₂×C₁₆	C₂²×C₄	1	4	yes	—
62	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₈	C₂×C₈	1	4	yes	—
64	C₂²⋊C₈	C₂×C₈	1	8	yes	—
65	C₄×C₈	C₂×C₈	1	2	yes	—
66	C₄×C₈	C₂²×C₄	1	2	yes	—
67	C₃₂	—	2	8	no	—
68	#41	—	4	2	no	—
69	#41	—	4	2	no	—
70	C₂×Q₈	C₂×C₁₆	1	2	yes	—
71	C₂×D₄	C₂×C₁₆	1	4	yes	—
72	C₂×D₄	C₂×C₁₆	1	4	yes	—
73	C₂×D₄	C₂×C₁₆	1	2	yes	—
74	C₂²×C₄	C₂×C₁₆	1	2	yes	—
75	C₂²×C₄	C₂²×C₈	1	4	yes	—
76	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
77	M₁₆	—	4	2	no	—
78	M₁₆	—	4	4	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₈	C₂²×C₈	1	2	yes	—
86	C₂×C₈	C₂²×C₈	1	4	yes	—
87	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
88	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
89	C₂²⋊C₄	C₂×C₁₆	1	8	yes	—
90	C₄²	C₂×C₁₆	1	1	yes	characteristic
91	C₄²	C₂×C₁₆	1	1	yes	characteristic
92	C₄²	C₂²×C₈	1	2	yes	—
93	C₁₆	—	2	4	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	2	2	no	—
96	C₁₆	H(5,𝔽₂)	1	2	yes	—
97	C₂³	C₂²×C₁₆	1	4	yes	—
98	C₂³	C₂²×C₁₆	1	1	yes	characteristic
99	Q₈	—	4	2	no	—
100	D₄	—	4	4	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
109	C₂×C₄	C₂²×C₁₆	1	2	yes	—
110	C₂×C₄	C₂²×C₁₆	1	2	yes	—
111	C₂×C₄	C₂²×C₁₆	1	4	yes	—
112	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	4	no	—
115	C₈	—	2	1	no	—
116	C₈	#518	1	1	yes	characteristic
117	C₈	#568	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	—	2	8	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	4	no	—
122	C₂²	C₂³×C₁₆	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	4	no	—
127	C₄	#2972	1	1	yes	characteristic
128	C₄	#3407	1	1	yes	characteristic
129	C₂	—	4	1	no	—
130	C₂	—	4	4	no	—
131	C₂	#33486	1	1	yes	characteristic
132	C₂	#33974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33434

C₂

yes

maximal, characteristic

#13645

C₂

yes

maximal, complemented

#13640

C₂

yes

maximal, complemented

#13554

C₂

yes

maximal, complemented

C₂×#504

C₂²

yes

—

#2931

C₄

yes

characteristic

C₂×M₆₄

C₂²

yes

—

C₂²×C₃₂

C₂²

yes

—

#2169

C₄

yes

—

D₄×C₁₆

C₂²

yes

—

#2150

C₄

yes

characteristic

#2113

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄⋊C₃₂

C₂²

yes

—

C₂²⋊C₃₂

C₂²

yes

—

C₂²⋊C₃₂

C₂²

yes

—

#1397

C₂²

yes

—

C₂×#131

C₂×C₄

yes

—

#532

C₈

yes

characteristic

#504

—

#504

—

C₂×M₃₂

C₂×C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

#436

C₈

yes

—

C₈×D₄

C₂×C₄

yes

—

#423

C₈

yes

characteristic

#408

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂³

yes

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₂²⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₂×#41

C₂×C₈

yes

—

#131

—

#131

—

C₂×M₁₆

C₂×C₈

yes

—

C₂×M₁₆

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

—

C₄⋊Q₈

C₁₆

yes

characteristic

#127

C₁₆

yes

characteristic

#124

C₁₆

yes

—

#121

C₁₆

yes

—

C₄×D₄

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂⁴

yes

characteristic

C₄⋊C₈

C₂×C₈

yes

—

C₂²⋊C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₃₂

—

#41

—

#41

—

C₂×Q₈

C₂×C₁₆

yes

—

C₂×D₄

C₂×C₁₆

yes

—

C₂×D₄

C₂×C₁₆

yes

—

C₂×D₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂²×C₈

yes

—

C₂²×C₄

C₂²×C₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

C₂³×C₄

yes

characteristic

C₄⋊C₄

C₂×C₁₆

yes

—

C₂²⋊C₄

C₂×C₁₆

yes

—

C₄²

C₂×C₁₆

yes

characteristic

C₄²

C₂×C₁₆

yes

characteristic

C₄²

C₂²×C₈

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

H(5,𝔽₂)

yes

—

C₂³

C₂²×C₁₆

yes

—

C₂³

C₂²×C₁₆

yes

characteristic

Q₈

—

100

D₄

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×C₁₆

yes

characteristic

109

C₂×C₄

C₂²×C₁₆

yes

—

110

C₂×C₄

C₂²×C₁₆

yes

—

111

C₂×C₄

C₂²×C₁₆

yes

—

112

C₂×C₄

C₂³×C₈

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#518

yes

characteristic

117

C₈

#568

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×C₁₆

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#2972

yes

characteristic

128

C₄

#3407

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

#33486

yes

characteristic

132

C₂

#33974

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	#532	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂×C₁₆

C₂⁴

commutator, socle

C₂²

C₂³×C₁₆

C₈

G ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₁₆ ◁ G

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13645	C₂	semidirect
3. #13640	C₂	semidirect
4. #13554	C₂	semidirect

H ≅ G/N

type

2. #13645

C₂

semidirect

3. #13640

C₂

semidirect

4. #13554

C₂

semidirect

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

subgroup	index
#33434	2
#13645	2
#13640	2
#13554	2

subgroup

index

Subgroups — #512804

Conjugacy classes of subgroups

Complements

Maximal subgroups