Group #512807 — subgroups

Subgroups — #512807

Order 512.

308 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 240 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	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#13660	C₂	1	8	yes	maximal, complemented
3	#13640	C₂	1	4	yes	maximal
4	#13639	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
6	Q₈×C₁₆	C₄	1	4	yes	—
7	Q₈×C₁₆	C₂²	1	4	yes	—
8	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	C₄⋊C₃₂	C₂²	1	4	yes	—
13	C₄⋊C₃₂	C₂²	1	8	yes	—
14	C₂²⋊C₃₂	C₂²	1	4	yes	—
15	#1397	C₂²	1	8	yes	—
16	C₄×C₃₂	C₂²	1	4	yes	—
17	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	C₂³	1	1	yes	characteristic
21	C₈×Q₈	C₈	1	4	yes	—
22	C₈×Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₈	C₈	1	2	yes	—
24	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃₂	C₂³	1	8	yes	—
28	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
29	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
30	C₄×C₁₆	C₂×C₄	1	2	yes	—
31	C₄×C₁₆	C₂×C₄	1	4	yes	—
32	C₄×C₁₆	C₂³	1	2	yes	—
33	C₄×C₁₆	C₂³	1	4	yes	—
34	C₂²×Q₈	C₁₆	1	1	yes	characteristic
35	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₈	C₂×C₈	1	2	yes	—
37	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₄×Q₈	C₁₆	1	4	yes	—
39	C₄×Q₈	C₂×C₈	1	4	yes	—
40	C₂×C₄⋊C₄	C₁₆	1	2	yes	—
41	C₂×C₄⋊C₄	C₁₆	1	1	yes	characteristic
42	C₂×C₄²	C₂×C₈	1	2	yes	—
43	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
44	C₂×C₁₆	—	2	2	no	—
45	C₂×C₁₆	—	4	1	no	—
46	C₂×C₁₆	—	4	2	no	—
47	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₁₆	C₂²×C₄	1	2	yes	—
49	C₂×C₁₆	C₂²×C₄	1	4	yes	—
50	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₈	C₂×C₈	1	8	yes	—
52	C₄⋊C₈	C₂×C₈	1	4	yes	—
53	C₄×C₈	C₂×C₈	1	4	yes	—
54	C₄×C₈	C₂×C₈	1	2	yes	—
55	C₄×C₈	C₂²×C₄	1	2	yes	—
56	C₄×C₈	C₂²×C₄	1	4	yes	—
57	C₃₂	—	2	8	no	—
58	C₂×Q₈	—	2	4	no	—
59	C₂×Q₈	C₂×C₁₆	1	4	yes	—
60	C₂²×C₄	C₂×C₁₆	1	2	yes	—
61	C₂²×C₄	C₂×C₁₆	1	2	yes	—
62	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
63	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
64	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₈	C₂²×C₈	1	2	yes	—
71	C₂×C₈	C₂²×C₈	1	4	yes	—
72	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
73	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
74	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
75	C₄²	C₂×C₁₆	1	4	yes	—
76	C₄²	C₂×C₁₆	1	2	yes	—
77	C₄²	C₂²×C₈	1	4	yes	—
78	C₄²	C₂²×C₈	1	2	yes	—
79	C₁₆	—	2	1	no	—
80	C₁₆	—	2	2	no	—
81	C₁₆	—	2	4	no	—
82	C₁₆	2₋¹⁺⁴	1	2	yes	—
83	C₂³	C₂²×C₁₆	1	1	yes	characteristic
84	Q₈	—	2	8	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	M₆₄	1	4	yes	—
91	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
92	C₂×C₄	C₂²×C₁₆	1	4	yes	—
93	C₂×C₄	C₂²×C₁₆	1	2	yes	—
94	C₂×C₄	C₂²×C₁₆	1	4	yes	—
95	C₂×C₄	C₂²×C₁₆	1	2	yes	—
96	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	4	no	—
99	C₈	—	2	1	no	—
100	C₈	#519	1	1	yes	characteristic
101	C₈	#568	1	1	yes	characteristic
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₂×M₆₄	1	2	yes	—
105	C₂²	C₂³×C₁₆	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	2	4	no	—
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	C₂×M₆₄	1	4	yes	—
111	C₄	#2982	1	1	yes	characteristic
112	C₄	#3407	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	#33496	1	1	yes	characteristic
115	C₂	C₂²×M₆₄	1	1	yes	characteristic
116	C₂	#33974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

#13660

C₂

yes

maximal, complemented

#13640

C₂

yes

maximal

#13639

C₂

yes

maximal

C₂×C₈×Q₈

C₄

yes

characteristic

Q₈×C₁₆

C₄

yes

—

Q₈×C₁₆

C₂²

yes

—

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

—

C₂²⋊C₃₂

C₂²

yes

—

#1397

C₂²

yes

—

C₄×C₃₂

C₂²

yes

—

C₂×C₄×Q₈

C₈

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

C₈×Q₈

C₈

yes

—

C₈×Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₈

C₈

yes

—

C₂×C₄⋊C₈

C₈

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₃₂

C₂³

yes

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₄×C₁₆

C₂³

yes

—

C₂²×Q₈

C₁₆

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₄×Q₈

C₁₆

yes

—

C₄×Q₈

C₂×C₈

yes

—

C₂×C₄⋊C₄

C₁₆

yes

—

C₂×C₄⋊C₄

C₁₆

yes

characteristic

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

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

—

C₄⋊C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

—

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

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

—

C₄⋊C₄

C₂×C₁₆

yes

—

C₄²

C₂×C₁₆

yes

—

C₄²

C₂×C₁₆

yes

—

C₄²

C₂²×C₈

yes

—

C₄²

C₂²×C₈

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

2₋¹⁺⁴

yes

—

C₂³

C₂²×C₁₆

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

M₆₄

yes

—

C₂×C₄

C₂²×C₁₆

yes

characteristic

C₂×C₄

C₂²×C₁₆

yes

—

C₂×C₄

C₂²×C₁₆

yes

—

C₂×C₄

C₂²×C₁₆

yes

—

C₂×C₄

C₂²×C₁₆

yes

—

C₂×C₄

C₂³×C₈

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

#519

yes

characteristic

101

C₈

#568

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×M₆₄

yes

—

105

C₂²

C₂³×C₁₆

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₂×M₆₄

yes

—

111

C₄

#2982

yes

characteristic

112

C₄

#3407

yes

characteristic

113

C₂

—

114

C₂

#33496

yes

characteristic

115

C₂

C₂²×M₆₄

yes

characteristic

116

C₂

#33974

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂³	C₂²×C₁₆
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

prime radical

C₂³

C₂²×C₁₆

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13660	C₂	semidirect

H ≅ G/N

type

2. #13660

C₂

semidirect

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

subgroup	index
C₂×Q₈×C₁₆	2
#13660	2
#13640	2
#13639	2

subgroup

index

C₂×Q₈×C₁₆

#13660

#13640

#13639

Subgroups — #512807

Conjugacy classes of subgroups

Complements

Maximal subgroups