Group #153000 — subgroups

Subgroups — #153000

Order 512.

224 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 114 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	#33474	C₂	1	1	yes	maximal, characteristic
2	#13627	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
4	#7470	C₂	1	4	yes	maximal
5	#2960	C₄	1	1	yes	characteristic
6	C₂×M₆₄	C₂²	1	2	yes	—
7	C₂²×C₃₂	C₂²	1	2	yes	—
8	#2183	—	2	2	no	—
9	Q₈×C₁₆	—	2	2	no	—
10	#2151	C₄	1	2	yes	—
11	#2150	C₄	1	1	yes	characteristic
12	#2108	C₂²	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄⋊C₃₂	—	2	2	no	—
15	C₄×C₃₂	—	2	2	no	—
16	#533	C₈	1	1	yes	characteristic
17	C₂×M₃₂	C₂×C₄	1	2	yes	—
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂³	1	1	yes	characteristic
20	#446	—	2	2	no	—
21	C₈×Q₈	—	2	2	no	—
22	#424	C₈	1	2	yes	—
23	#423	C₈	1	1	yes	characteristic
24	#405	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	M₆₄	—	2	4	no	—
27	C₂×C₃₂	—	2	2	no	—
28	C₂×C₃₂	—	2	2	no	—
29	C₂×C₃₂	—	2	2	no	—
30	C₂×C₃₂	Q₈	1	4	yes	complemented
31	C₄⋊C₁₆	—	2	4	no	—
32	C₄⋊C₁₆	—	2	2	no	—
33	#346	—	2	2	no	—
34	C₄×C₁₆	—	2	2	no	—
35	C₄×C₁₆	D₄	1	2	yes	—
36	C₄×C₁₆	D₄	1	2	yes	—
37	C₂×M₁₆	C₂×C₈	1	2	yes	—
38	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₄⋊Q₈	C₁₆	1	2	yes	—
41	#125	C₁₆	1	2	yes	—
42	C₂²⋊Q₈	—	2	2	no	—
43	C₄×Q₈	—	2	2	no	—
44	#117	C₂×C₈	1	2	yes	—
45	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
46	M₃₂	—	2	4	no	—
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
50	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
51	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
52	C₂×C₁₆	C₂×Q₈	1	2	yes	—
53	C₂×C₁₆	#41	1	2	yes	—
54	C₄⋊C₈	—	2	4	no	—
55	C₄⋊C₈	—	2	2	no	—
56	#97	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	C₂²⋊C₄	1	2	yes	—
59	C₄×C₈	M₁₆	1	2	yes	—
60	C₃₂	—	2	4	no	—
61	C₃₂	—	4	2	no	—
62	C₂×Q₈	—	2	2	no	—
63	C₂²×C₄	M₃₂	1	2	yes	—
64	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	C₂²⋊C₈	1	2	yes	—
69	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈	C₄×Q₈	1	2	yes	—
71	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₈	#131	1	2	yes	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	C₂²⋊C₈	1	2	yes	—
79	C₄²	C₂×C₁₆	1	1	yes	characteristic
80	C₄²	C₂×C₁₆	1	1	yes	characteristic
81	C₁₆	—	2	2	no	—
82	C₁₆	—	4	2	no	—
83	C₁₆	C₂²⋊Q₈	1	2	yes	—
84	C₁₆	#124	1	1	yes	characteristic
85	C₁₆	C₄⋊Q₈	1	1	yes	characteristic
86	C₂³	C₂×M₃₂	1	1	yes	characteristic
87	Q₈	—	4	4	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
91	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
92	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₂×C₄	C₈×Q₈	1	2	yes	—
94	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
95	C₂×C₄	#504	1	2	yes	—
96	C₈	—	2	2	no	—
97	C₈	—	4	2	no	—
98	C₈	#391	1	1	yes	characteristic
99	C₈	#408	1	1	yes	characteristic
100	C₈	#446	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
103	C₂²	#2183	1	2	yes	—
104	C₄	—	2	2	no	—
105	C₄	—	4	4	no	—
106	C₄	#1401	1	4	yes	—
107	C₄	#1984	1	1	yes	characteristic
108	C₄	#2113	1	1	yes	characteristic
109	C₄	Q₈×C₁₆	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	#13263	1	1	yes	characteristic
112	C₂	C₂×#1401	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33474

C₂

yes

maximal, characteristic

#13627

C₂

yes

maximal, characteristic

C₂×C₄×C₃₂

C₂

yes

maximal, characteristic

#7470

C₂

yes

maximal

#2960

C₄

yes

characteristic

C₂×M₆₄

C₂²

yes

—

C₂²×C₃₂

C₂²

yes

—

#2183

—

Q₈×C₁₆

—

#2151

C₄

yes

—

#2150

C₄

yes

characteristic

#2108

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄⋊C₃₂

—

C₄×C₃₂

—

#533

C₈

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₈

yes

—

#423

C₈

yes

characteristic

#405

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

Q₈

yes

complemented

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

C₂×M₁₆

C₂×C₈

yes

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₄⋊Q₈

C₁₆

yes

—

#125

C₁₆

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

M₁₆

yes

—

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂²×C₄

M₃₂

yes

—

C₂²×C₄

C₂²×C₈

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

C₂×M₁₆

yes

characteristic

C₂×C₈

#131

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₁₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊C₈

yes

—

C₄²

C₂×C₁₆

yes

characteristic

C₄²

C₂×C₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₂²⋊Q₈

yes

—

C₁₆

#124

yes

characteristic

C₁₆

C₄⋊Q₈

yes

characteristic

C₂³

C₂×M₃₂

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₁₆

yes

—

C₂×C₄

C₂²⋊C₁₆

yes

—

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₄

C₈×Q₈

yes

—

C₂×C₄

C₂²×C₁₆

yes

characteristic

C₂×C₄

#504

yes

—

C₈

—

C₈

—

C₈

yes

C₈

yes

100

C₈

#446

yes

—

101

C₂²

—

102

C₂²

C₂×C₂²⋊C₁₆

yes

characteristic

103

C₂²

#2183

yes

—

104

C₄

—

105

C₄

—

106

C₄

#1401

yes

—

107

C₄

#1984

yes

characteristic

108

C₄

#2113

yes

characteristic

109

C₄

Q₈×C₁₆

yes

—

110

C₂

—

111

C₂

#13263

yes

characteristic

112

C₂

C₂×#1401

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

C₂×D₄

commutator

C₂×C₄

C₂²×C₁₆

Frattini, cosocle

C₂²×C₁₆

C₂³

socle

C₂²

C₂×C₂²⋊C₁₆

prime radical

C₂³

C₂×M₃₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
30. C₂×C₃₂	Q₈	semidirect

H ≅ G/N

type

30. C₂×C₃₂

Q₈

semidirect

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

subgroup	index
#33474	2
#13627	2
C₂×C₄×C₃₂	2
#7470	2

subgroup

index

#33474

#13627

C₂×C₄×C₃₂

#7470

Subgroups — #153000

Conjugacy classes of subgroups

Complements

Maximal subgroups