Group #129121 — subgroups

Subgroups — #129121

Order 512.

237 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 140 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	#12656	C₂	1	2	yes	maximal
2	#12541	C₂	1	1	yes	maximal, characteristic
3	#12011	C₂	1	1	yes	maximal, characteristic
4	#12002	C₂	1	1	yes	maximal, characteristic
5	#7162	C₂	1	2	yes	maximal
6	C₂×#364	C₂²	1	2	yes	—
7	C₂×#364	C₂²	1	2	yes	—
8	C₂×#347	C₄	1	2	yes	—
9	C₂×#347	C₄	1	2	yes	—
10	C₄×M₃₂	—	2	2	no	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	#1726	C₂²	1	1	yes	characteristic
14	#1376	—	2	2	no	—
15	#1373	C₄	1	4	yes	complemented
16	#1369	C₄	1	4	yes	complemented
17	C₂×M₃₂	—	2	2	no	—
18	C₂×M₃₂	—	2	2	no	—
19	C₂×M₃₂	C₂×C₄	1	2	yes	—
20	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂×#97	C₂×C₄	1	2	yes	—
23	C₂×#97	Q₈	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	2	yes	—
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	#376	Q₈	1	1	yes	characteristic
27	#364	—	2	2	no	—
28	#364	—	2	2	no	—
29	#364	C₂×C₄	1	4	yes	—
30	#364	C₂×C₄	1	4	yes	—
31	#347	—	2	4	no	—
32	#347	C₈	1	8	yes	—
33	#346	—	2	1	no	—
34	#346	—	4	2	no	—
35	#346	C₂×C₄	1	2	yes	—
36	C₄×C₁₆	—	2	1	no	—
37	C₄×C₁₆	—	4	2	no	—
38	C₄×C₁₆	C₂×C₄	1	2	yes	—
39	C₂²×C₈	C₄⋊C₄	1	2	yes	—
40	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄²	C₄⋊C₄	1	2	yes	—
43	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
44	M₃₂	—	2	2	no	—
45	M₃₂	—	2	2	no	—
46	M₃₂	—	2	4	no	—
47	M₃₂	—	2	4	no	—
48	M₃₂	—	4	2	no	—
49	M₃₂	—	4	2	no	—
50	M₃₂	C₂×C₈	1	4	yes	—
51	M₃₂	C₂×C₈	1	4	yes	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	—	4	2	no	—
55	C₂×C₁₆	C₄²	1	2	yes	—
56	C₂×C₁₆	C₄²	1	2	yes	—
57	#97	—	4	1	no	—
58	#97	C₂×C₈	1	4	yes	—
59	#97	M₁₆	1	4	yes	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₄×C₈	#41	1	1	yes	characteristic
65	C₄×C₈	#41	1	1	yes	characteristic
66	#95	—	4	1	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₄⋊C₈	1	4	yes	—
73	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₈	#117	1	1	yes	characteristic
75	C₂×C₈	C₄×D₄	1	1	yes	characteristic
76	C₂×C₈	C₄×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂²×C₈	1	2	yes	—
78	C₂×C₈	C₂×M₁₆	1	2	yes	—
79	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₈	#137	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	—	2	1	no	—
83	C₄²	C₄⋊C₈	1	4	yes	—
84	C₄²	C₄×Q₈	1	1	yes	characteristic
85	C₄²	C₄×Q₈	1	1	yes	characteristic
86	C₁₆	—	2	2	no	—
87	C₁₆	—	2	4	no	—
88	C₁₆	—	4	4	no	—
89	C₁₆	—	4	4	no	—
90	C₁₆	C₄×C₈	1	4	yes	—
91	C₂³	#430	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	—	8	1	no	—
97	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
99	C₂×C₄	#428	1	2	yes	—
100	C₂×C₄	#430	1	1	yes	characteristic
101	C₈	—	2	4	no	—
102	C₈	C₂×C₄×C₈	1	1	yes	characteristic
103	C₈	C₄×M₁₆	1	1	yes	characteristic
104	C₈	#431	1	2	yes	—
105	C₈	#440	1	2	yes	—
106	C₈	#441	1	1	yes	characteristic
107	C₈	#442	1	1	yes	characteristic
108	C₂²	—	4	1	no	—
109	C₂²	#1565	1	2	yes	—
110	C₂²	#1778	1	1	yes	characteristic
111	C₄	—	4	4	no	—
112	C₄	—	4	1	no	—
113	C₄	—	4	1	no	—
114	C₄	#1565	1	2	yes	—
115	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
116	C₄	#1776	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	—	2	1	no	—
119	C₂	#8275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12656

C₂

yes

maximal

#12541

C₂

yes

maximal, characteristic

#12011

C₂

yes

maximal, characteristic

#12002

C₂

yes

maximal, characteristic

#7162

C₂

yes

maximal

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

C₂×#347

C₄

yes

—

C₂×#347

C₄

yes

—

C₄×M₃₂

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1376

—

#1373

C₄

yes

complemented

#1369

C₄

yes

complemented

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×#97

C₂×C₄

yes

—

C₂×#97

Q₈

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

Q₈

yes

—

—

C₂×C₄

yes

—

#364

C₂×C₄

yes

—

#347

—

#347

C₈

yes

—

#346

—

#346

—

#346

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×Q₈

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂×C₈

yes

—

M₃₂

C₂×C₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

#97

—

#97

C₂×C₈

yes

—

#97

M₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₂²×C₈

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄²

—

C₄²

—

C₄²

C₄⋊C₈

yes

—

C₄²

C₄×Q₈

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

—

C₂³

#430

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

—

C₂×C₄

#428

yes

—

100

C₂×C₄

#430

yes

characteristic

101

C₈

—

102

C₈

C₂×C₄×C₈

yes

characteristic

103

C₈

C₄×M₁₆

yes

characteristic

104

C₈

#431

yes

—

105

C₈

#440

yes

—

106

C₈

#441

yes

characteristic

107

C₈

#442

yes

characteristic

108

C₂²

—

109

C₂²

#1565

yes

—

110

C₂²

#1778

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#1565

yes

—

115

C₄

C₄×C₄⋊C₈

yes

characteristic

116

C₄

#1776

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

#8275

yes

characteristic

	subgroup	G/H
center	C₈	#442
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8275
prime radical	C₂³	#430
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#442

commutator

C₈

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
15. #1373	C₄	semidirect
16. #1369	C₄	semidirect

H ≅ G/N

type

15. #1373

C₄

semidirect

16. #1369

C₄

semidirect

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

subgroup	index
#12656	2
#12541	2
#12011	2
#12002	2
#7162	2

subgroup

index

Subgroups — #129121

Conjugacy classes of subgroups

Complements

Maximal subgroups