Group #9630 — subgroups

Subgroups — #9630

Order 256.

155 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 77 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	#1848	C₂	1	1	yes	maximal, characteristic
2	#1770	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	2	yes	maximal
4	#1564	C₂	1	1	yes	maximal, characteristic
5	#1303	C₂	1	2	yes	maximal
6	#431	C₂²	1	2	yes	—
7	C₂×#108	—	2	1	no	—
8	C₂×#108	—	2	1	no	—
9	#423	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#97	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#336	—	2	1	no	—
15	#335	—	2	4	no	—
16	#335	C₄	1	2	yes	complemented
17	#334	C₄	1	2	yes	complemented
18	C₂×M₁₆	—	2	2	no	—
19	C₂²×C₈	—	2	1	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄	—	2	1	no	—
23	C₂×C₄²	—	2	1	no	—
24	C₂×C₄²	C₂³	1	1	yes	characteristic
25	#108	—	4	2	no	—
26	#108	—	4	2	no	—
27	C₄⋊C₈	—	2	2	no	—
28	C₄⋊C₈	—	2	1	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₄⋊C₈	D₄	1	2	yes	—
32	C₂²⋊C₈	—	2	2	no	—
33	#97	—	4	2	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈	Q₈	1	1	yes	characteristic
38	C₄×C₈	Q₈	1	1	yes	characteristic
39	#95	Q₈	1	2	yes	—
40	C₂²×C₄	—	2	2	no	—
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄	#41	1	1	yes	characteristic
44	M₁₆	—	4	2	no	—
45	M₁₆	—	4	2	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	—	4	2	no	—
53	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈	D₈	1	2	yes	—
58	C₂×C₈	Q₁₆	1	2	yes	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄²	—	4	1	no	—
61	C₄²	—	4	1	no	—
62	C₄²	C₂²×C₄	1	1	yes	characteristic
63	C₄²	C₂×D₄	1	1	yes	characteristic
64	C₄²	C₂×Q₈	1	1	yes	characteristic
65	C₄²	#41	1	1	yes	characteristic
66	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄	#121	1	1	yes	characteristic
76	C₂×C₄	#125	1	1	yes	characteristic
77	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄	#137	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	2	1	no	—
84	C₈	—	4	4	no	—
85	C₈	—	4	2	no	—
86	C₈	—	8	2	no	—
87	C₈	#107	1	2	yes	—
88	C₈	#107	1	2	yes	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#389	1	1	yes	characteristic
92	C₂²	#474	1	1	yes	characteristic
93	C₂²	#477	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	2	1	no	—
97	C₄	—	8	2	no	—
98	C₄	C₂×#107	1	1	yes	characteristic
99	C₄	#428	1	1	yes	characteristic
100	C₄	#444	1	2	yes	—
101	C₄	#459	1	1	yes	characteristic
102	C₄	#462	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#1946	1	1	yes	characteristic
106	C₂	#1947	1	1	yes	characteristic
107	C₂	#1953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1848

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal, characteristic

#1571

C₂

yes

maximal

#1564

C₂

yes

maximal, characteristic

#1303

C₂

yes

maximal

#431

C₂²

yes

—

C₂×#108

—

C₂×#108

—

#423

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

C₄

yes

complemented

#334

C₄

yes

complemented

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

D₄

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

#95

Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

D₈

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₂²⋊Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#107

yes

—

C₈

#107

yes

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#444

yes

—

101

C₄

#459

yes

characteristic

102

C₄

#462

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#1946

yes

characteristic

106

C₂

#1947

yes

characteristic

107

C₂

#1953

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#389
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#389

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
16. #335	C₄	semidirect
17. #334	C₄	semidirect

H ≅ G/N

type

16. #335

C₄

semidirect

17. #334

C₄

semidirect

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

subgroup	index
#1848	2
#1770	2
#1571	2
#1564	2
#1303	2

subgroup

index

Subgroups — #9630

Conjugacy classes of subgroups

Complements

Maximal subgroups