Group #13048925 — subgroups

Subgroups — Q₈×C₉×C₃⋊D₃ #13048925

Order 1296.

438 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₃²×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₉×C₃⋊D₃	C₂	1	3	yes	maximal
3	C₉×#634	C₂	1	3	yes	maximal, complemented
4	C₃×Q₈×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic
5	D₃×Q₈×C₉	—	3	4	no	maximal
6	C₂×C₉×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃²×C₄×C₉	C₂²	1	3	yes	—
8	C₉×C₃⋊Dic₃	C₂²	1	3	yes	—
9	C₃³×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃×C₄×C₃⋊D₃	C₂×C₃	1	3	yes	—
11	C₃×#634	C₂×C₃	1	3	yes	—
12	C₃×Q₈×C₉	—	2	4	no	—
13	C₃×Q₈×C₉	D₃	1	4	yes	complemented
14	C₄×D₃×C₉	—	3	12	no	—
15	C₉×Dic₆	—	3	12	no	—
16	C₂×C₃²×C₉	C₂³	1	1	yes	characteristic
17	C₉×C₃⋊D₃	Q₈	1	2	yes	direct product factor
18	Q₈×C₃⋊D₃	C₉	1	1	yes	characteristic, direct product factor
19	C₃×D₃×Q₈	—	3	4	no	—
20	C₂×Q₈×C₉	—	9	1	no	—
21	C₂×C₃×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
22	C₃³×C₄	C₂²×C₃	1	3	yes	—
23	C₃×C₃⋊Dic₃	C₂²×C₃	1	3	yes	—
24	C₂×D₃×C₉	—	3	4	no	—
25	C₃×C₄×C₉	—	2	12	no	—
26	C₃×C₄×C₉	C₂×D₃	1	12	yes	—
27	C₉×Dic₃	—	3	12	no	—
28	C₃²×C₉	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
29	C₃²×Q₈	—	2	4	no	—
30	C₃²×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
31	C₃²×Q₈	C₃×D₃	1	4	yes	—
32	C₄×C₃⋊D₃	C₂×C₉	1	3	yes	—
33	#634	C₂×C₉	1	3	yes	complemented
34	C₃×C₄×D₃	—	3	12	no	—
35	C₃×Dic₆	—	3	12	no	—
36	Q₈×C₉	—	2	4	no	—
37	Q₈×C₉	—	9	3	no	—
38	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic, direct product factor
39	C₂×C₄×C₉	—	9	3	no	—
40	C₂×C₃³	C₂³×C₃	1	1	yes	characteristic
41	C₃×C₃⋊D₃	C₃×Q₈	1	2	yes	—
42	C₂×C₃×C₉	—	2	4	no	—
43	C₂×C₃×C₉	C₂²×D₃	1	4	yes	—
44	D₃×C₉	—	3	8	no	—
45	C₂×C₃×Q₈	—	9	1	no	—
46	D₃×Q₈	—	3	4	no	—
47	C₂×C₃⋊D₃	C₂²×C₉	1	1	yes	characteristic
48	C₂×C₃×D₃	—	3	4	no	—
49	C₃²×C₄	—	2	12	no	—
50	C₃²×C₄	C₂²×C₉	1	3	yes	—
51	C₃²×C₄	C₂×C₃×D₃	1	12	yes	—
52	C₃⋊Dic₃	C₂²×C₉	1	3	yes	—
53	C₃×Dic₃	—	3	12	no	—
54	C₂²×C₉	—	9	1	no	—
55	C₄×C₉	—	2	12	no	—
56	C₄×C₉	—	9	3	no	—
57	C₄×C₉	C₂×C₃⋊D₃	1	3	yes	—
58	C₃³	C₂×C₃×Q₈	1	1	yes	characteristic
59	C₃×C₉	—	2	4	no	—
60	C₃×C₉	D₃×Q₈	1	4	yes	complemented
61	C₃×Q₈	—	2	4	no	—
62	C₃×Q₈	—	9	3	no	—
63	C₃×Q₈	D₃×C₉	1	4	yes	complemented
64	C₃×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
65	C₂×C₃×C₄	—	9	3	no	—
66	C₄×D₃	—	3	12	no	—
67	Dic₆	—	3	12	no	—
68	C₂×C₃²	—	2	4	no	—
69	C₂×C₃²	C₂³×C₉	1	1	yes	characteristic
70	C₂×C₃²	C₂²×C₃×D₃	1	4	yes	—
71	C₃⋊D₃	Q₈×C₉	1	2	yes	direct product factor
72	C₃×D₃	—	3	8	no	—
73	C₂×C₉	—	2	4	no	—
74	C₂×C₉	—	9	2	no	—
75	C₂×C₉	C₂²×C₃⋊D₃	1	1	yes	characteristic
76	C₂×Q₈	—	9	1	no	Sylow
77	C₂²×C₃	—	9	1	no	—
78	C₂×D₃	—	3	4	no	—
79	C₃×C₄	—	2	12	no	—
80	C₃×C₄	—	9	3	no	—
81	C₃×C₄	C₂×D₃×C₉	1	12	yes	—
82	C₃×C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
83	Dic₃	—	3	12	no	—
84	C₃²	—	2	4	no	—
85	C₃²	C₂×Q₈×C₉	1	1	yes	characteristic, complemented
86	C₃²	C₃×D₃×Q₈	1	4	yes	—
87	C₉	—	2	4	no	—
88	C₉	Q₈×C₃⋊D₃	1	1	yes	characteristic, direct product factor
89	Q₈	—	9	3	no	—
90	Q₈	C₉×C₃⋊D₃	1	1	yes	characteristic, direct product factor
91	C₂×C₄	—	9	3	no	—
92	C₂×C₃	—	2	4	no	—
93	C₂×C₃	—	9	2	no	—
94	C₂×C₃	C₂²×D₃×C₉	1	4	yes	—
95	C₂×C₃	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
96	D₃	—	3	8	no	—
97	C₂²	—	9	1	no	—
98	C₄	—	9	3	no	—
99	C₄	C₂×C₉×C₃⋊D₃	1	3	yes	—
100	C₃	—	2	4	no	—
101	C₃	D₃×Q₈×C₉	1	4	yes	complemented
102	C₃	C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
103	C₂	—	9	2	no	—
104	C₂	C₂²×C₉×C₃⋊D₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×Q₈×C₉

C₂

yes

maximal, characteristic, complemented

C₄×C₉×C₃⋊D₃

C₂

yes

maximal

C₉×#634

C₂

yes

maximal, complemented

C₃×Q₈×C₃⋊D₃

C₃

yes

maximal, characteristic

D₃×Q₈×C₉

—

maximal

C₂×C₉×C₃⋊D₃

C₂²

yes

characteristic

C₃²×C₄×C₉

C₂²

yes

—

C₉×C₃⋊Dic₃

C₂²

yes

—

C₃³×Q₈

C₂×C₃

yes

characteristic

C₃×C₄×C₃⋊D₃

C₂×C₃

yes

—

C₃×#634

C₂×C₃

yes

—

C₃×Q₈×C₉

—

C₃×Q₈×C₉

D₃

yes

complemented

C₄×D₃×C₉

—

C₉×Dic₆

—

C₂×C₃²×C₉

C₂³

yes

characteristic

C₉×C₃⋊D₃

Q₈

yes

direct product factor

Q₈×C₃⋊D₃

C₉

yes

characteristic, direct product factor

C₃×D₃×Q₈

—

C₂×Q₈×C₉

—

C₂×C₃×C₃⋊D₃

C₂²×C₃

yes

characteristic

C₃³×C₄

C₂²×C₃

yes

—

C₃×C₃⋊Dic₃

C₂²×C₃

yes

—

C₂×D₃×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×D₃

yes

—

C₉×Dic₃

—

C₃²×C₉

C₂×Q₈

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

C₂×C₉

yes

characteristic, complemented

C₃²×Q₈

C₃×D₃

yes

—

C₄×C₃⋊D₃

C₂×C₉

yes

—

#634

C₂×C₉

yes

complemented

C₃×C₄×D₃

—

C₃×Dic₆

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

C₃⋊D₃

yes

characteristic, direct product factor

C₂×C₄×C₉

—

C₂×C₃³

C₂³×C₃

yes

characteristic

C₃×C₃⋊D₃

C₃×Q₈

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²×D₃

yes

—

D₃×C₉

—

C₂×C₃×Q₈

—

D₃×Q₈

—

C₂×C₃⋊D₃

C₂²×C₉

yes

characteristic

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

C₂²×C₉

yes

—

C₃²×C₄

C₂×C₃×D₃

yes

—

C₃⋊Dic₃

C₂²×C₉

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₃⋊D₃

yes

—

C₃³

C₂×C₃×Q₈

yes

characteristic

C₃×C₉

—

C₃×C₉

D₃×Q₈

yes

complemented

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

D₃×C₉

yes

complemented

C₃×Q₈

C₃×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

—

C₄×D₃

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

C₂³×C₉

yes

characteristic

C₂×C₃²

C₂²×C₃×D₃

yes

—

C₃⋊D₃

Q₈×C₉

yes

direct product factor

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×C₃⋊D₃

yes

characteristic

C₂×Q₈

—

Sylow

C₂²×C₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×D₃×C₉

yes

—

C₃×C₄

C₂×C₃×C₃⋊D₃

yes

—

Dic₃

—

C₃²

—

C₃²

C₂×Q₈×C₉

yes

characteristic, complemented

C₃²

C₃×D₃×Q₈

yes

—

C₉

—

C₉

Q₈×C₃⋊D₃

yes

characteristic, direct product factor

Q₈

—

Q₈

C₉×C₃⋊D₃

yes

characteristic, direct product factor

C₂×C₄

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂²×D₃×C₉

yes

—

C₂×C₃

C₂²×C₃×C₃⋊D₃

yes

characteristic

D₃

—

C₂²

—

C₄

—

C₄

C₂×C₉×C₃⋊D₃

yes

—

100

C₃

—

101

C₃

D₃×Q₈×C₉

yes

complemented

102

C₃

C₃×Q₈×C₃⋊D₃

yes

characteristic

103

C₂

—

104

C₂

C₂²×C₉×C₃⋊D₃

yes

characteristic

	subgroup	G/H
center	C₂×C₉	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₉
Frattini	C₂×C₃	C₂²×C₃×C₃⋊D₃
Fitting	C₃²×Q₈×C₉	C₂
socle, cosocle	C₂×C₃³	C₂³×C₃
prime radical	C₂×C₃×C₃⋊D₃	C₂²×C₃
hypercenter	Q₈×C₉	C₃⋊D₃
hypocenter	C₃²	C₂×Q₈×C₉
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₉ ◁ Q₈×C₉

subgroup

G/H

center

C₂×C₉

C₂²×C₃⋊D₃

commutator

C₂×C₃²

C₂³×C₉

Frattini

C₂×C₃

C₂²×C₃×C₃⋊D₃

Fitting

C₃²×Q₈×C₉

C₂

socle, cosocle

C₂×C₃³

C₂³×C₃

prime radical

C₂×C₃×C₃⋊D₃

C₂²×C₃

hypercenter

Q₈×C₉

C₃⋊D₃

hypocenter

C₃²

C₂×Q₈×C₉

LCS

G ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₉ ◁ Q₈×C₉

N	H ≅ G/N	type
1. C₃²×Q₈×C₉	C₂	semidirect
3. C₉×#634	C₂	semidirect
13. C₃×Q₈×C₉	D₃	semidirect
17. C₉×C₃⋊D₃	Q₈	direct (N × H)
18. Q₈×C₃⋊D₃	C₉	direct (N × H)
28. C₃²×C₉	C₂×Q₈	semidirect
30. C₃²×Q₈	C₂×C₉	semidirect
33. #634	C₂×C₉	semidirect
38. Q₈×C₉	C₃⋊D₃	direct (N × H)
60. C₃×C₉	D₃×Q₈	semidirect
63. C₃×Q₈	D₃×C₉	semidirect
71. C₃⋊D₃	Q₈×C₉	direct (N × H)
85. C₃²	C₂×Q₈×C₉	semidirect
88. C₉	Q₈×C₃⋊D₃	direct (N × H)
90. Q₈	C₉×C₃⋊D₃	direct (N × H)
101. C₃	D₃×Q₈×C₉	semidirect

H ≅ G/N

type

1. C₃²×Q₈×C₉

C₂

semidirect

3. C₉×#634

C₂

semidirect

13. C₃×Q₈×C₉

D₃

semidirect

17. C₉×C₃⋊D₃

Q₈

direct (N × H)

18. Q₈×C₃⋊D₃

C₉

direct (N × H)

28. C₃²×C₉

C₂×Q₈

semidirect

30. C₃²×Q₈

C₂×C₉

semidirect

33. #634

C₂×C₉

semidirect

38. Q₈×C₉

C₃⋊D₃

direct (N × H)

60. C₃×C₉

D₃×Q₈

semidirect

63. C₃×Q₈

D₃×C₉

semidirect

71. C₃⋊D₃

Q₈×C₉

direct (N × H)

85. C₃²

C₂×Q₈×C₉

semidirect

88. C₉

Q₈×C₃⋊D₃

direct (N × H)

90. Q₈

C₉×C₃⋊D₃

direct (N × H)

101. C₃

D₃×Q₈×C₉

semidirect

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

subgroup	index
C₃²×Q₈×C₉	2
C₄×C₉×C₃⋊D₃	2
C₉×#634	2
C₃×Q₈×C₃⋊D₃	3
D₃×Q₈×C₉	3

subgroup

index

C₃²×Q₈×C₉

C₄×C₉×C₃⋊D₃

C₉×#634

C₃×Q₈×C₃⋊D₃

D₃×Q₈×C₉

Subgroups — Q₈×C₉×C₃⋊D₃ #13048925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C9×C3⋊D3 #13048925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉×C₃⋊D₃ #13048925