Subgroups — C₃×D₃×SL(2,𝔽₃) #88976

Order 432.

125 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×SL(2,𝔽₃)	—	3	1	no	maximal
4	D₃×SL(2,𝔽₃)	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃²×D₃	—	4	1	no	maximal
6	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₄×D₃	—	3	1	no	—
8	C₃×Dic₆	—	3	1	no	—
9	C₃×SL(2,𝔽₃)	—	2	1	no	—
10	C₃×SL(2,𝔽₃)	—	2	3	no	—
11	C₃×SL(2,𝔽₃)	D₃	1	1	yes	characteristic, direct product factor
12	C₃×SL(2,𝔽₃)	C₂×C₃	1	3	yes	complemented
13	C₂×C₃³	—	4	1	no	—
14	C₃²×D₃	—	4	2	no	—
15	C₂×C₃×Q₈	—	3	1	no	—
16	D₃×Q₈	C₃²	1	1	yes	characteristic, complemented
17	C₂×SL(2,𝔽₃)	—	3	3	no	—
18	C₂²×C₃²	—	12	1	no	—
19	C₂×C₃×D₃	—	4	3	no	—
20	C₂×C₃×D₃	A₄	1	1	yes	characteristic
21	C₃²×C₄	—	3	1	no	—
22	C₃×Dic₃	—	3	1	no	—
23	C₃³	—	4	1	no	Sylow
24	C₃×Q₈	—	2	1	no	—
25	C₃×Q₈	—	9	1	no	—
26	C₃×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
27	C₃×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
28	C₂×C₃×C₄	—	9	1	no	—
29	C₄×D₃	—	3	1	no	—
30	Dic₆	—	3	1	no	—
31	SL(2,𝔽₃)	—	2	3	no	—
32	SL(2,𝔽₃)	C₃×D₃	1	3	yes	direct product factor
33	C₂×C₃²	—	4	1	no	—
34	C₂×C₃²	—	4	3	no	—
35	C₂×C₃²	—	8	3	no	—
36	C₂×C₃²	—	8	1	no	—
37	C₂×C₃²	—	12	2	no	—
38	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
39	C₃×D₃	—	4	6	no	—
40	C₃×D₃	SL(2,𝔽₃)	1	2	yes	direct product factor
41	C₂×Q₈	—	3	1	no	Sylow
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	—	12	3	no	—
44	C₂×D₃	C₃×A₄	1	1	yes	characteristic
45	C₃×C₄	—	3	1	no	—
46	C₃×C₄	—	3	1	no	—
47	C₃×C₄	—	6	1	no	—
48	C₃×C₄	—	9	1	no	—
49	Dic₃	—	3	1	no	—
50	C₃²	—	4	1	no	—
51	C₃²	—	4	3	no	—
52	C₃²	—	8	3	no	—
53	C₃²	—	8	1	no	—
54	C₃²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
55	Q₈	—	9	1	no	—
56	Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
57	C₂×C₄	—	9	1	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	3	2	no	—
60	C₂×C₃	—	4	3	no	—
61	C₂×C₃	—	8	3	no	—
62	C₂×C₃	—	12	6	no	—
63	C₂×C₃	D₃×A₄	1	1	yes	characteristic
64	C₂×C₃	C₂×C₃×A₄	1	1	yes	characteristic
65	D₃	C₃×SL(2,𝔽₃)	1	2	yes	direct product factor
66	C₂²	—	3	1	no	—
67	C₄	—	3	1	no	—
68	C₄	—	9	1	no	—
69	C₃	—	2	1	no	—
70	C₃	—	4	3	no	—
71	C₃	—	8	3	no	—
72	C₃	D₃×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
73	C₃	C₂×C₃×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
74	C₂	—	3	2	no	—
75	C₂	C₃×D₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×A₄
commutator, cosocle, hypocenter	C₃×Q₈	C₂×C₃²
Frattini	C₂	C₃×D₃×A₄
Fitting	C₃²×Q₈	C₂×C₃
socle	C₂×C₃²	C₂×A₄
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂×C₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃²×SL(2,𝔽₃)	C₂	semidirect
2. C₃×D₃×Q₈	C₃	semidirect
4. D₃×SL(2,𝔽₃)	C₃	direct (N × H)
6. C₃²×Q₈	C₂×C₃	semidirect
11. C₃×SL(2,𝔽₃)	D₃	direct (N × H)
12. C₃×SL(2,𝔽₃)	C₂×C₃	semidirect
16. D₃×Q₈	C₃²	semidirect
26. C₃×Q₈	C₃×D₃	semidirect
27. C₃×Q₈	C₂×C₃²	semidirect
32. SL(2,𝔽₃)	C₃×D₃	direct (N × H)
40. C₃×D₃	SL(2,𝔽₃)	direct (N × H)
54. C₃²	C₂×SL(2,𝔽₃)	semidirect
56. Q₈	C₃²×D₃	semidirect
65. D₃	C₃×SL(2,𝔽₃)	direct (N × H)
72. C₃	D₃×SL(2,𝔽₃)	direct (N × H)
73. C₃	C₂×C₃×SL(2,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×SL(2,𝔽₃)	2
C₃×D₃×Q₈	3
C₂×C₃×SL(2,𝔽₃)	3
D₃×SL(2,𝔽₃)	3
C₂×C₃²×D₃	4

Subgroups — C3×D3×SL(2,𝔽3) #88976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×SL(2,𝔽₃) #88976