Subgroups — C₃×C₃⋊GL(2,𝔽₃) #88970

Order 432.

117 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⋊GL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×GL(2,𝔽₃)	—	3	3	no	maximal
4	C₃×#215	—	3	1	no	maximal
5	C₂×C₃×C₃⋊D₃	—	4	1	no	maximal
6	C₃²×Q₈	D₃	1	1	yes	characteristic, complemented
7	C₃×D₁₂	—	3	1	no	—
8	C₃×SL(2,𝔽₃)	—	2	1	no	—
9	C₃×SL(2,𝔽₃)	—	2	3	no	—
10	C₃×SL(2,𝔽₃)	D₃	1	3	yes	complemented
11	C₃×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
12	C₃×C₃⋊C₈	—	3	1	no	—
13	C₂×C₃³	—	4	1	no	—
14	C₃×C₃⋊D₃	—	4	2	no	—
15	GL(2,𝔽₃)	—	3	3	no	—
16	C₃×SD₁₆	—	9	1	no	—
17	#215	—	3	1	no	—
18	C₂×C₃⋊D₃	—	4	1	no	—
19	C₂×C₃×D₃	—	6	1	no	—
20	C₂×C₃×D₃	—	12	3	no	—
21	C₃²×C₄	—	3	1	no	—
22	C₃³	—	4	1	no	Sylow
23	C₃×Q₈	—	2	1	no	—
24	C₃×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
25	C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
26	C₃×D₄	—	9	1	no	—
27	D₁₂	—	3	1	no	—
28	SL(2,𝔽₃)	—	2	3	no	—
29	SL(2,𝔽₃)	C₃×D₃	1	3	yes	complemented
30	C₃×C₈	—	9	1	no	—
31	C₃⋊C₈	—	3	1	no	—
32	C₂×C₃²	—	4	1	no	—
33	C₂×C₃²	—	4	3	no	—
34	C₂×C₃²	—	8	3	no	—
35	C₂×C₃²	—	8	1	no	—
36	C₂×C₃²	S₄	1	1	yes	characteristic
37	C₃⋊D₃	—	4	2	no	—
38	C₃×D₃	—	12	6	no	—
39	C₃×D₃	—	12	1	no	—
40	SD₁₆	—	9	1	no	Sylow
41	C₂²×C₃	—	18	1	no	—
42	C₂×D₃	—	6	1	no	—
43	C₂×D₃	—	12	3	no	—
44	C₃×C₄	—	3	1	no	—
45	C₃×C₄	—	3	1	no	—
46	C₃×C₄	—	6	1	no	—
47	C₃²	—	4	1	no	—
48	C₃²	—	4	3	no	—
49	C₃²	—	8	3	no	—
50	C₃²	—	8	1	no	—
51	C₃²	GL(2,𝔽₃)	1	1	yes	characteristic, complemented
52	Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
53	D₄	—	9	1	no	—
54	C₈	—	9	1	no	—
55	C₂×C₃	—	2	1	no	—
56	C₂×C₃	—	4	3	no	—
57	C₂×C₃	—	8	3	no	—
58	C₂×C₃	—	36	1	no	—
59	C₂×C₃	C₃×S₄	1	1	yes	characteristic
60	C₂×C₃	C₃⋊S₄	1	1	yes	characteristic
61	D₃	—	12	6	no	—
62	D₃	—	12	1	no	—
63	C₂²	—	18	1	no	—
64	C₄	—	3	1	no	—
65	C₃	—	2	1	no	—
66	C₃	—	4	3	no	—
67	C₃	—	8	3	no	—
68	C₃	C₃×GL(2,𝔽₃)	1	1	yes	characteristic, complemented
69	C₃	C₃⋊GL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
70	C₂	—	36	1	no	—
71	C₂	C₃×C₃⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	C₃⋊S₄
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₃)	C₂×C₃
Frattini	C₂	C₃×C₃⋊S₄
Fitting	C₃²×Q₈	D₃
socle	C₂×C₃²	S₄
LCS	G ▷ C₃×SL(2,𝔽₃)
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⋊GL(2,𝔽₃)	C₃	direct (N × H)
6. C₃²×Q₈	D₃	semidirect
10. C₃×SL(2,𝔽₃)	D₃	semidirect
11. C₃×SL(2,𝔽₃)	C₂×C₃	semidirect
24. C₃×Q₈	C₃×D₃	semidirect
25. C₃×Q₈	C₃⋊D₃	semidirect
29. SL(2,𝔽₃)	C₃×D₃	semidirect
51. C₃²	GL(2,𝔽₃)	semidirect
52. Q₈	C₃×C₃⋊D₃	semidirect
68. C₃	C₃×GL(2,𝔽₃)	semidirect
69. C₃	C₃⋊GL(2,𝔽₃)	direct (N × H)

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⋊GL(2,𝔽₃)	3
C₃×GL(2,𝔽₃)	3
C₃×#215	3
C₂×C₃×C₃⋊D₃	4

Subgroups — C3×C3⋊GL(2,𝔽3) #88970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊GL(2,𝔽₃) #88970