Subgroups — (C₃⋊GL(2,𝔽₃))⋊C₃ #88601

Order 432.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊GL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#215	—	3	1	no	maximal
4	C₂×(C₃⋊D₃)⋊C₃	—	4	1	no	maximal
5	C₃²×Q₈	D₃	1	1	yes	characteristic, complemented
6	C₃×D₁₂	—	3	1	no	—
7	C₃×SL(2,𝔽₃)	—	2	1	no	—
8	C₃×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₃⋊C₈	—	3	1	no	—
10	C₂×H(3,𝔽₃)	—	4	1	no	—
11	(C₃⋊D₃)⋊C₃	—	4	2	no	—
12	GL(2,𝔽₃)	—	9	1	no	—
13	C₃×SD₁₆	—	9	1	no	—
14	#215	—	3	1	no	—
15	C₂×C₃⋊D₃	—	4	1	no	—
16	C₂×C₃×D₃	—	6	1	no	—
17	C₃²×C₄	—	3	1	no	—
18	H(3,𝔽₃)	—	4	1	no	Sylow
19	C₃×Q₈	—	3	1	no	—
20	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
21	C₃×D₄	—	9	1	no	—
22	D₁₂	—	3	1	no	—
23	SL(2,𝔽₃)	—	3	1	no	—
24	SL(2,𝔽₃)	—	6	1	no	—
25	C₃×C₈	—	9	1	no	—
26	C₃⋊C₈	—	3	1	no	—
27	C₂×C₃²	—	4	1	no	—
28	C₂×C₃²	—	8	1	no	—
29	C₂×C₃²	S₄	1	1	yes	characteristic
30	C₃⋊D₃	—	4	2	no	—
31	C₃×D₃	—	12	1	no	—
32	SD₁₆	—	9	1	no	Sylow
33	C₂²×C₃	—	18	1	no	—
34	C₂×D₃	—	6	1	no	—
35	C₂×D₃	—	36	1	no	—
36	C₃×C₄	—	3	1	no	—
37	C₃×C₄	—	3	1	no	—
38	C₃×C₄	—	6	1	no	—
39	C₃²	—	4	1	no	—
40	C₃²	—	8	1	no	—
41	C₃²	GL(2,𝔽₃)	1	1	yes	characteristic, complemented
42	Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
43	D₄	—	9	1	no	—
44	C₈	—	9	1	no	—
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	12	1	no	—
47	C₂×C₃	—	24	1	no	—
48	C₂×C₃	—	36	1	no	—
49	C₂×C₃	C₃×S₄	1	1	yes	characteristic
50	D₃	—	12	1	no	—
51	D₃	—	36	2	no	—
52	C₂²	—	18	1	no	—
53	C₄	—	3	1	no	—
54	C₃	—	3	1	no	—
55	C₃	—	12	1	no	—
56	C₃	—	24	1	no	—
57	C₃	C₃×GL(2,𝔽₃)	1	1	yes	characteristic
58	C₂	—	36	1	no	—
59	C₂	(C₃⋊S₄)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	(C₃⋊S₄)⋊C₃
commutator, cosocle, hypocenter	C₃×SL(2,𝔽₃)	C₂×C₃
Frattini, socle	C₂×C₃	C₃×S₄
Fitting	C₃²×Q₈	D₃
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. Q₈⋊H(3,𝔽₃)	C₂	semidirect
2. C₃⋊GL(2,𝔽₃)	C₃	semidirect
5. C₃²×Q₈	D₃	semidirect
8. C₃×SL(2,𝔽₃)	C₂×C₃	semidirect
41. C₃²	GL(2,𝔽₃)	semidirect
42. Q₈	(C₃⋊D₃)⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈⋊H(3,𝔽₃)	2
C₃⋊GL(2,𝔽₃)	3
C₃×#215	3
C₂×(C₃⋊D₃)⋊C₃	4

Subgroups — (C3⋊GL(2,𝔽3))⋊C3 #88601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃⋊GL(2,𝔽₃))⋊C₃ #88601