Subgroups — (C₃⋊#226)⋊C₃ #88600

Order 432.

54 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
2	C₃⋊#226	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#216	—	3	1	no	maximal
4	(C₃⋊Dic₃)⋊C₃	—	4	1	no	maximal
5	C₃²×Q₈	D₃	1	1	yes	characteristic
6	C₃×Dic₆	—	3	1	no	—
7	C₃×SL(2,𝔽₃)	—	2	1	no	—
8	C₃×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	C₃×C₃⋊C₈	—	3	1	no	—
10	C₂×H(3,𝔽₃)	—	4	1	no	—
11	#226	—	9	1	no	—
12	C₃×Q₁₆	—	9	1	no	—
13	#216	—	3	1	no	—
14	C₃²×C₄	—	3	1	no	—
15	C₃⋊Dic₃	—	4	1	no	—
16	C₃×Dic₃	—	6	1	no	—
17	H(3,𝔽₃)	—	4	1	no	Sylow
18	C₃×Q₈	—	3	1	no	—
19	C₃×Q₈	—	9	1	no	—
20	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
21	Dic₆	—	3	1	no	—
22	SL(2,𝔽₃)	—	3	1	no	—
23	SL(2,𝔽₃)	—	6	1	no	—
24	C₃×C₈	—	9	1	no	—
25	C₃⋊C₈	—	3	1	no	—
26	C₂×C₃²	—	4	1	no	—
27	C₂×C₃²	—	8	1	no	—
28	C₂×C₃²	S₄	1	1	yes	characteristic
29	Q₁₆	—	9	1	no	Sylow
30	C₃×C₄	—	3	1	no	—
31	C₃×C₄	—	3	1	no	—
32	C₃×C₄	—	6	1	no	—
33	C₃×C₄	—	18	1	no	—
34	Dic₃	—	6	1	no	—
35	Dic₃	—	36	1	no	—
36	C₃²	—	4	1	no	—
37	C₃²	—	8	1	no	—
38	C₃²	#226	1	1	yes	characteristic, complemented
39	Q₈	—	9	1	no	—
40	Q₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
41	C₈	—	9	1	no	—
42	C₂×C₃	—	3	1	no	—
43	C₂×C₃	—	12	1	no	—
44	C₂×C₃	—	24	1	no	—
45	C₂×C₃	C₃×S₄	1	1	yes	characteristic
46	C₄	—	3	1	no	—
47	C₄	—	18	1	no	—
48	C₃	—	3	1	no	—
49	C₃	—	12	1	no	—
50	C₃	—	24	1	no	—
51	C₃	C₃×#226	1	1	yes	characteristic
52	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
prime radical	Q₈⋊H(3,𝔽₃)	C₂
LCS	G ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

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

N	H ≅ G/N	type
2. C₃⋊#226	C₃	semidirect
38. C₃²	#226	semidirect

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