Subgroups — C₃⋊(C₃⋊#226) #88972

Order 432.

158 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
2	C₃⋊#226	—	3	12	no	maximal
3	#3756	—	3	1	no	maximal
4	C₃⋊(C₃⋊Dic₃)	—	4	1	no	maximal
5	C₃²×Q₈	D₃	1	1	yes	characteristic
6	#634	—	3	1	no	—
7	C₃×SL(2,𝔽₃)	D₃	1	12	yes	—
8	C₃⋊(C₃⋊C₈)	—	3	1	no	—
9	C₂×C₃³	—	4	1	no	—
10	#226	—	9	9	no	—
11	#216	—	9	4	no	—
12	C₃²×C₄	—	3	1	no	—
13	C₃⋊Dic₃	—	6	1	no	—
14	C₃⋊Dic₃	—	12	12	no	—
15	C₃³	—	4	1	no	Sylow
16	C₃×Q₈	C₃⋊D₃	1	4	yes	—
17	Dic₆	—	9	4	no	—
18	SL(2,𝔽₃)	C₃⋊D₃	1	9	yes	—
19	C₃⋊C₈	—	9	4	no	—
20	C₂×C₃²	—	4	12	no	—
21	C₂×C₃²	S₄	1	1	yes	characteristic
22	Q₁₆	—	27	1	no	Sylow
23	C₃×C₄	—	3	4	no	—
24	Dic₃	—	18	4	no	—
25	Dic₃	—	36	9	no	—
26	C₃²	—	4	12	no	—
27	C₃²	#226	1	1	yes	characteristic, complemented
28	Q₈	—	27	1	no	—
29	Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
30	C₈	—	27	1	no	—
31	C₂×C₃	—	4	9	no	—
32	C₂×C₃	C₃⋊S₄	1	4	yes	—
33	C₄	—	3	1	no	—
34	C₄	—	54	1	no	—
35	C₃	—	4	9	no	—
36	C₃	C₃⋊#226	1	4	yes	complemented
37	C₂	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic

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

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

N	H ≅ G/N	type
27. C₃²	#226	semidirect
36. C₃	C₃⋊#226	semidirect

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