Subgroups — #6196

Order 216.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1084	C₂	1	1	yes	maximal, characteristic
2	(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal
3	#627	—	3	2	no	maximal
4	C₂×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₃⋊Dic₃	D₃	1	2	yes	—
6	C₃×Dic₃	—	3	2	no	—
7	C₃×Dic₃	—	3	2	no	—
8	C₃×Dic₃	—	6	1	no	—
9	H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
10	Dic₆	—	9	1	no	—
11	Dic₆	—	9	2	no	—
12	C₂×C₃²	—	2	1	no	—
13	C₂×C₃²	C₂×D₃	1	2	yes	—
14	C₃×C₄	—	9	2	no	—
15	C₃×C₄	—	9	1	no	—
16	Dic₃	—	3	2	no	—
17	Dic₃	—	3	2	no	—
18	Dic₃	—	6	1	no	—
19	Dic₃	—	9	2	no	—
20	C₃²	—	2	1	no	—
21	C₃²	Dic₆	1	2	yes	complemented
22	Q₈	—	27	1	no	Sylow
23	C₂×C₃	—	3	2	no	—
24	C₂×C₃	—	6	1	no	—
25	C₂×C₃	D₃²	1	1	yes	characteristic
26	C₄	—	9	1	no	—
27	C₄	—	9	2	no	—
28	C₃	—	3	2	no	—
29	C₃	—	6	1	no	—
30	C₃	#627	1	1	yes	characteristic
31	C₂	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, Fitting, prime radical, cosocle	C₂×H(3,𝔽₃)	C₂²
Frattini, socle	C₂×C₃	D₃²
hypocenter	H(3,𝔽₃)	Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂

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

N	H ≅ G/N	type
9. H(3,𝔽₃)	Q₈	semidirect
21. C₃²	Dic₆	semidirect

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