Subgroups — #4207

Order 162.

39 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#737	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊D₉	C₃	1	1	yes	maximal, characteristic, complemented
3	AGL(1,ℤ/9)	—	3	2	no	maximal
4	(C₃⋊D₃)⋊C₃	—	3	1	no	maximal
5	C₃×D₉	—	3	1	no	maximal
6	3₋¹⁺²	—	2	1	no	—
7	3₋¹⁺²	—	2	2	no	—
8	3₋¹⁺²	D₃	1	2	yes	complemented
9	H(3,𝔽₃)	D₃	1	1	yes	characteristic
10	C₃×C₉	—	2	1	no	—
11	C₃×C₉	D₃	1	1	yes	characteristic, complemented
12	C₃×C₉	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃⋊D₃	—	3	1	no	—
14	C₃×D₃	—	9	1	no	—
15	D₉	—	3	1	no	—
16	D₉	—	3	2	no	—
17	C₃²	—	2	1	no	—
18	C₃²	C₃×D₃	1	1	yes	characteristic
19	C₃²	C₃⋊D₃	1	1	yes	characteristic
20	C₉	—	2	2	no	—
21	C₉	—	2	1	no	—
22	C₉	C₃×D₃	1	1	yes	characteristic
23	C₉	C₃×D₃	1	2	yes	—
24	C₂×C₃	—	27	1	no	—
25	D₃	—	9	1	no	—
26	D₃	—	27	1	no	—
27	C₃	—	3	1	no	—
28	C₃	—	3	1	no	—
29	C₃	—	6	1	no	—
30	C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
31	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₉	C₂×C₃
Frattini, socle	C₃	C₃×C₃⋊D₃
Fitting	#737	C₂
LCS	G ▷ C₃×C₉
UCS	C₁

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

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