Subgroups — #6894

Order 243.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉	C₃	1	1	yes	maximal, characteristic
3	C₉⋊C₉	C₃	1	6	yes	maximal, complemented
4	C₉⋊C₉	C₃	1	3	yes	maximal, complemented
5	C₃²⋊C₉	C₃	1	2	yes	maximal
6	C₃³	C₃²	1	1	yes	characteristic
7	3₋¹⁺²	—	3	3	no	—
8	C₃×C₉	—	3	3	no	—
9	C₃×C₉	C₃²	1	6	yes	—
10	C₃×C₉	C₃²	1	2	yes	—
11	C₃×C₉	C₃²	1	3	yes	—
12	C₃×C₉	C₃²	1	1	yes	characteristic
13	C₃²	—	3	1	no	—
14	C₃²	—	3	2	no	—
15	C₃²	3₋¹⁺²	1	3	yes	—
16	C₃²	C₃³	1	1	yes	characteristic
17	C₉	—	3	2	no	—
18	C₉	—	3	3	no	—
19	C₉	—	3	6	no	—
20	C₉	3₋¹⁺²	1	3	yes	complemented
21	C₃	—	3	3	no	—
22	C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
23	C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
24	C₃	#737	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃³
prime radical	C₃³	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃² ▷ 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.