Subgroups — #10624607

Order 729.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²	C₃	1	2	yes	maximal
2	#6891	C₃	1	2	yes	maximal
3	#6860	C₃	1	9	yes	maximal, complemented
4	C₃⁴	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²	—	3	2	no	—
6	C₃×3₋¹⁺²	—	3	2	no	—
7	C₃×3₋¹⁺²	—	3	18	no	—
8	C₃×3₋¹⁺²	—	3	2	no	—
9	C₃×3₋¹⁺²	C₃²	1	6	yes	—
10	C₃²×C₉	—	3	2	no	—
11	C₃²⋊C₉	—	3	2	no	—
12	C₃²⋊C₉	—	3	2	no	—
13	C₃²⋊C₉	C₃²	1	6	yes	—
14	C₃³	—	3	6	no	—
15	C₃³	—	9	2	no	—
16	C₃³	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃³	H(3,𝔽₃)	1	2	yes	—
18	C₃³	C₃³	1	1	yes	characteristic
19	3₋¹⁺²	—	3	18	no	—
20	3₋¹⁺²	—	3	18	no	—
21	3₋¹⁺²	—	3	18	no	—
22	3₋¹⁺²	—	9	6	no	—
23	C₃×C₉	—	3	18	no	—
24	C₃×C₉	—	3	6	no	—
25	C₃×C₉	—	3	6	no	—
26	C₃²	—	3	4	no	—
27	C₃²	—	3	2	no	—
28	C₃²	—	3	2	no	—
29	C₃²	—	3	2	no	—
30	C₃²	—	3	2	no	—
31	C₃²	—	3	2	no	—
32	C₃²	—	9	9	no	—
33	C₃²	AΣL(1,𝔽₂₇)	1	6	yes	—
34	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₉	—	3	18	no	—
36	C₉	—	9	6	no	—
37	C₃	—	3	6	no	—
38	C₃	—	9	1	no	—
39	C₃	—	9	1	no	—
40	C₃	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
41	C₃	#6912	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³
prime radical	C₃⁴	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃³ ▷ C₃² ▷ C₁

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

N	H ≅ G/N	type
3. #6860	C₃	semidirect

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