Subgroups — #10624567

Order 729.

262 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	C₃×C₃²⋊C₉	C₃	1	2	yes	maximal
3	#6855	C₃	1	9	yes	maximal, complemented
4	C₃⁴	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²	—	3	6	no	—
6	C₃×3₋¹⁺²	—	3	18	no	—
7	C₃²×C₉	C₃²	1	6	yes	—
8	C₃²×C₉	C₃²	1	6	yes	—
9	C₃²⋊C₉	—	3	6	no	—
10	C₃³	—	3	6	no	—
11	C₃³	—	3	6	no	—
12	C₃³	H(3,𝔽₃)	1	3	yes	—
13	C₃³	C₃³	1	1	yes	characteristic
14	3₋¹⁺²	—	3	54	no	—
15	C₃×C₉	—	3	6	no	—
16	C₃×C₉	—	3	12	no	—
17	C₃×C₉	—	3	6	no	—
18	C₃×C₉	—	3	18	no	—
19	C₃×C₉	3₋¹⁺²	1	18	yes	complemented
20	C₃²	—	3	6	no	—
21	C₃²	—	3	6	no	—
22	C₃²	—	9	9	no	—
23	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
24	C₃²	C₃×3₋¹⁺²	1	6	yes	—
25	C₃²	#737	1	6	yes	—
26	C₉	—	3	18	no	—
27	C₉	—	9	6	no	—
28	C₃	—	9	3	no	—
29	C₃	#6891	1	2	yes	—
30	C₃	#6893	1	2	yes	—
31	C₃	#6896	1	9	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃³
prime radical	C₃⁴	C₃²
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

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

N	H ≅ G/N	type
3. #6855	C₃	semidirect
19. C₃×C₉	3₋¹⁺²	semidirect

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