Subgroups — #10624813

Order 729.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6895	C₃	1	2	yes	maximal, complemented
2	#6893	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal, complemented
5	H(3,𝔽₃)⋊C₉	C₃	1	3	yes	maximal
6	C₃×3₋¹⁺²	C₃²	1	2	yes	—
7	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
8	C₃²×C₉	C₃²	1	3	yes	—
9	C₃²×C₉	C₃²	1	1	yes	characteristic
10	C₉⋊C₉	—	3	6	no	—
11	C₃²⋊C₉	—	3	3	no	—
12	C₃²⋊C₉	C₃²	1	6	yes	—
13	C₉²	—	3	3	no	—
14	C₃³	—	3	1	no	—
15	C₃³	C₃³	1	1	yes	characteristic
16	3₋¹⁺²	—	3	6	no	—
17	H(3,𝔽₃)	—	3	3	no	—
18	C₃×C₉	—	3	3	no	—
19	C₃×C₉	—	3	3	no	—
20	C₃×C₉	—	3	9	no	—
21	C₃×C₉	—	3	2	no	—
22	C₃×C₉	—	3	6	no	—
23	C₃×C₉	H(3,𝔽₃)	1	3	yes	—
24	C₃²	—	3	3	no	—
25	C₃²	—	9	1	no	—
26	C₃²	—	9	3	no	—
27	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃²	#737	1	3	yes	—
29	C₉	—	3	3	no	—
30	C₉	—	3	9	no	—
31	C₉	—	9	2	no	—
32	C₉	—	9	6	no	—
33	C₃	—	3	3	no	—
34	C₃	—	27	1	no	—
35	C₃	#6893	1	1	yes	characteristic
36	C₃	#6908	1	3	yes	—

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³
prime radical	C₃×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ 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.