Subgroups — C₃×#731 #6905

Order 243.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
2	C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	#731	C₃	1	9	yes	maximal, direct product factor
5	C₃³	—	3	1	no	—
6	C₃³	C₃²	1	1	yes	characteristic
7	3₋¹⁺²	—	3	4	no	—
8	3₋¹⁺²	C₃²	1	6	yes	complemented
9	H(3,𝔽₃)	—	3	2	no	—
10	H(3,𝔽₃)	C₃²	1	3	yes	—
11	C₃×C₉	—	3	2	no	—
12	C₃×C₉	—	3	1	no	—
13	C₃×C₉	—	3	2	no	—
14	C₃×C₉	C₃²	1	3	yes	complemented
15	C₃²	—	3	3	no	—
16	C₃²	—	3	3	no	—
17	C₃²	—	9	3	no	—
18	C₃²	H(3,𝔽₃)	1	2	yes	—
19	C₃²	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²	C₃³	1	1	yes	characteristic
21	C₉	—	3	3	no	—
22	C₉	—	3	6	no	—
23	C₃	—	3	2	no	—
24	C₃	—	3	1	no	—
25	C₃	—	9	3	no	—
26	C₃	#731	1	3	yes	direct product factor
27	C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic

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