Subgroups — #6866

Order 243.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊C₉	C₃	1	3	yes	maximal, complemented
3	C₃³	—	3	9	no	—
4	C₃³	—	3	1	no	—
5	C₃³	—	3	2	no	—
6	C₃³	C₉	1	3	yes	complemented
7	C₃³	C₃²	1	1	yes	characteristic
8	C₃×C₉	—	3	3	no	—
9	C₃²	—	3	9	no	—
10	C₃²	—	3	18	no	—
11	C₃²	—	3	3	no	—
12	C₃²	—	3	9	no	—
13	C₃²	—	3	3	no	—
14	C₃²	C₃×C₉	1	1	yes	characteristic
15	C₃²	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃²	3₋¹⁺²	1	2	yes	—
17	C₉	—	9	3	no	—
18	C₃	—	3	9	no	—
19	C₃	—	3	1	no	—
20	C₃	—	3	2	no	—
21	C₃	C₃²⋊C₉	1	1	yes	characteristic
22	C₃	AΣL(1,𝔽₂₇)	1	3	yes	—

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

N	H ≅ G/N	type
2. C₃²⋊C₉	C₃	semidirect
6. C₃³	C₉	semidirect

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

subgroup	index
C₃⁴	3
C₃²⋊C₉	3