Finite Group DatabasebetaFinite Group DatabasebetaOrder 81.
13 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | 3−1+2 | C3 | 1 | 3 | yes | maximal |
| 2 | C3×C9 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C32 | C32 | 1 | 1 | yes | characteristic |
| 4 | C9 | — | 3 | 1 | no | — |
| 5 | C9 | — | 3 | 3 | no | — |
| 6 | C3 | — | 3 | 1 | no | — |
| 7 | C3 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | H(3,𝔽3) |
| commutator, Frattini, prime radical, cosocle | C32 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 3−1+2 | 3 |
| C3×C9 | 3 |