Finite Group DatabasebetaFinite Group DatabasebetaOrder 81.
14 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | 3−1+2 | C3 | 1 | 2 | yes | maximal, complemented |
| 2 | H(3,𝔽3) | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×C9 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C32 | — | 3 | 1 | no | — |
| 5 | C32 | C32 | 1 | 1 | yes | characteristic |
| 6 | C9 | — | 3 | 1 | no | — |
| 7 | C9 | — | 3 | 2 | no | — |
| 8 | C3 | — | 3 | 1 | no | — |
| 9 | C3 | — | 9 | 1 | no | — |
| 10 | C3 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32 |
| prime radical | H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 3−1+2 | C3 | semidirect |
| 3. C3×C9 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 3−1+2 | 3 |
| H(3,𝔽3) | 3 |
| C3×C9 | 3 |