Finite Group DatabasebetaFinite Group DatabasebetaOrder 81.
56 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C33 | C3 | 1 | 4 | yes | maximal, complemented |
| 2 | H(3,𝔽3) | C3 | 1 | 9 | yes | maximal, direct product factor |
| 3 | C32 | — | 3 | 12 | no | — |
| 4 | C32 | C32 | 1 | 1 | yes | characteristic |
| 5 | C32 | C32 | 1 | 12 | yes | complemented |
| 6 | C3 | — | 3 | 12 | no | — |
| 7 | C3 | H(3,𝔽3) | 1 | 3 | yes | direct product factor |
| 8 | C3 | C33 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32 |
| commutator, Frattini, cosocle | C3 | C33 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33 | C3 | semidirect |
| 2. H(3,𝔽3) | C3 | direct (N × H) |
| 5. C32 | C32 | semidirect |
| 7. C3 | H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33 | 3 |
| H(3,𝔽3) | 3 |