Finite Group DatabasebetaFinite Group DatabasebetaOrder 243.
240 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #737 | C3 | 1 | 27 | yes | maximal, direct product factor |
| 2 | C3×3−1+2 | C3 | 1 | 8 | yes | maximal, complemented |
| 3 | C3×H(3,𝔽3) | C3 | 1 | 1 | yes | maximal, characteristic |
| 4 | C32×C9 | C3 | 1 | 4 | yes | maximal, complemented |
| 5 | C33 | C32 | 1 | 4 | yes | — |
| 6 | 3−1+2 | C32 | 1 | 72 | yes | complemented |
| 7 | H(3,𝔽3) | C32 | 1 | 9 | yes | — |
| 8 | C3×C9 | C32 | 1 | 8 | yes | — |
| 9 | C3×C9 | C32 | 1 | 1 | yes | characteristic |
| 10 | C3×C9 | C32 | 1 | 36 | yes | complemented |
| 11 | C32 | — | 3 | 12 | no | — |
| 12 | C32 | C33 | 1 | 12 | yes | — |
| 13 | C32 | C33 | 1 | 1 | yes | characteristic |
| 14 | C9 | C33 | 1 | 24 | yes | — |
| 15 | C9 | C33 | 1 | 3 | yes | — |
| 16 | C3 | — | 3 | 12 | no | — |
| 17 | C3 | #737 | 1 | 3 | yes | direct product factor |
| 18 | C3 | C34 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C32 |
| commutator, Frattini, cosocle | C3 | C34 |
| socle | C32 | C33 |
| prime radical | C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #737 | C3 | direct (N × H) |
| 2. C3×3−1+2 | C3 | semidirect |
| 4. C32×C9 | C3 | semidirect |
| 6. 3−1+2 | C32 | semidirect |
| 10. C3×C9 | C32 | semidirect |
| 17. C3 | #737 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #737 | 3 |
| C3×3−1+2 | 3 |
| C3×H(3,𝔽3) | 3 |
| C32×C9 | 3 |