Finite Group DatabasebetaFinite Group DatabasebetaOrder 729.
214 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C33×C9 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #6891 | C3 | 1 | 3 | yes | maximal |
| 3 | #6868 | C3 | 1 | 9 | yes | maximal, complemented |
| 4 | C34 | C32 | 1 | 1 | yes | characteristic |
| 5 | C3×3−1+2 | — | 3 | 3 | no | — |
| 6 | C32×C9 | — | 3 | 1 | no | — |
| 7 | C32×C9 | — | 3 | 2 | no | — |
| 8 | C32×C9 | — | 3 | 9 | no | — |
| 9 | C32×C9 | C32 | 1 | 3 | yes | — |
| 10 | C32⋊C9 | — | 3 | 6 | no | — |
| 11 | C32⋊C9 | C32 | 1 | 9 | yes | — |
| 12 | C33 | — | 3 | 3 | no | — |
| 13 | C33 | — | 3 | 6 | no | — |
| 14 | C33 | — | 3 | 3 | no | — |
| 15 | C33 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 16 | C33 | H(3,𝔽3) | 1 | 2 | yes | — |
| 17 | C33 | C33 | 1 | 1 | yes | characteristic |
| 18 | 3−1+2 | — | 9 | 9 | no | — |
| 19 | C3×C9 | — | 3 | 9 | no | — |
| 20 | C3×C9 | — | 3 | 18 | no | — |
| 21 | C3×C9 | — | 3 | 3 | no | — |
| 22 | C3×C9 | — | 3 | 6 | no | — |
| 23 | C3×C9 | — | 3 | 9 | no | — |
| 24 | C3×C9 | 3−1+2 | 1 | 9 | yes | complemented |
| 25 | C32 | — | 3 | 27 | no | — |
| 26 | C32 | — | 3 | 1 | no | — |
| 27 | C32 | — | 3 | 2 | no | — |
| 28 | C32 | — | 3 | 2 | no | — |
| 29 | C32 | — | 3 | 1 | no | — |
| 30 | C32 | — | 3 | 2 | no | — |
| 31 | C32 | — | 3 | 2 | no | — |
| 32 | C32 | — | 3 | 2 | no | — |
| 33 | C32 | #732 | 1 | 3 | yes | — |
| 34 | C32 | #733 | 1 | 6 | yes | — |
| 35 | C32 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 36 | C32 | C3×3−1+2 | 1 | 3 | yes | — |
| 37 | C9 | — | 3 | 9 | no | — |
| 38 | C9 | — | 9 | 9 | no | — |
| 39 | C3 | — | 3 | 3 | no | — |
| 40 | C3 | — | 3 | 6 | no | — |
| 41 | C3 | — | 3 | 3 | no | — |
| 42 | C3 | #6891 | 1 | 1 | yes | characteristic |
| 43 | C3 | C3×#732 | 1 | 1 | yes | characteristic |
| 44 | C3 | C3×#733 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33 |
| prime radical | C34 | C32 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #6868 | C3 | semidirect |
| 24. C3×C9 | 3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×C9 | 3 |
| #6891 | 3 |
| #6868 | 3 |