Finite Group DatabasebetaFinite Group DatabasebetaOrder 729.
157 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33 |
| prime radical | #6856 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#737 | C3 | semidirect |
| 2. #6894 | C3 | semidirect |
| 3. C9⋊H(3,𝔽3) | C3 | semidirect |
| 4. C9×3−1+2 | C3 | semidirect |
| 5. #6860 | C3 | semidirect |
| 6. #6860 | C3 | semidirect |
| 7. #6859 | C3 | semidirect |
| 8. #6858 | C3 | semidirect |
| 9. #6858 | C3 | semidirect |
| 10. #6857 | C3 | semidirect |
| 11. #6857 | C3 | semidirect |
| 12. #6857 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#737 | 3 |
| #6894 | 3 |
| C9⋊H(3,𝔽3) | 3 |
| C9×3−1+2 | 3 |
| #6860 | 3 |
| #6859 | 3 |
| #6858 | 3 |
| #6857 | 3 |
| #6856 | 3 |