Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
542 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C34 | C33 |
| prime radical | C3×#6856 | C3 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#6894 | C3 | semidirect |
| 2. C3×#6893 | C3 | semidirect |
| 3. C3×#6860 | C3 | semidirect |
| 4. C3×#6860 | C3 | semidirect |
| 5. C3×#6859 | C3 | semidirect |
| 6. C3×#6858 | C3 | semidirect |
| 7. C3×#6858 | C3 | semidirect |
| 8. C3×#6857 | C3 | semidirect |
| 9. C3×#6857 | C3 | semidirect |
| 10. C3×#6857 | C3 | semidirect |
| 12. #10624558 | C3 | semidirect |
| 13. #10624555 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6894 | 3 |
| C3×#6893 | 3 |
| C3×#6860 | 3 |
| C3×#6859 | 3 |
| C3×#6858 | 3 |
| C3×#6857 | 3 |
| C3×#6856 | 3 |
| #10624558 | 3 |
| #10624555 | 3 |