Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
291 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 47 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×#732 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| prime radical | #10624734 | C3 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624766 | C3 | semidirect |
| 3. #10624688 | C3 | semidirect |
| 4. C3×#6855 | C3 | semidirect |
| 5. #10624491 | C3 | semidirect |
| 6. #10624490 | C3 | semidirect |
| 7. #10624490 | C3 | semidirect |
| 8. #10624489 | C3 | semidirect |
| 9. #10624488 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624766 | 3 |
| #10624734 | 3 |
| #10624688 | 3 |
| C3×#6855 | 3 |
| #10624491 | 3 |
| #10624490 | 3 |
| #10624489 | 3 |
| #10624488 | 3 |