Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
347 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 56 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 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C34 | C33 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624906 | C3 | semidirect |
| 2. #10624825 | C3 | semidirect |
| 3. #10624822 | C3 | semidirect |
| 4. #10624812 | C3 | semidirect |
| 5. #10624812 | C3 | semidirect |
| 6. #10624807 | C3 | semidirect |
| 9. C3×#6867 | C3 | semidirect |
| 10. #10624614 | C3 | semidirect |
| 12. #10624600 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624906 | 3 |
| #10624825 | 3 |
| #10624822 | 3 |
| #10624812 | 3 |
| #10624807 | 3 |
| #10624805 | 3 |
| C3×H(3,𝔽3)⋊C9 | 3 |
| C3×#6867 | 3 |
| #10624614 | 3 |
| #10624602 | 3 |
| #10624600 | 3 |
| #10624593 | 3 |