Finite Group DatabasebetaFinite Group DatabasebetaOrder 2187.
497 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C3×H(3,𝔽3) |
| commutator | C33 | C32×C9 |
| Frattini, cosocle | C32×C9 | C33 |
| socle | C32 | C9×H(3,𝔽3) |
| prime radical | C3×#6908 | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C32×C27 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C33×C27 | C3 | semidirect |
| 3. C3×#6873 | C3 | semidirect |
| 4. C3×#6872 | C3 | semidirect |
| 5. C3×#6869 | C3 | semidirect |
| 6. #10624684 | C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#6908 | 3 |
| C33×C27 | 3 |
| C3×#6873 | 3 |
| C3×#6872 | 3 |
| C3×#6869 | 3 |
| #10624684 | 3 |