Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
280 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C3⋊D27 |
| commutator, cosocle | C2×C3×C27 | C23 |
| Frattini | C2×C9 | C22×C3⋊D3 |
| Fitting | C3×D4×C27 | C2 |
| socle | C2×C32 | C22×D9 |
| hypercenter | D4 | C3⋊D27 |
| hypocenter | C3×C27 | #41 |
| LCS | G ▷ C2×C3×C27 ▷ C3×C27 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×D4×C27 | C2 | semidirect |
| 2. #10619677 | C2 | semidirect |
| 3. C2×C3⋊Dic27 | C2 | semidirect |
| 4. C4×C3⋊D27 | C2 | semidirect |
| 5. #10619657 | C2 | semidirect |
| 18. D4×C27 | D3 | semidirect |
| 39. C3×C27 | #41 | semidirect |
| 79. C27 | #237 | semidirect |
| 80. C3×D4 | D27 | semidirect |
| 115. D4 | C3⋊D27 | semidirect |
| 129. C3 | #88401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D4×C27 | 2 |
| #10619677 | 2 |
| C2×C3⋊Dic27 | 2 |
| C4×C3⋊D27 | 2 |
| #10619657 | 2 |
| #88742 | 3 |
| #88401 | 3 |