Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
184 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×D27 |
| commutator, cosocle | C2×C27 | C23×C3 |
| Frattini | C2×C9 | C22×C3×D3 |
| Fitting | C3×D4×C27 | C2 |
| socle | C2×C32 | C22×D9 |
| hypercenter | C3×D4 | D27 |
| hypocenter | C27 | C3×#41 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×D4×C27 | C2 | semidirect |
| 2. C3×#6171 | C2 | semidirect |
| 3. C2×C3×Dic27 | C2 | semidirect |
| 4. C3×C4×D27 | C2 | semidirect |
| 5. C3×Dic54 | C2 | semidirect |
| 7. #88401 | C3 | direct (N × H) |
| 19. D4×C27 | C2×C3 | semidirect |
| 20. #6171 | C2×C3 | semidirect |
| 21. C2×Dic27 | C2×C3 | semidirect |
| 22. C4×D27 | C2×C3 | semidirect |
| 23. Dic54 | C2×C3 | semidirect |
| 41. C3×C27 | #41 | semidirect |
| 77. C27 | C3×#41 | semidirect |
| 81. C3×D4 | D27 | semidirect |
| 115. D4 | C3×D27 | semidirect |
| 132. C3 | #88401 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D4×C27 | 2 |
| C3×#6171 | 2 |
| C2×C3×Dic27 | 2 |
| C3×C4×D27 | 2 |
| C3×Dic54 | 2 |
| C3×#3699 | 3 |
| #88401 | 3 |