Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
676 conjugacy classes of subgroups; only the 283 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 283 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | D27 |
| commutator, hypocenter | C27 | C23×C8 |
| Frattini | C4×C9 | C23×D3 |
| Fitting | C22×C8×C27 | C2 |
| socle | C23×C3 | C4×D9 |
| prime radical | C23×D27 | C4 |
| cosocle | C4×C27 | C24 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C22×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C8×C27 | C2 | semidirect |
| 3. C22×C27⋊C8 | C2 | semidirect |
| 4. C2×C8×D27 | C2 | direct (N × H) |
| 11. C2×C8×C27 | C22 | semidirect |
| 12. C2×C27⋊C8 | C22 | semidirect |
| 13. C8×D27 | C22 | direct (N × H) |
| 19. C22×D27 | C8 | direct (N × H) |
| 26. C8×C27 | C23 | semidirect |
| 27. C27⋊C8 | C23 | semidirect |
| 37. C22×C27 | C2×C8 | semidirect |
| 39. C2×D27 | C2×C8 | direct (N × H) |
| 59. C2×C27 | C22×C8 | semidirect |
| 61. D27 | C22×C8 | direct (N × H) |
| 80. C22×C8 | D27 | direct (N × H) |
| 81. C27 | C23×C8 | semidirect |
| 102. C2×C8 | C2×D27 | direct (N × H) |
| 120. C8 | C22×D27 | direct (N × H) |
| 126. C22 | C8×D27 | direct (N × H) |
| 133. C2 | C2×C8×D27 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D27 | 2 |
| C22×C8×C27 | 2 |
| C22×C27⋊C8 | 2 |
| C2×C8×D27 | 2 |
| C22×C8×D9 | 3 |