Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
996 conjugacy classes of subgroups; only the 411 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 411 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | D27 |
| commutator, hypocenter | C27 | C22×C42 |
| Frattini | C22×C9 | C23×D3 |
| Fitting | C2×C42×C27 | C2 |
| socle | C23×C3 | C22×D9 |
| prime radical | C23×D27 | C22 |
| cosocle | C22×C27 | C24 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C42×C27 | C2 | semidirect |
| 3. C2×C4×Dic27 | C2 | semidirect |
| 4. C42×D27 | C2 | direct (N × H) |
| 9. C2×C4×D27 | C4 | direct (N × H) |
| 11. C42×C27 | C22 | semidirect |
| 12. C4×Dic27 | C22 | semidirect |
| 20. C2×C4×C27 | C2×C4 | semidirect |
| 22. C2×Dic27 | C2×C4 | semidirect |
| 24. C4×D27 | C2×C4 | direct (N × H) |
| 35. C2×D27 | C42 | direct (N × H) |
| 37. C4×C27 | C22×C4 | semidirect |
| 38. Dic27 | C22×C4 | semidirect |
| 52. C2×C27 | C2×C42 | semidirect |
| 54. D27 | C2×C42 | direct (N × H) |
| 71. C2×C42 | D27 | direct (N × H) |
| 72. C27 | C22×C42 | semidirect |
| 90. C42 | C2×D27 | direct (N × H) |
| 104. C2×C4 | C4×D27 | direct (N × H) |
| 114. C4 | C2×C4×D27 | direct (N × H) |
| 117. C2 | C42×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 |
| C2×C42×C27 | 2 |
| C2×C4×Dic27 | 2 |
| C42×D27 | 2 |
| C2×C42×D9 | 3 |