Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
996 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D27 |
| commutator, cosocle | C22×C27 | C24 |
| Frattini | C22×C9 | C23×D3 |
| Fitting | C27×#127 | C2 |
| socle | C22×C3 | C23×D9 |
| hypercenter | #127 | D27 |
| hypocenter | C27 | C2×#127 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×D27 | C2 | semidirect |
| 2. C27×#127 | C2 | semidirect |
| 3. #11720189 | C2 | semidirect |
| 4. #11720123 | C2 | semidirect |
| 5. C42×D27 | C2 | semidirect |
| 8. C2×D4×C27 | C22 | semidirect |
| 9. C2×#6171 | C22 | semidirect |
| 11. C2×D108 | C22 | semidirect |
| 13. C42×C27 | C22 | semidirect |
| 14. C4×Dic27 | C22 | semidirect |
| 29. C4×D27 | D4 | semidirect |
| 44. C4×C27 | C2×D4 | semidirect |
| 45. Dic27 | C2×D4 | semidirect |
| 65. D27 | #127 | direct (N × H) |
| 84. #127 | D27 | direct (N × H) |
| 86. C27 | C2×#127 | semidirect |
| 105. C2×D4 | C2×D27 | semidirect |
| 108. C42 | C2×D27 | semidirect |
| 137. C4 | D4×D27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×D27 | 2 |
| C27×#127 | 2 |
| #11720189 | 2 |
| #11720123 | 2 |
| C42×D27 | 2 |
| D9×#127 | 3 |