Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
856 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 856 conjugacy classes of subgroups — too many to draw. Only its 163 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 | D42 |
| 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. #11720187 | C2 | semidirect |
| 4. #11720186 | C2 | semidirect |
| 5. C4×D108 | C2 | semidirect |
| 7. C23×D27 | C22 | semidirect |
| 8. C2×D4×C27 | C22 | semidirect |
| 9. C2×D4×C27 | C22 | semidirect |
| 10. C2×#6171 | C22 | semidirect |
| 13. C2×C4×D27 | C22 | semidirect |
| 14. C42×C27 | C22 | semidirect |
| 15. #88372 | C22 | semidirect |
| 16. (C2×D27)⋊C4 | C22 | semidirect |
| 17. C4⋊Dic27 | C22 | semidirect |
| 33. D108 | D4 | semidirect |
| 53. C2×D27 | C2×D4 | semidirect |
| 55. C4×C27 | C2×D4 | semidirect |
| 101. #127 | D27 | semidirect |
| 105. C27 | D42 | semidirect |
| 126. C2×D4 | C2×D27 | semidirect |
| 127. C2×D4 | C2×D27 | semidirect |
| 132. C42 | C2×D27 | semidirect |
| 169. 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 |
| #11720187 | 2 |
| #11720186 | 2 |
| C4×D108 | 2 |
| #10589819 | 3 |