Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
504 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 131 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×#126 | C2 |
| socle | C22×C3 | C23×D9 |
| hypercenter | #126 | D27 |
| hypocenter | C27 | #560 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C27×#126 | C2 | semidirect |
| 2. #11720144 | C2 | semidirect |
| 3. #11720143 | C2 | semidirect |
| 4. #11720133 | C2 | semidirect |
| 5. #11720131 | C2 | semidirect |
| 6. #11720125 | C2 | semidirect |
| 7. #11720123 | C2 | semidirect |
| 13. C2×D108 | C22 | semidirect |
| 14. C2×C4×D27 | C22 | semidirect |
| 15. C4⋊C4×C27 | C22 | semidirect |
| 17. C42×C27 | C22 | semidirect |
| 20. (C2×D27)⋊C4 | C22 | semidirect |
| 21. Dic27⋊C4 | C22 | semidirect |
| 119. #126 | D27 | semidirect |
| 124. C27 | #560 | semidirect |
| 150. C4⋊C4 | C2×D27 | semidirect |
| 155. C42 | C2×D27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C27×#126 | 2 |
| #11720144 | 2 |
| #11720143 | 2 |
| #11720133 | 2 |
| #11720131 | 2 |
| #11720125 | 2 |
| #11720123 | 2 |
| #10589813 | 3 |