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×#123 | C2 |
| socle | C22×C3 | C23×D9 |
| hypercenter | #123 | D27 |
| hypocenter | C27 | #560 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C27×#123 | C2 | semidirect |
| 2. #11720189 | C2 | semidirect |
| 3. #11720186 | C2 | semidirect |
| 4. #11720179 | C2 | semidirect |
| 5. #11720147 | C2 | semidirect |
| 6. #11720144 | C2 | semidirect |
| 7. #11720133 | C2 | semidirect |
| 8. #11720132 | C2 | semidirect |
| 9. #11720131 | C2 | semidirect |
| 13. C22×C4×C27 | C22 | semidirect |
| 16. C2×#6171 | C22 | semidirect |
| 19. C2×D108 | C22 | semidirect |
| 20. C2×C4×D27 | C22 | semidirect |
| 21. C4⋊C4×C27 | C22 | semidirect |
| 23. C22⋊C4×C27 | C22 | semidirect |
| 28. (C2×D27)⋊C4 | C22 | semidirect |
| 29. C4⋊Dic27 | C22 | semidirect |
| 30. Dic27⋊C4 | C22 | semidirect |
| 31. C4×Dic27 | C22 | semidirect |
| 177. #123 | D27 | semidirect |
| 183. C27 | #560 | semidirect |
| 219. C22×C4 | C2×D27 | semidirect |
| 222. C4⋊C4 | C2×D27 | semidirect |
| 229. C22⋊C4 | C2×D27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C27×#123 | 2 |
| #11720189 | 2 |
| #11720186 | 2 |
| #11720179 | 2 |
| #11720147 | 2 |
| #11720144 | 2 |
| #11720133 | 2 |
| #11720132 | 2 |
| #11720131 | 2 |
| #10589768 | 3 |