Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
620 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 620 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 | #532 |
| 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×#88401 | C2 | semidirect |
| 2. C27×#127 | C2 | semidirect |
| 3. #11720187 | C2 | semidirect |
| 4. #11720185 | C2 | semidirect |
| 5. D4×Dic27 | C2 | semidirect |
| 6. C42×D27 | C2 | semidirect |
| 7. #11720118 | C2 | semidirect |
| 41. C4×D27 | D4 | semidirect |
| 63. C4×C27 | #41 | semidirect |
| 118. #127 | D27 | semidirect |
| 123. C27 | #532 | semidirect |
| 193. C4 | #88401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88401 | 2 |
| C27×#127 | 2 |
| #11720187 | 2 |
| #11720185 | 2 |
| D4×Dic27 | 2 |
| C42×D27 | 2 |
| #11720118 | 2 |
| #10589818 | 3 |