Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
624 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 155 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 | #550 |
| 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. #11720189 | C2 | semidirect |
| 4. #11720188 | C2 | semidirect |
| 5. #11720187 | C2 | semidirect |
| 6. D4×Dic27 | C2 | semidirect |
| 7. C4×D108 | C2 | semidirect |
| 8. C4×Dic54 | C2 | semidirect |
| 51. Dic54 | D4 | semidirect |
| 79. C4×C27 | #41 | semidirect |
| 146. #127 | D27 | semidirect |
| 152. C27 | #550 | semidirect |
| 240. 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 |
| #11720189 | 2 |
| #11720188 | 2 |
| #11720187 | 2 |
| D4×Dic27 | 2 |
| C4×D108 | 2 |
| C4×Dic54 | 2 |
| #10589820 | 3 |