Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1416 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #157⋊D3 |
| commutator, cosocle | C3×C3⋊Dic3 | C24 |
| Frattini | C2 | C22×#157⋊D3 |
| Fitting, socle | C23×C33 | D4 |
| hypocenter | C33 | C22×D8 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#6295 | C2 | semidirect |
| 2. C22×C3×#622 | C2 | semidirect |
| 3. C2×#88941 | C2 | direct (N × H) |
| 8. C2×#6295 | C22 | semidirect |
| 9. C2×C3×#622 | C22 | semidirect |
| 10. #88941 | C22 | direct (N × H) |
| 13. C22×#622 | D3 | semidirect |
| 21. #6295 | C23 | semidirect |
| 22. C3×#622 | C23 | semidirect |
| 34. C2×#622 | C2×D3 | semidirect |
| 36. C22×C33 | D8 | semidirect |
| 65. #622 | C22×D3 | semidirect |
| 67. C2×C33 | C2×D8 | semidirect |
| 86. C22×C32 | D24 | semidirect |
| 103. C33 | C22×D8 | semidirect |
| 128. C2×C32 | C2×D24 | semidirect |
| 146. C22×C3 | #3774 | semidirect |
| 160. C32 | C22×D24 | semidirect |
| 174. C2×C3 | C2×#3774 | semidirect |
| 181. C22 | #88941 | direct (N × H) |
| 187. C3 | C22×#3774 | semidirect |
| 189. C2 | C2×#88941 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6295 | 2 |
| C22×C3×#622 | 2 |
| C2×#88941 | 2 |
| C22×#3774 | 3 |
| C22×D24 | 9 |