Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
648 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 162 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 | D3×#157 |
| commutator, hypocenter | C33 | C2×C4×C8 |
| Frattini | C22 | C2×D3×#157 |
| Fitting, socle, prime radical | C23×C33 | C2×C4 |
| cosocle | C2×C3×C3⋊Dic3 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. Dic3×#622 | C2 | direct (N × H) |
| 9. C2×C3⋊#622 | C4 | semidirect |
| 11. C2×C3×#622 | C4 | semidirect |
| 21. C2×C3⋊(C3⋊Dic3) | C8 | semidirect |
| 27. C2×C32×Dic3 | C8 | semidirect |
| 29. C3⋊#622 | C2×C4 | semidirect |
| 30. C3×#622 | C2×C4 | semidirect |
| 41. C2×#622 | Dic3 | direct (N × H) |
| 51. C3⋊(C3⋊Dic3) | C2×C8 | semidirect |
| 55. C32×Dic3 | C2×C8 | semidirect |
| 83. #622 | C2×Dic3 | direct (N × H) |
| 85. C2×C33 | C4×C8 | semidirect |
| 131. C33 | C2×C4×C8 | semidirect |
| 151. C2×Dic3 | #622 | direct (N × H) |
| 163. C2×C32 | C8×Dic3 | semidirect |
| 198. Dic3 | C2×#622 | direct (N × H) |
| 201. C32 | C2×C8×Dic3 | semidirect |
| 221. C2×C3 | C4×#622 | semidirect |
| 236. C3 | C2×C4×#622 | semidirect |
| 237. C2 | Dic3×#622 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3⋊#622 | 2 |
| C22×C3×#622 | 2 |
| C2×Dic3×C3⋊Dic3 | 2 |
| Dic3×#622 | 2 |
| C2×C4×#622 | 3 |
| C2×C8×Dic3 | 9 |