Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2834 conjugacy classes of subgroups; only the 508 normal subgroups are shown.
This group has 2834 conjugacy classes of subgroups — too many to draw. Only its 508 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 | D33 |
| commutator, hypocenter | C33 | C22×C42 |
| Frattini | C22 | C2×D33 |
| Fitting, socle | C23×C33 | C23 |
| prime radical | C23×C3⋊(C3⋊D3) | C22 |
| cosocle | C22×C33 | C24 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#88813 | C2 | semidirect |
| 3. C2×C3×Dic32 | C2 | semidirect |
| 4. #11722900 | C2 | direct (N × H) |
| 9. C2×#6289 | C4 | semidirect |
| 11. #88813 | C22 | semidirect |
| 12. C3×Dic32 | C22 | semidirect |
| 18. C2×Dic32 | D3 | semidirect |
| 23. C2×C3×C3⋊Dic3 | C2×C4 | semidirect |
| 25. C2×C32×Dic3 | C2×C4 | semidirect |
| 27. #6289 | C2×C4 | semidirect |
| 46. Dic32 | C2×D3 | semidirect |
| 49. C2×C3⋊(C3⋊D3) | C42 | semidirect |
| 51. C3×C3⋊Dic3 | C22×C4 | semidirect |
| 52. C32×Dic3 | C22×C4 | semidirect |
| 73. C2×C3⋊Dic3 | C4×D3 | semidirect |
| 82. C2×C3×Dic3 | C4×D3 | semidirect |
| 87. C2×C33 | C2×C42 | semidirect |
| 89. C3⋊(C3⋊D3) | C2×C42 | semidirect |
| 125. C3⋊Dic3 | C2×C4×D3 | semidirect |
| 129. C3×Dic3 | C2×C4×D3 | semidirect |
| 134. C33 | C22×C42 | semidirect |
| 156. C2×Dic3 | #624 | semidirect |
| 166. C2×C32 | C42×D3 | semidirect |
| 200. Dic3 | C2×#624 | semidirect |
| 203. C32 | C2×C42×D3 | semidirect |
| 217. C2×C3 | C4×#624 | semidirect |
| 231. C3 | C2×C4×#624 | semidirect |
| 233. C2 | #11722900 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#6289 | 2 |
| C2×#88813 | 2 |
| C2×C3×Dic32 | 2 |
| #11722900 | 2 |
| C2×C4×#624 | 3 |