Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2834 conjugacy classes of subgroups; only the 871 normal subgroups are shown.
This group has 2834 conjugacy classes of subgroups — too many to draw. Only its 871 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1113 |
| commutator, hypocenter | C33 | C22×C42 |
| Frattini | C22 | C22×#1113 |
| Fitting, socle, prime radical | C24×C33 | C22 |
| cosocle | C22×C33 | C24 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#88813 | C2 | direct (N × H) |
| 5. C22×C3×C3⋊Dic3 | C4 | semidirect |
| 7. #88813 | C22 | direct (N × H) |
| 14. C2×C3×C3⋊Dic3 | C2×C4 | semidirect |
| 20. C22×C3⋊Dic3 | Dic3 | semidirect |
| 27. C22×C33 | C42 | semidirect |
| 30. C3×C3⋊Dic3 | C22×C4 | semidirect |
| 43. C2×C3⋊Dic3 | C2×Dic3 | semidirect |
| 50. C2×C33 | C2×C42 | semidirect |
| 69. C22×C32 | C4×Dic3 | semidirect |
| 73. C3⋊Dic3 | C22×Dic3 | semidirect |
| 78. C33 | C22×C42 | semidirect |
| 98. C2×C32 | C2×C4×Dic3 | semidirect |
| 112. C22×C3 | Dic32 | semidirect |
| 121. C32 | C22×C4×Dic3 | semidirect |
| 131. C2×C3 | C2×Dic32 | semidirect |
| 134. C22 | #88813 | direct (N × H) |
| 140. C3 | C22×Dic32 | semidirect |
| 141. C2 | C2×#88813 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C3⋊Dic3 | 2 |
| C2×#88813 | 2 |
| C22×Dic32 | 3 |