Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
792 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 792 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | C3⋊#157 |
| commutator, hypocenter | C33 | C43 |
| Frattini | C22 | C22×C3⋊#157 |
| Fitting | C33×C42 | C4 |
| socle | C22×C33 | C22×C4 |
| prime radical, cosocle | C22×C3×C3⋊D3 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C4×C3⋊#157 | C4 | direct (N × H) |
| 7. C33×C42 | C4 | semidirect |
| 8. C3×C4×C3⋊Dic3 | C4 | semidirect |
| 24. C32×C42 | Dic3 | semidirect |
| 25. C4×C3⋊Dic3 | Dic3 | semidirect |
| 29. C3⋊#157 | C42 | direct (N × H) |
| 30. C33×C4 | C42 | semidirect |
| 31. C3×C3⋊Dic3 | C42 | semidirect |
| 54. C3×C42 | #157 | semidirect |
| 65. C32×C4 | C4×Dic3 | semidirect |
| 66. C3⋊Dic3 | C4×Dic3 | semidirect |
| 70. C33 | C43 | semidirect |
| 90. C42 | C3⋊#157 | direct (N × H) |
| 99. C3×C4 | C4×#157 | semidirect |
| 104. C32 | C42×Dic3 | semidirect |
| 119. C4 | C4×C3⋊#157 | direct (N × H) |
| 122. C3 | C42×#157 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C3⋊#157 | 2 |
| C3×C42×C3⋊D3 | 2 |
| C42×#157 | 3 |
| C42×Dic3 | 9 |