Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
352 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C4×C8 |
| Fitting, cosocle | C2×C33×C4 | C22 |
| socle, prime radical | C22×C33 | C2×C4 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C3⋊(C3⋊(C3⋊C8)) | C4 | semidirect |
| 9. C3×C3⋊(C3⋊C8) | C4 | semidirect |
| 20. C32×Dic3 | C8 | semidirect |
| 31. C3⋊(C3⋊C8) | Dic3 | direct (N × H) |
| 52. C3×Dic3 | C3⋊C8 | semidirect |
| 54. C33 | C4×C8 | semidirect |
| 84. Dic3 | C3⋊(C3⋊C8) | direct (N × H) |
| 86. C32 | C4×C3⋊C8 | semidirect |
| 87. C32 | C8×Dic3 | semidirect |
| 103. C3 | Dic3×C3⋊C8 | semidirect |
| 104. C3 | C4×C3⋊(C3⋊C8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊(C3⋊(C3⋊C8)) | 2 |
| C2×C3×C3⋊(C3⋊C8) | 2 |
| C32×C4×Dic3 | 2 |
| C4×C3⋊(C3⋊C8) | 3 |
| Dic3×C3⋊C8 | 3 |