Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
118 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 23 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 | D3×S4 |
| commutator, hypocenter | C3×A4 | C2×C4 |
| Fitting, socle | C23×C3 | C2×D3 |
| prime radical | C2×C3×S4 | C2 |
| cosocle | C2×C3×A4 | C22 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Dic3×A4 | C2 | semidirect |
| 3. C3⋊SL(2,ℤ/4) | C2 | semidirect |
| 7. C3×S4 | C4 | semidirect |
| 12. C22×Dic3 | D3 | semidirect |
| 21. C3×A4 | C2×C4 | semidirect |
| 29. S4 | Dic3 | direct (N × H) |
| 54. C22×C3 | C4×D3 | semidirect |
| 57. A4 | C2×Dic3 | semidirect |
| 65. Dic3 | S4 | direct (N × H) |
| 90. C22 | D3×Dic3 | semidirect |
| 98. C3 | C4×S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×S4 | 2 |
| Dic3×A4 | 2 |
| C3⋊SL(2,ℤ/4) | 2 |
| C4×S4 | 3 |
| D4×Dic3 | 3 |
| D3×Dic3 | 4 |