Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
138 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×A4 |
| commutator, hypocenter | C22×C3 | C2×C3×C4 |
| Frattini | C2 | C2×D3×A4 |
| Fitting | C22×C3×C4 | C2×C3 |
| socle, cosocle | C23×C3 | C22×C3 |
| prime radical | C2×D3×A4 | C2 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C4×A4 | C2 | semidirect |
| 3. Dic3×A4 | C2 | semidirect |
| 4. C22×C4×D3 | C3 | semidirect |
| 7. D3×A4 | C4 | direct (N × H) |
| 11. C22×C3×C4 | C2×C3 | semidirect |
| 12. C22×Dic3 | C2×C3 | semidirect |
| 19. C4×A4 | D3 | direct (N × H) |
| 21. C3×A4 | C2×C4 | semidirect |
| 29. C22×D3 | C3×C4 | semidirect |
| 42. C4×D3 | A4 | direct (N × H) |
| 51. C22×C4 | C3×D3 | semidirect |
| 55. C22×C3 | C2×C3×C4 | semidirect |
| 63. A4 | C4×D3 | direct (N × H) |
| 68. C3×C4 | C2×A4 | semidirect |
| 70. Dic3 | C2×A4 | semidirect |
| 93. D3 | C4×A4 | direct (N × H) |
| 102. C22 | C3×C4×D3 | semidirect |
| 106. C4 | D3×A4 | direct (N × H) |
| 109. C3 | C2×C4×A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×A4 | 2 |
| C3×C4×A4 | 2 |
| Dic3×A4 | 2 |
| C22×C4×D3 | 3 |
| C2×C4×A4 | 3 |
| C3×C4×D3 | 4 |