Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
366 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×A4 |
| commutator, cosocle, hypocenter | C22×C3 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | C2×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×A4 | C2 | semidirect |
| 2. C2×D3×A4 | C2 | direct (N × H) |
| 3. C24×D3 | C3 | semidirect |
| 6. C2×C3×A4 | C22 | semidirect |
| 7. D3×A4 | C22 | direct (N × H) |
| 8. C24×C3 | C2×C3 | semidirect |
| 12. C23×D3 | C2×C3 | semidirect |
| 15. C22×A4 | D3 | direct (N × H) |
| 18. C3×A4 | C23 | semidirect |
| 23. C23×C3 | C22×C3 | semidirect |
| 31. C22×D3 | A4 | direct (N × H) |
| 32. C22×D3 | C22×C3 | semidirect |
| 35. C2×A4 | C2×D3 | direct (N × H) |
| 42. C24 | C3×D3 | semidirect |
| 50. C22×C3 | C2×A4 | semidirect |
| 51. C22×C3 | C23×C3 | semidirect |
| 57. C2×D3 | C2×A4 | direct (N × H) |
| 59. A4 | C22×D3 | direct (N × H) |
| 72. C23 | C2×C3×D3 | semidirect |
| 78. C2×C3 | C22×A4 | semidirect |
| 80. D3 | C22×A4 | direct (N × H) |
| 91. C22 | D3×A4 | direct (N × H) |
| 92. C22 | C22×C3×D3 | semidirect |
| 95. C3 | C23×A4 | semidirect |
| 100. C2 | C2×D3×A4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×A4 | 2 |
| C2×D3×A4 | 2 |
| C24×D3 | 3 |
| C23×A4 | 3 |
| C22×C3×D3 | 4 |