Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
215 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C22×D3 |
| commutator, cosocle | C22×C3 | C23×C3 |
| Frattini | C22 | C22×C3×D3 |
| Fitting | C22×C32×C4 | C2 |
| socle | C22×C32 | C23 |
| hypercenter | C22×C3×C4 | D3 |
| hypocenter | C3 | C3×#121 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C32×C4 | C2 | semidirect |
| 2. C2×C3×#67 | C2 | semidirect |
| 3. C2×C3×D12 | C2 | semidirect |
| 4. C3×(C2×D3)⋊C4 | C2 | semidirect |
| 5. C3×C4⋊Dic3 | C2 | semidirect |
| 7. #930 | C3 | direct (N × H) |
| 19. C22×C3×C4 | D3 | semidirect |
| 20. C22×C3×C4 | C2×C3 | semidirect |
| 21. C2×#67 | C2×C3 | semidirect |
| 22. C2×D12 | C2×C3 | semidirect |
| 25. (C2×D3)⋊C4 | C2×C3 | semidirect |
| 26. C4⋊Dic3 | C2×C3 | semidirect |
| 29. C22×C32 | D4 | semidirect |
| 35. C32×C4 | D4 | semidirect |
| 65. C22×C4 | C3×D3 | semidirect |
| 79. C22×C3 | D12 | semidirect |
| 80. C22×C3 | C3×D4 | semidirect |
| 91. C3×C4 | #67 | semidirect |
| 92. C3×C4 | C3×D4 | semidirect |
| 94. C32 | #121 | semidirect |
| 122. C22 | C3×D12 | semidirect |
| 126. C4 | C3×#67 | semidirect |
| 128. C3 | #930 | direct (N × H) |
| 129. C3 | C3×#121 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×C4 | 2 |
| C2×C3×#67 | 2 |
| C2×C3×D12 | 2 |
| C3×(C2×D3)⋊C4 | 2 |
| C3×C4⋊Dic3 | 2 |
| C3×#121 | 3 |
| #930 | 3 |