Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
312 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C23×D3 |
| commutator, cosocle | C2×C3 | C24×C3 |
| Frattini | C2 | C23×C3×D3 |
| Fitting | C32×#41 | C2 |
| socle | C2×C32 | C24 |
| hypercenter | C3×#41 | D3 |
| hypocenter | C3 | C3×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#41 | C2 | semidirect |
| 2. C3×D3×Q8 | C2 | semidirect |
| 3. C3×#237 | C2 | semidirect |
| 4. C3×#235 | C2 | semidirect |
| 5. C2×C3×Dic6 | C2 | semidirect |
| 7. #1010 | C3 | direct (N × H) |
| 20. C3×#41 | D3 | semidirect |
| 21. C3×#41 | C2×C3 | semidirect |
| 24. D3×Q8 | C2×C3 | semidirect |
| 25. #237 | C2×C3 | semidirect |
| 26. #235 | C2×C3 | semidirect |
| 27. C2×Dic6 | C2×C3 | semidirect |
| 60. #41 | C3×D3 | semidirect |
| 76. C32 | 2−1+4 | semidirect |
| 100. C3 | #1010 | direct (N × H) |
| 101. C3 | C3×2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#41 | 2 |
| C3×D3×Q8 | 2 |
| C3×#237 | 2 |
| C3×#235 | 2 |
| C2×C3×Dic6 | 2 |
| C3×2−1+4 | 3 |
| #1010 | 3 |