Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
146 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D32 |
| commutator, socle | C2×C32 | C22×C4 |
| Fitting | C2×C32×C4 | C22 |
| prime radical | C22×C3×D3 | C4 |
| cosocle | C32×C4 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C2×M16 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #3748 | C2 | semidirect |
| 3. C3×#208 | C2 | semidirect |
| 4. #3711 | C2 | semidirect |
| 5. D3×C3⋊C8 | C2 | semidirect |
| 15. C3⋊(C3⋊C8) | C22 | semidirect |
| 16. C3×C3⋊C8 | C22 | semidirect |
| 24. #208 | D3 | direct (N × H) |
| 56. C3⋊C8 | C2×D3 | semidirect |
| 60. C3×D3 | M16 | semidirect |
| 85. C32 | C2×M16 | semidirect |
| 104. D3 | #208 | direct (N × H) |
| 114. C3 | D3×M16 | semidirect |
| 115. C3 | C2×#208 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×D3 | 2 |
| #3748 | 2 |
| C3×#208 | 2 |
| #3711 | 2 |
| D3×C3⋊C8 | 2 |
| C2×#208 | 3 |
| D3×M16 | 3 |