Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
134 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 50 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 | C32×C8 | C22 |
| prime radical | #3798 | C2 |
| cosocle | C32×C4 | C23 |
| hypercenter | C8 | D32 |
| hypocenter | C32 | #131 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #3743 | C2 | semidirect |
| 3. C3×ΓL(1,𝔽25) | C2 | semidirect |
| 4. C3×D3×C8 | C2 | semidirect |
| 5. #3711 | C2 | semidirect |
| 6. #3710 | C2 | semidirect |
| 7. D3×C3⋊C8 | C2 | semidirect |
| 31. ΓL(1,𝔽25) | D3 | semidirect |
| 35. D3×C8 | D3 | semidirect |
| 97. C32 | #131 | semidirect |
| 127. C3 | #901 | semidirect |
| 128. C3 | #907 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3798 | 2 |
| #3743 | 2 |
| C3×ΓL(1,𝔽25) | 2 |
| C3×D3×C8 | 2 |
| #3711 | 2 |
| #3710 | 2 |
| D3×C3⋊C8 | 2 |
| #907 | 3 |
| #901 | 3 |