Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | A4×D15 |
| commutator, hypocenter | C3×C5×Q8 | C3×C4 |
| Fitting, cosocle | C2×C3×C5×Q8 | C2×C3 |
| socle | C22×C3×C5 | C2×A4 |
| prime radical | C2×C3×C5×SL(2,𝔽3) | C2 |
| LCS | G ▷ C3×C5×Q8 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×Dic15 | C3 | semidirect |
| 5. C3×C5×SL(2,𝔽3) | C4 | semidirect |
| 17. C3×C5×Q8 | C3×C4 | semidirect |
| 23. C5×SL(2,𝔽3) | Dic3 | semidirect |
| 33. C3×SL(2,𝔽3) | Dic5 | semidirect |
| 40. Dic15 | SL(2,𝔽3) | direct (N × H) |
| 49. C5×Q8 | C3×Dic3 | semidirect |
| 66. C3×Q8 | C3×Dic5 | semidirect |
| 72. SL(2,𝔽3) | Dic15 | direct (N × H) |
| 86. C3×C5 | C4×SL(2,𝔽3) | semidirect |
| 100. Q8 | C3×Dic15 | semidirect |
| 113. C5 | Dic3×SL(2,𝔽3) | semidirect |
| 121. C3 | Dic5×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C5×SL(2,𝔽3) | 2 |
| Q8×Dic15 | 3 |
| Dic5×SL(2,𝔽3) | 3 |
| C2×C3×Dic15 | 4 |
| Dic3×SL(2,𝔽3) | 5 |