Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
166 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 25 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 | C2 | S4×D15 |
| commutator, cosocle, hypocenter | C3×C5×SL(2,𝔽3) | C22 |
| Fitting | C3×C5×Q8 | C2×D3 |
| socle | C2×C3×C5 | C2×S4 |
| LCS | G ▷ C3×C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊(C5⋊(#41⋊C3)) | C2 | semidirect |
| 2. C3⋊(C5⋊#226) | C2 | semidirect |
| 3. C3×C5×GL(2,𝔽3) | C2 | semidirect |
| 9. #6821 | D3 | semidirect |
| 14. C5×GL(2,𝔽3) | D3 | semidirect |
| 26. C3×GL(2,𝔽3) | D5 | semidirect |
| 64. GL(2,𝔽3) | D15 | semidirect |
| 120. C3×C5 | #985 | semidirect |
| 149. C5 | #64197 | semidirect |
| 157. C3 | #91892 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊(C5⋊(#41⋊C3)) | 2 |
| C3⋊(C5⋊#226) | 2 |
| C3×C5×GL(2,𝔽3) | 2 |
| #91892 | 3 |
| #91805 | 3 |
| D3×Dic15 | 4 |
| #64197 | 5 |