Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
2374 conjugacy classes of subgroups; only the 300 normal subgroups are shown.
This group has 2374 conjugacy classes of subgroups — too many to draw. Only its 300 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24×C5 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24×C5 |
| Frattini | C2 | C23×C5×S4 |
| Fitting | C23×C5×Q8 | D3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C24×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C5×SL(2,𝔽3) | C2 | semidirect |
| 2. C22×C5×GL(2,𝔽3) | C2 | direct (N × H) |
| 5. C22×C5×SL(2,𝔽3) | C22 | semidirect |
| 6. C2×C5×GL(2,𝔽3) | C22 | direct (N × H) |
| 7. C23×GL(2,𝔽3) | C5 | direct (N × H) |
| 8. C23×C5×Q8 | D3 | semidirect |
| 16. C2×C5×SL(2,𝔽3) | C23 | semidirect |
| 17. C5×GL(2,𝔽3) | C23 | direct (N × H) |
| 18. C23×SL(2,𝔽3) | C2×C5 | semidirect |
| 19. C22×GL(2,𝔽3) | C2×C5 | direct (N × H) |
| 22. C22×C5×Q8 | C2×D3 | semidirect |
| 34. C5×SL(2,𝔽3) | C24 | semidirect |
| 36. C22×SL(2,𝔽3) | C22×C5 | semidirect |
| 37. C2×GL(2,𝔽3) | C22×C5 | direct (N × H) |
| 43. C2×C5×Q8 | C22×D3 | semidirect |
| 51. C23×Q8 | C5×D3 | semidirect |
| 63. C2×SL(2,𝔽3) | C23×C5 | semidirect |
| 64. GL(2,𝔽3) | C23×C5 | direct (N × H) |
| 67. C23×C5 | GL(2,𝔽3) | direct (N × H) |
| 71. C5×Q8 | C23×D3 | semidirect |
| 79. C22×Q8 | C2×C5×D3 | semidirect |
| 93. SL(2,𝔽3) | C24×C5 | semidirect |
| 96. C22×C5 | C2×GL(2,𝔽3) | direct (N × H) |
| 105. C2×Q8 | C22×C5×D3 | semidirect |
| 119. C2×C5 | C22×GL(2,𝔽3) | direct (N × H) |
| 123. C23 | C5×GL(2,𝔽3) | direct (N × H) |
| 127. Q8 | C23×C5×D3 | semidirect |
| 136. C5 | C23×GL(2,𝔽3) | direct (N × H) |
| 139. C22 | C2×C5×GL(2,𝔽3) | direct (N × H) |
| 145. C2 | C22×C5×GL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C5×SL(2,𝔽3) | 2 |
| C22×C5×GL(2,𝔽3) | 2 |
| C23×C5×SD16 | 3 |
| C24×C5×D3 | 4 |
| C23×GL(2,𝔽3) | 5 |