Finite Group DatabasebetaFinite Group DatabasebetaOrder 24256.
240 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D379 |
| commutator, socle, cosocle | C22×C379 | C24 |
| Fitting | C4×D4×C379 | C2 |
| hypercenter | C4×D4 | D379 |
| hypocenter | C379 | #540 |
| LCS | G ▷ C22×C379 ▷ C379 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #27131327 | C2 | semidirect |
| 2. #27131322 | C2 | semidirect |
| 3. #27131314 | C2 | semidirect |
| 4. #27131313 | C2 | semidirect |
| 5. #27131306 | C2 | semidirect |
| 6. #27131305 | C2 | semidirect |
| 7. #27131304 | C2 | semidirect |
| 8. #27131302 | C2 | semidirect |
| 9. C4×D1516 | C2 | semidirect |
| 10. #27131288 | C2 | semidirect |
| 11. C4×D4×C379 | C2 | semidirect |
| 79. C379 | #540 | semidirect |
| 90. C4×D4 | D379 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #27131327 | 2 |
| #27131322 | 2 |
| #27131314 | 2 |
| #27131313 | 2 |
| #27131306 | 2 |
| #27131305 | 2 |
| #27131304 | 2 |
| #27131302 | 2 |
| C4×D1516 | 2 |
| #27131288 | 2 |
| C4×D4×C379 | 2 |
| #540 | 379 |