Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1119 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×M16 | C2 | semidirect |
| 3. C2×#418 | C2 | semidirect |
| 4. #2280 | C2 | semidirect |
| 5. #1895 | C2 | semidirect |
| 6. #1893 | C2 | semidirect |
| 7. #1736 | C2 | semidirect |
| 14. C22×M16 | C22 | semidirect |
| 15. C23×C8 | C22 | semidirect |
| 19. C2×#121 | C4 | semidirect |
| 26. C2×#102 | C22 | semidirect |
| 28. #380 | C22 | semidirect |
| 29. #380 | C22 | semidirect |
| 33. #340 | C22 | semidirect |
| 62. C2×M16 | D4 | semidirect |
| 149. C2×C8 | C2×D4 | semidirect |
| 238. C22 | #418 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C22×#121 | 2 |
| C2×#418 | 2 |
| #2280 | 2 |
| #1895 | 2 |
| #1893 | 2 |
| #1736 | 2 |