Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
357 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3227 | C2 | semidirect |
| 3. C2×#486 | C2 | semidirect |
| 4. #2931 | C2 | semidirect |
| 5. #1697 | C2 | semidirect |
| 6. #1677 | C2 | semidirect |
| 7. #1564 | C2 | semidirect |
| 8. #1548 | C2 | semidirect |
| 9. #1545 | C2 | semidirect |
| 23. #476 | C22 | semidirect |
| 27. #436 | C22 | semidirect |
| 29. C8×D4 | C22 | semidirect |
| 37. #335 | C22 | semidirect |
| 38. #334 | C22 | semidirect |
| 39. #333 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3227 | 2 |
| #3205 | 2 |
| C2×#486 | 2 |
| #2931 | 2 |
| #1697 | 2 |
| #1677 | 2 |
| #1564 | 2 |
| #1548 | 2 |
| #1545 | 2 |