Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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 | #3036 | 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 |
| 2. C2×#476 | C2 | semidirect |
| 4. #2972 | C2 | semidirect |
| 5. #1677 | C2 | semidirect |
| 6. #1671 | C2 | semidirect |
| 7. #1570 | C2 | semidirect |
| 8. #1548 | C2 | semidirect |
| 9. #1540 | C2 | semidirect |
| 14. #498 | C22 | semidirect |
| 15. #486 | C22 | semidirect |
| 33. #334 | C22 | semidirect |
| 34. #333 | C22 | semidirect |
| 35. #332 | C22 | semidirect |
| 36. #331 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3227 | 2 |
| C2×#476 | 2 |
| #3036 | 2 |
| #2972 | 2 |
| #1677 | 2 |
| #1671 | 2 |
| #1570 | 2 |
| #1548 | 2 |
| #1540 | 2 |