Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
339 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3100 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2981 | C2 | semidirect |
| 3. #2972 | C2 | semidirect |
| 4. #1586 | C2 | semidirect |
| 5. #1581 | C2 | semidirect |
| 6. #1580 | C2 | semidirect |
| 7. C8×D8 | C2 | semidirect |
| 8. #1572 | C2 | semidirect |
| 9. #1490 | C2 | semidirect |
| 10. #1484 | C2 | semidirect |
| 11. #1453 | C2 | semidirect |
| 20. #436 | C22 | semidirect |
| 25. C8×D4 | C22 | semidirect |
| 33. #335 | C22 | semidirect |
| 34. #334 | C22 | semidirect |
| 37. D4⋊C8 | C22 | semidirect |
| 38. D4⋊C8 | C22 | semidirect |
| 39. #322 | C22 | semidirect |
| 40. #322 | C22 | semidirect |
| 41. #322 | C22 | semidirect |
| 42. C82 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3100 | 2 |
| #2981 | 2 |
| #2972 | 2 |
| #1586 | 2 |
| #1581 | 2 |
| #1580 | 2 |
| C8×D8 | 2 |
| #1572 | 2 |
| #1490 | 2 |
| #1484 | 2 |
| #1453 | 2 |