Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
393 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 163 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 | C22×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. D4×M16 | C2 | semidirect |
| 3. #1594 | C2 | semidirect |
| 4. #1586 | C2 | semidirect |
| 5. #1580 | C2 | semidirect |
| 6. #1566 | C2 | semidirect |
| 7. #1486 | C2 | semidirect |
| 8. #1451 | C2 | semidirect |
| 18. #436 | C22 | semidirect |
| 20. C8×D4 | C22 | semidirect |
| 28. #334 | C22 | semidirect |
| 29. #334 | C22 | semidirect |
| 30. D4⋊C8 | C22 | semidirect |
| 31. D4⋊C8 | C22 | semidirect |
| 32. #322 | C22 | semidirect |
| 33. #322 | C22 | semidirect |
| 94. D8 | M16 | semidirect |
| 129. D4 | C2×M16 | semidirect |
| 155. C8 | C2×M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| D4×M16 | 2 |
| #1594 | 2 |
| #1586 | 2 |
| #1580 | 2 |
| #1566 | 2 |
| #1486 | 2 |
| #1451 | 2 |