Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1683 conjugacy classes of subgroups; only the 583 normal subgroups are shown.
This group has 1683 conjugacy classes of subgroups — too many to draw. Only its 583 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#135 | C2 | semidirect |
| 2. C2×#532 | C2 | semidirect |
| 3. #3147 | C2 | direct (N × H) |
| 4. C2×#495 | C2 | semidirect |
| 5. C2×#494 | C2 | semidirect |
| 6. C2×#493 | C2 | semidirect |
| 7. C2×#492 | C2 | semidirect |
| 8. C22×C4×C8 | C2 | semidirect |
| 11. C2×#135 | C22 | semidirect |
| 17. #532 | C22 | semidirect |
| 24. #495 | C22 | semidirect |
| 25. #494 | C22 | semidirect |
| 26. #493 | C22 | semidirect |
| 27. #492 | C22 | semidirect |
| 31. C2×C4×C8 | C22 | semidirect |
| 49. C22×C8 | D4 | semidirect |
| 102. C2×C8 | C2×D4 | semidirect |
| 132. C2×C4 | #135 | semidirect |
| 155. C4 | C2×#135 | semidirect |
| 159. C2 | #3147 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#135 | 2 |
| C2×#532 | 2 |
| #3147 | 2 |
| C2×#495 | 2 |
| C2×#494 | 2 |
| C2×#493 | 2 |
| C2×#492 | 2 |
| C22×C4×C8 | 2 |