Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1631 conjugacy classes of subgroups; only the 583 normal subgroups are shown.
This group has 1631 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, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ 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×#137 | C2 | semidirect |
| 3. C22×C4⋊Q8 | C2 | semidirect |
| 4. #3152 | C2 | direct (N × H) |
| 5. C2×#497 | C2 | semidirect |
| 6. C2×#494 | C2 | semidirect |
| 7. C2×#492 | C2 | semidirect |
| 8. C2×C4×M16 | C2 | semidirect |
| 12. C2×#137 | C22 | semidirect |
| 23. C2×C4⋊Q8 | C22 | semidirect |
| 30. #497 | C22 | semidirect |
| 31. #494 | C22 | semidirect |
| 32. #492 | C22 | semidirect |
| 33. C4×M16 | C22 | semidirect |
| 57. C2×M16 | D4 | semidirect |
| 123. M16 | C2×D4 | semidirect |
| 161. C2×C4 | #137 | semidirect |
| 184. C4 | C2×#137 | semidirect |
| 188. C2 | #3152 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#137 | 2 |
| C2×#532 | 2 |
| C22×C4⋊Q8 | 2 |
| #3152 | 2 |
| C2×#497 | 2 |
| C2×#494 | 2 |
| C2×#492 | 2 |
| C2×C4×M16 | 2 |