Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
453 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 227 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C22 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. C4×M16 | C4 | semidirect |
| 16. C2×#97 | C4 | semidirect |
| 18. C2×C4×C8 | C4 | semidirect |
| 23. #344 | C4 | direct (N × H) |
| 79. M16 | C42 | semidirect |
| 84. C2×C8 | C42 | semidirect |
| 146. C4 | #344 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×M16 | 2 |
| C4×#117 | 2 |
| C2×#344 | 2 |