Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1247 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1247 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#343 | C2 | direct (N × H) |
| 8. C22×M16 | C4 | semidirect |
| 15. #343 | C22 | direct (N × H) |
| 31. C2×M16 | C2×C4 | semidirect |
| 71. M16 | C22×C4 | semidirect |
| 116. C22 | #343 | direct (N × H) |
| 131. C2 | C2×#343 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C23×C4⋊C4 | 2 |
| C2×#343 | 2 |