Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1491 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1491 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 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2659 | C2 | direct (N × H) |
| 3. C2×#398 | C2 | semidirect |
| 4. C2×#395 | C2 | semidirect |
| 17. #398 | C22 | semidirect |
| 18. #395 | C22 | semidirect |
| 31. C23×C4 | Q8 | semidirect |
| 77. C22×C4 | C2×Q8 | semidirect |
| 141. C2 | #2659 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Q8 | 2 |
| #2659 | 2 |
| C2×#398 | 2 |
| C2×#395 | 2 |
| C2×#380 | 2 |
| C22×#95 | 2 |