Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
787 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 787 conjugacy classes of subgroups — too many to draw. Only its 175 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 | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#419 | C2 | semidirect |
| 3. C2×#419 | C2 | semidirect |
| 6. #2296 | C2 | semidirect |
| 7. #2042 | C2 | semidirect |
| 8. #1728 | C2 | semidirect |
| 25. C2×#103 | C22 | semidirect |
| 26. C2×#103 | C22 | semidirect |
| 29. #389 | C22 | semidirect |
| 32. #336 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#124 | 2 |
| C2×#419 | 2 |
| C2×#418 | 2 |
| #2296 | 2 |
| #2042 | 2 |
| #1728 | 2 |