Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
729 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 729 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 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#424 | C2 | semidirect |
| 3. C2×#419 | C2 | semidirect |
| 5. C22×#101 | C2 | semidirect |
| 7. #2046 | C2 | direct (N × H) |
| 8. C2×#341 | C2 | semidirect |
| 22. #424 | C22 | semidirect |
| 25. #419 | C22 | semidirect |
| 31. C2×#101 | C22 | semidirect |
| 34. #341 | C22 | semidirect |
| 266. C2 | #2046 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SD16 | 2 |
| C2×#424 | 2 |
| C2×#419 | 2 |
| C2×#418 | 2 |
| C22×#101 | 2 |
| C22×#100 | 2 |
| #2046 | 2 |
| C2×#341 | 2 |