Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1147 conjugacy classes of subgroups; only the 519 normal subgroups are shown.
This group has 1147 conjugacy classes of subgroups — too many to draw. Only its 519 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2401 | C2 | direct (N × H) |
| 2. C2×#401 | C2 | semidirect |
| 3. C2×#400 | C2 | semidirect |
| 6. C2×C4×C4⋊C4 | C2 | semidirect |
| 8. C2×#376 | C2 | semidirect |
| 16. #401 | C22 | semidirect |
| 17. #400 | C22 | semidirect |
| 20. C4×C4⋊C4 | C22 | semidirect |
| 22. #376 | C22 | semidirect |
| 137. C2 | #2401 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2401 | 2 |
| C2×#401 | 2 |
| C2×#400 | 2 |
| C2×#397 | 2 |
| C2×#396 | 2 |
| C2×C4×C4⋊C4 | 2 |
| C2×C4×C22⋊C4 | 2 |
| C2×#376 | 2 |