Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2215 conjugacy classes of subgroups; only the 823 normal subgroups are shown.
This group has 2215 conjugacy classes of subgroups — too many to draw. Only its 823 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 | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C22 ▷ 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. C22×C4×D4 | C2 | semidirect |
| 3. #2328 | C2 | direct (N × H) |
| 4. C2×#390 | C2 | semidirect |
| 5. C2×#386 | C2 | semidirect |
| 6. C2×#382 | C2 | semidirect |
| 7. C2×#376 | C2 | semidirect |
| 11. C2×#127 | C4 | semidirect |
| 14. C2×C4×D4 | C22 | semidirect |
| 21. #390 | C22 | semidirect |
| 22. #386 | C22 | semidirect |
| 23. #382 | C22 | semidirect |
| 24. #376 | C22 | semidirect |
| 31. C22×D4 | C2×C4 | semidirect |
| 32. C22×D4 | C2×C4 | semidirect |
| 43. #127 | C2×C4 | semidirect |
| 52. C2×C42 | C2×C4 | semidirect |
| 53. C2×C42 | D4 | semidirect |
| 71. C2×D4 | C22×C4 | semidirect |
| 72. C2×D4 | C22×C4 | semidirect |
| 96. C42 | C22×C4 | semidirect |
| 97. C42 | C2×D4 | semidirect |
| 152. C2 | #2328 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#127 | 2 |
| C22×C4×D4 | 2 |
| #2328 | 2 |
| C2×#390 | 2 |
| C2×#386 | 2 |
| C2×#382 | 2 |
| C2×#376 | 2 |