Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
341 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2950 | C2 | semidirect |
| 4. #2175 | C2 | semidirect |
| 5. C4×SD32 | C2 | semidirect |
| 6. #2158 | C2 | semidirect |
| 7. #2141 | C2 | semidirect |
| 9. #2108 | C2 | semidirect |
| 14. C2×SD32 | C4 | semidirect |
| 21. C4×Q16 | C22 | semidirect |
| 28. #367 | C22 | semidirect |
| 29. #366 | C22 | semidirect |
| 30. #358 | C22 | semidirect |
| 32. #346 | C22 | semidirect |
| 33. C4×C16 | C22 | semidirect |
| 38. C2×Q16 | C2×C4 | semidirect |
| 55. SD32 | C2×C4 | semidirect |
| 56. SD32 | C2×C4 | semidirect |
| 58. C2×C16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD32 | 2 |
| #2950 | 2 |
| #2948 | 2 |
| #2175 | 2 |
| C4×SD32 | 2 |
| #2158 | 2 |
| #2141 | 2 |
| #2140 | 2 |
| #2108 | 2 |