Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1211 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1211 conjugacy classes of subgroups — too many to draw. Only its 487 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#533 | C2 | semidirect |
| 3. #2861 | C2 | semidirect |
| 4. #2640 | C2 | semidirect |
| 5. #2592 | C2 | semidirect |
| 6. #2561 | C2 | semidirect |
| 7. #2310 | C2 | semidirect |
| 9. C4×#117 | C2 | semidirect |
| 13. C2×C4⋊Q8 | C22 | semidirect |
| 14. C2×#125 | C22 | semidirect |
| 20. C2×C4×Q8 | C22 | semidirect |
| 25. #397 | C22 | semidirect |
| 26. #393 | C22 | semidirect |
| 27. #391 | C22 | semidirect |
| 28. #391 | C22 | semidirect |
| 31. #388 | C22 | semidirect |
| 34. C4×C4⋊C4 | C22 | semidirect |
| 36. #376 | C22 | semidirect |
| 37. C43 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| C22×#124 | 2 |
| #2861 | 2 |
| #2640 | 2 |
| #2592 | 2 |
| #2561 | 2 |
| #2310 | 2 |
| #2309 | 2 |
| C4×#117 | 2 |