Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
984 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 984 conjugacy classes of subgroups — too many to draw. Only its 449 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 | C2×#120 | 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. #2848 | C2 | semidirect |
| 2. #2840 | C2 | semidirect |
| 3. #2804 | C2 | semidirect |
| 4. #2763 | C2 | semidirect |
| 5. #2563 | C2 | semidirect |
| 6. #2492 | C2 | semidirect |
| 7. #2409 | C2 | semidirect |
| 10. C23×Q8 | C22 | semidirect |
| 12. C2×C4⋊Q8 | C22 | semidirect |
| 16. #400 | C22 | semidirect |
| 17. #395 | C22 | semidirect |
| 21. #393 | C22 | semidirect |
| 22. #391 | C22 | semidirect |
| 23. #391 | C22 | semidirect |
| 30. #385 | C22 | semidirect |
| 31. #382 | C22 | semidirect |
| 33. #380 | C22 | semidirect |
| 36. C4×C4⋊C4 | C22 | semidirect |
| 51. #120 | Q8 | semidirect |
| 74. C24 | C2×Q8 | semidirect |
| 109. C22⋊C4 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2848 | 2 |
| #2840 | 2 |
| #2804 | 2 |
| #2763 | 2 |
| #2563 | 2 |
| #2492 | 2 |
| #2409 | 2 |
| #2375 | 2 |
| C4×#120 | 2 |