Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1089 conjugacy classes of subgroups; only the 519 normal subgroups are shown.
This group has 1089 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, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3458 | C2 | semidirect |
| 3. #3081 | C2 | semidirect |
| 4. #3077 | C2 | semidirect |
| 5. #2911 | C2 | semidirect |
| 6. #2905 | C2 | semidirect |
| 11. #544 | C22 | semidirect |
| 14. #533 | C22 | semidirect |
| 23. #478 | C22 | semidirect |
| 24. #476 | C22 | semidirect |
| 25. #474 | C22 | semidirect |
| 26. #428 | C22 | semidirect |
| 27. #428 | C22 | semidirect |
| 28. #427 | C22 | semidirect |
| 29. #424 | C22 | semidirect |
| 30. #423 | C22 | semidirect |
| 38. C22×D4 | Q8 | semidirect |
| 48. C4⋊Q8 | C23 | semidirect |
| 50. #125 | C23 | semidirect |
| 76. #107 | C23 | semidirect |
| 77. #106 | C23 | semidirect |
| 78. C4⋊C8 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3458 | 2 |
| C2×#518 | 2 |
| #3081 | 2 |
| #3077 | 2 |
| #2911 | 2 |
| #2905 | 2 |
| C2×#418 | 2 |