Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1103 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1103 conjugacy classes of subgroups — too many to draw. Only its 479 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 | C2×#535 | C2 |
| 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. #3140 | C2 | semidirect |
| 4. #3138 | C2 | semidirect |
| 5. #3136 | C2 | semidirect |
| 6. #2896 | C2 | semidirect |
| 7. C2×#418 | C2 | semidirect |
| 8. #2873 | C2 | semidirect |
| 13. #544 | C22 | semidirect |
| 18. #533 | C22 | semidirect |
| 25. #490 | C22 | semidirect |
| 26. #489 | C22 | semidirect |
| 27. #488 | C22 | semidirect |
| 28. #486 | C22 | semidirect |
| 29. #419 | C22 | semidirect |
| 31. #418 | C22 | semidirect |
| 32. #418 | C22 | semidirect |
| 33. #417 | C22 | semidirect |
| 34. #416 | C22 | semidirect |
| 35. C2×#102 | C22 | semidirect |
| 36. #405 | C22 | semidirect |
| 37. C4×M16 | C22 | semidirect |
| 58. C4⋊Q8 | C23 | semidirect |
| 62. #125 | C23 | semidirect |
| 83. #103 | C23 | semidirect |
| 85. #102 | C23 | semidirect |
| 86. #97 | C23 | semidirect |
| 87. 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×#535 | 2 |
| #3140 | 2 |
| #3138 | 2 |
| #3136 | 2 |
| #2896 | 2 |
| C2×#418 | 2 |
| #2873 | 2 |