Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
343 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 105 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 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3214 | C2 | semidirect |
| 2. #3205 | C2 | semidirect |
| 3. #3205 | C2 | semidirect |
| 4. #2981 | C2 | semidirect |
| 5. #1701 | C2 | semidirect |
| 6. #1687 | C2 | semidirect |
| 7. #1681 | C2 | semidirect |
| 8. #1667 | C2 | semidirect |
| 9. #1572 | C2 | semidirect |
| 10. #1555 | C2 | semidirect |
| 11. #1553 | C2 | semidirect |
| 23. #461 | C22 | semidirect |
| 27. #459 | C22 | semidirect |
| 31. #436 | C22 | semidirect |
| 33. C8×D4 | C22 | semidirect |
| 39. #335 | C22 | semidirect |
| 40. #334 | C22 | semidirect |
| 41. #332 | C22 | semidirect |
| 44. #331 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3214 | 2 |
| #3205 | 2 |
| #2981 | 2 |
| #1701 | 2 |
| #1687 | 2 |
| #1681 | 2 |
| #1667 | 2 |
| #1572 | 2 |
| #1555 | 2 |
| #1553 | 2 |