Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
348 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#459 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3414 | C2 | semidirect |
| 2. #3147 | C2 | semidirect |
| 3. C2×#443 | C2 | semidirect |
| 4. #2214 | C2 | semidirect |
| 5. #2213 | C2 | semidirect |
| 6. #2212 | C2 | semidirect |
| 7. C2×#364 | C2 | semidirect |
| 8. #2148 | C2 | semidirect |
| 21. #495 | C22 | semidirect |
| 22. #494 | C22 | semidirect |
| 23. #493 | C22 | semidirect |
| 32. #364 | C22 | semidirect |
| 33. #364 | C22 | semidirect |
| 34. #362 | C22 | semidirect |
| 35. #361 | C22 | semidirect |
| 48. #135 | D4 | semidirect |
| 51. C2×Q16 | D4 | semidirect |
| 57. C2×D8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3414 | 2 |
| #3147 | 2 |
| C2×#443 | 2 |
| #2214 | 2 |
| #2213 | 2 |
| #2212 | 2 |
| C2×#364 | 2 |
| #2148 | 2 |