Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
353 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 353 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, socle | C2 | C2×#459 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ 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. C2×#509 | C2 | semidirect |
| 3. #3155 | C2 | semidirect |
| 4. #2957 | C2 | semidirect |
| 5. #2213 | C2 | semidirect |
| 6. #2212 | C2 | semidirect |
| 7. #2153 | C2 | semidirect |
| 8. #2148 | C2 | semidirect |
| 9. C2×#361 | C2 | semidirect |
| 17. #510 | C22 | semidirect |
| 21. #509 | C22 | semidirect |
| 28. #495 | C22 | semidirect |
| 29. #493 | C22 | semidirect |
| 31. #443 | C22 | semidirect |
| 35. #364 | C22 | semidirect |
| 36. #362 | C22 | semidirect |
| 37. #361 | C22 | semidirect |
| 38. #361 | C22 | semidirect |
| 50. #135 | D4 | semidirect |
| 52. C2×Q16 | D4 | semidirect |
| 63. 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 |
| C2×#509 | 2 |
| #3155 | 2 |
| #2957 | 2 |
| #2213 | 2 |
| #2212 | 2 |
| #2153 | 2 |
| #2148 | 2 |
| C2×#361 | 2 |