Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
355 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 151 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 | #442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#444 | C2 | semidirect |
| 3. C2×#442 | C2 | semidirect |
| 4. AGL(1,ℤ/16) | C2 | direct (N × H) |
| 5. C2×#365 | C2 | semidirect |
| 6. C2×#361 | C2 | semidirect |
| 7. C2×#360 | C2 | semidirect |
| 8. C2×#347 | C2 | semidirect |
| 13. #509 | C4 | semidirect |
| 15. C2×SD32 | C4 | semidirect |
| 16. C2×D16 | C4 | semidirect |
| 20. #444 | C22 | semidirect |
| 22. #442 | C22 | semidirect |
| 28. #365 | C22 | semidirect |
| 29. #361 | C22 | semidirect |
| 30. #360 | C22 | semidirect |
| 31. #347 | C22 | semidirect |
| 37. #135 | C2×C4 | semidirect |
| 39. C2×Q16 | C2×C4 | semidirect |
| 46. C2×D8 | C2×C4 | semidirect |
| 60. SD32 | C2×C4 | semidirect |
| 61. D16 | C2×C4 | semidirect |
| 63. M32 | C2×C4 | semidirect |
| 67. C2×C16 | C2×C4 | semidirect |
| 97. Q16 | C22×C4 | semidirect |
| 105. D8 | C22×C4 | semidirect |
| 126. C16 | C22×C4 | semidirect |
| 186. C2 | AGL(1,ℤ/16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#509 | 2 |
| C2×#444 | 2 |
| C2×#442 | 2 |
| AGL(1,ℤ/16) | 2 |
| C2×#365 | 2 |
| C2×#361 | 2 |
| C2×#360 | 2 |
| C2×#347 | 2 |