Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2485 conjugacy classes of subgroups; only the 583 normal subgroups are shown.
This group has 2485 conjugacy classes of subgroups — too many to draw. Only its 583 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q8 | C2 | semidirect |
| 2. C22×#137 | C2 | semidirect |
| 4. #3004 | C2 | direct (N × H) |
| 5. C2×#451 | C2 | semidirect |
| 6. C2×#448 | C2 | semidirect |
| 7. C2×#419 | C2 | semidirect |
| 8. C2×#407 | C2 | semidirect |
| 16. C23×Q8 | C22 | semidirect |
| 20. C2×#137 | C22 | semidirect |
| 21. C22×Q16 | C22 | semidirect |
| 22. C22×SD16 | C22 | semidirect |
| 32. #451 | C22 | semidirect |
| 33. #448 | C22 | semidirect |
| 34. #419 | C22 | semidirect |
| 35. C2×#103 | C22 | semidirect |
| 36. #407 | C22 | semidirect |
| 54. C22×Q8 | D4 | semidirect |
| 76. C2×Q16 | C23 | semidirect |
| 78. C2×SD16 | C23 | semidirect |
| 97. #103 | C23 | semidirect |
| 122. C2×Q8 | C2×D4 | semidirect |
| 179. C23 | #137 | semidirect |
| 229. C22 | C2×#137 | semidirect |
| 241. C2 | #3004 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8 | 2 |
| C22×#137 | 2 |
| C2×#525 | 2 |
| #3004 | 2 |
| C2×#451 | 2 |
| C2×#448 | 2 |
| C2×#419 | 2 |
| C2×#407 | 2 |