Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
371 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C2×D8 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ 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 |
|---|---|---|
| 2. C2×C4×Q16 | C2 | semidirect |
| 4. C4×SD32 | C2 | direct (N × H) |
| 5. C2×#367 | C2 | semidirect |
| 6. C2×#358 | C2 | semidirect |
| 8. C2×C4×C16 | C2 | semidirect |
| 13. C2×SD32 | C4 | direct (N × H) |
| 18. C4×Q16 | C22 | semidirect |
| 26. #367 | C22 | semidirect |
| 27. #358 | C22 | semidirect |
| 29. C4×C16 | C22 | semidirect |
| 35. C2×Q16 | C2×C4 | semidirect |
| 53. SD32 | C2×C4 | direct (N × H) |
| 55. C2×C16 | C2×C4 | semidirect |
| 85. Q16 | C22×C4 | semidirect |
| 105. C16 | C22×C4 | semidirect |
| 125. C2×C4 | SD32 | direct (N × H) |
| 149. C4 | C2×SD32 | direct (N × H) |
| 153. C2 | C4×SD32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD32 | 2 |
| C2×C4×Q16 | 2 |
| C2×C4×D8 | 2 |
| C4×SD32 | 2 |
| C2×#367 | 2 |
| C2×#358 | 2 |
| C2×#357 | 2 |
| C2×C4×C16 | 2 |