Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1033 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 1033 conjugacy classes of subgroups — too many to draw. Only its 247 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#518 | C2 | semidirect |
| 3. C2×#418 | C2 | semidirect |
| 4. C2×#418 | C2 | semidirect |
| 5. #2309 | C2 | semidirect |
| 6. #1879 | C2 | semidirect |
| 7. #1735 | C2 | semidirect |
| 8. #1729 | C2 | semidirect |
| 13. C22×D8 | C4 | semidirect |
| 20. C2×C4×D4 | C22 | semidirect |
| 27. C2×#102 | C22 | semidirect |
| 28. C2×#102 | C22 | semidirect |
| 30. #390 | C22 | semidirect |
| 32. #340 | C22 | semidirect |
| 33. #336 | C22 | semidirect |
| 47. C22×D4 | C2×C4 | semidirect |
| 57. C2×D8 | C2×C4 | semidirect |
| 63. C22×C8 | C2×C4 | semidirect |
| 121. C2×D4 | C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C2×#518 | 2 |
| C2×#418 | 2 |
| #2309 | 2 |
| #1879 | 2 |
| #1735 | 2 |
| #1729 | 2 |