Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
751 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 751 conjugacy classes of subgroups — too many to draw. Only its 163 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 | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#470 | C2 | semidirect |
| 3. C2×#467 | C2 | semidirect |
| 5. #1894 | C2 | semidirect |
| 6. #1892 | C2 | semidirect |
| 7. #1809 | C2 | semidirect |
| 8. #1807 | C2 | semidirect |
| 10. C22×Q16 | C22 | semidirect |
| 21. C2×#107 | C22 | semidirect |
| 22. C2×#106 | C22 | semidirect |
| 23. C2×#103 | C22 | semidirect |
| 24. C2×#103 | C22 | semidirect |
| 30. #340 | C22 | semidirect |
| 31. #340 | C22 | semidirect |
| 186. C22 | #470 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C2×#470 | 2 |
| C2×#467 | 2 |
| #2848 | 2 |
| #1894 | 2 |
| #1892 | 2 |
| #1809 | 2 |
| #1807 | 2 |