Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
671 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 671 conjugacy classes of subgroups — too many to draw. Only its 211 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×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#428 | C2 | semidirect |
| 5. C4×#117 | C2 | semidirect |
| 6. #1926 | C2 | semidirect |
| 7. #1925 | C2 | semidirect |
| 8. #1736 | C2 | semidirect |
| 9. #1728 | C2 | semidirect |
| 19. C2×#107 | C22 | semidirect |
| 20. C2×#106 | C22 | semidirect |
| 22. #418 | C4 | semidirect |
| 29. C4×C4⋊C4 | C22 | semidirect |
| 30. C4×C4⋊C4 | C22 | semidirect |
| 34. #340 | C22 | semidirect |
| 35. #336 | C22 | semidirect |
| 85. #102 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#428 | 2 |
| C2×#418 | 2 |
| C2×#390 | 2 |
| C4×#117 | 2 |
| #1926 | 2 |
| #1925 | 2 |
| #1736 | 2 |
| #1728 | 2 |