Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3230 | C2 | semidirect |
| 5. #1700 | C2 | semidirect |
| 6. #1694 | C2 | semidirect |
| 7. #1679 | C2 | semidirect |
| 8. #1673 | C2 | semidirect |
| 9. #1574 | C2 | semidirect |
| 10. #1561 | C2 | semidirect |
| 11. #1549 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3230 | 2 |
| #3107 | 2 |
| #3039 | 2 |
| #2990 | 2 |
| #1700 | 2 |
| #1694 | 2 |
| #1679 | 2 |
| #1673 | 2 |
| #1574 | 2 |
| #1561 | 2 |
| #1549 | 2 |