Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
654 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 139 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 | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q16 | C2 | semidirect |
| 2. C2×#473 | C2 | semidirect |
| 5. #2047 | C2 | semidirect |
| 6. #2020 | C2 | semidirect |
| 7. #1979 | C2 | semidirect |
| 8. #1810 | C2 | semidirect |
| 9. #1784 | C2 | semidirect |
| 54. C2×Q16 | D4 | semidirect |
| 181. C22 | #473 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q16 | 2 |
| C2×#473 | 2 |
| #3004 | 2 |
| #2929 | 2 |
| #2047 | 2 |
| #2020 | 2 |
| #1979 | 2 |
| #1810 | 2 |
| #1784 | 2 |