Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
373 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 127 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 | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2512 | C2 | semidirect |
| 2. #2071 | C2 | semidirect |
| 3. #2067 | C2 | semidirect |
| 7. #1942 | C2 | semidirect |
| 8. #1925 | C2 | semidirect |
| 9. #1845 | C2 | semidirect |
| 10. C4×#107 | C2 | semidirect |
| 11. C8×C4⋊C4 | C2 | semidirect |
| 121. C4⋊C4 | D8 | semidirect |
| 183. C8 | #126 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2512 | 2 |
| #2071 | 2 |
| #2067 | 2 |
| #1970 | 2 |
| #1942 | 2 |
| #1925 | 2 |
| #1845 | 2 |
| C4×#107 | 2 |
| C8×C4⋊C4 | 2 |