Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
469 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 123 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 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3155 | C2 | semidirect |
| 2. C2×#472 | C2 | semidirect |
| 3. #3062 | C2 | semidirect |
| 4. #3061 | C2 | semidirect |
| 5. C2×#443 | C2 | semidirect |
| 6. #2037 | C2 | semidirect |
| 7. #2006 | C2 | semidirect |
| 8. #1970 | C2 | semidirect |
| 9. #1952 | C2 | semidirect |
| 10. #1907 | C2 | semidirect |
| 11. #1764 | C2 | semidirect |
| 94. #108 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3155 | 2 |
| C2×#472 | 2 |
| #3062 | 2 |
| #3061 | 2 |
| C2×#443 | 2 |
| #2037 | 2 |
| #2006 | 2 |
| #1970 | 2 |
| #1952 | 2 |
| #1907 | 2 |
| #1764 | 2 |