Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
443 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 443 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 |
| prime radical | #3060 | C2 |
| 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. #3156 | C2 | semidirect |
| 2. C2×#473 | C2 | semidirect |
| 3. #3063 | C2 | semidirect |
| 5. C2×#444 | C2 | semidirect |
| 6. #2038 | C2 | semidirect |
| 7. #2007 | C2 | semidirect |
| 8. #1971 | C2 | semidirect |
| 9. #1951 | C2 | semidirect |
| 10. #1908 | C2 | semidirect |
| 11. #1765 | C2 | semidirect |
| 93. #108 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3156 | 2 |
| C2×#473 | 2 |
| #3063 | 2 |
| #3060 | 2 |
| C2×#444 | 2 |
| #2038 | 2 |
| #2007 | 2 |
| #1971 | 2 |
| #1951 | 2 |
| #1908 | 2 |
| #1765 | 2 |