Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
501 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 501 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 | #2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#468 | C2 | semidirect |
| 2. C2×#465 | C2 | semidirect |
| 3. #2621 | C2 | semidirect |
| 5. #2097 | C2 | semidirect |
| 6. #2092 | C2 | semidirect |
| 7. #2014 | C2 | semidirect |
| 8. #2013 | C2 | semidirect |
| 9. #1729 | C2 | semidirect |
| 50. C2×M16 | D4 | semidirect |
| 53. C22×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#468 | 2 |
| C2×#465 | 2 |
| #2621 | 2 |
| #2619 | 2 |
| #2097 | 2 |
| #2092 | 2 |
| #2014 | 2 |
| #2013 | 2 |
| #1729 | 2 |