Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
509 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 509 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 | C2×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#470 | C2 | semidirect |
| 3. C2×#468 | C2 | semidirect |
| 4. #2850 | C2 | semidirect |
| 5. #2097 | C2 | semidirect |
| 6. #2092 | C2 | semidirect |
| 7. #2015 | C2 | semidirect |
| 8. #2012 | C2 | semidirect |
| 9. #1728 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#470 | 2 |
| C2×#469 | 2 |
| C2×#468 | 2 |
| #2850 | 2 |
| #2097 | 2 |
| #2092 | 2 |
| #2015 | 2 |
| #2012 | 2 |
| #1728 | 2 |