Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
459 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 459 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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#464 | C2 | semidirect |
| 3. #2295 | C2 | semidirect |
| 4. #2043 | C2 | semidirect |
| 5. #2042 | C2 | semidirect |
| 7. #2016 | C2 | semidirect |
| 8. #2015 | C2 | semidirect |
| 10. #1983 | C2 | semidirect |
| 11. #1831 | C2 | semidirect |
| 64. C4×Q8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#464 | 2 |
| #2858 | 2 |
| #2295 | 2 |
| #2043 | 2 |
| #2042 | 2 |
| #2041 | 2 |
| #2016 | 2 |
| #2015 | 2 |
| #2014 | 2 |
| #1983 | 2 |
| #1831 | 2 |