Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
761 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 761 conjugacy classes of subgroups — too many to draw. Only its 175 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 | C23×D4 | C22 |
| 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 |
|---|---|---|
| 2. C2×#424 | C2 | semidirect |
| 3. C2×#419 | C2 | semidirect |
| 5. #2310 | C2 | semidirect |
| 7. #2055 | C2 | semidirect |
| 8. #1735 | C2 | semidirect |
| 21. C2×C4⋊C8 | C22 | semidirect |
| 24. C2×#103 | C22 | semidirect |
| 27. #391 | C22 | semidirect |
| 32. #340 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SD16 | 2 |
| C2×#424 | 2 |
| C2×#419 | 2 |
| C2×#418 | 2 |
| #2310 | 2 |
| #2309 | 2 |
| #2055 | 2 |
| #1735 | 2 |