Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
248 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 63 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 | #327 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#349 | C2 | semidirect |
| 3. C2×H(3,ℤ/4) | C2 | semidirect |
| 4. #1339 | C2 | direct (N × H) |
| 5. C22×D8 | C4 | semidirect |
| 12. #469 | C4 | semidirect |
| 14. C2×#106 | C4 | semidirect |
| 17. #349 | C22 | semidirect |
| 19. H(3,ℤ/4) | C22 | semidirect |
| 26. C2×D8 | C2×C4 | semidirect |
| 52. #106 | C2×C4 | semidirect |
| 174. C2 | #1339 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#469 | 2 |
| C2×#349 | 2 |
| C2×H(3,ℤ/4) | 2 |
| #1339 | 2 |