Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
719 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 719 conjugacy classes of subgroups — too many to draw. Only its 271 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#428 | C2 | semidirect |
| 4. #1809 | C2 | semidirect |
| 5. #1808 | C2 | semidirect |
| 6. #1807 | C2 | semidirect |
| 7. #1735 | C2 | semidirect |
| 8. C2×#340 | C2 | semidirect |
| 11. C2×#131 | C4 | semidirect |
| 44. C2×M16 | C2×C4 | semidirect |
| 49. C22×C8 | C2×C4 | semidirect |
| 98. M16 | C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#131 | 2 |
| C2×#428 | 2 |
| #2311 | 2 |
| #1809 | 2 |
| #1808 | 2 |
| #1807 | 2 |
| #1735 | 2 |
| C2×#340 | 2 |