Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1383 conjugacy classes of subgroups; only the 999 normal subgroups are shown.
This group has 1383 conjugacy classes of subgroups — too many to draw. Only its 999 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 | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #2913 | C2 | direct (N × H) |
| 4. C2×#428 | C2 | semidirect |
| 5. C2×#427 | C2 | semidirect |
| 6. C22×#107 | C2 | semidirect |
| 8. C2×#131 | C4 | semidirect |
| 18. #428 | C22 | semidirect |
| 19. #427 | C22 | semidirect |
| 23. C2×#107 | C22 | semidirect |
| 33. #131 | C2×C4 | semidirect |
| 34. C2×M16 | C2×C4 | semidirect |
| 36. C22×C8 | C2×C4 | semidirect |
| 73. M16 | C22×C4 | semidirect |
| 76. C2×C8 | C22×C4 | semidirect |
| 124. C2 | #2913 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#131 | 2 |
| C2×#520 | 2 |
| #2913 | 2 |
| C2×#428 | 2 |
| C2×#427 | 2 |
| C22×#107 | 2 |