Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1347 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 1347 conjugacy classes of subgroups — too many to draw. Only its 991 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 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #2913 | C2 | semidirect |
| 5. #2912 | C2 | semidirect |
| 6. #2911 | C2 | semidirect |
| 7. C2×#428 | C2 | semidirect |
| 9. #568 | C4 | semidirect |
| 26. #428 | C22 | semidirect |
| 27. #428 | C22 | semidirect |
| 28. #428 | C22 | semidirect |
| 29. #427 | C22 | semidirect |
| 30. C2×#107 | C22 | semidirect |
| 31. C2×#106 | C22 | semidirect |
| 40. #131 | C2×C4 | semidirect |
| 41. C2×M16 | C2×C4 | semidirect |
| 93. M16 | C22×C4 | semidirect |
| 94. C2×C8 | C22×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#568 | 2 |
| #3428 | 2 |
| C2×#520 | 2 |
| #2913 | 2 |
| #2912 | 2 |
| #2911 | 2 |
| C2×#428 | 2 |