Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1543 conjugacy classes of subgroups; only the 943 normal subgroups are shown.
This group has 1543 conjugacy classes of subgroups — too many to draw. Only its 943 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C22 |
| commutator | C2 | C22×C4×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C25 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#336 | C2 | direct (N × H) |
| 8. C23×C8 | C4 | semidirect |
| 13. #336 | C22 | direct (N × H) |
| 17. C23×C4 | C8 | semidirect |
| 28. C22×C8 | C2×C4 | semidirect |
| 48. C22×C4 | C2×C8 | semidirect |
| 59. C2×C8 | C22×C4 | semidirect |
| 82. C2×C4 | C22×C8 | semidirect |
| 88. C22 | #336 | direct (N × H) |
| 103. C2 | C2×#336 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8 | 2 |
| C23×C42 | 2 |
| C2×#336 | 2 |