Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
607 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 607 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #379 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1742 | C2 | semidirect |
| 5. #1729 | C2 | semidirect |
| 18. #413 | C4 | semidirect |
| 22. #408 | C4 | semidirect |
| 47. C2×M16 | C2×C4 | semidirect |
| 48. C2×M16 | C2×C4 | semidirect |
| 55. C22×C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3428 | 2 |
| C2×#413 | 2 |
| C2×#408 | 2 |
| #1742 | 2 |
| #1729 | 2 |