Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
577 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 243 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 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1986 | C2 | semidirect |
| 11. #533 | C4 | semidirect |
| 13. C2×C4⋊Q8 | C4 | semidirect |
| 14. C2×#125 | C4 | semidirect |
| 34. C2×M16 | Q8 | semidirect |
| 101. M16 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| C2×C4×M16 | 2 |
| #2285 | 2 |
| #1986 | 2 |
| #1733 | 2 |