Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
597 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1503 | C2 | direct (N × H) |
| 8. #533 | C4 | semidirect |
| 10. C2×C4⋊Q8 | C4 | semidirect |
| 11. C2×C4⋊Q8 | C4 | semidirect |
| 16. C2×C4×Q8 | C4 | semidirect |
| 39. C4⋊Q8 | C2×C4 | semidirect |
| 40. C4⋊Q8 | C2×C4 | semidirect |
| 52. C4×Q8 | C2×C4 | semidirect |
| 203. C2 | #1503 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| C2×#431 | 2 |
| C2×C4×C4⋊C4 | 2 |
| #1503 | 2 |
| C2×#328 | 2 |