Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
719 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 719 conjugacy classes of subgroups — too many to draw. Only its 267 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×C4⋊C4 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#430 | C2 | semidirect |
| 4. #1811 | C2 | semidirect |
| 5. #1810 | C2 | semidirect |
| 6. #1738 | C2 | semidirect |
| 16. C2×#108 | C22 | semidirect |
| 21. #341 | C22 | semidirect |
| 146. C22 | #430 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C2×#430 | 2 |
| C2×#407 | 2 |
| #1811 | 2 |
| #1810 | 2 |
| #1738 | 2 |