Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
180 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2023 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3163 | C2 | semidirect |
| 5. #1386 | C2 | semidirect |
| 6. #1384 | C2 | semidirect |
| 7. #1378 | C2 | semidirect |
| 30. C2×Q16 | Q8 | semidirect |
| 32. C2×D8 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3163 | 2 |
| #2145 | 2 |
| C2×#360 | 2 |
| #1976 | 2 |
| #1386 | 2 |
| #1384 | 2 |
| #1378 | 2 |