Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
759 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 759 conjugacy classes of subgroups — too many to draw. Only its 343 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #2873 | C2 | semidirect |
| 4. C4×#101 | C2 | semidirect |
| 5. C4×#100 | C2 | semidirect |
| 6. C2×#343 | C2 | semidirect |
| 17. #413 | C4 | semidirect |
| 35. C2×M16 | C2×C4 | semidirect |
| 54. #101 | C2×C4 | semidirect |
| 55. #100 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#520 | 2 |
| C2×#413 | 2 |
| #2873 | 2 |
| C4×#101 | 2 |
| C4×#100 | 2 |
| C2×#343 | 2 |