Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
163 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 63 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 | #357 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M64 | C2 | semidirect |
| 3. C2×#368 | C2 | semidirect |
| 4. #1419 | C2 | direct (N × H) |
| 15. M64 | C22 | semidirect |
| 18. #368 | C22 | semidirect |
| 101. C2 | #1419 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D16 | 2 |
| C2×M64 | 2 |
| C2×#368 | 2 |
| #1419 | 2 |