Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
231 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #102 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C22×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#346 | C2 | semidirect |
| 3. C2×#334 | C2 | semidirect |
| 4. #1333 | C2 | direct (N × H) |
| 17. #346 | C22 | semidirect |
| 19. #334 | C22 | semidirect |
| 24. C2×D8 | C8 | semidirect |
| 62. D8 | C2×C8 | semidirect |
| 126. C2 | #1333 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| C2×#346 | 2 |
| C2×#334 | 2 |
| #1333 | 2 |