Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
591 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#431 | C2 | semidirect |
| 3. C2×#376 | C2 | semidirect |
| 4. #1500 | C2 | direct (N × H) |
| 5. C2×#328 | C2 | semidirect |
| 9. #529 | C4 | semidirect |
| 12. C2×#125 | C4 | semidirect |
| 13. C2×#124 | C4 | semidirect |
| 17. C2×C4×Q8 | C4 | semidirect |
| 19. C2×C4×D4 | C4 | semidirect |
| 24. #431 | C22 | semidirect |
| 29. #376 | C22 | semidirect |
| 32. #328 | C22 | semidirect |
| 46. #125 | C2×C4 | semidirect |
| 48. #124 | C2×C4 | semidirect |
| 60. C4×Q8 | C2×C4 | semidirect |
| 64. C4×D4 | C2×C4 | semidirect |
| 241. C2 | #1500 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#529 | 2 |
| C2×#431 | 2 |
| C2×#376 | 2 |
| #1500 | 2 |
| C2×#328 | 2 |
| C2×#327 | 2 |