Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
539 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 127 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 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#510 | C2 | semidirect |
| 3. C22×Q32 | C2 | semidirect |
| 4. C2×#473 | C2 | semidirect |
| 5. #2195 | C2 | direct (N × H) |
| 6. C2×#362 | C2 | semidirect |
| 7. C2×#359 | C2 | semidirect |
| 8. C2×#350 | C2 | semidirect |
| 24. #510 | C22 | semidirect |
| 28. C2×Q32 | C22 | semidirect |
| 33. #473 | C22 | semidirect |
| 36. #362 | C22 | semidirect |
| 37. #359 | C22 | semidirect |
| 38. #350 | C22 | semidirect |
| 258. C2 | #2195 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#578 | 2 |
| C2×#510 | 2 |
| C22×Q32 | 2 |
| C2×#473 | 2 |
| #2195 | 2 |
| C2×#362 | 2 |
| C2×#359 | 2 |
| C2×#350 | 2 |