Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
511 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 143 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 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#487 | C2 | semidirect |
| 4. C2×#462 | C2 | semidirect |
| 5. #1644 | C2 | direct (N × H) |
| 6. C2×#333 | C2 | semidirect |
| 7. C2×Q8⋊C8 | C2 | semidirect |
| 24. #487 | C22 | semidirect |
| 28. #462 | C22 | semidirect |
| 36. #333 | C22 | semidirect |
| 37. Q8⋊C8 | C22 | semidirect |
| 113. C2×D4 | Q16 | semidirect |
| 176. D4 | C2×Q16 | semidirect |
| 227. C2 | #1644 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#554 | 2 |
| C2×#487 | 2 |
| C2×#474 | 2 |
| C2×#462 | 2 |
| #1644 | 2 |
| C2×#333 | 2 |
| C2×Q8⋊C8 | 2 |
| C2×D4⋊C8 | 2 |