Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
489 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 135 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×#500 | C2 | semidirect |
| 3. C2×#475 | C2 | semidirect |
| 5. #1650 | C2 | direct (N × H) |
| 6. C2×#332 | C2 | semidirect |
| 7. C2×Q8⋊C8 | C2 | semidirect |
| 24. #500 | C22 | semidirect |
| 26. #475 | C22 | semidirect |
| 36. #332 | C22 | semidirect |
| 37. Q8⋊C8 | C22 | semidirect |
| 227. C2 | #1650 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#556 | 2 |
| C2×#500 | 2 |
| C2×#475 | 2 |
| C2×#461 | 2 |
| #1650 | 2 |
| C2×#332 | 2 |
| C2×Q8⋊C8 | 2 |
| C2×D4⋊C8 | 2 |