Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
333 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2965 | C2 | semidirect |
| 3. #1594 | C2 | semidirect |
| 4. #1593 | C2 | semidirect |
| 5. #1592 | C2 | semidirect |
| 6. #1564 | C2 | semidirect |
| 7. #1476 | C2 | semidirect |
| 8. C2×#322 | C2 | semidirect |
| 95. C2×C8 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2965 | 2 |
| C4×#135 | 2 |
| #1594 | 2 |
| #1593 | 2 |
| #1592 | 2 |
| #1564 | 2 |
| #1476 | 2 |
| C2×#322 | 2 |