Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
231 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 65 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 | #390 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1726 | C2 | semidirect |
| 3. #1568 | C2 | semidirect |
| 4. #1530 | C2 | semidirect |
| 5. #1505 | C2 | semidirect |
| 7. #500 | C4 | semidirect |
| 8. #493 | C4 | semidirect |
| 9. #486 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3169 | 2 |
| #1726 | 2 |
| #1568 | 2 |
| #1530 | 2 |
| #1505 | 2 |