Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
669 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 669 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#384 | C2 | semidirect |
| 4. #1445 | C2 | semidirect |
| 5. #1442 | C2 | semidirect |
| 6. #1441 | C2 | semidirect |
| 9. #1434 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#397 | 2 |
| C2×#388 | 2 |
| C2×#384 | 2 |
| #1445 | 2 |
| #1442 | 2 |
| #1441 | 2 |
| #1439 | 2 |
| #1434 | 2 |