Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
309 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3170 | C2 | semidirect |
| 4. #1684 | C2 | semidirect |
| 5. #1683 | C2 | semidirect |
| 6. #1676 | C2 | semidirect |
| 7. #1675 | C2 | semidirect |
| 8. #1670 | C2 | semidirect |
| 9. #1669 | C2 | semidirect |
| 10. C2×#335 | C2 | semidirect |
| 11. #1542 | C2 | semidirect |
| 103. #41 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3170 | 2 |
| #3076 | 2 |
| C8×#41 | 2 |
| #1684 | 2 |
| #1683 | 2 |
| #1676 | 2 |
| #1675 | 2 |
| #1670 | 2 |
| #1669 | 2 |
| C2×#335 | 2 |
| #1542 | 2 |