Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
311 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 111 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 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3395 | C2 | semidirect |
| 4. #1713 | C2 | semidirect |
| 5. #1694 | C2 | semidirect |
| 6. #1670 | C2 | semidirect |
| 9. #1648 | C2 | semidirect |
| 10. #1591 | C2 | semidirect |
| 112. SD16 | Q16 | semidirect |
| 150. Q8 | C2×Q16 | semidirect |
| 172. C8 | C2×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3395 | 2 |
| #3386 | 2 |
| D4×Q16 | 2 |
| #1713 | 2 |
| #1694 | 2 |
| #1670 | 2 |
| #1669 | 2 |
| #1649 | 2 |
| #1648 | 2 |
| #1591 | 2 |
| C8×SD16 | 2 |