Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
806 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#1416 | C2 | direct (N × H) |
| 8. C22×Q32 | C4 | semidirect |
| 16. #1416 | C22 | direct (N × H) |
| 27. C2×Q32 | C2×C4 | semidirect |
| 59. Q32 | C22×C4 | semidirect |
| 114. C22 | #1416 | direct (N × H) |
| 124. C2 | C2×#1416 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q32 | 2 |
| C23×C32 | 2 |
| C22×#366 | 2 |
| C2×#1416 | 2 |