Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
343 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 57. C22⋊C16 | Q8 | semidirect |
| 58. C22⋊C16 | Q8 | semidirect |
| 61. #341 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30300 | 2 |
| #12920 | 2 |
| #12919 | 2 |
| #11270 | 2 |
| #7259 | 2 |
| #7186 | 2 |