Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
542 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 170 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 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1713 | C2 | semidirect |
| 4. #12447 | C2 | direct (N × H) |
| 5. C2×#1370 | C2 | semidirect |
| 6. C2×#1365 | C2 | semidirect |
| 18. #1713 | C22 | semidirect |
| 24. #1370 | C22 | semidirect |
| 25. #1365 | C22 | semidirect |
| 73. C2×D8 | Q16 | semidirect |
| 139. D8 | C2×Q16 | semidirect |
| 225. C2 | #12447 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2226 | 2 |
| C2×#1713 | 2 |
| C2×C8×D8 | 2 |
| #12447 | 2 |
| C2×#1370 | 2 |
| C2×#1365 | 2 |