Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
138 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12067 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | #346 | C23 |
| prime radical | #2263 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1313 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12068 | C2 | semidirect |
| 3. #12067 | C2 | semidirect |
| 4. #7457 | C2 | semidirect |
| 5. #7390 | C2 | semidirect |
| 6. #7389 | C2 | semidirect |
| 7. #7333 | C2 | semidirect |
| 29. SD64 | C8 | semidirect |
| 30. D32 | C8 | semidirect |
| 50. Q32 | C2×C8 | semidirect |
| 54. D16 | C2×C8 | semidirect |
| 71. C32 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13658 | 2 |
| #12068 | 2 |
| #12067 | 2 |
| #7457 | 2 |
| #7390 | 2 |
| #7389 | 2 |
| #7333 | 2 |