Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
250 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | #1331 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1397 | C2 | semidirect |
| 3. C2×#1373 | C2 | semidirect |
| 4. #7388 | C2 | direct (N × H) |
| 17. #1397 | C22 | semidirect |
| 19. #1373 | C22 | semidirect |
| 188. C2 | #7388 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2178 | 2 |
| C2×#1397 | 2 |
| C2×#1373 | 2 |
| #7388 | 2 |