Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
389 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23 | #442 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #12088 | C2 | semidirect |
| 5. #12079 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33673 | 2 |
| #17972 | 2 |
| #17897 | 2 |
| #12088 | 2 |
| #12079 | 2 |
| #12031 | 2 |
| #11668 | 2 |
| #11653 | 2 |
| #11416 | 2 |