Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
978 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 978 conjugacy classes of subgroups — too many to draw. Only its 370 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 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 19. #1747 | C4 | semidirect |
| 99. #97 | C4⋊C4 | semidirect |
| 101. C4×C8 | C4⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20444 | 2 |
| #20436 | 2 |
| C2×#1747 | 2 |
| #11338 | 2 |
| #11337 | 2 |
| #11336 | 2 |
| #11335 | 2 |