Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
636 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 182 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 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#493 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ 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 |
|---|---|---|
| 1. #33733 | C2 | semidirect |
| 2. #19603 | C2 | semidirect |
| 3. #12521 | C2 | semidirect |
| 4. #12515 | C2 | semidirect |
| 5. #12509 | C2 | semidirect |
| 6. #12503 | C2 | semidirect |
| 7. C2×C8×C16 | C2 | semidirect |
| 108. C2×C8 | D16 | semidirect |
| 124. C16 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33733 | 2 |
| #19603 | 2 |
| #12521 | 2 |
| #12515 | 2 |
| #12509 | 2 |
| #12503 | 2 |
| C2×C8×C16 | 2 |