Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
818 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 818 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, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33242 | C2 | semidirect |
| 4. C2×#1683 | C2 | semidirect |
| 5. #10278 | C2 | semidirect |
| 6. #10274 | C2 | semidirect |
| 7. #9701 | C2 | semidirect |
| 8. #8274 | C2 | semidirect |
| 75. #335 | D4 | semidirect |
| 98. #127 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33242 | 2 |
| #28521 | 2 |
| C8×#127 | 2 |
| C2×#1683 | 2 |
| #10278 | 2 |
| #10274 | 2 |
| #9701 | 2 |
| #8274 | 2 |