Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
628 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 628 conjugacy classes of subgroups — too many to draw. Only its 138 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 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2234 | C2 | semidirect |
| 2. #24778 | C2 | semidirect |
| 3. #13476 | C2 | semidirect |
| 4. #13380 | C2 | semidirect |
| 5. #13304 | C2 | semidirect |
| 6. #13235 | C2 | semidirect |
| 7. #13189 | C2 | semidirect |
| 8. #13165 | C2 | semidirect |
| 9. #12921 | C2 | semidirect |
| 76. #366 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2234 | 2 |
| #24778 | 2 |
| #13476 | 2 |
| #13380 | 2 |
| #13304 | 2 |
| #13235 | 2 |
| #13189 | 2 |
| #13165 | 2 |
| #12921 | 2 |