Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
475 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 98 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 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12920 | C2 | semidirect |
| 5. #11256 | C2 | semidirect |
| 6. #7258 | C2 | semidirect |
| 7. #7183 | C2 | semidirect |
| 27. #1347 | C4 | semidirect |
| 66. C22⋊C16 | C2×C4 | semidirect |
| 72. #340 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22181 | 2 |
| #12991 | 2 |
| #12920 | 2 |
| C2×#1347 | 2 |
| #11256 | 2 |
| #7258 | 2 |
| #7183 | 2 |