Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
319 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12013 | C2 | semidirect |
| 3. #11745 | C2 | semidirect |
| 5. #11655 | C2 | semidirect |
| 7. #8283 | C2 | semidirect |
| 17. #1710 | C4 | semidirect |
| 18. #1709 | C4 | semidirect |
| 20. #1707 | C4 | semidirect |
| 21. #1706 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19600 | 2 |
| #12013 | 2 |
| #11745 | 2 |
| #11744 | 2 |
| #11655 | 2 |
| #11654 | 2 |
| #8283 | 2 |