Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
553 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 553 conjugacy classes of subgroups — too many to draw. Only its 146 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 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#459 |
| 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. #33660 | C2 | semidirect |
| 2. #19603 | C2 | semidirect |
| 3. C2×C8×D8 | C2 | semidirect |
| 4. #12445 | C2 | semidirect |
| 5. #12424 | C2 | semidirect |
| 6. #12416 | C2 | semidirect |
| 7. #12012 | C2 | semidirect |
| 8. #11947 | C2 | semidirect |
| 20. #1713 | C22 | semidirect |
| 21. #1712 | C22 | semidirect |
| 24. #1706 | C22 | semidirect |
| 32. #1370 | C22 | semidirect |
| 33. #1367 | C22 | semidirect |
| 34. #1364 | C22 | semidirect |
| 35. #1361 | C22 | semidirect |
| 188. D8 | #135 | semidirect |
| 199. C2×C8 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33660 | 2 |
| #19603 | 2 |
| C2×C8×D8 | 2 |
| #12445 | 2 |
| #12424 | 2 |
| #12416 | 2 |
| #12012 | 2 |
| #11947 | 2 |