Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
239 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 118 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 | C4×C4⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1770 |
| prime radical | C23 | #436 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ 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 |
|---|---|---|
| 15. #1370 | C4 | semidirect |
| 16. #1370 | C4 | semidirect |
| 17. #1312 | C4 | semidirect |
| 18. #1312 | C4 | semidirect |
| 136. C8 | #347 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12666 | 2 |
| #12004 | 2 |
| #11402 | 2 |
| #8283 | 2 |
| #7168 | 2 |