Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
722 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 722 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 26. #1713 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22270 | 2 |
| #22248 | 2 |
| #22247 | 2 |
| C2×#1713 | 2 |
| #9835 | 2 |
| #9834 | 2 |
| #9831 | 2 |
| #8817 | 2 |
| #7845 | 2 |