Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
650 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 178 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 | #2541 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #550 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 114. #108 | Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28699 | 2 |
| #19633 | 2 |
| C2×#1713 | 2 |
| #19208 | 2 |
| #17958 | 2 |
| #10155 | 2 |
| #9801 | 2 |
| C8×#108 | 2 |