Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
494 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#121 | Q8 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10864 | C2 | semidirect |
| 3. #8595 | C2 | semidirect |
| 5. #7045 | C2 | semidirect |
| 153. C2×D4 | #108 | semidirect |
| 222. C4⋊C4 | #108 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#121 | 2 |
| #10864 | 2 |
| #8595 | 2 |
| #7616 | 2 |
| #7045 | 2 |