Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2694 conjugacy classes of subgroups; only the 842 normal subgroups are shown.
This group has 2694 conjugacy classes of subgroups — too many to draw. Only its 842 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23357 | C2 | direct (N × H) |
| 3. C2×#1870 | C2 | semidirect |
| 4. C2×#1839 | C2 | semidirect |
| 14. #1870 | C22 | semidirect |
| 15. #1839 | C22 | semidirect |
| 67. C2×C42 | M16 | semidirect |
| 107. C42 | C2×M16 | semidirect |
| 151. C2 | #23357 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#127 | 2 |
| #23357 | 2 |
| C2×#1870 | 2 |
| C2×#1839 | 2 |