Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
336 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 336 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 | C22 | #1802 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ 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 |
|---|---|---|
| 3. #12004 | C2 | semidirect |
| 4. #8814 | C2 | semidirect |
| 5. #7278 | C2 | semidirect |
| 13. C8×Q16 | C4 | semidirect |
| 15. C8×D8 | C4 | semidirect |
| 125. Q16 | #107 | semidirect |
| 129. D8 | #107 | semidirect |
| 201. C8 | #360 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#135 | 2 |
| #12816 | 2 |
| #12004 | 2 |
| #8814 | 2 |
| #7278 | 2 |