Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
401 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 98 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 | #1864 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ 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 |
|---|---|---|
| 2. #12903 | C2 | semidirect |
| 5. #9631 | C2 | semidirect |
| 6. #7278 | C2 | semidirect |
| 7. #7168 | C2 | semidirect |
| 24. #1331 | C4 | semidirect |
| 60. C4×C16 | C2×C4 | semidirect |
| 63. #335 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22670 | 2 |
| #12903 | 2 |
| #12818 | 2 |
| #11593 | 2 |
| #9631 | 2 |
| #7278 | 2 |
| #7168 | 2 |