Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
421 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 112 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 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ 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. #13282 | C2 | semidirect |
| 4. #13276 | C2 | semidirect |
| 6. #7274 | C2 | semidirect |
| 7. #7171 | C2 | semidirect |
| 21. #1675 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22658 | 2 |
| C2×#1675 | 2 |
| #13282 | 2 |
| #13276 | 2 |
| C2×D4⋊C16 | 2 |
| #7274 | 2 |
| #7171 | 2 |