Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
343 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 343 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 | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ 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. #12909 | C2 | semidirect |
| 5. #9627 | C2 | semidirect |
| 6. #7277 | C2 | semidirect |
| 7. #7167 | C2 | semidirect |
| 24. #1332 | C4 | semidirect |
| 25. #1332 | C4 | semidirect |
| 26. #1331 | C4 | semidirect |
| 27. #1331 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22688 | 2 |
| #12909 | 2 |
| #12815 | 2 |
| #11591 | 2 |
| #9627 | 2 |
| #7277 | 2 |
| #7167 | 2 |