Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
439 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 94 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 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | #3054 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ 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. C2×#1453 | C2 | semidirect |
| 3. #12791 | C2 | semidirect |
| 4. C2×#1336 | C2 | semidirect |
| 5. C2×#1335 | C2 | semidirect |
| 6. #7182 | C2 | semidirect |
| 7. #7179 | C2 | semidirect |
| 82. #322 | D4 | semidirect |
| 84. C82 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21549 | 2 |
| C2×#1453 | 2 |
| #12791 | 2 |
| C2×#1336 | 2 |
| C2×#1335 | 2 |
| #7182 | 2 |
| #7179 | 2 |