Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1727 conjugacy classes of subgroups.
The subgroup lattice has 421 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #23172 | C2 | semidirect |
| 4. #23171 | C2 | semidirect |
| 5. #23170 | C2 | semidirect |
| 13. #3166 | C4 | semidirect |
| 14. #3165 | C4 | semidirect |
| 15. #3165 | C4 | semidirect |
| 77. #501 | C2×C4 | semidirect |
| 78. #501 | C2×C4 | semidirect |
| 79. #500 | C2×C4 | semidirect |
| 80. #500 | C2×C4 | semidirect |
| 81. #499 | C2×C4 | semidirect |
| 82. #499 | C2×C4 | semidirect |
| 83. #498 | C2×C4 | semidirect |
| 91. #430 | Q8 | semidirect |
| 222. #108 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60663 | 2 |
| #60192 | 2 |
| #23172 | 2 |
| #23171 | 2 |
| #23170 | 2 |
| C2×#1986 | 2 |
| #21165 | 2 |
| #21042 | 2 |
| C4×#430 | 2 |
| #20445 | 2 |
| C4⋊C4×M16 | 2 |