Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
488 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10054 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28504 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ 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 |
|---|---|---|
| 1. #33637 | C2 | semidirect |
| 3. #19652 | C2 | semidirect |
| 4. #13271 | C2 | semidirect |
| 5. C2×#1357 | C2 | semidirect |
| 6. #11602 | C2 | semidirect |
| 7. #7180 | C2 | semidirect |
| 44. #498 | D4 | semidirect |
| 79. #364 | D4 | semidirect |
| 83. C82 | D4 | semidirect |
| 93. C2×Q16 | SD16 | semidirect |
| 110. C2×D8 | SD16 | semidirect |
| 144. #108 | SD16 | semidirect |
| 150. #106 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33637 | 2 |
| #28504 | 2 |
| #19652 | 2 |
| #13271 | 2 |
| C2×#1357 | 2 |
| #11602 | 2 |
| #7180 | 2 |