Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1960 conjugacy classes of subgroups; only the 1022 normal subgroups are shown.
This group has 1960 conjugacy classes of subgroups — too many to draw. Only its 1022 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C2×H(3,𝔽4) | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23346 | C2 | semidirect |
| 3. #23306 | C2 | semidirect |
| 4. C8×#124 | C2 | semidirect |
| 5. C8×#120 | C2 | semidirect |
| 6. C24×C8 | C22 | semidirect |
| 8. C2×C8×Q8 | C22 | semidirect |
| 14. #1984 | C22 | semidirect |
| 16. #1923 | C22 | semidirect |
| 18. #1850 | C22 | semidirect |
| 19. #1831 | C22 | semidirect |
| 20. #1779 | C22 | semidirect |
| 21. C8×C22⋊C4 | C22 | semidirect |
| 22. C42×C8 | C22 | semidirect |
| 27. H(3,𝔽4) | C8 | direct (N × H) |
| 63. #124 | C2×C8 | semidirect |
| 64. #120 | C2×C8 | semidirect |
| 86. C24 | C22×C8 | semidirect |
| 90. C2×Q8 | C22×C8 | semidirect |
| 113. C22⋊C4 | C22×C8 | semidirect |
| 117. C42 | C22×C8 | semidirect |
| 140. C8 | H(3,𝔽4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×H(3,𝔽4) | 2 |
| #23346 | 2 |
| #23306 | 2 |
| C8×#124 | 2 |
| C8×#120 | 2 |