Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2583 conjugacy classes of subgroups; only the 635 normal subgroups are shown.
This group has 2583 conjugacy classes of subgroups — too many to draw. Only its 635 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q8 | C2 | semidirect |
| 2. C2×D4×Q8 | C2 | semidirect |
| 3. #2848 | C2 | semidirect |
| 4. #2434 | C2 | semidirect |
| 5. #2409 | C2 | semidirect |
| 6. Q8×C22⋊C4 | C2 | semidirect |
| 11. C23×Q8 | C22 | semidirect |
| 14. C2×C4⋊Q8 | C22 | semidirect |
| 16. C2×C22⋊Q8 | C22 | semidirect |
| 18. C2×C4×Q8 | C22 | semidirect |
| 20. #395 | C22 | semidirect |
| 21. #393 | C22 | semidirect |
| 22. #391 | C22 | semidirect |
| 35. C22×Q8 | D4 | semidirect |
| 48. #120 | Q8 | direct (N × H) |
| 63. C24 | C2×Q8 | semidirect |
| 76. C2×D4 | C2×Q8 | semidirect |
| 94. C22⋊C4 | C2×Q8 | semidirect |
| 105. C23 | C22×Q8 | semidirect |
| 111. Q8 | #120 | direct (N × H) |
| 124. C2×C4 | C22×Q8 | semidirect |
| 133. C22 | D4×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8 | 2 |
| C2×D4×Q8 | 2 |
| #2848 | 2 |
| #2434 | 2 |
| #2409 | 2 |
| Q8×C22⋊C4 | 2 |
| C4×#120 | 2 |