Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2663 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 2663 conjugacy classes of subgroups — too many to draw. Only its 647 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q8 | C2 | semidirect |
| 2. C23×SD16 | C2 | semidirect |
| 4. C2×#448 | C2 | direct (N × H) |
| 5. C22×#103 | C2 | semidirect |
| 6. C22×C22⋊C8 | C2 | semidirect |
| 11. C23×Q8 | C22 | semidirect |
| 17. C22×SD16 | C22 | semidirect |
| 23. #448 | C22 | direct (N × H) |
| 24. C2×#103 | C22 | semidirect |
| 25. C2×C22⋊C8 | C22 | semidirect |
| 35. C22×Q8 | D4 | semidirect |
| 37. C22×Q8 | C23 | semidirect |
| 53. C2×SD16 | C23 | semidirect |
| 61. #103 | C23 | semidirect |
| 62. C22⋊C8 | C23 | semidirect |
| 69. C24 | SD16 | semidirect |
| 79. C2×Q8 | C2×D4 | semidirect |
| 118. C23 | C2×SD16 | semidirect |
| 125. Q8 | C22×D4 | semidirect |
| 150. C22 | #448 | direct (N × H) |
| 153. C22 | C22×SD16 | semidirect |
| 163. C2 | C2×#448 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8 | 2 |
| C23×SD16 | 2 |
| C22×#121 | 2 |
| C2×#448 | 2 |
| C22×#103 | 2 |
| C22×C22⋊C8 | 2 |