Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1375 conjugacy classes of subgroups; only the 783 normal subgroups are shown.
This group has 1375 conjugacy classes of subgroups — too many to draw. Only its 783 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 | D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C4×D4 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#131 | C2 | semidirect |
| 3. C2×#443 | C2 | direct (N × H) |
| 4. C22×#108 | C2 | semidirect |
| 5. C22×#104 | C2 | semidirect |
| 6. C22×C4×C8 | C2 | semidirect |
| 8. C2×#135 | C4 | semidirect |
| 10. C22×Q16 | C4 | semidirect |
| 11. C22×SD16 | C4 | semidirect |
| 12. C22×D8 | C4 | semidirect |
| 14. C2×#131 | C22 | semidirect |
| 21. #443 | C22 | direct (N × H) |
| 22. C2×#108 | C22 | semidirect |
| 23. C2×#104 | C22 | semidirect |
| 24. C2×C4×C8 | C22 | semidirect |
| 35. #135 | C2×C4 | semidirect |
| 37. C2×Q16 | C2×C4 | semidirect |
| 38. C2×SD16 | C2×C4 | semidirect |
| 39. C2×D8 | C2×C4 | semidirect |
| 41. #131 | C23 | semidirect |
| 57. #108 | C23 | semidirect |
| 58. #104 | C23 | semidirect |
| 59. C4×C8 | C23 | semidirect |
| 84. Q16 | C22×C4 | semidirect |
| 85. SD16 | C22×C4 | semidirect |
| 86. D8 | C22×C4 | semidirect |
| 137. C22 | #443 | direct (N × H) |
| 150. C2 | C2×#443 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#135 | 2 |
| C22×#131 | 2 |
| C2×#443 | 2 |
| C22×#108 | 2 |
| C22×#104 | 2 |
| C22×C4×C8 | 2 |