Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
615 conjugacy classes of subgroups; only the 327 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 327 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C42×Q8 | C2 | semidirect |
| 4. C4×#106 | C2 | semidirect |
| 5. C4×#103 | C2 | semidirect |
| 7. C42×C8 | C2 | semidirect |
| 12. C4×SD16 | C4 | direct (N × H) |
| 32. C4×Q8 | C2×C4 | semidirect |
| 46. #106 | C2×C4 | semidirect |
| 47. #103 | C2×C4 | semidirect |
| 50. C4×C8 | C2×C4 | semidirect |
| 67. SD16 | C42 | direct (N × H) |
| 81. C42 | SD16 | direct (N × H) |
| 89. Q8 | C2×C42 | semidirect |
| 107. C8 | C2×C42 | semidirect |
| 118. C4 | C4×SD16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×SD16 | 2 |
| C42×Q8 | 2 |
| C42×D4 | 2 |
| C4×#106 | 2 |
| C4×#103 | 2 |
| C4×#102 | 2 |
| C42×C8 | 2 |