Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2751 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 2751 conjugacy classes of subgroups — too many to draw. Only its 1039 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 | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ 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 |
|---|---|---|
| 2. C23×M16 | C2 | semidirect |
| 3. C23×C4⋊C4 | C2 | semidirect |
| 4. C2×#417 | C2 | direct (N × H) |
| 5. C22×#103 | C2 | semidirect |
| 6. C22×#102 | C2 | semidirect |
| 9. C22×#41 | C4 | semidirect |
| 18. C22×M16 | C22 | semidirect |
| 23. C22×C4⋊C4 | C22 | semidirect |
| 24. #417 | C22 | direct (N × H) |
| 25. C2×#103 | C22 | semidirect |
| 26. C2×#102 | C22 | semidirect |
| 32. C2×#41 | C2×C4 | semidirect |
| 36. C22×Q8 | C2×C4 | semidirect |
| 42. C22×D4 | C2×C4 | semidirect |
| 57. C2×M16 | C23 | semidirect |
| 64. C2×C4⋊C4 | C23 | semidirect |
| 65. #103 | C23 | semidirect |
| 66. #102 | C23 | semidirect |
| 78. #41 | C22×C4 | semidirect |
| 82. C2×Q8 | C22×C4 | semidirect |
| 89. C2×D4 | C22×C4 | semidirect |
| 132. Q8 | C23×C4 | semidirect |
| 136. D4 | C23×C4 | semidirect |
| 161. C22 | #417 | direct (N × H) |
| 175. C2 | C2×#417 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C23×M16 | 2 |
| C23×C4⋊C4 | 2 |
| C2×#417 | 2 |
| C22×#103 | 2 |
| C22×#102 | 2 |