Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
630 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#381 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1827 | C2 | semidirect |
| 5. #1733 | C2 | semidirect |
| 6. C2×H(3,ℤ/4) | C2 | semidirect |
| 14. C4×#41 | C4 | semidirect |
| 16. C2×#117 | C4 | semidirect |
| 27. #339 | C22 | semidirect |
| 28. H(3,ℤ/4) | C22 | semidirect |
| 67. #117 | C2×C4 | semidirect |
| 80. C2×C42 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3428 | 2 |
| C2×#413 | 2 |
| C2×#410 | 2 |
| #1827 | 2 |
| #1733 | 2 |
| C2×H(3,ℤ/4) | 2 |