Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
727 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 727 conjugacy classes of subgroups — too many to draw. Only its 247 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #2057 | C2 | direct (N × H) |
| 37. C22×C8 | Q8 | semidirect |
| 52. C2×C4⋊C4 | Q8 | semidirect |
| 95. C2×C8 | C2×Q8 | semidirect |
| 105. C4⋊C4 | C2×Q8 | semidirect |
| 169. C2 | #2057 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊Q8 | 2 |
| C22×#107 | 2 |
| C2×#389 | 2 |
| #2057 | 2 |
| C2×#340 | 2 |
| C2×#336 | 2 |