Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
305 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 123 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 | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 61. #106 | Q8 | semidirect |
| 62. #106 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2584 | 2 |
| #2088 | 2 |
| #2059 | 2 |
| #1985 | 2 |
| #1944 | 2 |
| #1920 | 2 |
| #1919 | 2 |