Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1395 conjugacy classes of subgroups; only the 823 normal subgroups are shown.
This group has 1395 conjugacy classes of subgroups — too many to draw. Only its 823 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 | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#441 | C2 | direct (N × H) |
| 8. C22×Q16 | C4 | semidirect |
| 18. #441 | C22 | direct (N × H) |
| 30. C2×Q16 | C2×C4 | semidirect |
| 63. Q16 | C22×C4 | semidirect |
| 93. C22 | #441 | direct (N × H) |
| 103. C2 | C2×#441 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q16 | 2 |
| C22×C4×Q8 | 2 |
| C2×#441 | 2 |
| C22×#106 | 2 |
| C22×#103 | 2 |
| C22×#97 | 2 |