Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1415 conjugacy classes of subgroups; only the 863 normal subgroups are shown.
This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 863 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle, prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ 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×C4×Q16 | C2 | direct (N × H) |
| 8. C22×Q16 | C4 | direct (N × H) |
| 18. C4×Q16 | C22 | direct (N × H) |
| 31. C2×Q16 | C2×C4 | direct (N × H) |
| 59. C22×C4 | Q16 | direct (N × H) |
| 66. Q16 | C22×C4 | direct (N × H) |
| 90. C2×C4 | C2×Q16 | direct (N × H) |
| 99. C22 | C4×Q16 | direct (N × H) |
| 108. C4 | C22×Q16 | direct (N × H) |
| 111. C2 | C2×C4×Q16 | 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×C4×Q16 | 2 |
| C22×#107 | 2 |
| C22×#103 | 2 |
| C22×C4×C8 | 2 |