Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
106 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C3×C4 |
| Frattini, cosocle | C4⋊Q8 | C22×C3 |
| prime radical | C3×2−1+4 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#356 | C2 | semidirect |
| 3. C3×#333 | C2 | semidirect |
| 4. #1405 | C3 | direct (N × H) |
| 13. #356 | C2×C3 | semidirect |
| 14. #333 | C2×C3 | semidirect |
| 99. C3 | #1405 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#456 | 2 |
| C3×#356 | 2 |
| C3×#333 | 2 |
| #1405 | 3 |