Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
264 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #394 |
| prime radical | C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#424 | C2 | semidirect |
| 3. C3×#416 | C2 | semidirect |
| 4. C3×#410 | C2 | semidirect |
| 5. C3×H(3,ℤ/4) | C2 | semidirect |
| 6. #2053 | C3 | direct (N × H) |
| 20. #424 | C2×C3 | semidirect |
| 21. #416 | C2×C3 | semidirect |
| 22. #410 | C2×C3 | semidirect |
| 23. H(3,ℤ/4) | C2×C3 | semidirect |
| 144. C3 | #2053 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#575 | 2 |
| C3×#424 | 2 |
| C3×#416 | 2 |
| C3×#410 | 2 |
| C3×H(3,ℤ/4) | 2 |
| #2053 | 3 |