Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
180 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C4⋊C4 |
| prime radical | C23×C3 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#335 | C2 | semidirect |
| 4. C3×#334 | C2 | semidirect |
| 5. #1564 | C3 | direct (N × H) |
| 18. #335 | C2×C3 | semidirect |
| 19. #334 | C2×C3 | semidirect |
| 25. C2×C3×C8 | C8 | semidirect |
| 65. C2×C8 | C3×C8 | semidirect |
| 116. C3 | #1564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#431 | 2 |
| C2×C3×C4×C8 | 2 |
| C3×#335 | 2 |
| C3×#334 | 2 |
| #1564 | 3 |