Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
168 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#475 | C2 | semidirect |
| 4. C3×#446 | C2 | semidirect |
| 5. C3×#335 | C2 | semidirect |
| 6. C3×#332 | C2 | semidirect |
| 8. #1686 | C3 | direct (N × H) |
| 25. #475 | C2×C3 | semidirect |
| 27. #446 | C2×C3 | semidirect |
| 28. #335 | C2×C3 | semidirect |
| 29. #332 | C2×C3 | semidirect |
| 85. C3×C8 | SD16 | semidirect |
| 137. C8 | C3×SD16 | semidirect |
| 154. C3 | #1686 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#492 | 2 |
| C3×#475 | 2 |
| C3×#460 | 2 |
| C3×#446 | 2 |
| C3×#335 | 2 |
| C3×#332 | 2 |
| C3×#331 | 2 |
| #1686 | 3 |