Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
406 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 92 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 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C3×#537 | C2 | semidirect |
| 3. C2×C3×#104 | C2 | semidirect |
| 4. C3×#390 | C2 | semidirect |
| 5. C3×#343 | C2 | semidirect |
| 6. #2006 | C3 | direct (N × H) |
| 21. C2×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 22. #537 | C2×C3 | semidirect |
| 23. C2×#104 | C2×C3 | semidirect |
| 24. #390 | C2×C3 | semidirect |
| 25. #343 | C2×C3 | semidirect |
| 38. C2×C3×D4 | D4 | semidirect |
| 46. C3×M16 | D4 | semidirect |
| 108. C2×D4 | C3×D4 | semidirect |
| 116. M16 | C3×D4 | semidirect |
| 195. C3 | #2006 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×AGL(1,ℤ/8) | 2 |
| C3×#537 | 2 |
| C2×C3×#104 | 2 |
| C3×#390 | 2 |
| C3×#343 | 2 |
| #2006 | 3 |