Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
404 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 208 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 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C2×C3×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×D4×Q8 | C2 | semidirect |
| 2. C3×#500 | C2 | semidirect |
| 3. C3×#474 | C2 | semidirect |
| 4. C3×C8×Q8 | C2 | semidirect |
| 6. Q8×D8 | C3 | direct (N × H) |
| 18. D4×Q8 | C2×C3 | semidirect |
| 19. #500 | C2×C3 | semidirect |
| 20. #474 | C2×C3 | semidirect |
| 21. C8×Q8 | C2×C3 | semidirect |
| 28. C3×D8 | Q8 | direct (N × H) |
| 49. C3×Q8 | D8 | direct (N × H) |
| 51. C3×D4 | C2×Q8 | semidirect |
| 58. C3×C8 | C2×Q8 | semidirect |
| 64. D8 | C3×Q8 | direct (N × H) |
| 83. Q8 | C3×D8 | direct (N × H) |
| 85. D4 | C2×C3×Q8 | semidirect |
| 92. C8 | C2×C3×Q8 | semidirect |
| 106. C3 | Q8×D8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D4×Q8 | 2 |
| C3×#500 | 2 |
| C3×#474 | 2 |
| C3×C8×Q8 | 2 |
| C3×C4×D8 | 2 |
| Q8×D8 | 3 |