Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
370 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 370 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×D4 | C23 |
| 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×Q82 | C2 | semidirect |
| 3. C3×#498 | C2 | semidirect |
| 5. C3×#475 | C2 | semidirect |
| 6. C3×C8×Q8 | C2 | semidirect |
| 8. Q8×SD16 | C3 | direct (N × H) |
| 25. Q82 | C2×C3 | semidirect |
| 27. #498 | C2×C3 | semidirect |
| 29. #475 | C2×C3 | semidirect |
| 30. C8×Q8 | C2×C3 | semidirect |
| 39. C3×SD16 | Q8 | direct (N × H) |
| 71. C3×Q8 | SD16 | direct (N × H) |
| 72. C3×Q8 | C2×Q8 | semidirect |
| 83. C3×C8 | C2×Q8 | semidirect |
| 91. SD16 | C3×Q8 | direct (N × H) |
| 118. Q8 | C3×SD16 | direct (N × H) |
| 119. Q8 | C2×C3×Q8 | semidirect |
| 130. C8 | C2×C3×Q8 | semidirect |
| 146. C3 | Q8×SD16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q82 | 2 |
| C3×D4×Q8 | 2 |
| C3×#498 | 2 |
| C3×#476 | 2 |
| C3×#475 | 2 |
| C3×C8×Q8 | 2 |
| C3×C4×SD16 | 2 |
| Q8×SD16 | 3 |