Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
512 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C24×C3 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×Q16 | C2 | semidirect |
| 4. C3×#467 | C2 | direct (N × H) |
| 5. C2×C3×#107 | C2 | semidirect |
| 6. C2×C3×#103 | C2 | semidirect |
| 7. C2×#467 | C3 | direct (N × H) |
| 11. C2×C3×Q16 | C22 | semidirect |
| 21. C3×#107 | C22 | semidirect |
| 22. C3×#103 | C22 | semidirect |
| 23. C22×Q16 | C2×C3 | semidirect |
| 26. #467 | C2×C3 | direct (N × H) |
| 27. C2×#107 | C2×C3 | semidirect |
| 28. C2×#103 | C2×C3 | semidirect |
| 55. C2×Q16 | C22×C3 | semidirect |
| 65. #107 | C22×C3 | semidirect |
| 66. #103 | C22×C3 | semidirect |
| 69. C23×C3 | Q16 | semidirect |
| 118. C22×C3 | C2×Q16 | semidirect |
| 129. C23 | C3×Q16 | semidirect |
| 151. C2×C3 | #467 | direct (N × H) |
| 160. C22 | C2×C3×Q16 | semidirect |
| 169. C3 | C2×#467 | direct (N × H) |
| 171. C2 | C3×#467 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Q16 | 2 |
| C23×C3×C8 | 2 |
| C2×C3×C22⋊Q8 | 2 |
| C3×#467 | 2 |
| C2×C3×#107 | 2 |
| C2×C3×#103 | 2 |
| C2×#467 | 3 |