Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
474 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 116 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 | C22×C4 | C23×C3 |
| prime radical | C2×C3×#127 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×SD16 | C2 | semidirect |
| 3. C2×C3×#102 | C2 | semidirect |
| 4. C3×#391 | C2 | semidirect |
| 5. C3×#336 | C2 | semidirect |
| 6. #2014 | C3 | direct (N × H) |
| 19. C22×SD16 | C2×C3 | semidirect |
| 21. C2×#102 | C2×C3 | semidirect |
| 22. #391 | C2×C3 | semidirect |
| 23. #336 | C2×C3 | semidirect |
| 28. C2×C3×Q8 | D4 | semidirect |
| 45. C2×C3×C8 | D4 | semidirect |
| 80. C2×C3×C4 | SD16 | semidirect |
| 90. C2×Q8 | C3×D4 | semidirect |
| 107. C2×C8 | C3×D4 | semidirect |
| 145. C2×C4 | C3×SD16 | semidirect |
| 172. C3 | #2014 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×SD16 | 2 |
| C2×C3×#127 | 2 |
| C2×C3×#102 | 2 |
| C3×#391 | 2 |
| C3×#336 | 2 |
| #2014 | 3 |