Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#532 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #3153 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #532 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#532 | C2 | semidirect |
| 2. C2×#951 | C2 | semidirect |
| 3. C2×#949 | C2 | semidirect |
| 4. C2×#875 | C2 | semidirect |
| 5. #5086 | C2 | semidirect |
| 6. #5074 | C2 | semidirect |
| 7. #5059 | C2 | semidirect |
| 8. C4×#208 | C2 | semidirect |
| 13. C3×C4⋊Q8 | C22 | semidirect |
| 14. C3×#127 | C22 | semidirect |
| 15. C3×#124 | C22 | semidirect |
| 21. C2×#216 | C22 | semidirect |
| 23. C2×#214 | C22 | semidirect |
| 30. #803 | C22 | semidirect |
| 31. C4×C3⋊C8 | C22 | semidirect |
| 34. #532 | D3 | semidirect |
| 68. #208 | D4 | semidirect |
| 80. C4⋊Q8 | C2×D3 | semidirect |
| 81. #127 | C2×D3 | semidirect |
| 85. #124 | C2×D3 | semidirect |
| 207. C3 | #3153 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#532 | 2 |
| C2×#951 | 2 |
| C2×#949 | 2 |
| C2×#875 | 2 |
| #5086 | 2 |
| #5074 | 2 |
| #5059 | 2 |
| C4×#208 | 2 |
| #3153 | 3 |