Finite Group DatabasebetaFinite Group DatabasebetaOrder 3072.
At least 7597 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has at least 7597 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#1020 |
| commutator, cosocle | C23×C22⋊A4 | C23 |
| Fitting | C26×D4 | D3 |
| socle | C27 | C22×D3 |
| hypercenter | C22×D4 | #1020 |
| hypocenter | C22⋊A4 | #392 |
| LCS | G ▷ C23×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72341369 | C2 | semidirect |
| 2. C2×#11715572 | C2 | semidirect |
| 3. C22×D4×C22⋊A4 | C2 | semidirect |
| 4. C22×#87524 | C2 | semidirect |
| 5. C2×#11715569 | C2 | semidirect |
| 11. C26×D4 | D3 | semidirect |
| 12. C23×C22⋊A4 | D4 | semidirect |
| 14. C22×#1020 | D4 | semidirect |
| 15. C2×C4×C22⋊A4 | D4 | semidirect |
| 16. C2×#5936 | D4 | semidirect |
| 24. C27 | #67 | semidirect |
| 26. C24×D4 | S4 | semidirect |
| 27. C25×C4 | #67 | semidirect |
| 40. C22⋊A4 | #392 | semidirect |
| 46. C25 | GL(2,ℤ/4) | semidirect |
| 48. C22×D4 | #1020 | semidirect |
| 49. C23×C4 | GL(2,ℤ/4) | semidirect |
| 50. C24 | #5223 | semidirect |
| 63. C23 | #87524 | semidirect |
| 65. C2×C4 | #87524 | semidirect |
| 66. C22 | #11711747 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72341369 | 2 |
| C2×#11715572 | 2 |
| C22×D4×C22⋊A4 | 2 |
| C22×#87524 | 2 |
| #72240113 | 3 |
| C2×#11715569 | 2 |
| #11711747 | 4 |