Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
346 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#87604 |
| commutator, socle | C22×C97 | C23×C4 |
| Fitting, prime radical | #121×C97 | C4 |
| cosocle | C2×Dic97 | C24 |
| hypercenter | #121 | #87604 |
| hypocenter | C97 | #2995 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20261787 | C2 | semidirect |
| 2. #20261770 | C2 | semidirect |
| 3. #20261766 | C2 | semidirect |
| 4. #20261766 | C2 | semidirect |
| 5. #20261759 | C2 | semidirect |
| 6. #20261755 | C2 | semidirect |
| 7. #20261741 | C2 | semidirect |
| 8. #20261738 | C2 | semidirect |
| 115. C97 | #2995 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261787 | 2 |
| #20261770 | 2 |
| #20261766 | 2 |
| #20261759 | 2 |
| #20261755 | 2 |
| #20261741 | 2 |
| #20261738 | 2 |
| #20261266 | 2 |
| #2995 | 97 |