Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
552 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 214 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 | C4×Q8×C97 | C4 |
| prime radical, cosocle | C22×D97 | C24 |
| hypercenter | C4×Q8 | #87604 |
| hypocenter | C97 | #2351 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. Q8×Dic97 | C4 | semidirect |
| 15. C4×Dic194 | C4 | semidirect |
| 16. #15690775 | C4 | semidirect |
| 19. C4×Q8×C97 | C4 | semidirect |
| 46. Q8×C97 | C4⋊C4 | semidirect |
| 57. Dic194 | C4⋊C4 | semidirect |
| 77. C97 | #2351 | semidirect |
| 88. C4×Q8 | #87604 | semidirect |
| 118. Q8 | C4⋊#87604 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×#87604 | 2 |
| #20261686 | 2 |
| #20261679 | 2 |
| C4×C4⋊#87604 | 2 |
| C4×Q8×D97 | 2 |
| #2351 | 97 |