Finite Group DatabasebetaFinite Group DatabasebetaOrder 1472.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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×D23 |
| commutator, socle, cosocle | C22×C23 | C24 |
| Fitting | C23×C22⋊Q8 | C2 |
| hypercenter | C22⋊Q8 | D23 |
| hypocenter | C23 | #538 |
| LCS | G ▷ C22×C23 ▷ C23 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊Q8 | C2 | semidirect |
| 2. #10625118 | C2 | semidirect |
| 3. #10625101 | C2 | semidirect |
| 4. C2×(C2×D23)⋊C4 | C2 | semidirect |
| 5. #10625069 | C2 | semidirect |
| 6. #10625066 | C2 | semidirect |
| 7. #10625065 | C2 | semidirect |
| 8. #10625053 | C2 | semidirect |
| 9. (C22⋊D23)⋊C4 | C2 | semidirect |
| 11. C2×Q8×C23 | C22 | semidirect |
| 17. C2×D92 | C22 | semidirect |
| 18. C2×C4×D23 | C22 | semidirect |
| 19. C4⋊C4×C23 | C22 | semidirect |
| 20. C4⋊C4×C23 | C22 | semidirect |
| 26. (C2×D23)⋊C4 | C22 | semidirect |
| 27. (C2×D23)⋊C4 | C22 | semidirect |
| 28. C4⋊Dic23 | C22 | semidirect |
| 29. Dic23⋊C4 | C22 | semidirect |
| 30. C4×Dic23 | C22 | semidirect |
| 78. C22⋊Q8 | D23 | semidirect |
| 84. C23 | #538 | semidirect |
| 86. C2×Q8 | C2×D23 | semidirect |
| 96. C4⋊C4 | C2×D23 | semidirect |
| 97. C4⋊C4 | C2×D23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊Q8 | 2 |
| #10625118 | 2 |
| #10625101 | 2 |
| C2×(C2×D23)⋊C4 | 2 |
| #10625069 | 2 |
| #10625066 | 2 |
| #10625065 | 2 |
| #10625053 | 2 |
| (C22⋊D23)⋊C4 | 2 |
| #538 | 23 |