Finite Group DatabasebetaFinite Group DatabasebetaOrder 368.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8×C23 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D92 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×D23 | — | 2 | 1 | no | — |
| 5 | C4×C23 | — | 2 | 1 | no | — |
| 6 | C4×C23 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C23 | D4 | 1 | 1 | yes | characteristic |
| 8 | D23 | — | 4 | 1 | no | — |
| 9 | C23 | SD16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | SD16 | — | 23 | 1 | no | maximal, Sylow |
| 11 | Q8 | D23 | 1 | 1 | yes | characteristic, complemented |
| 12 | D4 | — | 23 | 1 | no | — |
| 13 | C8 | — | 23 | 1 | no | — |
| 14 | C22 | — | 46 | 1 | no | — |
| 15 | C4 | — | 2 | 1 | no | — |
| 16 | C4 | C2×D23 | 1 | 1 | yes | characteristic |
| 17 | C2 | — | 92 | 1 | no | — |
| 18 | C2 | C22⋊D23 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D23 |
| commutator, cosocle | C4×C23 | C22 |
| Frattini | C4 | C2×D23 |
| Fitting | Q8×C23 | C2 |
| socle | C2×C23 | D4 |
| prime radical | D92 | C2 |
| hypercenter | Q8 | D23 |
| hypocenter | C23 | SD16 |
| LCS | G ▷ C4×C23 ▷ C2×C23 ▷ C23 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×C23 | C2 | semidirect |
| 3. C23⋊C8 | C2 | semidirect |
| 9. C23 | SD16 | semidirect |
| 11. Q8 | D23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C23 | 2 |
| D92 | 2 |
| C23⋊C8 | 2 |
| SD16 | 23 |