Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1195 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C2×C22⋊A4 | — | 4 | 3 | no | maximal |
| 3 | #997 | — | 4 | 16 | no | maximal |
| 4 | C22⋊SL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 5 | C2×H(5,𝔽2) | — | 3 | 16 | no | — |
| 6 | C22×#41 | — | 3 | 2 | no | — |
| 7 | C23×D4 | — | 3 | 3 | no | — |
| 8 | C22⋊A4 | — | 4 | 3 | no | — |
| 9 | C25 | — | 3 | 1 | no | — |
| 10 | C25 | A4 | 1 | 3 | yes | complemented |
| 11 | H(5,𝔽2) | — | 3 | 32 | no | — |
| 12 | H(5,𝔽2) | — | 3 | 48 | no | — |
| 13 | H(5,𝔽2) | A4 | 1 | 16 | yes | complemented |
| 14 | C2×#41 | — | 3 | 48 | no | — |
| 15 | C22×Q8 | A4 | 1 | 2 | yes | complemented |
| 16 | C22×D4 | — | 3 | 6 | no | — |
| 17 | C22×D4 | — | 3 | 72 | no | — |
| 18 | C23×C4 | — | 3 | 3 | no | — |
| 19 | C2×A4 | — | 16 | 1 | no | — |
| 20 | C2×A4 | — | 16 | 12 | no | — |
| 21 | SL(2,𝔽3) | — | 16 | 8 | no | — |
| 22 | C24 | — | 3 | 12 | no | — |
| 23 | C24 | — | 3 | 3 | no | — |
| 24 | C24 | — | 3 | 12 | no | — |
| 25 | C24 | — | 4 | 3 | no | — |
| 26 | C24 | — | 12 | 3 | no | — |
| 27 | C24 | — | 12 | 3 | no | — |
| 28 | C24 | — | 12 | 1 | no | — |
| 29 | #41 | — | 3 | 96 | no | — |
| 30 | #41 | — | 3 | 32 | no | — |
| 31 | C2×Q8 | — | 3 | 8 | no | — |
| 32 | C2×D4 | — | 3 | 72 | no | — |
| 33 | C2×D4 | — | 3 | 144 | no | — |
| 34 | C2×D4 | — | 3 | 48 | no | — |
| 35 | C22×C4 | — | 3 | 36 | no | — |
| 36 | C22×C4 | — | 3 | 2 | no | — |
| 37 | A4 | — | 16 | 12 | no | — |
| 38 | A4 | — | 16 | 1 | no | — |
| 39 | C23 | — | 3 | 18 | no | — |
| 40 | C23 | — | 3 | 12 | no | — |
| 41 | C23 | — | 3 | 16 | no | — |
| 42 | C23 | — | 6 | 9 | no | — |
| 43 | C23 | — | 6 | 3 | no | — |
| 44 | C23 | — | 12 | 12 | no | — |
| 45 | C23 | — | 12 | 12 | no | — |
| 46 | C23 | — | 12 | 12 | no | — |
| 47 | C23 | — | 12 | 12 | no | — |
| 48 | C23 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 49 | C23 | C22⋊A4 | 1 | 12 | yes | complemented |
| 50 | Q8 | — | 3 | 8 | no | — |
| 51 | Q8 | C22⋊A4 | 1 | 8 | yes | complemented |
| 52 | D4 | — | 3 | 96 | no | — |
| 53 | C2×C4 | — | 3 | 12 | no | — |
| 54 | C2×C4 | — | 3 | 48 | no | — |
| 55 | C2×C3 | — | 64 | 1 | no | — |
| 56 | C22 | — | 3 | 1 | no | — |
| 57 | C22 | — | 3 | 1 | no | — |
| 58 | C22 | — | 3 | 12 | no | — |
| 59 | C22 | — | 4 | 12 | no | — |
| 60 | C22 | — | 6 | 18 | no | — |
| 61 | C22 | — | 6 | 18 | no | — |
| 62 | C22 | — | 12 | 12 | no | — |
| 63 | C22 | — | 12 | 16 | no | — |
| 64 | C22 | #997 | 1 | 1 | yes | characteristic, complemented |
| 65 | C4 | — | 3 | 8 | no | — |
| 66 | C3 | — | 64 | 1 | no | Sylow |
| 67 | C2 | — | 3 | 1 | no | — |
| 68 | C2 | — | 3 | 1 | no | — |
| 69 | C2 | — | 6 | 12 | no | — |
| 70 | C2 | C22⋊(C22⋊A4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C22⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×H(5,𝔽2) | C3 |
| socle | C23 | C22⋊A4 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×H(5,𝔽2) | C3 | semidirect |
| 10. C25 | A4 | semidirect |
| 13. H(5,𝔽2) | A4 | semidirect |
| 15. C22×Q8 | A4 | semidirect |
| 49. C23 | C22⋊A4 | semidirect |
| 51. Q8 | C22⋊A4 | semidirect |
| 64. C22 | #997 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(5,𝔽2) | 3 |
| C2×C22⋊A4 | 4 |
| #997 | 4 |
| C22⋊SL(2,𝔽3) | 4 |