Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
138 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #384⋊C3 | C2 | 1 | 2 | yes | maximal |
| 2 | C4×C42⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #2868 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | Q8×A4 | — | 4 | 1 | no | maximal |
| 5 | C2×C42⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 7 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 8 | #384 | — | 3 | 2 | no | — |
| 9 | #384 | C2×C3 | 1 | 2 | yes | — |
| 10 | C43 | C2×C3 | 1 | 1 | yes | characteristic |
| 11 | C4×A4 | — | 4 | 1 | no | — |
| 12 | C4×A4 | — | 4 | 2 | no | — |
| 13 | C42⋊C3 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×Q8 | — | 3 | 1 | no | — |
| 15 | C22×Q8 | A4 | 1 | 1 | yes | characteristic |
| 16 | C4⋊Q8 | — | 6 | 2 | no | — |
| 17 | C4⋊Q8 | — | 6 | 2 | no | — |
| 18 | C4⋊Q8 | — | 6 | 2 | no | — |
| 19 | C4⋊Q8 | — | 6 | 2 | no | — |
| 20 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 21 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 22 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 23 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 24 | C2×C42 | — | 3 | 1 | no | — |
| 25 | C2×C42 | — | 3 | 1 | no | — |
| 26 | C2×C42 | C22×C3 | 1 | 1 | yes | characteristic |
| 27 | C2×A4 | — | 4 | 1 | no | — |
| 28 | C3×Q8 | — | 16 | 1 | no | — |
| 29 | C2×Q8 | — | 6 | 1 | no | — |
| 30 | C2×Q8 | — | 6 | 1 | no | — |
| 31 | C2×Q8 | — | 6 | 1 | no | — |
| 32 | C2×Q8 | — | 6 | 2 | no | — |
| 33 | C2×Q8 | — | 6 | 1 | no | — |
| 34 | C2×Q8 | — | 6 | 2 | no | — |
| 35 | C22×C4 | — | 3 | 1 | no | — |
| 36 | C22×C4 | — | 3 | 1 | no | — |
| 37 | C22×C4 | — | 3 | 2 | no | — |
| 38 | C22×C4 | C2×A4 | 1 | 1 | yes | characteristic |
| 39 | C22×C4 | C2×A4 | 1 | 2 | yes | — |
| 40 | C4⋊C4 | — | 6 | 4 | no | — |
| 41 | C4⋊C4 | — | 6 | 4 | no | — |
| 42 | C4⋊C4 | — | 6 | 4 | no | — |
| 43 | C4⋊C4 | — | 6 | 4 | no | — |
| 44 | C42 | — | 3 | 1 | no | — |
| 45 | C42 | — | 3 | 4 | no | — |
| 46 | C42 | — | 3 | 2 | no | — |
| 47 | C42 | — | 3 | 2 | no | — |
| 48 | C42 | C3×Q8 | 1 | 1 | yes | characteristic, complemented |
| 49 | A4 | — | 4 | 1 | no | — |
| 50 | C3×C4 | — | 16 | 2 | no | — |
| 51 | C3×C4 | — | 16 | 1 | no | — |
| 52 | C23 | C22×A4 | 1 | 1 | yes | characteristic |
| 53 | Q8 | — | 4 | 1 | no | — |
| 54 | Q8 | — | 12 | 1 | no | — |
| 55 | Q8 | — | 12 | 1 | no | — |
| 56 | Q8 | — | 12 | 2 | no | — |
| 57 | Q8 | — | 12 | 1 | no | — |
| 58 | C2×C4 | — | 3 | 2 | no | — |
| 59 | C2×C4 | — | 3 | 2 | no | — |
| 60 | C2×C4 | — | 3 | 2 | no | — |
| 61 | C2×C4 | — | 3 | 4 | no | — |
| 62 | C2×C4 | — | 3 | 1 | no | — |
| 63 | C2×C4 | — | 3 | 1 | no | — |
| 64 | C2×C4 | — | 3 | 1 | no | — |
| 65 | C2×C4 | — | 3 | 1 | no | — |
| 66 | C2×C4 | — | 6 | 2 | no | — |
| 67 | C2×C4 | — | 6 | 2 | no | — |
| 68 | C2×C4 | — | 6 | 4 | no | — |
| 69 | C2×C3 | — | 16 | 1 | no | — |
| 70 | C22 | — | 3 | 1 | no | — |
| 71 | C22 | — | 3 | 1 | no | — |
| 72 | C22 | Q8×A4 | 1 | 1 | yes | characteristic |
| 73 | C4 | — | 3 | 4 | no | — |
| 74 | C4 | — | 3 | 2 | no | — |
| 75 | C4 | — | 3 | 2 | no | — |
| 76 | C4 | — | 3 | 1 | no | — |
| 77 | C4 | — | 4 | 2 | no | — |
| 78 | C4 | — | 12 | 2 | no | — |
| 79 | C4 | #127⋊C3 | 1 | 1 | yes | characteristic |
| 80 | C3 | — | 16 | 1 | no | Sylow |
| 81 | C2 | — | 3 | 1 | no | — |
| 82 | C2 | — | 3 | 1 | no | — |
| 83 | C2 | C2×#127⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#127⋊C3 |
| commutator, cosocle | C2×C42 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2868 | C3 |
| prime radical | C2×C42⋊C3 | C22 |
| hypercenter | C4 | #127⋊C3 |
| hypocenter | C42 | C3×Q8 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #2868 | C3 | semidirect |
| 13. C42⋊C3 | Q8 | semidirect |
| 48. C42 | C3×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #384⋊C3 | 2 |
| C4×C42⋊C3 | 2 |
| #2868 | 3 |
| Q8×A4 | 4 |