Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
139 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C42×Q8 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C22⋊SL(2,𝔽3) | — | 4 | 1 | no | maximal |
| 3 | C2×C42⋊C3 | — | 4 | 1 | no | maximal |
| 4 | #95⋊C3 | — | 4 | 3 | no | maximal |
| 5 | C2×C4×Q8 | — | 3 | 1 | no | — |
| 6 | C4×C4⋊C4 | — | 3 | 3 | no | — |
| 7 | C43 | — | 3 | 1 | no | — |
| 8 | C42⋊C3 | — | 4 | 1 | no | — |
| 9 | C22×Q8 | A4 | 1 | 1 | yes | characteristic |
| 10 | C4×Q8 | — | 3 | 8 | no | — |
| 11 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 12 | C2×C42 | — | 3 | 3 | no | — |
| 13 | C2×C42 | — | 3 | 3 | no | — |
| 14 | C2×C42 | A4 | 1 | 1 | yes | characteristic |
| 15 | #95 | — | 3 | 1 | no | — |
| 16 | #95 | A4 | 1 | 3 | yes | — |
| 17 | C2×A4 | — | 16 | 1 | no | — |
| 18 | SL(2,𝔽3) | — | 16 | 4 | no | — |
| 19 | C2×Q8 | — | 3 | 4 | no | — |
| 20 | C22×C4 | — | 3 | 1 | no | — |
| 21 | C22×C4 | — | 3 | 1 | no | — |
| 22 | C22×C4 | — | 3 | 3 | no | — |
| 23 | C4⋊C4 | — | 3 | 12 | no | — |
| 24 | C42 | — | 3 | 1 | no | — |
| 25 | C42 | — | 3 | 12 | no | — |
| 26 | C42 | — | 6 | 6 | no | — |
| 27 | C42 | SL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 28 | A4 | — | 16 | 1 | no | — |
| 29 | C23 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 30 | Q8 | — | 3 | 4 | no | — |
| 31 | Q8 | C42⋊C3 | 1 | 4 | yes | complemented |
| 32 | C2×C4 | — | 3 | 2 | no | — |
| 33 | C2×C4 | — | 3 | 6 | no | — |
| 34 | C2×C4 | — | 3 | 2 | no | — |
| 35 | C2×C4 | — | 3 | 1 | no | — |
| 36 | C2×C4 | — | 3 | 1 | no | — |
| 37 | C2×C4 | — | 3 | 6 | no | — |
| 38 | C2×C4 | — | 6 | 6 | no | — |
| 39 | C2×C3 | — | 64 | 1 | no | — |
| 40 | C22 | — | 3 | 1 | no | — |
| 41 | C22 | — | 3 | 1 | no | — |
| 42 | C22 | C22⋊SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 43 | C4 | — | 3 | 2 | no | — |
| 44 | C4 | — | 3 | 2 | no | — |
| 45 | C4 | — | 3 | 4 | no | — |
| 46 | C4 | — | 6 | 6 | no | — |
| 47 | C3 | — | 64 | 1 | no | Sylow |
| 48 | C2 | — | 3 | 1 | no | — |
| 49 | C2 | — | 3 | 1 | no | — |
| 50 | C2 | C22⋊(C42⋊C3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C22⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C42×Q8 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ C42×Q8 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C42×Q8 | C3 | semidirect |
| 27. C42 | SL(2,𝔽3) | semidirect |
| 31. Q8 | C42⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C42×Q8 | 3 |
| C22⋊SL(2,𝔽3) | 4 |
| C2×C42⋊C3 | 4 |
| #95⋊C3 | 4 |