Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
306 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #543⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×#543 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C23×A4 | — | 4 | 1 | no | maximal |
| 4 | C22×SL(2,𝔽3) | — | 4 | 4 | no | maximal |
| 5 | #543 | — | 3 | 4 | no | — |
| 6 | #543 | — | 3 | 1 | no | — |
| 7 | #543 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C22⋊Q8 | — | 3 | 4 | no | — |
| 9 | C22×C22⋊C4 | — | 3 | 1 | no | — |
| 10 | C22×A4 | — | 4 | 4 | no | — |
| 11 | C22×A4 | — | 4 | 2 | no | — |
| 12 | C22×A4 | — | 4 | 1 | no | — |
| 13 | C2×SL(2,𝔽3) | — | 4 | 8 | no | — |
| 14 | C2×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 15 | C25 | A4 | 1 | 1 | yes | characteristic |
| 16 | C22×Q8 | A4 | 1 | 4 | yes | complemented |
| 17 | C23×C4 | — | 3 | 2 | no | — |
| 18 | C22⋊Q8 | — | 3 | 12 | no | — |
| 19 | C22⋊Q8 | — | 3 | 4 | no | — |
| 20 | C22⋊Q8 | — | 3 | 8 | no | — |
| 21 | C22⋊Q8 | — | 3 | 4 | no | — |
| 22 | C22⋊Q8 | — | 3 | 4 | no | — |
| 23 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 24 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 25 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 26 | C2×C22⋊C4 | — | 3 | 2 | no | — |
| 27 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 28 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 29 | C23×C3 | — | 16 | 1 | no | — |
| 30 | C2×A4 | — | 4 | 2 | no | — |
| 31 | C2×A4 | — | 4 | 4 | no | — |
| 32 | C2×A4 | — | 4 | 1 | no | — |
| 33 | SL(2,𝔽3) | — | 4 | 4 | no | — |
| 34 | C24 | — | 3 | 1 | no | — |
| 35 | C24 | — | 3 | 1 | no | — |
| 36 | C24 | — | 3 | 2 | no | — |
| 37 | C24 | — | 4 | 4 | no | — |
| 38 | C24 | SL(2,𝔽3) | 1 | 2 | yes | complemented |
| 39 | C24 | C2×A4 | 1 | 1 | yes | characteristic |
| 40 | C2×Q8 | — | 3 | 4 | no | — |
| 41 | C2×Q8 | — | 4 | 8 | no | — |
| 42 | C2×Q8 | C2×A4 | 1 | 4 | yes | complemented |
| 43 | C22×C4 | — | 3 | 4 | no | — |
| 44 | C22×C4 | — | 3 | 2 | no | — |
| 45 | C22×C4 | — | 3 | 4 | no | — |
| 46 | C22×C4 | — | 3 | 2 | no | — |
| 47 | C22×C4 | — | 12 | 4 | no | — |
| 48 | C4⋊C4 | — | 3 | 12 | no | — |
| 49 | C4⋊C4 | — | 3 | 4 | no | — |
| 50 | C22⋊C4 | — | 3 | 4 | no | — |
| 51 | C22⋊C4 | — | 3 | 8 | no | — |
| 52 | C22⋊C4 | — | 3 | 4 | no | — |
| 53 | C22×C3 | — | 16 | 2 | no | — |
| 54 | C22×C3 | — | 16 | 4 | no | — |
| 55 | C22×C3 | — | 16 | 1 | no | — |
| 56 | A4 | — | 4 | 1 | no | — |
| 57 | C23 | — | 3 | 2 | no | — |
| 58 | C23 | — | 3 | 2 | no | — |
| 59 | C23 | — | 3 | 2 | no | — |
| 60 | C23 | — | 3 | 1 | no | — |
| 61 | C23 | — | 3 | 1 | no | — |
| 62 | C23 | — | 3 | 2 | no | — |
| 63 | C23 | — | 3 | 1 | no | — |
| 64 | C23 | — | 4 | 4 | no | — |
| 65 | C23 | — | 4 | 2 | no | — |
| 66 | C23 | — | 6 | 4 | no | — |
| 67 | C23 | — | 12 | 2 | no | — |
| 68 | C23 | — | 12 | 4 | no | — |
| 69 | C23 | C2×SL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 70 | C23 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 71 | Q8 | — | 4 | 4 | no | — |
| 72 | Q8 | — | 12 | 4 | no | — |
| 73 | C2×C4 | — | 3 | 4 | no | — |
| 74 | C2×C4 | — | 3 | 4 | no | — |
| 75 | C2×C4 | — | 6 | 8 | no | — |
| 76 | C2×C4 | — | 12 | 2 | no | — |
| 77 | C2×C4 | — | 12 | 4 | no | — |
| 78 | C2×C4 | — | 12 | 2 | no | — |
| 79 | C2×C3 | — | 16 | 1 | no | — |
| 80 | C2×C3 | — | 16 | 2 | no | — |
| 81 | C2×C3 | — | 16 | 4 | no | — |
| 82 | C22 | — | 3 | 1 | no | — |
| 83 | C22 | — | 3 | 2 | no | — |
| 84 | C22 | — | 3 | 1 | no | — |
| 85 | C22 | — | 4 | 1 | no | — |
| 86 | C22 | — | 6 | 2 | no | — |
| 87 | C22 | — | 6 | 2 | no | — |
| 88 | C22 | — | 6 | 4 | no | — |
| 89 | C22 | — | 6 | 4 | no | — |
| 90 | C22 | — | 12 | 2 | no | — |
| 91 | C22 | — | 12 | 1 | no | — |
| 92 | C22 | — | 12 | 2 | no | — |
| 93 | C22 | C22⋊SL(2,𝔽3) | 1 | 2 | yes | — |
| 94 | C22 | #997 | 1 | 4 | yes | — |
| 95 | C22 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 96 | C4 | — | 6 | 4 | no | — |
| 97 | C4 | — | 6 | 4 | no | — |
| 98 | C3 | — | 16 | 1 | no | Sylow |
| 99 | C2 | — | 6 | 1 | no | — |
| 100 | C2 | — | 6 | 1 | no | — |
| 101 | C2 | — | 6 | 2 | no | — |
| 102 | C2 | #543⋊C3 | 1 | 4 | yes | direct product factor |
| 103 | C2 | C2×C22⋊SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 104 | C2 | C2×#997 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | C22⋊A4 |
| commutator, cosocle, hypocenter | #543 | C2×C3 |
| Frattini | C22 | C2×C22⋊A4 |
| Fitting | C2×#543 | C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #543⋊C3 | C2 | direct (N × H) |
| 2. C2×#543 | C3 | semidirect |
| 7. #543 | C2×C3 | semidirect |
| 16. C22×Q8 | A4 | semidirect |
| 38. C24 | SL(2,𝔽3) | semidirect |
| 42. C2×Q8 | C2×A4 | semidirect |
| 69. C23 | C2×SL(2,𝔽3) | semidirect |
| 102. C2 | #543⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #543⋊C3 | 2 |
| C2×#543 | 3 |
| C23×A4 | 4 |
| C22×SL(2,𝔽3) | 4 |