Finite Group DatabasebetaFinite Group DatabasebetaOrder 3888.
468 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 23 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 | #14519512 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊#10620193 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#10620192 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13049788 | — | 3 | 1 | no | maximal |
| 5 | #11758168 | C4 | 1 | 1 | yes | characteristic |
| 6 | C3⋊#66472 | C4 | 1 | 1 | yes | characteristic |
| 7 | C3×#66472 | C22 | 1 | 1 | yes | characteristic |
| 8 | #10620198 | — | 3 | 1 | no | — |
| 9 | #10620193 | — | 3 | 1 | no | — |
| 10 | #10620192 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3×C3⋊(C3⋊(C3⋊D3)) | C2×C4 | 1 | 1 | yes | characteristic |
| 12 | C32×C3⋊(C3⋊D3) | C8 | 1 | 2 | yes | complemented |
| 13 | D3×AGL(1,𝔽9) | — | 9 | 2 | no | maximal |
| 14 | C3⋊#1113 | — | 12 | 2 | no | — |
| 15 | C3⋊D32 | — | 3 | 1 | no | — |
| 16 | #66472 | — | 3 | 1 | no | — |
| 17 | #66472 | C2×D3 | 1 | 1 | yes | characteristic |
| 18 | C35 | C2×C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | D3×#157 | — | 9 | 2 | no | — |
| 20 | C3⋊AGL(1,𝔽9) | — | 9 | 2 | no | — |
| 21 | C3×AGL(1,𝔽9) | — | 9 | 2 | no | — |
| 22 | C3×AGL(1,𝔽9) | — | 18 | 4 | no | — |
| 23 | C3⋊(C3⋊(C3⋊D3)) | C4×D3 | 1 | 1 | yes | characteristic |
| 24 | C3×C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 25 | C3×C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 26 | C3×C3⋊(C3⋊D3) | — | 24 | 4 | no | — |
| 27 | C32×C3⋊D3 | — | 3 | 2 | no | — |
| 28 | C32×C3⋊D3 | — | 12 | 2 | no | — |
| 29 | C32×C3⋊D3 | — | 24 | 2 | no | — |
| 30 | C2×AGL(1,𝔽9) | — | 27 | 2 | no | — |
| 31 | #1113 | — | 36 | 4 | no | — |
| 32 | D3×C3⋊D3 | — | 9 | 2 | no | — |
| 33 | D3×C3⋊D3 | — | 36 | 2 | no | — |
| 34 | C3⋊#157 | — | 9 | 2 | no | — |
| 35 | C3⋊#157 | — | 36 | 2 | no | — |
| 36 | C3×#157 | — | 9 | 2 | no | — |
| 37 | C3×#157 | — | 18 | 4 | no | — |
| 38 | C34 | — | 4 | 2 | no | — |
| 39 | C34 | — | 8 | 4 | no | — |
| 40 | C34 | — | 8 | 2 | no | — |
| 41 | C34 | — | 16 | 4 | no | — |
| 42 | C34 | D3×C8 | 1 | 1 | yes | characteristic, complemented |
| 43 | C2×#157 | — | 27 | 2 | no | — |
| 44 | AGL(1,𝔽9) | — | 9 | 2 | no | — |
| 45 | AGL(1,𝔽9) | — | 18 | 4 | no | — |
| 46 | AGL(1,𝔽9) | — | 27 | 2 | no | — |
| 47 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 48 | C3⋊(C3⋊D3) | — | 24 | 4 | no | — |
| 49 | C3⋊(C3⋊D3) | AGL(1,𝔽9) | 1 | 2 | yes | complemented |
| 50 | C3×C3⋊D3 | — | 9 | 2 | no | — |
| 51 | C3×C3⋊D3 | — | 12 | 2 | no | — |
| 52 | C3×C3⋊D3 | — | 12 | 8 | no | — |
| 53 | C3×C3⋊D3 | — | 18 | 4 | no | — |
| 54 | C3×C3⋊D3 | — | 24 | 8 | no | — |
| 55 | C3×C3⋊D3 | — | 36 | 4 | no | — |
| 56 | C3×C3⋊D3 | — | 36 | 2 | no | — |
| 57 | C3×C3⋊D3 | — | 72 | 8 | no | — |
| 58 | C3×C3⋊D3 | — | 72 | 4 | no | — |
| 59 | C32×D3 | — | 9 | 2 | no | — |
| 60 | C32×D3 | — | 36 | 2 | no | — |
| 61 | C32×D3 | — | 72 | 2 | no | — |
| 62 | D3×C8 | — | 81 | 1 | no | — |
| 63 | C2×C3⋊D3 | — | 27 | 2 | no | — |
| 64 | D32 | — | 108 | 4 | no | — |
| 65 | D32 | — | 108 | 2 | no | — |
| 66 | #157 | — | 9 | 2 | no | — |
| 67 | #157 | — | 18 | 4 | no | — |
| 68 | #157 | — | 27 | 2 | no | — |
| 69 | #157 | — | 108 | 2 | no | — |
| 70 | C33 | — | 2 | 4 | no | — |
| 71 | C33 | — | 4 | 2 | no | — |
| 72 | C33 | — | 4 | 4 | no | — |
| 73 | C33 | — | 4 | 2 | no | — |
| 74 | C33 | — | 8 | 4 | no | — |
| 75 | C33 | — | 8 | 8 | no | — |
| 76 | C33 | — | 8 | 8 | no | — |
| 77 | C33 | — | 8 | 4 | no | — |
| 78 | C33 | — | 8 | 2 | no | — |
| 79 | C33 | — | 16 | 4 | no | — |
| 80 | C33 | — | 16 | 4 | no | — |
| 81 | C33 | — | 16 | 4 | no | — |
| 82 | C33 | — | 16 | 8 | no | — |
| 83 | C33 | — | 16 | 24 | no | — |
| 84 | C33 | — | 16 | 8 | no | — |
| 85 | C33 | — | 16 | 8 | no | — |
| 86 | C33 | C2×AGL(1,𝔽9) | 1 | 2 | yes | complemented |
| 87 | C4×D3 | — | 81 | 1 | no | — |
| 88 | C3×C8 | — | 81 | 1 | no | — |
| 89 | C3⋊C8 | — | 81 | 1 | no | — |
| 90 | C2×C32 | — | 27 | 2 | no | — |
| 91 | C3⋊D3 | — | 3 | 2 | no | — |
| 92 | C3⋊D3 | — | 9 | 2 | no | — |
| 93 | C3⋊D3 | — | 12 | 2 | no | — |
| 94 | C3⋊D3 | — | 18 | 4 | no | — |
| 95 | C3⋊D3 | — | 24 | 2 | no | — |
| 96 | C3⋊D3 | — | 36 | 4 | no | — |
| 97 | C3⋊D3 | — | 36 | 2 | no | — |
| 98 | C3⋊D3 | — | 72 | 8 | no | — |
| 99 | C3⋊D3 | — | 72 | 4 | no | — |
| 100 | C3×D3 | — | 36 | 2 | no | — |
| 101 | C3×D3 | — | 36 | 8 | no | — |
| 102 | C3×D3 | — | 72 | 8 | no | — |
| 103 | C3×D3 | — | 108 | 2 | no | — |
| 104 | C3×D3 | — | 216 | 4 | no | — |
| 105 | C2×C8 | — | 243 | 1 | no | Sylow |
| 106 | C2×D3 | — | 324 | 2 | no | — |
| 107 | C2×D3 | — | 81 | 1 | no | — |
| 108 | C3×C4 | — | 81 | 1 | no | — |
| 109 | Dic3 | — | 81 | 1 | no | — |
| 110 | C32 | — | 2 | 4 | no | — |
| 111 | C32 | — | 4 | 4 | no | — |
| 112 | C32 | — | 4 | 2 | no | — |
| 113 | C32 | — | 4 | 2 | no | — |
| 114 | C32 | — | 8 | 2 | no | — |
| 115 | C32 | — | 8 | 8 | no | — |
| 116 | C32 | — | 8 | 8 | no | — |
| 117 | C32 | — | 8 | 4 | no | — |
| 118 | C32 | — | 8 | 4 | no | — |
| 119 | C32 | — | 16 | 8 | no | — |
| 120 | C32 | — | 16 | 4 | no | — |
| 121 | C32 | — | 16 | 24 | no | — |
| 122 | C32 | — | 16 | 4 | no | — |
| 123 | C32 | — | 16 | 4 | no | — |
| 124 | C32 | — | 16 | 8 | no | — |
| 125 | C32 | — | 16 | 8 | no | — |
| 126 | C32 | D3×AGL(1,𝔽9) | 1 | 2 | yes | complemented |
| 127 | C2×C4 | — | 243 | 1 | no | — |
| 128 | C8 | — | 81 | 1 | no | — |
| 129 | C8 | — | 243 | 1 | no | — |
| 130 | C2×C3 | — | 81 | 1 | no | — |
| 131 | C2×C3 | — | 108 | 2 | no | — |
| 132 | D3 | — | 9 | 2 | no | — |
| 133 | D3 | — | 36 | 2 | no | — |
| 134 | D3 | — | 72 | 2 | no | — |
| 135 | D3 | — | 108 | 2 | no | — |
| 136 | D3 | — | 216 | 4 | no | — |
| 137 | C22 | — | 243 | 1 | no | — |
| 138 | C4 | — | 81 | 1 | no | — |
| 139 | C4 | — | 243 | 1 | no | — |
| 140 | C3 | — | 4 | 2 | no | — |
| 141 | C3 | — | 8 | 2 | no | — |
| 142 | C3 | — | 8 | 4 | no | — |
| 143 | C3 | — | 16 | 4 | no | — |
| 144 | C3 | #13049788 | 1 | 1 | yes | characteristic, complemented |
| 145 | C2 | — | 27 | 2 | no | — |
| 146 | C2 | — | 81 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C35 | C2×C8 |
| prime radical | #11758168 | C4 |
| cosocle | C3×#66472 | C22 |
| LCS | G ▷ C35 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3⋊#10620193 | C2 | semidirect |
| 3. C3×#10620192 | C2 | semidirect |
| 10. #10620192 | D3 | semidirect |
| 12. C32×C3⋊(C3⋊D3) | C8 | semidirect |
| 18. C35 | C2×C8 | semidirect |
| 42. C34 | D3×C8 | semidirect |
| 49. C3⋊(C3⋊D3) | AGL(1,𝔽9) | semidirect |
| 86. C33 | C2×AGL(1,𝔽9) | semidirect |
| 126. C32 | D3×AGL(1,𝔽9) | semidirect |
| 144. C3 | #13049788 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14519512 | 2 |
| C3⋊#10620193 | 2 |
| C3×#10620192 | 2 |
| #13049788 | 3 |
| D3×AGL(1,𝔽9) | 9 |