Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
262 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 26 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 | C3⋊D3×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11724155 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10593842 | — | 3 | 1 | no | maximal |
| 4 | C32×#41⋊C3 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #88989 | — | 4 | 1 | no | maximal |
| 6 | C3⋊D3×SL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3⋊(C3⋊(#41⋊C3)) | C22 | 1 | 1 | yes | characteristic |
| 8 | #41×C3⋊D3 | D3 | 1 | 1 | yes | characteristic |
| 9 | #64286 | — | 3 | 1 | no | — |
| 10 | C2×#3788 | — | 3 | 1 | no | — |
| 11 | D3×#41⋊C3 | — | 6 | 2 | no | — |
| 12 | #63785 | — | 3 | 2 | no | — |
| 13 | #63783 | — | 3 | 2 | no | — |
| 14 | C3×C4×C3⋊D3 | — | 4 | 1 | no | — |
| 15 | C32×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C3⋊#622 | — | 4 | 2 | no | — |
| 17 | #5423 | — | 9 | 1 | no | maximal |
| 18 | C32×#41 | Dic3 | 1 | 1 | yes | characteristic |
| 19 | #3832 | C2×D3 | 1 | 1 | yes | characteristic |
| 20 | Q8×C3⋊D3 | Dic3 | 1 | 1 | yes | characteristic |
| 21 | #3830 | — | 3 | 1 | no | — |
| 22 | D4×C3⋊D3 | — | 3 | 1 | no | — |
| 23 | #3828 | — | 3 | 1 | no | — |
| 24 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 25 | C3×#41⋊C3 | — | 2 | 2 | no | — |
| 26 | C3×#41⋊C3 | — | 4 | 2 | no | — |
| 27 | C22⋊#157 | — | 6 | 1 | no | — |
| 28 | C2×#622 | — | 6 | 1 | no | — |
| 29 | C4⋊#157 | — | 6 | 1 | no | — |
| 30 | C4×#157 | — | 3 | 2 | no | — |
| 31 | #3788 | — | 3 | 2 | no | — |
| 32 | #3788 | — | 6 | 1 | no | — |
| 33 | D3×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 34 | C3⋊(#41⋊C3) | — | 6 | 2 | no | — |
| 35 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 36 | C33×C4 | — | 4 | 1 | no | — |
| 37 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 38 | D3×#41 | — | 6 | 2 | no | — |
| 39 | C2×#41⋊C3 | — | 9 | 1 | no | — |
| 40 | GU(2,𝔽3) | — | 9 | 2 | no | — |
| 41 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 42 | C2×#157 | — | 6 | 2 | no | — |
| 43 | C32×Q8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 44 | C32×D4 | — | 3 | 1 | no | — |
| 45 | C2×C32×C4 | — | 3 | 1 | no | — |
| 46 | #638 | — | 6 | 1 | no | — |
| 47 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 48 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 49 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 50 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 51 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 52 | C4×C3⋊D3 | S4 | 1 | 1 | yes | characteristic |
| 53 | #634 | — | 3 | 1 | no | — |
| 54 | C3×C4×D3 | — | 24 | 2 | no | — |
| 55 | C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 56 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 57 | #622 | — | 6 | 2 | no | — |
| 58 | #421 | — | 27 | 1 | no | Sylow |
| 59 | C2×C33 | — | 4 | 1 | no | — |
| 60 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 61 | C3×#41 | — | 2 | 2 | no | — |
| 62 | #239 | — | 6 | 2 | no | — |
| 63 | D3×Q8 | — | 6 | 2 | no | — |
| 64 | #237 | — | 18 | 2 | no | — |
| 65 | D3×D4 | — | 18 | 2 | no | — |
| 66 | #235 | — | 18 | 2 | no | — |
| 67 | C2×C4×D3 | — | 18 | 2 | no | — |
| 68 | #41⋊C3 | — | 4 | 2 | no | — |
| 69 | #41⋊C3 | — | 9 | 1 | no | — |
| 70 | #41⋊C3 | #157 | 1 | 1 | yes | characteristic, complemented |
| 71 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 72 | #208 | — | 36 | 1 | no | — |
| 73 | C22×C32 | — | 3 | 1 | no | — |
| 74 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 75 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 76 | C2×C3⋊D3 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 77 | C2×C3×D3 | — | 24 | 2 | no | — |
| 78 | #157 | — | 12 | 2 | no | — |
| 79 | C32×C4 | — | 3 | 1 | no | — |
| 80 | C32×C4 | — | 8 | 2 | no | — |
| 81 | C32×C4 | — | 16 | 2 | no | — |
| 82 | C32×C4 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 83 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 84 | C3⋊Dic3 | C2×S4 | 1 | 1 | yes | characteristic |
| 85 | C3×Dic3 | — | 24 | 2 | no | — |
| 86 | C2×#41 | — | 9 | 1 | no | — |
| 87 | C2×M16 | — | 27 | 1 | no | — |
| 88 | #117 | — | 27 | 1 | no | — |
| 89 | #104 | — | 27 | 2 | no | — |
| 90 | #104 | — | 27 | 2 | no | — |
| 91 | C33 | — | 4 | 1 | no | Sylow |
| 92 | C22×D3 | — | 18 | 2 | no | — |
| 93 | C3×Q8 | — | 2 | 2 | no | — |
| 94 | C3×D4 | — | 6 | 2 | no | — |
| 95 | C2×C3×C4 | — | 6 | 2 | no | — |
| 96 | C2×C3×C4 | — | 36 | 1 | no | — |
| 97 | #67 | — | 18 | 4 | no | — |
| 98 | C2×Dic3 | — | 18 | 2 | no | — |
| 99 | D12 | — | 18 | 2 | no | — |
| 100 | C4×D3 | — | 6 | 2 | no | — |
| 101 | C4×D3 | — | 18 | 2 | no | — |
| 102 | C4×D3 | — | 18 | 2 | no | — |
| 103 | C4×D3 | — | 18 | 2 | no | — |
| 104 | Dic6 | — | 18 | 2 | no | — |
| 105 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 106 | SL(2,𝔽3) | C2×#157 | 1 | 1 | yes | characteristic |
| 107 | C3⋊C8 | — | 36 | 2 | no | — |
| 108 | C2×C32 | — | 6 | 1 | no | — |
| 109 | C2×C32 | — | 8 | 2 | no | — |
| 110 | C2×C32 | — | 16 | 2 | no | — |
| 111 | C2×C32 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 112 | C3⋊D3 | — | 2 | 1 | no | — |
| 113 | C3⋊D3 | — | 6 | 1 | no | — |
| 114 | C3×D3 | — | 24 | 4 | no | — |
| 115 | #41 | — | 9 | 1 | no | — |
| 116 | #41 | — | 27 | 1 | no | — |
| 117 | #41 | — | 27 | 1 | no | — |
| 118 | #41 | C3⋊#157 | 1 | 1 | yes | characteristic |
| 119 | C2×Q8 | — | 9 | 1 | no | — |
| 120 | C2×D4 | — | 27 | 1 | no | — |
| 121 | C22×C4 | — | 27 | 1 | no | — |
| 122 | M16 | — | 27 | 2 | no | — |
| 123 | M16 | — | 54 | 1 | no | — |
| 124 | C2×C8 | — | 54 | 1 | no | — |
| 125 | C4⋊C4 | — | 54 | 1 | no | — |
| 126 | C22⋊C4 | — | 54 | 1 | no | — |
| 127 | C42 | — | 27 | 2 | no | — |
| 128 | C22×C3 | — | 6 | 2 | no | — |
| 129 | C22×C3 | — | 36 | 1 | no | — |
| 130 | C2×D3 | — | 6 | 2 | no | — |
| 131 | C2×D3 | — | 18 | 2 | no | — |
| 132 | C2×D3 | — | 36 | 4 | no | — |
| 133 | C3×C4 | — | 2 | 2 | no | — |
| 134 | C3×C4 | — | 4 | 1 | no | — |
| 135 | C3×C4 | — | 6 | 2 | no | — |
| 136 | C3×C4 | — | 16 | 2 | no | — |
| 137 | C3×C4 | — | 36 | 1 | no | — |
| 138 | Dic3 | — | 6 | 2 | no | — |
| 139 | Dic3 | — | 18 | 2 | no | — |
| 140 | C32 | — | 8 | 2 | no | — |
| 141 | C32 | — | 16 | 2 | no | — |
| 142 | C32 | #5423 | 1 | 1 | yes | characteristic, complemented |
| 143 | C23 | — | 27 | 1 | no | — |
| 144 | Q8 | — | 27 | 1 | no | — |
| 145 | Q8 | C2×C3⋊#157 | 1 | 1 | yes | characteristic |
| 146 | D4 | — | 3 | 1 | no | — |
| 147 | D4 | — | 27 | 1 | no | — |
| 148 | D4 | — | 54 | 1 | no | — |
| 149 | C2×C4 | — | 3 | 1 | no | — |
| 150 | C2×C4 | — | 9 | 1 | no | — |
| 151 | C2×C4 | — | 27 | 1 | no | — |
| 152 | C2×C4 | — | 27 | 1 | no | — |
| 153 | C2×C4 | — | 27 | 1 | no | — |
| 154 | C2×C4 | — | 27 | 1 | no | — |
| 155 | C2×C4 | — | 54 | 2 | no | — |
| 156 | C8 | — | 54 | 2 | no | — |
| 157 | C2×C3 | — | 2 | 2 | no | — |
| 158 | C2×C3 | — | 4 | 1 | no | — |
| 159 | C2×C3 | — | 12 | 2 | no | — |
| 160 | C2×C3 | — | 16 | 2 | no | — |
| 161 | C2×C3 | — | 72 | 1 | no | — |
| 162 | D3 | — | 6 | 4 | no | — |
| 163 | D3 | — | 36 | 2 | no | — |
| 164 | C22 | — | 3 | 1 | no | — |
| 165 | C22 | — | 9 | 1 | no | — |
| 166 | C22 | — | 27 | 1 | no | — |
| 167 | C22 | — | 108 | 1 | no | — |
| 168 | C4 | — | 3 | 1 | no | — |
| 169 | C4 | — | 9 | 1 | no | — |
| 170 | C4 | — | 27 | 1 | no | — |
| 171 | C4 | — | 108 | 2 | no | — |
| 172 | C4 | A4⋊#157 | 1 | 1 | yes | characteristic |
| 173 | C3 | — | 2 | 2 | no | — |
| 174 | C3 | — | 4 | 1 | no | — |
| 175 | C3 | — | 16 | 2 | no | — |
| 176 | C2 | — | 6 | 1 | no | — |
| 177 | C2 | — | 18 | 1 | no | — |
| 178 | C2 | — | 54 | 1 | no | — |
| 179 | C2 | C2×A4⋊#157 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11724155 | C2 | semidirect |
| 4. C32×#41⋊C3 | C4 | semidirect |
| 6. C3⋊D3×SL(2,𝔽3) | C4 | semidirect |
| 70. #41⋊C3 | #157 | semidirect |
| 142. C32 | #5423 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×#41⋊C3 | 2 |
| #11724155 | 2 |
| #10593842 | 3 |
| #88989 | 4 |
| #5423 | 9 |