Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
536 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 15 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×C22⋊A4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10593847 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×C22⋊A4 | C4 | 1 | 1 | yes | characteristic, complemented |
| 4 | A4×#157 | — | 4 | 1 | no | maximal |
| 5 | C24×C3⋊D3 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | D3×C22⋊A4 | — | 6 | 2 | no | — |
| 7 | C2×C22⋊#157 | — | 3 | 1 | no | — |
| 8 | C2×C22⋊#157 | — | 3 | 1 | no | — |
| 9 | A4×C3⋊D3 | — | 4 | 1 | no | — |
| 10 | A4×C3⋊D3 | — | 8 | 2 | no | — |
| 11 | #379⋊C3 | — | 9 | 1 | no | maximal |
| 12 | C24×C32 | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C23×C3⋊D3 | — | 3 | 1 | no | — |
| 14 | C23×C3⋊D3 | — | 3 | 1 | no | — |
| 15 | C23×C3⋊D3 | — | 6 | 2 | no | — |
| 16 | C23×C3⋊D3 | — | 6 | 2 | no | — |
| 17 | C3×C22⋊A4 | — | 2 | 2 | no | — |
| 18 | C3×C22⋊A4 | — | 4 | 2 | no | — |
| 19 | C22×#157 | — | 3 | 1 | no | — |
| 20 | C22×#157 | A4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C22⋊#157 | — | 6 | 2 | no | — |
| 22 | C22⋊#157 | — | 6 | 2 | no | — |
| 23 | C32×A4 | — | 4 | 1 | no | — |
| 24 | C32×A4 | — | 8 | 2 | no | — |
| 25 | C3×#157 | — | 16 | 1 | no | — |
| 26 | C24×D3 | — | 6 | 2 | no | — |
| 27 | C2×C22⋊A4 | — | 9 | 1 | no | — |
| 28 | C23×C32 | — | 3 | 1 | no | — |
| 29 | C23×C32 | — | 6 | 2 | no | — |
| 30 | C22×C3⋊D3 | — | 2 | 2 | no | — |
| 31 | C22×C3⋊D3 | — | 3 | 2 | no | — |
| 32 | C22×C3⋊D3 | — | 3 | 2 | no | — |
| 33 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 34 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 35 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 36 | C22×C3⋊D3 | — | 6 | 4 | no | — |
| 37 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 38 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 39 | C22×C3⋊D3 | — | 6 | 4 | no | — |
| 40 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 41 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 42 | C22×C3⋊D3 | C2×A4 | 1 | 1 | yes | characteristic |
| 43 | C2×#157 | — | 6 | 1 | no | — |
| 44 | C2×#157 | — | 6 | 1 | no | — |
| 45 | C2×#157 | — | 6 | 2 | no | — |
| 46 | D3×A4 | — | 24 | 2 | no | — |
| 47 | D3×A4 | — | 24 | 8 | no | — |
| 48 | #379 | — | 9 | 1 | no | Sylow |
| 49 | C3×C3⋊D3 | — | 16 | 1 | no | — |
| 50 | C24×C3 | — | 2 | 2 | no | — |
| 51 | C23×D3 | — | 18 | 2 | no | — |
| 52 | C23×D3 | — | 18 | 2 | no | — |
| 53 | C23×D3 | — | 18 | 8 | no | — |
| 54 | C23×D3 | — | 18 | 8 | no | — |
| 55 | C22⋊A4 | — | 4 | 2 | no | — |
| 56 | C22⋊A4 | #157 | 1 | 1 | yes | characteristic, complemented |
| 57 | C4×A4 | — | 36 | 1 | no | — |
| 58 | C22×C32 | — | 2 | 2 | no | — |
| 59 | C22×C32 | — | 3 | 2 | no | — |
| 60 | C22×C32 | — | 6 | 2 | no | — |
| 61 | C22×C32 | — | 6 | 2 | no | — |
| 62 | C22×C32 | C4×A4 | 1 | 1 | yes | characteristic |
| 63 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 64 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 65 | C2×C3⋊D3 | — | 3 | 2 | no | — |
| 66 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 67 | C2×C3⋊D3 | — | 6 | 4 | no | — |
| 68 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 69 | C2×C3⋊D3 | — | 6 | 4 | no | — |
| 70 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 71 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 72 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 73 | C3×A4 | — | 8 | 2 | no | — |
| 74 | C3×A4 | — | 8 | 8 | no | — |
| 75 | C3×A4 | — | 16 | 8 | no | — |
| 76 | C3×A4 | — | 16 | 2 | no | — |
| 77 | #157 | — | 4 | 1 | no | — |
| 78 | #157 | — | 12 | 1 | no | — |
| 79 | C25 | — | 9 | 1 | no | — |
| 80 | C2×C22⋊C4 | — | 27 | 1 | no | — |
| 81 | C2×C22⋊C4 | — | 27 | 1 | no | — |
| 82 | C33 | — | 16 | 1 | no | Sylow |
| 83 | C23×C3 | — | 6 | 2 | no | — |
| 84 | C23×C3 | — | 6 | 8 | no | — |
| 85 | C22×D3 | — | 6 | 2 | no | — |
| 86 | C22×D3 | — | 6 | 8 | no | — |
| 87 | C22×D3 | — | 18 | 2 | no | — |
| 88 | C22×D3 | — | 18 | 4 | no | — |
| 89 | C22×D3 | — | 18 | 4 | no | — |
| 90 | C22×D3 | — | 18 | 8 | no | — |
| 91 | C22×D3 | — | 18 | 8 | no | — |
| 92 | C22×D3 | — | 18 | 8 | no | — |
| 93 | C22×D3 | — | 18 | 16 | no | — |
| 94 | C22×D3 | — | 18 | 8 | no | — |
| 95 | C22×D3 | — | 18 | 8 | no | — |
| 96 | C22×D3 | — | 18 | 8 | no | — |
| 97 | C22×D3 | — | 18 | 16 | no | — |
| 98 | C2×A4 | — | 36 | 1 | no | — |
| 99 | C2×A4 | — | 72 | 2 | no | — |
| 100 | C2×C32 | — | 3 | 1 | no | — |
| 101 | C2×C32 | — | 6 | 2 | no | — |
| 102 | C3⋊D3 | — | 3 | 1 | no | — |
| 103 | C3⋊D3 | — | 6 | 2 | no | — |
| 104 | C3⋊D3 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 105 | C3×D3 | — | 96 | 2 | no | — |
| 106 | C24 | — | 27 | 1 | no | — |
| 107 | C24 | — | 27 | 1 | no | — |
| 108 | C24 | — | 54 | 2 | no | — |
| 109 | C24 | — | 54 | 2 | no | — |
| 110 | C24 | C3×#157 | 1 | 1 | yes | characteristic, complemented |
| 111 | C22×C4 | — | 9 | 1 | no | — |
| 112 | C22×C4 | — | 27 | 1 | no | — |
| 113 | C22⋊C4 | — | 54 | 2 | no | — |
| 114 | C22⋊C4 | — | 54 | 2 | no | — |
| 115 | C22×C3 | — | 2 | 8 | no | — |
| 116 | C22×C3 | — | 2 | 2 | no | — |
| 117 | C22×C3 | — | 6 | 4 | no | — |
| 118 | C22×C3 | — | 6 | 8 | no | — |
| 119 | C22×C3 | — | 6 | 8 | no | — |
| 120 | C2×D3 | — | 18 | 2 | no | — |
| 121 | C2×D3 | — | 18 | 4 | no | — |
| 122 | C2×D3 | — | 18 | 2 | no | — |
| 123 | C2×D3 | — | 18 | 8 | no | — |
| 124 | C2×D3 | — | 18 | 8 | no | — |
| 125 | C2×D3 | — | 18 | 8 | no | — |
| 126 | C2×D3 | — | 18 | 16 | no | — |
| 127 | C2×D3 | — | 18 | 8 | no | — |
| 128 | C2×D3 | — | 18 | 16 | no | — |
| 129 | C2×D3 | — | 18 | 8 | no | — |
| 130 | A4 | — | 4 | 1 | no | — |
| 131 | A4 | — | 8 | 2 | no | — |
| 132 | A4 | — | 16 | 8 | no | — |
| 133 | A4 | — | 16 | 2 | no | — |
| 134 | C3×C4 | — | 144 | 1 | no | — |
| 135 | C32 | — | 32 | 2 | no | — |
| 136 | C32 | — | 64 | 2 | no | — |
| 137 | C32 | #379⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 138 | C23 | — | 3 | 1 | no | — |
| 139 | C23 | — | 6 | 2 | no | — |
| 140 | C23 | — | 9 | 1 | no | — |
| 141 | C23 | — | 18 | 2 | no | — |
| 142 | C23 | — | 27 | 2 | no | — |
| 143 | C23 | — | 27 | 2 | no | — |
| 144 | C23 | — | 27 | 1 | no | — |
| 145 | C23 | — | 54 | 2 | no | — |
| 146 | C23 | — | 54 | 2 | no | — |
| 147 | C23 | — | 54 | 2 | no | — |
| 148 | C23 | — | 54 | 2 | no | — |
| 149 | C23 | — | 54 | 4 | no | — |
| 150 | C23 | — | 54 | 2 | no | — |
| 151 | C23 | — | 54 | 2 | no | — |
| 152 | C23 | — | 54 | 4 | no | — |
| 153 | C2×C4 | — | 54 | 1 | no | — |
| 154 | C2×C4 | — | 54 | 1 | no | — |
| 155 | C2×C4 | — | 54 | 2 | no | — |
| 156 | C2×C3 | — | 6 | 2 | no | — |
| 157 | C2×C3 | — | 6 | 8 | no | — |
| 158 | C2×C3 | — | 144 | 1 | no | — |
| 159 | D3 | — | 6 | 2 | no | — |
| 160 | D3 | — | 18 | 2 | no | — |
| 161 | D3 | — | 18 | 8 | no | — |
| 162 | C22 | — | 2 | 2 | no | — |
| 163 | C22 | — | 3 | 2 | no | — |
| 164 | C22 | — | 6 | 2 | no | — |
| 165 | C22 | — | 6 | 2 | no | — |
| 166 | C22 | — | 27 | 1 | no | — |
| 167 | C22 | — | 27 | 1 | no | — |
| 168 | C22 | — | 27 | 2 | no | — |
| 169 | C22 | — | 54 | 2 | no | — |
| 170 | C22 | — | 54 | 2 | no | — |
| 171 | C22 | — | 54 | 2 | no | — |
| 172 | C22 | — | 54 | 4 | no | — |
| 173 | C22 | — | 54 | 2 | no | — |
| 174 | C22 | — | 54 | 2 | no | — |
| 175 | C22 | — | 54 | 4 | no | — |
| 176 | C22 | A4×#157 | 1 | 1 | yes | characteristic |
| 177 | C4 | — | 36 | 1 | no | — |
| 178 | C4 | — | 108 | 1 | no | — |
| 179 | C3 | — | 2 | 2 | no | — |
| 180 | C3 | — | 16 | 1 | no | — |
| 181 | C3 | — | 64 | 2 | no | — |
| 182 | C2 | — | 3 | 1 | no | — |
| 183 | C2 | — | 6 | 2 | no | — |
| 184 | C2 | — | 9 | 1 | no | — |
| 185 | C2 | — | 27 | 1 | no | — |
| 186 | C2 | — | 54 | 2 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C24×C32 | C3×C4 |
| Frattini | C22 | A4×#157 |
| socle | C22×C32 | C4×A4 |
| prime radical | C3⋊D3×C22⋊A4 | C2 |
| cosocle | C24×C3⋊D3 | C2×C3 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10593847 | C3 | semidirect |
| 3. C32×C22⋊A4 | C4 | semidirect |
| 12. C24×C32 | C3×C4 | semidirect |
| 20. C22×#157 | A4 | semidirect |
| 56. C22⋊A4 | #157 | semidirect |
| 110. C24 | C3×#157 | semidirect |
| 137. C32 | #379⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×C22⋊A4 | 2 |
| #10593847 | 3 |
| A4×#157 | 4 |
| #379⋊C3 | 9 |