Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
640 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 640 conjugacy classes of subgroups — too many to draw. Only its 87 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 | C42×C5×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | A4×#685 | C2 | 1 | 3 | yes | maximal |
| 3 | A4×#683 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C22×#4022 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 5 | C22×D5×A4 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×C4×C5×A4 | C22 | 1 | 3 | yes | — |
| 7 | C2×A4×Dic5 | C22 | 1 | 3 | yes | — |
| 8 | C3×#4022 | — | 4 | 1 | no | maximal |
| 9 | A4×#126 | — | 5 | 1 | no | maximal, Hall |
| 10 | C22×C42×C5 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×#685 | C2×C3 | 1 | 3 | yes | — |
| 12 | C22×#683 | C2×C3 | 1 | 3 | yes | complemented |
| 13 | C2×#4022 | — | 3 | 1 | no | — |
| 14 | C2×#4022 | — | 3 | 1 | no | — |
| 15 | C2×#4022 | — | 3 | 3 | no | — |
| 16 | C2×#4022 | — | 3 | 3 | no | — |
| 17 | C22×C5×A4 | C23 | 1 | 1 | yes | characteristic |
| 18 | C2×D5×A4 | — | 2 | 3 | no | — |
| 19 | C4×C5×A4 | — | 2 | 3 | no | — |
| 20 | A4×Dic5 | — | 2 | 3 | no | — |
| 21 | C3×C42×C5 | — | 4 | 1 | no | — |
| 22 | C3×#685 | — | 4 | 3 | no | — |
| 23 | C3×#683 | — | 4 | 3 | no | — |
| 24 | A4×C4⋊C4 | — | 5 | 3 | no | — |
| 25 | A4×C22⋊C4 | — | 5 | 3 | no | — |
| 26 | C42×A4 | D5 | 1 | 1 | yes | characteristic, complemented |
| 27 | C24×D5 | C22×C3 | 1 | 1 | yes | characteristic |
| 28 | C23×C4×C5 | C22×C3 | 1 | 3 | yes | — |
| 29 | C23×Dic5 | C22×C3 | 1 | 3 | yes | — |
| 30 | C2×C42×C5 | — | 3 | 1 | no | — |
| 31 | C2×C42×C5 | — | 3 | 3 | no | — |
| 32 | C2×#685 | — | 3 | 3 | no | — |
| 33 | C2×#685 | — | 3 | 3 | no | — |
| 34 | C2×#685 | — | 3 | 3 | no | — |
| 35 | C2×#685 | — | 3 | 3 | no | — |
| 36 | C2×#683 | — | 3 | 3 | no | — |
| 37 | C2×#683 | — | 3 | 3 | no | — |
| 38 | C2×#683 | — | 3 | 6 | no | — |
| 39 | #4022 | — | 3 | 2 | no | — |
| 40 | #4022 | — | 3 | 6 | no | — |
| 41 | #4022 | — | 3 | 6 | no | — |
| 42 | #4022 | — | 3 | 1 | no | — |
| 43 | #4022 | — | 3 | 3 | no | — |
| 44 | #4022 | — | 3 | 3 | no | — |
| 45 | #4022 | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 46 | C22×#126 | — | 5 | 1 | no | Sylow |
| 47 | C22×C3×D5 | — | 4 | 1 | no | — |
| 48 | C2×C5×A4 | #41 | 1 | 3 | yes | — |
| 49 | D5×A4 | — | 4 | 1 | no | — |
| 50 | C2×C3×C4×C5 | — | 4 | 3 | no | — |
| 51 | C2×C3×Dic5 | — | 4 | 3 | no | — |
| 52 | C23×A4 | — | 5 | 1 | no | — |
| 53 | C2×C4×A4 | — | 5 | 3 | no | — |
| 54 | C2×C4×A4 | C2×D5 | 1 | 3 | yes | — |
| 55 | C3×#126 | — | 20 | 1 | no | — |
| 56 | C24×C5 | C23×C3 | 1 | 1 | yes | characteristic |
| 57 | C23×D5 | — | 2 | 3 | no | — |
| 58 | C23×D5 | — | 3 | 1 | no | — |
| 59 | C23×D5 | — | 3 | 1 | no | — |
| 60 | C23×D5 | — | 6 | 3 | no | — |
| 61 | C22×C4×C5 | — | 2 | 3 | no | — |
| 62 | C22×C4×C5 | — | 3 | 3 | no | — |
| 63 | C22×C4×C5 | — | 3 | 3 | no | — |
| 64 | C22×C4×C5 | — | 6 | 3 | no | — |
| 65 | C22×Dic5 | — | 2 | 3 | no | — |
| 66 | C22×Dic5 | — | 3 | 3 | no | — |
| 67 | C22×Dic5 | — | 3 | 3 | no | — |
| 68 | C22×Dic5 | — | 6 | 3 | no | — |
| 69 | C42×C5 | — | 3 | 2 | no | — |
| 70 | C42×C5 | — | 3 | 3 | no | — |
| 71 | C42×C5 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 72 | #685 | — | 3 | 6 | no | — |
| 73 | #685 | — | 3 | 3 | no | — |
| 74 | #685 | — | 3 | 3 | no | — |
| 75 | #685 | — | 3 | 3 | no | — |
| 76 | #685 | C2×A4 | 1 | 3 | yes | — |
| 77 | #683 | — | 3 | 6 | no | — |
| 78 | #683 | — | 3 | 3 | no | — |
| 79 | #683 | — | 3 | 6 | no | — |
| 80 | #683 | C2×A4 | 1 | 3 | yes | complemented |
| 81 | C2×#126 | — | 15 | 1 | no | — |
| 82 | C2×#126 | — | 15 | 1 | no | — |
| 83 | C2×#126 | — | 15 | 3 | no | — |
| 84 | C2×#126 | — | 15 | 3 | no | — |
| 85 | C22×C4⋊C4 | — | 5 | 3 | no | — |
| 86 | C22×C22⋊C4 | — | 5 | 3 | no | — |
| 87 | C22×C42 | C3×D5 | 1 | 1 | yes | characteristic, complemented |
| 88 | C22×C3×C5 | — | 4 | 1 | no | — |
| 89 | C2×C3×D5 | — | 8 | 3 | no | — |
| 90 | C5×A4 | #126 | 1 | 1 | yes | characteristic, complemented |
| 91 | C3×C4×C5 | — | 8 | 3 | no | — |
| 92 | C3×Dic5 | — | 8 | 3 | no | — |
| 93 | C22×A4 | — | 10 | 3 | no | — |
| 94 | C22×A4 | C22×D5 | 1 | 1 | yes | characteristic |
| 95 | C4×A4 | — | 2 | 3 | no | — |
| 96 | C4×A4 | — | 10 | 3 | no | — |
| 97 | C3×C4⋊C4 | — | 20 | 3 | no | — |
| 98 | C3×C22⋊C4 | — | 20 | 3 | no | — |
| 99 | C3×C42 | — | 4 | 1 | no | — |
| 100 | C23×C5 | — | 3 | 3 | no | — |
| 101 | C23×C5 | — | 3 | 1 | no | — |
| 102 | C23×C5 | C3×#41 | 1 | 3 | yes | — |
| 103 | C22×D5 | — | 3 | 1 | no | — |
| 104 | C22×D5 | — | 4 | 1 | no | — |
| 105 | C22×D5 | — | 6 | 3 | no | — |
| 106 | C22×D5 | — | 6 | 3 | no | — |
| 107 | C22×D5 | — | 6 | 3 | no | — |
| 108 | C22×D5 | — | 6 | 3 | no | — |
| 109 | C22×D5 | — | 12 | 2 | no | — |
| 110 | C22×D5 | — | 12 | 3 | no | — |
| 111 | C22×D5 | C22×A4 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C5 | — | 3 | 3 | no | — |
| 113 | C2×C4×C5 | — | 6 | 3 | no | — |
| 114 | C2×C4×C5 | — | 6 | 3 | no | — |
| 115 | C2×C4×C5 | — | 6 | 3 | no | — |
| 116 | C2×C4×C5 | — | 6 | 3 | no | — |
| 117 | C2×C4×C5 | C22×A4 | 1 | 3 | yes | — |
| 118 | C2×Dic5 | — | 3 | 3 | no | — |
| 119 | C2×Dic5 | — | 6 | 3 | no | — |
| 120 | C2×Dic5 | — | 6 | 3 | no | — |
| 121 | C2×Dic5 | — | 6 | 3 | no | — |
| 122 | C2×Dic5 | — | 6 | 3 | no | — |
| 123 | C2×Dic5 | C22×A4 | 1 | 3 | yes | — |
| 124 | C25 | — | 5 | 1 | no | — |
| 125 | C23×C4 | — | 5 | 3 | no | — |
| 126 | C23×C4 | C2×C3×D5 | 1 | 3 | yes | — |
| 127 | #126 | — | 5 | 1 | no | — |
| 128 | #126 | — | 15 | 3 | no | — |
| 129 | #126 | — | 15 | 3 | no | — |
| 130 | #126 | — | 15 | 6 | no | — |
| 131 | #126 | — | 15 | 2 | no | — |
| 132 | #126 | — | 15 | 6 | no | — |
| 133 | #126 | — | 15 | 1 | no | — |
| 134 | C2×C4⋊C4 | — | 15 | 3 | no | — |
| 135 | C2×C4⋊C4 | — | 15 | 6 | no | — |
| 136 | C2×C4⋊C4 | — | 15 | 3 | no | — |
| 137 | C2×C22⋊C4 | — | 15 | 3 | no | — |
| 138 | C2×C22⋊C4 | — | 15 | 3 | no | — |
| 139 | C2×C22⋊C4 | — | 15 | 3 | no | — |
| 140 | C2×C22⋊C4 | — | 15 | 3 | no | — |
| 141 | C2×C42 | — | 3 | 1 | no | — |
| 142 | C2×C42 | — | 3 | 3 | no | — |
| 143 | C2×C3×C5 | — | 4 | 3 | no | — |
| 144 | C3×D5 | — | 16 | 1 | no | — |
| 145 | C23×C3 | — | 20 | 1 | no | — |
| 146 | C2×A4 | — | 20 | 1 | no | — |
| 147 | C2×A4 | #709 | 1 | 3 | yes | — |
| 148 | C2×C3×C4 | — | 4 | 3 | no | — |
| 149 | C2×C3×C4 | — | 20 | 3 | no | — |
| 150 | C22×C5 | — | 3 | 3 | no | — |
| 151 | C22×C5 | — | 3 | 2 | no | — |
| 152 | C22×C5 | — | 3 | 3 | no | — |
| 153 | C22×C5 | — | 3 | 3 | no | — |
| 154 | C22×C5 | C3×#126 | 1 | 1 | yes | characteristic, complemented |
| 155 | C22×C5 | C23×A4 | 1 | 1 | yes | characteristic |
| 156 | C2×D5 | — | 2 | 3 | no | — |
| 157 | C2×D5 | — | 6 | 3 | no | — |
| 158 | C2×D5 | — | 12 | 1 | no | — |
| 159 | C2×D5 | — | 12 | 3 | no | — |
| 160 | C2×D5 | — | 12 | 1 | no | — |
| 161 | C2×D5 | — | 12 | 3 | no | — |
| 162 | C4×C5 | — | 2 | 3 | no | — |
| 163 | C4×C5 | — | 6 | 3 | no | — |
| 164 | Dic5 | — | 2 | 3 | no | — |
| 165 | Dic5 | — | 6 | 3 | no | — |
| 166 | C24 | — | 10 | 3 | no | — |
| 167 | C24 | — | 15 | 1 | no | — |
| 168 | C24 | — | 15 | 1 | no | — |
| 169 | C24 | — | 30 | 3 | no | — |
| 170 | C24 | C22×C3×D5 | 1 | 1 | yes | characteristic |
| 171 | C22×C4 | — | 2 | 3 | no | — |
| 172 | C22×C4 | — | 3 | 3 | no | — |
| 173 | C22×C4 | — | 3 | 3 | no | — |
| 174 | C22×C4 | — | 6 | 3 | no | — |
| 175 | C22×C4 | — | 10 | 3 | no | — |
| 176 | C22×C4 | — | 15 | 3 | no | — |
| 177 | C22×C4 | — | 15 | 3 | no | — |
| 178 | C22×C4 | — | 30 | 3 | no | — |
| 179 | C4⋊C4 | — | 5 | 3 | no | — |
| 180 | C4⋊C4 | — | 15 | 6 | no | — |
| 181 | C4⋊C4 | — | 15 | 6 | no | — |
| 182 | C4⋊C4 | — | 15 | 3 | no | — |
| 183 | C22⋊C4 | — | 5 | 3 | no | — |
| 184 | C22⋊C4 | — | 15 | 3 | no | — |
| 185 | C22⋊C4 | — | 15 | 3 | no | — |
| 186 | C22⋊C4 | — | 15 | 6 | no | — |
| 187 | C22⋊C4 | — | 15 | 3 | no | — |
| 188 | C42 | — | 3 | 2 | no | — |
| 189 | C42 | — | 3 | 3 | no | — |
| 190 | C42 | D5×A4 | 1 | 1 | yes | characteristic, complemented |
| 191 | C3×C5 | — | 4 | 1 | no | Hall |
| 192 | C22×C3 | — | 4 | 1 | no | — |
| 193 | C22×C3 | — | 40 | 3 | no | — |
| 194 | A4 | #4022 | 1 | 1 | yes | characteristic, direct product factor |
| 195 | C3×C4 | — | 8 | 3 | no | — |
| 196 | C3×C4 | — | 40 | 3 | no | — |
| 197 | C2×C5 | — | 3 | 3 | no | — |
| 198 | C2×C5 | — | 3 | 1 | no | — |
| 199 | C2×C5 | A4×#41 | 1 | 3 | yes | — |
| 200 | D5 | — | 4 | 1 | no | — |
| 201 | D5 | — | 12 | 1 | no | — |
| 202 | C23 | — | 3 | 3 | no | — |
| 203 | C23 | — | 3 | 1 | no | — |
| 204 | C23 | — | 5 | 1 | no | — |
| 205 | C23 | — | 15 | 1 | no | — |
| 206 | C23 | — | 20 | 1 | no | — |
| 207 | C23 | — | 30 | 3 | no | — |
| 208 | C23 | — | 30 | 3 | no | — |
| 209 | C23 | — | 30 | 3 | no | — |
| 210 | C23 | — | 30 | 3 | no | — |
| 211 | C23 | — | 60 | 3 | no | — |
| 212 | C23 | — | 60 | 2 | no | — |
| 213 | C23 | C3×#709 | 1 | 3 | yes | — |
| 214 | C2×C4 | — | 3 | 3 | no | — |
| 215 | C2×C4 | — | 5 | 3 | no | — |
| 216 | C2×C4 | — | 6 | 3 | no | — |
| 217 | C2×C4 | — | 6 | 3 | no | — |
| 218 | C2×C4 | — | 6 | 3 | no | — |
| 219 | C2×C4 | — | 6 | 3 | no | — |
| 220 | C2×C4 | — | 15 | 3 | no | — |
| 221 | C2×C4 | — | 30 | 3 | no | — |
| 222 | C2×C4 | — | 30 | 3 | no | — |
| 223 | C2×C4 | — | 30 | 3 | no | — |
| 224 | C2×C4 | — | 30 | 3 | no | — |
| 225 | C2×C4 | C2×D5×A4 | 1 | 3 | yes | — |
| 226 | C2×C3 | — | 4 | 3 | no | — |
| 227 | C2×C3 | — | 80 | 1 | no | — |
| 228 | C5 | A4×#126 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 229 | C22 | — | 3 | 3 | no | — |
| 230 | C22 | — | 3 | 2 | no | — |
| 231 | C22 | — | 3 | 3 | no | — |
| 232 | C22 | — | 3 | 3 | no | — |
| 233 | C22 | — | 10 | 3 | no | — |
| 234 | C22 | — | 30 | 3 | no | — |
| 235 | C22 | — | 60 | 1 | no | — |
| 236 | C22 | — | 60 | 3 | no | — |
| 237 | C22 | — | 60 | 1 | no | — |
| 238 | C22 | — | 60 | 3 | no | — |
| 239 | C22 | C3×#4022 | 1 | 1 | yes | characteristic, complemented |
| 240 | C22 | C22×D5×A4 | 1 | 1 | yes | characteristic |
| 241 | C4 | — | 2 | 3 | no | — |
| 242 | C4 | — | 6 | 3 | no | — |
| 243 | C4 | — | 10 | 3 | no | — |
| 244 | C4 | — | 30 | 3 | no | — |
| 245 | C3 | — | 4 | 1 | no | Sylow |
| 246 | C2 | — | 3 | 1 | no | — |
| 247 | C2 | — | 3 | 3 | no | — |
| 248 | C2 | — | 20 | 1 | no | — |
| 249 | C2 | — | 60 | 1 | no | — |
| 250 | C2 | A4×#709 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C42×C5×A4 | C2 | semidirect |
| 3. A4×#683 | C2 | semidirect |
| 4. C22×#4022 | C3 | semidirect |
| 10. C22×C42×C5 | C2×C3 | semidirect |
| 12. C22×#683 | C2×C3 | semidirect |
| 26. C42×A4 | D5 | semidirect |
| 45. #4022 | A4 | direct (N × H) |
| 71. C42×C5 | C2×A4 | semidirect |
| 80. #683 | C2×A4 | semidirect |
| 87. C22×C42 | C3×D5 | semidirect |
| 90. C5×A4 | #126 | semidirect |
| 154. C22×C5 | C3×#126 | semidirect |
| 190. C42 | D5×A4 | semidirect |
| 194. A4 | #4022 | direct (N × H) |
| 228. C5 | A4×#126 | semidirect |
| 239. C22 | C3×#4022 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C42×C5×A4 | 2 |
| A4×#685 | 2 |
| A4×#683 | 2 |
| C22×#4022 | 3 |
| C3×#4022 | 4 |
| A4×#126 | 5 |