Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
252 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 36 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 | A4×#700 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11756716 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | D5×A4⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10603917 | — | 3 | 1 | no | maximal, Hall |
| 5 | C4×D5×A4 | C22 | 1 | 1 | yes | characteristic |
| 6 | A4×#170 | — | 2 | 1 | no | — |
| 7 | A4⋊#170 | — | 2 | 1 | no | — |
| 8 | A4⋊#168 | C4 | 1 | 1 | yes | characteristic |
| 9 | C5×A4⋊C8 | C4 | 1 | 1 | yes | characteristic |
| 10 | #91162 | — | 4 | 1 | no | maximal |
| 11 | #73114 | — | 5 | 1 | no | maximal, Hall |
| 12 | C22×#700 | D3 | 1 | 1 | yes | characteristic |
| 13 | #65756 | — | 3 | 1 | no | — |
| 14 | C2×#3996 | — | 3 | 1 | no | — |
| 15 | C2×#3996 | — | 3 | 1 | no | — |
| 16 | D5×C22⋊C8 | — | 3 | 1 | no | — |
| 17 | #64896 | — | 3 | 1 | no | — |
| 18 | #64896 | — | 3 | 1 | no | — |
| 19 | C2×D5×A4 | Q8 | 1 | 1 | yes | characteristic |
| 20 | C4×C5×A4 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | #6758 | — | 4 | 1 | no | — |
| 22 | C3×#700 | — | 4 | 1 | no | — |
| 23 | A4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 24 | D5×C3⋊C8 | — | 4 | 1 | no | — |
| 25 | A4×M16 | — | 5 | 1 | no | — |
| 26 | #5409 | — | 5 | 1 | no | — |
| 27 | C2×A4⋊C8 | — | 5 | 1 | no | — |
| 28 | C22×C4×D5 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C22×#170 | — | 2 | 1 | no | — |
| 30 | C2×#700 | — | 3 | 1 | no | — |
| 31 | C2×#700 | — | 3 | 1 | no | — |
| 32 | C2×#700 | — | 3 | 1 | no | — |
| 33 | C2×#700 | — | 3 | 1 | no | — |
| 34 | C2×#700 | — | 6 | 1 | no | — |
| 35 | C2×C8×D5 | — | 3 | 1 | no | — |
| 36 | C2×C8×D5 | — | 3 | 1 | no | — |
| 37 | C22⋊#170 | — | 6 | 1 | no | — |
| 38 | #4003 | — | 6 | 1 | no | — |
| 39 | #3996 | — | 6 | 2 | no | — |
| 40 | #3996 | — | 6 | 1 | no | — |
| 41 | #3996 | — | 6 | 1 | no | — |
| 42 | C5×C22⋊C8 | — | 3 | 1 | no | — |
| 43 | C22⋊#168 | — | 3 | 1 | no | — |
| 44 | #3952 | — | 3 | 1 | no | — |
| 45 | #3952 | — | 3 | 1 | no | — |
| 46 | #1855 | — | 15 | 1 | no | Sylow |
| 47 | C2×C5×A4 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 48 | D5×A4 | — | 2 | 1 | no | — |
| 49 | C3×C4×D5 | — | 4 | 1 | no | — |
| 50 | C3⋊#170 | — | 8 | 1 | no | — |
| 51 | C3×#170 | — | 8 | 1 | no | — |
| 52 | C3⋊#168 | — | 4 | 1 | no | — |
| 53 | C5×C3⋊C8 | — | 4 | 1 | no | — |
| 54 | C2×C4×A4 | — | 5 | 1 | no | — |
| 55 | C8×A4 | — | 10 | 1 | no | — |
| 56 | A4⋊C8 | — | 5 | 1 | no | — |
| 57 | A4⋊C8 | — | 10 | 1 | no | — |
| 58 | A4⋊C8 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 59 | #822 | — | 20 | 1 | no | — |
| 60 | C23×D5 | Dic6 | 1 | 1 | yes | characteristic |
| 61 | C22×C4×C5 | C4×D3 | 1 | 1 | yes | characteristic |
| 62 | C22×Dic5 | #67 | 1 | 1 | yes | characteristic |
| 63 | C2×C4×D5 | — | 3 | 1 | no | — |
| 64 | C2×C4×D5 | — | 3 | 1 | no | — |
| 65 | C2×C4×D5 | — | 3 | 1 | no | — |
| 66 | C2×C4×D5 | — | 3 | 1 | no | — |
| 67 | C2×#170 | — | 6 | 1 | no | — |
| 68 | C2×#170 | — | 6 | 1 | no | — |
| 69 | C2×#170 | — | 6 | 1 | no | — |
| 70 | C2×#170 | — | 6 | 1 | no | — |
| 71 | #700 | — | 3 | 2 | no | — |
| 72 | #700 | — | 3 | 1 | no | — |
| 73 | #700 | — | 6 | 1 | no | — |
| 74 | #700 | — | 6 | 1 | no | — |
| 75 | #700 | — | 6 | 1 | no | — |
| 76 | #700 | — | 6 | 1 | no | — |
| 77 | #700 | — | 6 | 1 | no | — |
| 78 | #700 | S4 | 1 | 1 | yes | characteristic |
| 79 | C2×C5×C8 | — | 3 | 1 | no | — |
| 80 | C2×C5×C8 | — | 3 | 1 | no | — |
| 81 | C2×#168 | — | 3 | 1 | no | — |
| 82 | C2×#168 | — | 3 | 1 | no | — |
| 83 | C8×D5 | — | 6 | 1 | no | — |
| 84 | C8×D5 | — | 6 | 1 | no | — |
| 85 | C8×D5 | — | 6 | 1 | no | — |
| 86 | C8×D5 | — | 6 | 1 | no | — |
| 87 | C22×M16 | — | 5 | 1 | no | — |
| 88 | C2×#108 | — | 15 | 1 | no | — |
| 89 | C2×#108 | — | 15 | 1 | no | — |
| 90 | #407 | — | 15 | 1 | no | — |
| 91 | C2×C22⋊C8 | — | 15 | 1 | no | — |
| 92 | #341 | — | 15 | 1 | no | — |
| 93 | #341 | — | 15 | 1 | no | — |
| 94 | C2×C3×D5 | — | 4 | 1 | no | — |
| 95 | C5×A4 | #108 | 1 | 1 | yes | characteristic, complemented |
| 96 | C3×C4×C5 | — | 4 | 1 | no | — |
| 97 | C3×Dic5 | — | 4 | 1 | no | — |
| 98 | C22×A4 | — | 5 | 1 | no | — |
| 99 | C4×A4 | — | 5 | 1 | no | — |
| 100 | C4×A4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 101 | C3×M16 | — | 20 | 1 | no | — |
| 102 | #208 | — | 20 | 1 | no | — |
| 103 | C2×C3⋊C8 | — | 20 | 1 | no | — |
| 104 | C23×C5 | Dic3⋊C4 | 1 | 1 | yes | characteristic |
| 105 | C22×D5 | — | 2 | 1 | no | — |
| 106 | C22×D5 | — | 3 | 1 | no | — |
| 107 | C22×D5 | — | 3 | 1 | no | — |
| 108 | C22×D5 | — | 6 | 1 | no | — |
| 109 | C2×C4×C5 | — | 3 | 1 | no | — |
| 110 | C2×C4×C5 | — | 3 | 1 | no | — |
| 111 | C2×Dic5 | — | 3 | 1 | no | — |
| 112 | C2×Dic5 | — | 3 | 1 | no | — |
| 113 | C4×D5 | — | 3 | 1 | no | — |
| 114 | C4×D5 | — | 3 | 1 | no | — |
| 115 | C4×D5 | — | 3 | 1 | no | — |
| 116 | C4×D5 | — | 6 | 1 | no | — |
| 117 | C4×D5 | C2×S4 | 1 | 1 | yes | characteristic |
| 118 | #170 | — | 2 | 1 | no | — |
| 119 | #170 | — | 6 | 1 | no | — |
| 120 | #170 | — | 12 | 1 | no | — |
| 121 | #170 | — | 12 | 1 | no | — |
| 122 | C5×C8 | — | 6 | 1 | no | — |
| 123 | C5×C8 | — | 6 | 1 | no | — |
| 124 | #168 | — | 6 | 1 | no | — |
| 125 | #168 | — | 6 | 1 | no | — |
| 126 | C23×C4 | — | 5 | 1 | no | — |
| 127 | C2×M16 | — | 15 | 1 | no | — |
| 128 | C2×M16 | — | 15 | 1 | no | — |
| 129 | C2×M16 | — | 15 | 1 | no | — |
| 130 | C2×M16 | — | 15 | 1 | no | — |
| 131 | C2×M16 | — | 30 | 1 | no | — |
| 132 | C22×C8 | — | 10 | 1 | no | — |
| 133 | C22×C8 | — | 15 | 1 | no | — |
| 134 | C22×C8 | — | 15 | 1 | no | — |
| 135 | #108 | — | 30 | 1 | no | — |
| 136 | #108 | — | 30 | 1 | no | — |
| 137 | #108 | — | 30 | 2 | no | — |
| 138 | C22⋊C8 | — | 3 | 1 | no | — |
| 139 | C22⋊C8 | — | 15 | 1 | no | — |
| 140 | C22⋊C8 | — | 15 | 1 | no | — |
| 141 | C22⋊C8 | — | 15 | 1 | no | — |
| 142 | C22⋊C8 | — | 30 | 1 | no | — |
| 143 | C22⋊C8 | — | 30 | 1 | no | — |
| 144 | C2×C3×C5 | — | 4 | 1 | no | — |
| 145 | C3×D5 | — | 8 | 1 | no | — |
| 146 | C2×A4 | — | 10 | 1 | no | — |
| 147 | C2×A4 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 148 | C2×C3×C4 | — | 20 | 1 | no | — |
| 149 | C3×C8 | — | 40 | 1 | no | — |
| 150 | C3⋊C8 | — | 4 | 1 | no | — |
| 151 | C3⋊C8 | — | 20 | 1 | no | — |
| 152 | C3⋊C8 | — | 40 | 1 | no | — |
| 153 | C22×C5 | — | 3 | 1 | no | — |
| 154 | C22×C5 | — | 3 | 1 | no | — |
| 155 | C22×C5 | #822 | 1 | 1 | yes | characteristic, complemented |
| 156 | C2×D5 | — | 3 | 1 | no | — |
| 157 | C2×D5 | — | 6 | 1 | no | — |
| 158 | C2×D5 | — | 6 | 1 | no | — |
| 159 | C2×D5 | — | 6 | 1 | no | — |
| 160 | C2×D5 | — | 6 | 1 | no | — |
| 161 | C2×D5 | A4⋊Q8 | 1 | 1 | yes | characteristic |
| 162 | C4×C5 | — | 3 | 1 | no | — |
| 163 | C4×C5 | C4×S4 | 1 | 1 | yes | characteristic |
| 164 | Dic5 | — | 3 | 1 | no | — |
| 165 | Dic5 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 166 | C24 | — | 5 | 1 | no | — |
| 167 | C22×C4 | — | 5 | 1 | no | — |
| 168 | C22×C4 | — | 15 | 1 | no | — |
| 169 | C22×C4 | — | 15 | 1 | no | — |
| 170 | C22×C4 | — | 15 | 1 | no | — |
| 171 | C22×C4 | — | 15 | 1 | no | — |
| 172 | C22×C4 | D3×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 173 | M16 | — | 5 | 1 | no | — |
| 174 | M16 | — | 15 | 1 | no | — |
| 175 | M16 | — | 15 | 2 | no | — |
| 176 | M16 | — | 30 | 1 | no | — |
| 177 | M16 | — | 30 | 1 | no | — |
| 178 | M16 | — | 30 | 1 | no | — |
| 179 | M16 | — | 30 | 1 | no | — |
| 180 | M16 | — | 30 | 1 | no | — |
| 181 | C2×C8 | — | 3 | 1 | no | — |
| 182 | C2×C8 | — | 3 | 1 | no | — |
| 183 | C2×C8 | — | 15 | 1 | no | — |
| 184 | C2×C8 | — | 15 | 1 | no | — |
| 185 | C2×C8 | — | 30 | 1 | no | — |
| 186 | C2×C8 | — | 30 | 1 | no | — |
| 187 | C2×C8 | — | 30 | 1 | no | — |
| 188 | C2×C8 | — | 30 | 1 | no | — |
| 189 | C2×C8 | — | 30 | 1 | no | — |
| 190 | C2×C8 | — | 30 | 1 | no | — |
| 191 | C2×C8 | — | 30 | 1 | no | — |
| 192 | C2×C8 | — | 30 | 1 | no | — |
| 193 | C3×C5 | — | 4 | 1 | no | Hall |
| 194 | C22×C3 | — | 20 | 1 | no | — |
| 195 | A4 | #3996 | 1 | 1 | yes | characteristic, complemented |
| 196 | C3×C4 | — | 4 | 1 | no | — |
| 197 | C3×C4 | — | 20 | 1 | no | — |
| 198 | C2×C5 | — | 3 | 1 | no | — |
| 199 | C2×C5 | — | 3 | 1 | no | — |
| 200 | C2×C5 | SL(2,ℤ/4)⋊C4 | 1 | 1 | yes | characteristic |
| 201 | D5 | — | 2 | 1 | no | — |
| 202 | D5 | — | 6 | 1 | no | — |
| 203 | C23 | — | 10 | 1 | no | — |
| 204 | C23 | — | 15 | 1 | no | — |
| 205 | C23 | — | 15 | 1 | no | — |
| 206 | C23 | — | 30 | 1 | no | — |
| 207 | C23 | Dic3⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | — | 3 | 1 | no | — |
| 209 | C2×C4 | — | 3 | 1 | no | — |
| 210 | C2×C4 | — | 5 | 1 | no | — |
| 211 | C2×C4 | — | 15 | 1 | no | — |
| 212 | C2×C4 | — | 15 | 1 | no | — |
| 213 | C2×C4 | — | 15 | 1 | no | — |
| 214 | C2×C4 | — | 15 | 1 | no | — |
| 215 | C2×C4 | — | 15 | 1 | no | — |
| 216 | C2×C4 | — | 30 | 1 | no | — |
| 217 | C8 | — | 6 | 1 | no | — |
| 218 | C8 | — | 6 | 1 | no | — |
| 219 | C8 | — | 10 | 1 | no | — |
| 220 | C8 | — | 30 | 1 | no | — |
| 221 | C8 | — | 30 | 1 | no | — |
| 222 | C8 | — | 30 | 1 | no | — |
| 223 | C8 | — | 60 | 1 | no | — |
| 224 | C8 | — | 60 | 1 | no | — |
| 225 | C2×C3 | — | 4 | 1 | no | — |
| 226 | C2×C3 | — | 40 | 1 | no | — |
| 227 | C5 | #73114 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 228 | C22 | — | 3 | 1 | no | — |
| 229 | C22 | — | 3 | 1 | no | — |
| 230 | C22 | — | 5 | 1 | no | — |
| 231 | C22 | — | 15 | 1 | no | — |
| 232 | C22 | — | 30 | 1 | no | — |
| 233 | C22 | — | 30 | 1 | no | — |
| 234 | C22 | — | 30 | 1 | no | — |
| 235 | C22 | — | 30 | 1 | no | — |
| 236 | C22 | #91162 | 1 | 1 | yes | characteristic, complemented |
| 237 | C4 | — | 3 | 1 | no | — |
| 238 | C4 | — | 5 | 1 | no | — |
| 239 | C4 | — | 15 | 1 | no | — |
| 240 | C4 | AGL(1,𝔽5)×S4 | 1 | 1 | yes | characteristic |
| 241 | C3 | — | 4 | 1 | no | Sylow |
| 242 | C2 | — | 3 | 1 | no | — |
| 243 | C2 | — | 3 | 1 | no | — |
| 244 | C2 | — | 10 | 1 | no | — |
| 245 | C2 | — | 30 | 1 | no | — |
| 246 | C2 | SL(2,ℤ/4)⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 95. C5×A4 | #108 | semidirect |
| 155. C22×C5 | #822 | semidirect |
| 195. A4 | #3996 | semidirect |
| 227. C5 | #73114 | semidirect |
| 236. C22 | #91162 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#700 | 2 |
| #11756716 | 2 |
| D5×A4⋊C8 | 2 |
| #10603917 | 3 |
| #91162 | 4 |
| #73114 | 5 |