Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
386 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 38 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 | C2×#4147⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11756722 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10604974 | — | 3 | 1 | no | maximal, Hall |
| 4 | C22×C5×SL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #4147⋊C3 | — | 2 | 2 | no | — |
| 6 | C2×D5×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C5⋊(#41⋊C3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | #91886 | — | 2 | 2 | no | — |
| 9 | #91240 | — | 4 | 1 | no | maximal |
| 10 | #73156 | — | 5 | 1 | no | maximal, Hall |
| 11 | C2×#4147 | D3 | 1 | 1 | yes | characteristic |
| 12 | #65784 | — | 3 | 2 | no | — |
| 13 | #65782 | — | 3 | 2 | no | — |
| 14 | #65761 | — | 3 | 1 | no | — |
| 15 | #65745 | — | 3 | 1 | no | — |
| 16 | C2×C3×C22⋊D5 | — | 4 | 1 | no | — |
| 17 | C2×C5×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 18 | C2×C5×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | #6763 | — | 4 | 2 | no | — |
| 20 | D5×SL(2,𝔽3) | D4 | 1 | 2 | yes | — |
| 21 | C5⋊(#41⋊C3) | — | 2 | 1 | no | — |
| 22 | C2×2−1+4⋊C3 | — | 5 | 1 | no | — |
| 23 | #5420 | — | 5 | 2 | no | — |
| 24 | C22×C5×Q8 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 25 | #4147 | — | 2 | 2 | no | — |
| 26 | #4147 | — | 6 | 2 | no | — |
| 27 | C2×#713 | — | 3 | 1 | no | — |
| 28 | C2×#713 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 29 | C2×Q8×D5 | — | 3 | 1 | no | — |
| 30 | C2×Q8×D5 | C2×D3 | 1 | 1 | yes | characteristic |
| 31 | C2×#709 | — | 3 | 1 | no | — |
| 32 | C2×#709 | — | 3 | 1 | no | — |
| 33 | #4130 | — | 3 | 2 | no | — |
| 34 | D5⋊M16 | — | 3 | 2 | no | — |
| 35 | #4013 | — | 6 | 1 | no | — |
| 36 | #4011 | — | 6 | 1 | no | — |
| 37 | #4010 | — | 6 | 4 | no | — |
| 38 | #4009 | — | 6 | 2 | no | — |
| 39 | #4009 | — | 6 | 2 | no | — |
| 40 | #4005 | — | 6 | 1 | no | — |
| 41 | #3999 | — | 6 | 1 | no | — |
| 42 | #1796 | — | 15 | 1 | no | Sylow |
| 43 | C23×C3×C5 | — | 4 | 1 | no | — |
| 44 | C22×C3×D5 | — | 4 | 1 | no | — |
| 45 | C2×ΓL(1,𝔽16) | — | 8 | 2 | no | — |
| 46 | C3×C22⋊D5 | — | 8 | 2 | no | — |
| 47 | C2×C3×Dic5 | — | 4 | 1 | no | — |
| 48 | C5×SL(2,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 49 | 2−1+4⋊C3 | — | 10 | 2 | no | — |
| 50 | C2×#41⋊C3 | — | 5 | 1 | no | — |
| 51 | C22×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 52 | C22×SL(2,𝔽3) | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 53 | #859 | — | 10 | 2 | no | — |
| 54 | #834 | — | 20 | 1 | no | — |
| 55 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 56 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 57 | C2×C5×Q8 | — | 6 | 1 | no | — |
| 58 | C2×C5×Q8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 59 | C22×C4×C5 | — | 3 | 1 | no | — |
| 60 | C2×C22⋊D5 | — | 3 | 1 | no | — |
| 61 | C2×C22⋊D5 | S4 | 1 | 1 | yes | characteristic |
| 62 | #713 | — | 2 | 1 | no | — |
| 63 | #713 | — | 3 | 4 | no | — |
| 64 | #713 | — | 6 | 2 | no | — |
| 65 | #713 | — | 6 | 1 | no | — |
| 66 | Q8×D5 | — | 3 | 2 | no | — |
| 67 | Q8×D5 | — | 6 | 2 | no | — |
| 68 | Q8×D5 | — | 6 | 2 | no | — |
| 69 | Q8×D5 | #67 | 1 | 2 | yes | — |
| 70 | #709 | — | 6 | 2 | no | — |
| 71 | #709 | — | 6 | 2 | no | — |
| 72 | #709 | — | 6 | 2 | no | — |
| 73 | #709 | — | 6 | 2 | no | — |
| 74 | C2×D20 | — | 3 | 1 | no | — |
| 75 | C2×D20 | — | 3 | 1 | no | — |
| 76 | C2×C4×D5 | — | 3 | 1 | no | — |
| 77 | C2×C4×D5 | — | 3 | 1 | no | — |
| 78 | C2×C4×D5 | — | 6 | 1 | no | — |
| 79 | C2×Dic10 | — | 3 | 1 | no | — |
| 80 | C2×Dic10 | — | 3 | 1 | no | — |
| 81 | C22⋊AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 82 | #704 | — | 6 | 2 | no | — |
| 83 | C4⋊AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 84 | C4×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 85 | #700 | — | 6 | 2 | no | — |
| 86 | D5⋊C8 | — | 6 | 2 | no | — |
| 87 | C2×2−1+4 | — | 5 | 1 | no | — |
| 88 | #421 | — | 15 | 2 | no | — |
| 89 | #419 | — | 15 | 2 | no | — |
| 90 | #413 | — | 15 | 1 | no | — |
| 91 | #410 | — | 15 | 1 | no | — |
| 92 | C22×C3×C5 | — | 4 | 1 | no | — |
| 93 | C22×C3×C5 | — | 8 | 1 | no | — |
| 94 | C22×C3×C5 | — | 16 | 1 | no | — |
| 95 | C2×C3×D5 | — | 4 | 2 | no | — |
| 96 | C2×C3×D5 | — | 16 | 1 | no | — |
| 97 | ΓL(1,𝔽16) | — | 16 | 2 | no | — |
| 98 | C3×Dic5 | — | 8 | 1 | no | — |
| 99 | C2×C3×D4 | — | 20 | 1 | no | — |
| 100 | #41⋊C3 | — | 10 | 1 | no | — |
| 101 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 102 | C2×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 103 | C2×SL(2,𝔽3) | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 104 | #217 | — | 20 | 2 | no | — |
| 105 | C23×C5 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 106 | C22×D5 | — | 3 | 1 | no | — |
| 107 | C22×D5 | C2×S4 | 1 | 1 | yes | characteristic |
| 108 | C2×AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 109 | C5×Q8 | — | 3 | 1 | no | — |
| 110 | C5×Q8 | — | 3 | 2 | no | — |
| 111 | C5×Q8 | — | 6 | 1 | no | — |
| 112 | C5×Q8 | #217 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C5 | — | 3 | 1 | no | — |
| 114 | C2×C4×C5 | — | 3 | 1 | no | — |
| 115 | C2×C4×C5 | — | 6 | 1 | no | — |
| 116 | C2×C4×C5 | — | 6 | 1 | no | — |
| 117 | C22⋊D5 | — | 2 | 2 | no | — |
| 118 | C22⋊D5 | — | 6 | 2 | no | — |
| 119 | C2×Dic5 | — | 3 | 1 | no | — |
| 120 | C2×Dic5 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 121 | D20 | — | 3 | 2 | no | — |
| 122 | D20 | — | 6 | 1 | no | — |
| 123 | D20 | — | 6 | 2 | no | — |
| 124 | C4×D5 | — | 3 | 2 | no | — |
| 125 | C4×D5 | — | 3 | 2 | no | — |
| 126 | C4×D5 | — | 6 | 1 | no | — |
| 127 | C4×D5 | — | 6 | 1 | no | — |
| 128 | C4×D5 | — | 6 | 4 | no | — |
| 129 | Dic10 | — | 3 | 2 | no | — |
| 130 | Dic10 | — | 6 | 2 | no | — |
| 131 | Dic10 | — | 6 | 1 | no | — |
| 132 | #170 | — | 12 | 2 | no | — |
| 133 | 2−1+4 | — | 10 | 2 | no | — |
| 134 | 2−1+4 | — | 30 | 2 | no | — |
| 135 | C2×#41 | — | 5 | 1 | no | — |
| 136 | C2×#41 | — | 15 | 1 | no | — |
| 137 | C2×#41 | — | 15 | 1 | no | — |
| 138 | C2×#41 | — | 15 | 1 | no | — |
| 139 | C22×Q8 | — | 5 | 1 | no | — |
| 140 | C22×Q8 | — | 15 | 1 | no | — |
| 141 | C22×Q8 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic, complemented |
| 142 | C2×M16 | — | 15 | 2 | no | — |
| 143 | #117 | — | 15 | 2 | no | — |
| 144 | #104 | — | 30 | 4 | no | — |
| 145 | #103 | — | 30 | 2 | no | — |
| 146 | #103 | — | 30 | 2 | no | — |
| 147 | #101 | — | 30 | 1 | no | — |
| 148 | #100 | — | 30 | 1 | no | — |
| 149 | #99 | — | 30 | 1 | no | — |
| 150 | #99 | — | 30 | 1 | no | — |
| 151 | C2×C3×C5 | — | 4 | 1 | no | — |
| 152 | C2×C3×C5 | — | 8 | 1 | no | — |
| 153 | C2×C3×C5 | — | 16 | 1 | no | — |
| 154 | C3×D5 | — | 8 | 2 | no | — |
| 155 | C23×C3 | — | 4 | 1 | no | — |
| 156 | C23×C3 | — | 20 | 1 | no | — |
| 157 | C3×D4 | — | 40 | 2 | no | — |
| 158 | C2×C3×C4 | — | 20 | 1 | no | — |
| 159 | C2×Dic3 | — | 40 | 2 | no | — |
| 160 | SL(2,𝔽3) | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 161 | C22×C5 | — | 2 | 1 | no | — |
| 162 | C22×C5 | — | 4 | 1 | no | — |
| 163 | C22×C5 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 164 | C2×D5 | — | 4 | 1 | no | — |
| 165 | C2×D5 | — | 6 | 1 | no | — |
| 166 | C2×D5 | — | 12 | 1 | no | — |
| 167 | C2×D5 | GL(2,ℤ/4) | 1 | 2 | yes | — |
| 168 | AGL(1,𝔽5) | — | 24 | 2 | no | — |
| 169 | C4×C5 | — | 3 | 1 | no | — |
| 170 | C4×C5 | — | 3 | 1 | no | — |
| 171 | C4×C5 | — | 6 | 1 | no | — |
| 172 | Dic5 | — | 2 | 1 | no | — |
| 173 | Dic5 | — | 3 | 2 | no | — |
| 174 | #41 | — | 10 | 1 | no | — |
| 175 | #41 | — | 15 | 4 | no | — |
| 176 | #41 | — | 30 | 2 | no | — |
| 177 | #41 | — | 30 | 2 | no | — |
| 178 | #41 | — | 30 | 2 | no | — |
| 179 | #41 | — | 30 | 2 | no | — |
| 180 | #41 | — | 30 | 2 | no | — |
| 181 | #41 | — | 30 | 1 | no | — |
| 182 | C2×Q8 | — | 2 | 1 | no | — |
| 183 | C2×Q8 | — | 3 | 1 | no | — |
| 184 | C2×Q8 | — | 5 | 2 | no | — |
| 185 | C2×Q8 | — | 6 | 1 | no | — |
| 186 | C2×Q8 | — | 15 | 1 | no | — |
| 187 | C2×Q8 | — | 15 | 1 | no | — |
| 188 | C2×Q8 | — | 15 | 2 | no | — |
| 189 | C2×Q8 | — | 30 | 2 | no | — |
| 190 | C2×Q8 | — | 30 | 2 | no | — |
| 191 | C2×Q8 | C2×ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 192 | C2×D4 | — | 5 | 1 | no | — |
| 193 | C2×D4 | — | 15 | 1 | no | — |
| 194 | C2×D4 | — | 15 | 1 | no | — |
| 195 | C2×D4 | — | 15 | 1 | no | — |
| 196 | C22×C4 | — | 3 | 1 | no | — |
| 197 | C22×C4 | — | 15 | 1 | no | — |
| 198 | C22×C4 | — | 15 | 1 | no | — |
| 199 | C22×C4 | — | 30 | 1 | no | — |
| 200 | M16 | — | 30 | 2 | no | — |
| 201 | M16 | — | 30 | 2 | no | — |
| 202 | C2×C8 | — | 30 | 2 | no | — |
| 203 | C4⋊C4 | — | 30 | 2 | no | — |
| 204 | C22⋊C4 | — | 30 | 2 | no | — |
| 205 | C42 | — | 30 | 2 | no | — |
| 206 | C3×C5 | — | 4 | 1 | no | Hall |
| 207 | C22×C3 | — | 4 | 1 | no | — |
| 208 | C22×C3 | — | 8 | 1 | no | — |
| 209 | C22×C3 | — | 16 | 1 | no | — |
| 210 | C22×C3 | — | 20 | 2 | no | — |
| 211 | C22×C3 | — | 80 | 1 | no | — |
| 212 | C3×C4 | — | 40 | 1 | no | — |
| 213 | Dic3 | — | 80 | 2 | no | — |
| 214 | C2×C5 | — | 2 | 1 | no | — |
| 215 | C2×C5 | — | 4 | 1 | no | — |
| 216 | C2×C5 | #5432 | 1 | 1 | yes | characteristic |
| 217 | D5 | — | 2 | 2 | no | — |
| 218 | D5 | — | 12 | 1 | no | — |
| 219 | C23 | — | 5 | 1 | no | — |
| 220 | C23 | — | 15 | 1 | no | — |
| 221 | C23 | A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 222 | Q8 | — | 3 | 1 | no | — |
| 223 | Q8 | — | 3 | 2 | no | — |
| 224 | Q8 | — | 6 | 1 | no | — |
| 225 | Q8 | — | 15 | 2 | no | — |
| 226 | Q8 | — | 30 | 1 | no | — |
| 227 | Q8 | — | 30 | 2 | no | — |
| 228 | Q8 | #6763 | 1 | 1 | yes | characteristic |
| 229 | D4 | — | 10 | 2 | no | — |
| 230 | D4 | — | 15 | 2 | no | — |
| 231 | D4 | — | 30 | 1 | no | — |
| 232 | D4 | — | 30 | 2 | no | — |
| 233 | D4 | — | 30 | 2 | no | — |
| 234 | C2×C4 | — | 3 | 1 | no | — |
| 235 | C2×C4 | — | 3 | 1 | no | — |
| 236 | C2×C4 | — | 5 | 1 | no | — |
| 237 | C2×C4 | — | 6 | 1 | no | — |
| 238 | C2×C4 | — | 6 | 1 | no | — |
| 239 | C2×C4 | — | 15 | 2 | no | — |
| 240 | C2×C4 | — | 15 | 2 | no | — |
| 241 | C2×C4 | — | 15 | 1 | no | — |
| 242 | C2×C4 | — | 30 | 1 | no | — |
| 243 | C2×C4 | — | 30 | 1 | no | — |
| 244 | C2×C4 | — | 30 | 4 | no | — |
| 245 | C2×C4 | — | 60 | 2 | no | — |
| 246 | C8 | — | 60 | 2 | no | — |
| 247 | C2×C3 | — | 4 | 1 | no | — |
| 248 | C2×C3 | — | 8 | 1 | no | — |
| 249 | C2×C3 | — | 16 | 1 | no | — |
| 250 | C2×C3 | — | 40 | 2 | no | — |
| 251 | C5 | #73156 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 252 | C22 | — | 2 | 1 | no | — |
| 253 | C22 | — | 4 | 1 | no | — |
| 254 | C22 | — | 5 | 2 | no | — |
| 255 | C22 | — | 20 | 1 | no | — |
| 256 | C22 | — | 30 | 1 | no | — |
| 257 | C22 | — | 60 | 1 | no | — |
| 258 | C22 | C2×A4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 259 | C4 | — | 3 | 1 | no | — |
| 260 | C4 | — | 3 | 1 | no | — |
| 261 | C4 | — | 6 | 1 | no | — |
| 262 | C4 | — | 10 | 1 | no | — |
| 263 | C4 | — | 15 | 2 | no | — |
| 264 | C4 | — | 120 | 2 | no | — |
| 265 | C3 | — | 4 | 1 | no | Sylow |
| 266 | C2 | — | 2 | 1 | no | — |
| 267 | C2 | — | 4 | 1 | no | — |
| 268 | C2 | — | 10 | 2 | no | — |
| 269 | C2 | — | 60 | 1 | no | — |
| 270 | C2 | #11756729 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11756722 | C2 | semidirect |
| 4. C22×C5×SL(2,𝔽3) | C4 | semidirect |
| 7. C2×C5⋊(#41⋊C3) | C4 | semidirect |
| 24. C22×C5×Q8 | Dic3 | semidirect |
| 28. C2×#713 | Dic3 | semidirect |
| 52. C22×SL(2,𝔽3) | AGL(1,𝔽5) | semidirect |
| 141. C22×Q8 | ΓL(1,𝔽16) | semidirect |
| 251. C5 | #73156 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4147⋊C3 | 2 |
| #11756722 | 2 |
| #10604974 | 3 |
| #91240 | 4 |
| #73156 | 5 |