Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
510 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 67 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×#91955 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#4094⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×C5×GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11751618 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 5 | C2×#65112 | — | 3 | 1 | no | maximal, Hall |
| 6 | C2×C5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | #91955 | C4 | 1 | 2 | yes | complemented |
| 8 | #91955 | C22 | 1 | 2 | yes | — |
| 9 | C2×C5⋊GL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×C5⋊#226 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#6651 | — | 4 | 1 | no | maximal |
| 12 | #4094⋊C3 | C22 | 1 | 2 | yes | complemented |
| 13 | C5×GU(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 14 | C2×#5405 | — | 5 | 1 | no | maximal, Hall |
| 15 | C2×#4096 | — | 3 | 1 | no | — |
| 16 | C2×#4094 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×#4053 | — | 3 | 1 | no | — |
| 18 | C2×C5×#104 | — | 3 | 1 | no | — |
| 19 | C2×#3954 | — | 3 | 1 | no | — |
| 20 | C2×#3937 | — | 3 | 1 | no | — |
| 21 | C2×#3935 | — | 3 | 1 | no | — |
| 22 | #65112 | — | 3 | 4 | no | — |
| 23 | #65112 | — | 3 | 4 | no | — |
| 24 | C2×C4×D15 | — | 4 | 1 | no | — |
| 25 | C5×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C5×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×C5×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C5⋊GL(2,𝔽3) | C2×C4 | 1 | 2 | yes | complemented |
| 29 | C5⋊#226 | C2×C4 | 1 | 2 | yes | complemented |
| 30 | C2×C5×C3⋊C8 | — | 4 | 1 | no | — |
| 31 | C2×C3×#168 | — | 4 | 1 | no | — |
| 32 | #6651 | — | 4 | 4 | no | — |
| 33 | C2×#985 | — | 5 | 1 | no | — |
| 34 | C2×#131⋊C3 | — | 5 | 1 | no | — |
| 35 | C2×GU(2,𝔽3) | D5 | 1 | 1 | yes | characteristic, complemented |
| 36 | #5405 | — | 5 | 4 | no | — |
| 37 | C2×C5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 38 | C2×#709 | — | 3 | 1 | no | — |
| 39 | C2×C5×M16 | — | 3 | 1 | no | — |
| 40 | C2×C42×C5 | — | 3 | 1 | no | — |
| 41 | #4096 | — | 3 | 2 | no | — |
| 42 | #4096 | — | 3 | 2 | no | — |
| 43 | #4096 | — | 3 | 2 | no | — |
| 44 | #4096 | — | 3 | 2 | no | — |
| 45 | #4094 | — | 3 | 2 | no | — |
| 46 | #4094 | C2×D3 | 1 | 2 | yes | complemented |
| 47 | C2×#689 | — | 3 | 1 | no | — |
| 48 | C2×#688 | — | 3 | 1 | no | — |
| 49 | C2×#687 | — | 3 | 1 | no | — |
| 50 | C2×#686 | — | 3 | 1 | no | — |
| 51 | C2×#681 | — | 3 | 1 | no | — |
| 52 | C22×#168 | — | 3 | 1 | no | — |
| 53 | C22×#168 | — | 3 | 1 | no | — |
| 54 | #4053 | — | 3 | 4 | no | — |
| 55 | #4053 | — | 6 | 2 | no | — |
| 56 | C2×#676 | — | 6 | 1 | no | — |
| 57 | C2×C8×D5 | — | 6 | 1 | no | — |
| 58 | C5×#104 | — | 3 | 2 | no | — |
| 59 | C5×#104 | — | 3 | 2 | no | — |
| 60 | #3954 | — | 3 | 4 | no | — |
| 61 | #3937 | — | 3 | 4 | no | — |
| 62 | #3935 | — | 3 | 4 | no | — |
| 63 | C2×#444 | — | 15 | 1 | no | Sylow |
| 64 | C22×D15 | — | 4 | 1 | no | — |
| 65 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 66 | C2×Dic15 | — | 4 | 1 | no | — |
| 67 | C4×D15 | — | 4 | 2 | no | — |
| 68 | C4×D15 | — | 4 | 2 | no | — |
| 69 | C5×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C3×#168 | — | 4 | 2 | no | — |
| 71 | C5×C3⋊C8 | — | 4 | 2 | no | — |
| 72 | C2×#41⋊C3 | C2×D5 | 1 | 1 | yes | characteristic |
| 73 | #985 | — | 5 | 2 | no | — |
| 74 | #985 | — | 5 | 2 | no | — |
| 75 | C2×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 76 | C2×#226 | — | 5 | 1 | no | — |
| 77 | C2×ΓL(1,𝔽25) | — | 20 | 1 | no | — |
| 78 | #131⋊C3 | — | 5 | 2 | no | — |
| 79 | GU(2,𝔽3) | C2×D5 | 1 | 2 | yes | complemented |
| 80 | C5×#41 | — | 3 | 1 | no | — |
| 81 | C5×#41 | — | 3 | 1 | no | — |
| 82 | C5×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 83 | C5×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 84 | C2×C5×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 85 | C2×C5×D4 | — | 3 | 1 | no | — |
| 86 | C22×C4×C5 | — | 3 | 1 | no | — |
| 87 | C22×C4×C5 | — | 6 | 1 | no | — |
| 88 | C2×C22⋊D5 | — | 6 | 1 | no | — |
| 89 | #709 | — | 3 | 2 | no | — |
| 90 | #709 | — | 3 | 2 | no | — |
| 91 | #709 | — | 6 | 1 | no | — |
| 92 | #709 | — | 6 | 1 | no | — |
| 93 | C2×D20 | — | 3 | 1 | no | — |
| 94 | C2×C4×D5 | — | 6 | 1 | no | — |
| 95 | C2×Dic10 | — | 3 | 1 | no | — |
| 96 | C5×M16 | — | 3 | 2 | no | — |
| 97 | C5×M16 | — | 6 | 1 | no | — |
| 98 | C2×C5×C8 | — | 6 | 1 | no | — |
| 99 | C42×C5 | — | 3 | 2 | no | — |
| 100 | C42×C5 | — | 6 | 1 | no | — |
| 101 | #689 | — | 3 | 2 | no | — |
| 102 | #689 | — | 3 | 2 | no | — |
| 103 | #688 | — | 3 | 2 | no | — |
| 104 | #688 | — | 3 | 2 | no | — |
| 105 | #687 | — | 3 | 2 | no | — |
| 106 | #687 | — | 3 | 2 | no | — |
| 107 | #686 | — | 3 | 2 | no | — |
| 108 | #686 | — | 3 | 2 | no | — |
| 109 | #681 | — | 3 | 2 | no | — |
| 110 | #681 | — | 6 | 1 | no | — |
| 111 | C2×#168 | — | 3 | 2 | no | — |
| 112 | C2×#168 | — | 3 | 2 | no | — |
| 113 | C2×#168 | — | 3 | 1 | no | — |
| 114 | C2×#168 | — | 3 | 1 | no | — |
| 115 | C2×#168 | — | 3 | 1 | no | — |
| 116 | C2×#168 | — | 3 | 2 | no | — |
| 117 | C2×#168 | — | 3 | 2 | no | — |
| 118 | C2×#168 | S4 | 1 | 1 | yes | characteristic |
| 119 | #676 | — | 6 | 4 | no | — |
| 120 | C8×D5 | — | 6 | 4 | no | — |
| 121 | C2×#135 | — | 15 | 1 | no | — |
| 122 | C2×#131 | — | 5 | 1 | no | — |
| 123 | C2×#131 | — | 15 | 1 | no | — |
| 124 | #444 | — | 15 | 4 | no | — |
| 125 | #444 | — | 15 | 4 | no | — |
| 126 | C2×#108 | — | 15 | 1 | no | — |
| 127 | C2×#104 | — | 3 | 1 | no | — |
| 128 | C2×#104 | — | 15 | 1 | no | — |
| 129 | C2×#97 | — | 15 | 1 | no | — |
| 130 | C22×C3×C5 | — | 4 | 1 | no | — |
| 131 | C2×D15 | — | 4 | 2 | no | — |
| 132 | C2×D15 | — | 8 | 2 | no | — |
| 133 | C3×C4×C5 | — | 4 | 1 | no | — |
| 134 | C3×C4×C5 | — | 4 | 1 | no | — |
| 135 | Dic15 | — | 4 | 2 | no | — |
| 136 | C2×C4×D3 | — | 20 | 1 | no | — |
| 137 | #41⋊C3 | C4×D5 | 1 | 1 | yes | characteristic |
| 138 | #41⋊C3 | C22×D5 | 1 | 1 | yes | characteristic |
| 139 | C2×SL(2,𝔽3) | C4×D5 | 1 | 1 | yes | characteristic |
| 140 | GL(2,𝔽3) | — | 5 | 2 | no | — |
| 141 | #226 | — | 5 | 2 | no | — |
| 142 | C2×C3×C8 | — | 20 | 1 | no | — |
| 143 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 144 | ΓL(1,𝔽25) | — | 20 | 4 | no | — |
| 145 | C23×C5 | — | 3 | 1 | no | — |
| 146 | C22×D5 | — | 6 | 1 | no | — |
| 147 | C5×Q8 | — | 3 | 1 | no | — |
| 148 | C5×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 149 | C5×D4 | — | 3 | 1 | no | — |
| 150 | C5×D4 | — | 3 | 1 | no | — |
| 151 | C5×D4 | — | 6 | 1 | no | — |
| 152 | C2×C4×C5 | — | 3 | 1 | no | — |
| 153 | C2×C4×C5 | — | 3 | 1 | no | — |
| 154 | C2×C4×C5 | — | 3 | 1 | no | — |
| 155 | C2×C4×C5 | — | 3 | 1 | no | — |
| 156 | C2×C4×C5 | — | 3 | 1 | no | — |
| 157 | C2×C4×C5 | — | 6 | 2 | no | — |
| 158 | C2×C4×C5 | — | 12 | 2 | no | — |
| 159 | C2×C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 160 | C22⋊D5 | — | 6 | 2 | no | — |
| 161 | C22⋊D5 | — | 6 | 2 | no | — |
| 162 | C2×Dic5 | — | 6 | 1 | no | — |
| 163 | D20 | — | 3 | 2 | no | — |
| 164 | D20 | — | 6 | 1 | no | — |
| 165 | C4×D5 | — | 6 | 2 | no | — |
| 166 | C4×D5 | — | 6 | 2 | no | — |
| 167 | Dic10 | — | 3 | 2 | no | — |
| 168 | Dic10 | — | 6 | 1 | no | — |
| 169 | C5×C8 | — | 6 | 2 | no | — |
| 170 | #168 | — | 3 | 2 | no | — |
| 171 | #168 | — | 3 | 2 | no | — |
| 172 | #168 | — | 3 | 2 | no | — |
| 173 | #168 | C2×S4 | 1 | 2 | yes | — |
| 174 | C2×#41 | — | 15 | 1 | no | — |
| 175 | C2×#41 | D3×D5 | 1 | 1 | yes | characteristic |
| 176 | #135 | — | 15 | 2 | no | — |
| 177 | #135 | — | 15 | 2 | no | — |
| 178 | #135 | — | 15 | 2 | no | — |
| 179 | #135 | — | 15 | 2 | no | — |
| 180 | C2×Q16 | — | 15 | 1 | no | — |
| 181 | C2×SD16 | — | 15 | 1 | no | — |
| 182 | C2×SD16 | — | 15 | 1 | no | — |
| 183 | C2×D8 | — | 15 | 1 | no | — |
| 184 | #131 | — | 5 | 2 | no | — |
| 185 | #131 | — | 15 | 2 | no | — |
| 186 | #131 | — | 15 | 4 | no | — |
| 187 | #131 | — | 30 | 2 | no | — |
| 188 | C2×M16 | — | 3 | 1 | no | — |
| 189 | C2×M16 | — | 15 | 1 | no | — |
| 190 | C2×M16 | — | 30 | 1 | no | — |
| 191 | C22×C8 | — | 15 | 1 | no | — |
| 192 | C22×C8 | — | 15 | 1 | no | — |
| 193 | C22×C8 | — | 30 | 1 | no | — |
| 194 | C2×C42 | — | 3 | 1 | no | — |
| 195 | #108 | — | 15 | 4 | no | — |
| 196 | #104 | — | 3 | 2 | no | — |
| 197 | #104 | — | 3 | 2 | no | — |
| 198 | #104 | — | 15 | 4 | no | — |
| 199 | #97 | — | 15 | 4 | no | — |
| 200 | C2×C3×C5 | — | 4 | 1 | no | — |
| 201 | C2×C3×C5 | — | 4 | 2 | no | — |
| 202 | D15 | — | 8 | 2 | no | — |
| 203 | C22×D3 | — | 20 | 1 | no | — |
| 204 | C2×C3×C4 | — | 4 | 1 | no | — |
| 205 | C2×Dic3 | — | 20 | 1 | no | — |
| 206 | C4×D3 | — | 20 | 2 | no | — |
| 207 | C4×D3 | — | 20 | 2 | no | — |
| 208 | SL(2,𝔽3) | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 209 | C3×C8 | — | 20 | 2 | no | — |
| 210 | C3⋊C8 | — | 4 | 2 | no | — |
| 211 | C22×C5 | — | 3 | 1 | no | — |
| 212 | C22×C5 | — | 3 | 1 | no | — |
| 213 | C22×C5 | — | 6 | 2 | no | — |
| 214 | C22×C5 | C4×S4 | 1 | 1 | yes | characteristic |
| 215 | C2×D5 | — | 6 | 2 | no | — |
| 216 | C2×D5 | — | 12 | 2 | no | — |
| 217 | C4×C5 | — | 3 | 1 | no | — |
| 218 | C4×C5 | — | 3 | 1 | no | — |
| 219 | C4×C5 | — | 12 | 2 | no | — |
| 220 | C4×C5 | C4×S4 | 1 | 1 | yes | characteristic |
| 221 | C4×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 222 | Dic5 | — | 6 | 2 | no | — |
| 223 | #41 | — | 3 | 1 | no | — |
| 224 | #41 | — | 3 | 1 | no | — |
| 225 | #41 | — | 15 | 2 | no | — |
| 226 | #41 | — | 15 | 2 | no | — |
| 227 | #41 | — | 30 | 1 | no | — |
| 228 | #41 | — | 30 | 1 | no | — |
| 229 | #41 | #1200 | 1 | 1 | yes | characteristic |
| 230 | #41 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 231 | C2×Q8 | — | 15 | 1 | no | — |
| 232 | C2×Q8 | #1200 | 1 | 1 | yes | characteristic |
| 233 | C2×D4 | — | 3 | 1 | no | — |
| 234 | C2×D4 | — | 15 | 1 | no | — |
| 235 | C2×D4 | — | 30 | 1 | no | — |
| 236 | C22×C4 | — | 3 | 1 | no | — |
| 237 | C22×C4 | — | 6 | 1 | no | — |
| 238 | C22×C4 | — | 30 | 1 | no | — |
| 239 | Q16 | — | 15 | 2 | no | — |
| 240 | Q16 | — | 15 | 2 | no | — |
| 241 | SD16 | — | 15 | 2 | no | — |
| 242 | SD16 | — | 15 | 2 | no | — |
| 243 | SD16 | — | 15 | 2 | no | — |
| 244 | SD16 | — | 15 | 2 | no | — |
| 245 | D8 | — | 15 | 2 | no | — |
| 246 | D8 | — | 15 | 2 | no | — |
| 247 | M16 | — | 3 | 2 | no | — |
| 248 | M16 | — | 6 | 1 | no | — |
| 249 | M16 | — | 15 | 2 | no | — |
| 250 | M16 | — | 30 | 4 | no | — |
| 251 | M16 | — | 30 | 1 | no | — |
| 252 | C2×C8 | — | 5 | 1 | no | — |
| 253 | C2×C8 | — | 6 | 1 | no | — |
| 254 | C2×C8 | — | 15 | 2 | no | — |
| 255 | C2×C8 | — | 15 | 2 | no | — |
| 256 | C2×C8 | — | 15 | 2 | no | — |
| 257 | C2×C8 | — | 15 | 1 | no | — |
| 258 | C2×C8 | — | 15 | 1 | no | — |
| 259 | C2×C8 | — | 15 | 2 | no | — |
| 260 | C2×C8 | — | 15 | 1 | no | — |
| 261 | C2×C8 | — | 30 | 4 | no | — |
| 262 | C42 | — | 3 | 2 | no | — |
| 263 | C42 | — | 6 | 1 | no | — |
| 264 | C3×C5 | — | 4 | 1 | no | Hall |
| 265 | C22×C3 | — | 4 | 1 | no | — |
| 266 | C2×D3 | — | 20 | 2 | no | — |
| 267 | C2×D3 | — | 40 | 2 | no | — |
| 268 | C3×C4 | — | 4 | 1 | no | — |
| 269 | C3×C4 | — | 4 | 1 | no | — |
| 270 | Dic3 | — | 20 | 2 | no | — |
| 271 | C2×C5 | — | 6 | 1 | no | — |
| 272 | C2×C5 | — | 6 | 1 | no | — |
| 273 | C2×C5 | #5405 | 1 | 2 | yes | complemented |
| 274 | C2×C5 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 275 | D5 | — | 12 | 2 | no | — |
| 276 | C23 | — | 3 | 1 | no | — |
| 277 | C23 | — | 30 | 1 | no | — |
| 278 | Q8 | — | 3 | 1 | no | — |
| 279 | Q8 | — | 15 | 2 | no | — |
| 280 | Q8 | — | 30 | 1 | no | — |
| 281 | Q8 | C2×#1200 | 1 | 1 | yes | characteristic |
| 282 | D4 | — | 3 | 1 | no | — |
| 283 | D4 | — | 3 | 1 | no | — |
| 284 | D4 | — | 6 | 1 | no | — |
| 285 | D4 | — | 15 | 2 | no | — |
| 286 | D4 | — | 30 | 1 | no | — |
| 287 | D4 | — | 30 | 2 | no | — |
| 288 | D4 | — | 30 | 2 | no | — |
| 289 | C2×C4 | — | 3 | 1 | no | — |
| 290 | C2×C4 | — | 3 | 1 | no | — |
| 291 | C2×C4 | — | 3 | 1 | no | — |
| 292 | C2×C4 | — | 3 | 1 | no | — |
| 293 | C2×C4 | — | 3 | 1 | no | — |
| 294 | C2×C4 | — | 6 | 2 | no | — |
| 295 | C2×C4 | — | 12 | 2 | no | — |
| 296 | C2×C4 | — | 30 | 1 | no | — |
| 297 | C2×C4 | — | 30 | 2 | no | — |
| 298 | C2×C4 | — | 30 | 2 | no | — |
| 299 | C2×C4 | D5×S4 | 1 | 1 | yes | characteristic |
| 300 | C8 | — | 5 | 2 | no | — |
| 301 | C8 | — | 6 | 2 | no | — |
| 302 | C8 | — | 15 | 2 | no | — |
| 303 | C8 | — | 15 | 2 | no | — |
| 304 | C8 | — | 15 | 2 | no | — |
| 305 | C2×C3 | — | 4 | 2 | no | — |
| 306 | C2×C3 | — | 4 | 1 | no | — |
| 307 | D3 | — | 40 | 2 | no | — |
| 308 | C5 | C2×#5405 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 309 | C22 | — | 3 | 1 | no | — |
| 310 | C22 | — | 3 | 1 | no | — |
| 311 | C22 | — | 6 | 2 | no | — |
| 312 | C22 | — | 30 | 2 | no | — |
| 313 | C22 | — | 60 | 2 | no | — |
| 314 | C22 | #91901 | 1 | 1 | yes | characteristic |
| 315 | C4 | — | 3 | 1 | no | — |
| 316 | C4 | — | 3 | 1 | no | — |
| 317 | C4 | — | 12 | 2 | no | — |
| 318 | C4 | — | 30 | 2 | no | — |
| 319 | C4 | #91901 | 1 | 1 | yes | characteristic |
| 320 | C4 | C2×D5×S4 | 1 | 1 | yes | characteristic |
| 321 | C3 | — | 4 | 1 | no | Sylow |
| 322 | C2 | — | 6 | 1 | no | — |
| 323 | C2 | — | 6 | 1 | no | — |
| 324 | C2 | — | 60 | 2 | no | — |
| 325 | C2 | #11751618 | 1 | 2 | yes | direct product factor |
| 326 | C2 | C2×#91901 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4094⋊C3 | C2 | semidirect |
| 3. C2×C5×GU(2,𝔽3) | C2 | semidirect |
| 4. #11751618 | C2 | direct (N × H) |
| 7. #91955 | C4 | semidirect |
| 9. C2×C5⋊GL(2,𝔽3) | C4 | semidirect |
| 10. C2×C5⋊#226 | C4 | semidirect |
| 12. #4094⋊C3 | C22 | semidirect |
| 13. C5×GU(2,𝔽3) | C22 | semidirect |
| 16. C2×#4094 | D3 | semidirect |
| 28. C5⋊GL(2,𝔽3) | C2×C4 | semidirect |
| 29. C5⋊#226 | C2×C4 | semidirect |
| 35. C2×GU(2,𝔽3) | D5 | semidirect |
| 46. #4094 | C2×D3 | semidirect |
| 79. GU(2,𝔽3) | C2×D5 | semidirect |
| 273. C2×C5 | #5405 | semidirect |
| 308. C5 | C2×#5405 | semidirect |
| 325. C2 | #11751618 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#91955 | 2 |
| C2×#4094⋊C3 | 2 |
| C2×C5×GU(2,𝔽3) | 2 |
| #11751618 | 2 |
| C2×#65112 | 3 |
| C2×#6651 | 4 |
| C2×#5405 | 5 |