Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
262 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 37 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 | C4⋊(C5⋊GL(2,𝔽3)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×C4⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | SL(2,𝔽3)×#168 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10601814 | — | 3 | 1 | no | maximal, Hall |
| 5 | C2×C5⋊GL(2,𝔽3) | — | 2 | 1 | no | — |
| 6 | C2×C5×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 7 | C4×C5×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#6654 | — | 4 | 1 | no | maximal |
| 9 | #73022 | — | 5 | 1 | no | maximal, Hall |
| 10 | C5×#460 | — | 3 | 1 | no | — |
| 11 | #65362 | — | 3 | 1 | no | — |
| 12 | #65316 | — | 3 | 1 | no | — |
| 13 | Q8×#168 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #64764 | — | 3 | 1 | no | — |
| 15 | #64705 | — | 3 | 1 | no | — |
| 16 | #64699 | — | 3 | 1 | no | — |
| 17 | C2×D60 | — | 4 | 1 | no | — |
| 18 | C2×C5×D12 | — | 4 | 1 | no | — |
| 19 | C2×C5×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 20 | C5⋊GL(2,𝔽3) | — | 4 | 1 | no | — |
| 21 | C5×GL(2,𝔽3) | — | 4 | 1 | no | — |
| 22 | C2×C3×#168 | — | 4 | 1 | no | — |
| 23 | #6654 | — | 4 | 4 | no | — |
| 24 | C4⋊GL(2,𝔽3) | — | 5 | 1 | no | — |
| 25 | C4⋊GL(2,𝔽3) | D5 | 1 | 1 | yes | characteristic, complemented |
| 26 | C8×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 27 | C2×C5×SD16 | — | 6 | 1 | no | — |
| 28 | C5×#127 | — | 3 | 1 | no | — |
| 29 | C4×C5×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 30 | C2×#688 | — | 6 | 1 | no | — |
| 31 | C2×#686 | — | 6 | 1 | no | — |
| 32 | C2×#686 | — | 6 | 1 | no | — |
| 33 | #4020 | — | 3 | 1 | no | — |
| 34 | C5×C4⋊C8 | — | 3 | 1 | no | — |
| 35 | C5×#102 | — | 6 | 1 | no | — |
| 36 | #3941 | — | 6 | 1 | no | — |
| 37 | C4⋊#168 | — | 3 | 1 | no | — |
| 38 | C4⋊#168 | — | 3 | 1 | no | — |
| 39 | C4×#168 | — | 3 | 1 | no | — |
| 40 | #1668 | — | 15 | 1 | no | Sylow |
| 41 | C22×D15 | — | 8 | 1 | no | — |
| 42 | C22×C5×D3 | — | 8 | 1 | no | — |
| 43 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 44 | D60 | — | 4 | 2 | no | — |
| 45 | D60 | — | 8 | 1 | no | — |
| 46 | C5×D12 | — | 4 | 2 | no | — |
| 47 | C5×D12 | — | 8 | 1 | no | — |
| 48 | C5×SL(2,𝔽3) | D8 | 1 | 1 | yes | characteristic, complemented |
| 49 | C3×#168 | — | 4 | 2 | no | — |
| 50 | C2×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 51 | C2×GL(2,𝔽3) | — | 10 | 1 | no | — |
| 52 | C2×D24 | — | 20 | 1 | no | — |
| 53 | C4×SL(2,𝔽3) | C2×D5 | 1 | 1 | yes | characteristic |
| 54 | C2×C5×Q8 | D12 | 1 | 1 | yes | characteristic |
| 55 | C2×C5×D4 | — | 6 | 1 | no | — |
| 56 | C2×C5×D4 | — | 6 | 1 | no | — |
| 57 | C2×C5×D4 | — | 6 | 1 | no | — |
| 58 | C2×D20 | — | 6 | 1 | no | — |
| 59 | C2×D20 | — | 6 | 1 | no | — |
| 60 | C2×D20 | — | 6 | 1 | no | — |
| 61 | C5×SD16 | — | 12 | 1 | no | — |
| 62 | C5×SD16 | — | 12 | 1 | no | — |
| 63 | C2×C5×C8 | — | 6 | 1 | no | — |
| 64 | C5×C4⋊C4 | — | 3 | 1 | no | — |
| 65 | C42×C5 | — | 3 | 1 | no | — |
| 66 | #688 | — | 12 | 1 | no | — |
| 67 | #688 | — | 12 | 1 | no | — |
| 68 | #686 | — | 12 | 2 | no | — |
| 69 | #686 | — | 12 | 2 | no | — |
| 70 | C2×#168 | — | 3 | 1 | no | — |
| 71 | C2×#168 | — | 6 | 1 | no | — |
| 72 | C2×#168 | S4 | 1 | 1 | yes | characteristic |
| 73 | #493 | — | 15 | 1 | no | — |
| 74 | #460 | — | 3 | 1 | no | — |
| 75 | #460 | — | 15 | 1 | no | — |
| 76 | C8×Q8 | — | 5 | 1 | no | — |
| 77 | #335 | — | 15 | 1 | no | — |
| 78 | #331 | — | 15 | 1 | no | — |
| 79 | #331 | — | 15 | 1 | no | — |
| 80 | C22×C3×C5 | — | 4 | 1 | no | — |
| 81 | C2×D15 | — | 8 | 2 | no | — |
| 82 | C2×D15 | — | 16 | 1 | no | — |
| 83 | C2×D15 | — | 16 | 1 | no | — |
| 84 | C2×C5×D3 | — | 8 | 2 | no | — |
| 85 | C2×C5×D3 | — | 16 | 1 | no | — |
| 86 | C2×C5×D3 | — | 16 | 1 | no | — |
| 87 | C3×C4×C5 | — | 4 | 1 | no | — |
| 88 | C3×C4×C5 | — | 4 | 1 | no | — |
| 89 | C2×D12 | — | 4 | 1 | no | — |
| 90 | C2×D12 | — | 20 | 1 | no | — |
| 91 | C2×SL(2,𝔽3) | C22⋊D5 | 1 | 1 | yes | characteristic |
| 92 | GL(2,𝔽3) | — | 4 | 1 | no | — |
| 93 | GL(2,𝔽3) | — | 20 | 1 | no | — |
| 94 | C2×C3×C8 | — | 20 | 1 | no | — |
| 95 | D24 | — | 20 | 4 | no | — |
| 96 | C23×C5 | — | 12 | 1 | no | — |
| 97 | C22×D5 | — | 12 | 1 | no | — |
| 98 | C5×Q8 | — | 3 | 1 | no | — |
| 99 | C5×Q8 | D24 | 1 | 1 | yes | characteristic, complemented |
| 100 | C5×D4 | — | 12 | 1 | no | — |
| 101 | C5×D4 | — | 12 | 1 | no | — |
| 102 | C5×D4 | — | 12 | 1 | no | — |
| 103 | C5×D4 | — | 12 | 1 | no | — |
| 104 | C5×D4 | — | 24 | 1 | no | — |
| 105 | C2×C4×C5 | — | 3 | 1 | no | — |
| 106 | C2×C4×C5 | — | 3 | 1 | no | — |
| 107 | C2×C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 108 | D20 | — | 12 | 1 | no | — |
| 109 | D20 | — | 12 | 1 | no | — |
| 110 | D20 | — | 12 | 1 | no | — |
| 111 | D20 | — | 12 | 1 | no | — |
| 112 | D20 | — | 24 | 1 | no | — |
| 113 | C5×C8 | — | 12 | 1 | no | — |
| 114 | #168 | — | 6 | 1 | no | — |
| 115 | #168 | — | 12 | 1 | no | — |
| 116 | #168 | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 117 | C2×SD16 | — | 6 | 1 | no | — |
| 118 | C2×SD16 | — | 30 | 1 | no | — |
| 119 | C2×D8 | — | 30 | 1 | no | — |
| 120 | C2×D8 | — | 30 | 1 | no | — |
| 121 | #127 | — | 3 | 1 | no | — |
| 122 | #127 | — | 15 | 1 | no | — |
| 123 | C4×Q8 | D3×D5 | 1 | 1 | yes | characteristic |
| 124 | C4⋊C8 | — | 3 | 1 | no | — |
| 125 | C4⋊C8 | — | 15 | 1 | no | — |
| 126 | C4⋊C8 | — | 15 | 1 | no | — |
| 127 | #102 | — | 6 | 1 | no | — |
| 128 | #102 | — | 30 | 1 | no | — |
| 129 | C4×C8 | — | 15 | 1 | no | — |
| 130 | C2×C3×C5 | — | 4 | 1 | no | — |
| 131 | C2×C3×C5 | — | 4 | 1 | no | — |
| 132 | C2×C3×C5 | — | 4 | 1 | no | — |
| 133 | D15 | — | 16 | 2 | no | — |
| 134 | C5×D3 | — | 16 | 2 | no | — |
| 135 | C22×D3 | — | 8 | 1 | no | — |
| 136 | C22×D3 | — | 40 | 1 | no | — |
| 137 | C2×C3×C4 | — | 4 | 1 | no | — |
| 138 | D12 | — | 4 | 2 | no | — |
| 139 | D12 | — | 8 | 1 | no | — |
| 140 | D12 | — | 20 | 2 | no | — |
| 141 | D12 | — | 40 | 1 | no | — |
| 142 | SL(2,𝔽3) | #686 | 1 | 1 | yes | characteristic, complemented |
| 143 | C3×C8 | — | 20 | 2 | no | — |
| 144 | C22×C5 | — | 24 | 1 | no | — |
| 145 | C22×C5 | — | 24 | 1 | no | — |
| 146 | C22×C5 | — | 24 | 1 | no | — |
| 147 | C22×C5 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 148 | C2×D5 | — | 24 | 1 | no | — |
| 149 | C2×D5 | — | 24 | 1 | no | — |
| 150 | C2×D5 | — | 24 | 1 | no | — |
| 151 | C4×C5 | — | 3 | 1 | no | — |
| 152 | C4×C5 | — | 3 | 1 | no | — |
| 153 | C4×C5 | — | 6 | 1 | no | — |
| 154 | C4×C5 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 155 | C4×C5 | #986 | 1 | 1 | yes | characteristic |
| 156 | C2×Q8 | #1203 | 1 | 1 | yes | characteristic |
| 157 | C2×D4 | — | 6 | 1 | no | — |
| 158 | C2×D4 | — | 6 | 1 | no | — |
| 159 | C2×D4 | — | 6 | 1 | no | — |
| 160 | C2×D4 | — | 30 | 1 | no | — |
| 161 | C2×D4 | — | 30 | 1 | no | — |
| 162 | C2×D4 | — | 30 | 1 | no | — |
| 163 | SD16 | — | 12 | 1 | no | — |
| 164 | SD16 | — | 12 | 1 | no | — |
| 165 | SD16 | — | 60 | 1 | no | — |
| 166 | SD16 | — | 60 | 1 | no | — |
| 167 | D8 | — | 60 | 2 | no | — |
| 168 | D8 | — | 60 | 2 | no | — |
| 169 | C2×C8 | — | 5 | 1 | no | — |
| 170 | C2×C8 | — | 6 | 1 | no | — |
| 171 | C2×C8 | — | 15 | 1 | no | — |
| 172 | C2×C8 | — | 30 | 1 | no | — |
| 173 | C4⋊C4 | — | 3 | 1 | no | — |
| 174 | C42 | — | 3 | 1 | no | — |
| 175 | C3×C5 | — | 4 | 1 | no | Hall |
| 176 | C22×C3 | — | 4 | 1 | no | — |
| 177 | C2×D3 | — | 8 | 2 | no | — |
| 178 | C2×D3 | — | 16 | 1 | no | — |
| 179 | C2×D3 | — | 16 | 1 | no | — |
| 180 | C2×D3 | — | 40 | 2 | no | — |
| 181 | C2×D3 | — | 80 | 1 | no | — |
| 182 | C2×D3 | — | 80 | 1 | no | — |
| 183 | C3×C4 | — | 4 | 1 | no | — |
| 184 | C3×C4 | — | 4 | 1 | no | — |
| 185 | C2×C5 | — | 48 | 1 | no | — |
| 186 | C2×C5 | C4⋊GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 187 | C2×C5 | #5402 | 1 | 1 | yes | characteristic |
| 188 | C2×C5 | #5407 | 1 | 1 | yes | characteristic |
| 189 | D5 | — | 48 | 1 | no | — |
| 190 | C23 | — | 12 | 1 | no | — |
| 191 | C23 | — | 60 | 1 | no | — |
| 192 | Q8 | — | 3 | 1 | no | — |
| 193 | Q8 | #6654 | 1 | 1 | yes | characteristic, complemented |
| 194 | D4 | — | 12 | 1 | no | — |
| 195 | D4 | — | 12 | 1 | no | — |
| 196 | D4 | — | 12 | 1 | no | — |
| 197 | D4 | — | 12 | 1 | no | — |
| 198 | D4 | — | 24 | 1 | no | — |
| 199 | D4 | — | 60 | 1 | no | — |
| 200 | D4 | — | 60 | 1 | no | — |
| 201 | D4 | — | 60 | 1 | no | — |
| 202 | D4 | — | 60 | 1 | no | — |
| 203 | D4 | — | 120 | 1 | no | — |
| 204 | C2×C4 | — | 3 | 1 | no | — |
| 205 | C2×C4 | — | 3 | 1 | no | — |
| 206 | C2×C4 | D5×S4 | 1 | 1 | yes | characteristic |
| 207 | C8 | — | 5 | 2 | no | — |
| 208 | C8 | — | 12 | 1 | no | — |
| 209 | C8 | — | 30 | 1 | no | — |
| 210 | C8 | — | 60 | 1 | no | — |
| 211 | C2×C3 | — | 4 | 1 | no | — |
| 212 | C2×C3 | — | 4 | 1 | no | — |
| 213 | C2×C3 | — | 4 | 1 | no | — |
| 214 | D3 | — | 16 | 2 | no | — |
| 215 | D3 | — | 80 | 2 | no | — |
| 216 | C5 | #73022 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 217 | C22 | — | 24 | 1 | no | — |
| 218 | C22 | — | 24 | 1 | no | — |
| 219 | C22 | — | 24 | 1 | no | — |
| 220 | C22 | — | 120 | 1 | no | — |
| 221 | C22 | — | 120 | 1 | no | — |
| 222 | C22 | — | 120 | 1 | no | — |
| 223 | C22 | #91902 | 1 | 1 | yes | characteristic |
| 224 | C4 | — | 3 | 1 | no | — |
| 225 | C4 | — | 3 | 1 | no | — |
| 226 | C4 | — | 6 | 1 | no | — |
| 227 | C4 | #91894 | 1 | 1 | yes | characteristic |
| 228 | C4 | D5×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 229 | C3 | — | 4 | 1 | no | Sylow |
| 230 | C2 | — | 48 | 1 | no | — |
| 231 | C2 | — | 240 | 1 | no | — |
| 232 | C2 | #11751575 | 1 | 1 | yes | characteristic |
| 233 | C2 | #11751611 | 1 | 1 | yes | characteristic |
| 234 | C2 | #11751621 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4⋊(C5⋊GL(2,𝔽3)) | C2 | semidirect |
| 2. C5×C4⋊GL(2,𝔽3) | C2 | semidirect |
| 3. SL(2,𝔽3)×#168 | C2 | semidirect |
| 13. Q8×#168 | D3 | semidirect |
| 25. C4⋊GL(2,𝔽3) | D5 | semidirect |
| 48. C5×SL(2,𝔽3) | D8 | semidirect |
| 99. C5×Q8 | D24 | semidirect |
| 116. #168 | GL(2,𝔽3) | semidirect |
| 142. SL(2,𝔽3) | #686 | semidirect |
| 193. Q8 | #6654 | semidirect |
| 216. C5 | #73022 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4⋊(C5⋊GL(2,𝔽3)) | 2 |
| C5×C4⋊GL(2,𝔽3) | 2 |
| SL(2,𝔽3)×#168 | 2 |
| #10601814 | 3 |
| C2×#6654 | 4 |
| #73022 | 5 |