Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
496 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 79 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 | C3×C5×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11751163 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11746363 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10602744 | — | 3 | 1 | no | maximal, Hall |
| 5 | #92098 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C3×C5×#124 | — | 2 | 3 | no | — |
| 7 | C3×C5×#124 | C22 | 1 | 3 | yes | — |
| 8 | C3×C5×#120 | — | 2 | 3 | no | — |
| 9 | #91821 | — | 2 | 3 | no | — |
| 10 | #91120 | C22 | 1 | 3 | yes | complemented |
| 11 | #91089 | — | 2 | 3 | no | — |
| 12 | #71951 | — | 5 | 1 | no | maximal, Hall |
| 13 | C5×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #65351 | — | 3 | 3 | no | — |
| 15 | #64762 | — | 3 | 3 | no | — |
| 16 | C24×C3×C5 | — | 2 | 1 | no | — |
| 17 | C2×C3×C5×Q8 | — | 2 | 1 | no | — |
| 18 | C2×C3×C5×Q8 | C23 | 1 | 1 | yes | characteristic |
| 19 | C2×C3×C5×D4 | — | 2 | 3 | no | — |
| 20 | C2×C3×C5×D4 | D4 | 1 | 3 | yes | — |
| 21 | #6821 | — | 4 | 1 | no | — |
| 22 | Q8×D15 | — | 4 | 1 | no | — |
| 23 | #6817 | — | 2 | 3 | no | — |
| 24 | C3×C5×C22⋊C4 | — | 2 | 3 | no | — |
| 25 | C3×C5×C22⋊C4 | — | 2 | 6 | no | — |
| 26 | C3×C42×C5 | D4 | 1 | 3 | yes | — |
| 27 | #6716 | — | 4 | 3 | no | — |
| 28 | #6715 | — | 4 | 3 | no | — |
| 29 | #6710 | — | 2 | 3 | no | — |
| 30 | C3×H(3,𝔽4) | D5 | 1 | 1 | yes | characteristic, complemented |
| 31 | #5062 | — | 5 | 3 | no | — |
| 32 | #4542 | — | 5 | 3 | no | — |
| 33 | #4147 | — | 3 | 1 | no | — |
| 34 | C5×#124 | — | 2 | 3 | no | — |
| 35 | C5×#124 | C2×D3 | 1 | 3 | yes | — |
| 36 | C5×#120 | — | 2 | 3 | no | — |
| 37 | #4078 | — | 6 | 3 | no | — |
| 38 | #3968 | — | 3 | 3 | no | — |
| 39 | #3937 | — | 6 | 3 | no | — |
| 40 | #2202 | — | 15 | 1 | no | Sylow |
| 41 | C23×C3×C5 | — | 2 | 3 | no | — |
| 42 | C23×C3×C5 | — | 8 | 2 | no | — |
| 43 | C23×C3×C5 | — | 8 | 1 | no | — |
| 44 | C3×C5×Q8 | — | 4 | 1 | no | — |
| 45 | C3×C5×Q8 | — | 8 | 1 | no | — |
| 46 | C3×C5×D4 | — | 4 | 3 | no | — |
| 47 | C3×C5×D4 | — | 8 | 3 | no | — |
| 48 | C2×C3×C4×C5 | — | 2 | 3 | no | — |
| 49 | C2×C3×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 50 | #1220 | — | 4 | 1 | no | — |
| 51 | D60 | — | 2 | 3 | no | — |
| 52 | C4×D15 | — | 4 | 3 | no | — |
| 53 | Dic30 | — | 2 | 3 | no | — |
| 54 | C3⋊#168 | — | 4 | 3 | no | — |
| 55 | #1007 | — | 5 | 1 | no | — |
| 56 | C3×#124 | — | 2 | 3 | no | — |
| 57 | C3×#124 | C2×D5 | 1 | 3 | yes | — |
| 58 | C3×#120 | — | 2 | 3 | no | — |
| 59 | #932 | — | 10 | 3 | no | — |
| 60 | #836 | — | 5 | 3 | no | — |
| 61 | #805 | — | 10 | 3 | no | — |
| 62 | C24×C5 | — | 2 | 1 | no | — |
| 63 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 64 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 65 | C2×C5×D4 | — | 2 | 3 | no | — |
| 66 | C2×C5×D4 | #67 | 1 | 3 | yes | — |
| 67 | #713 | — | 12 | 1 | no | — |
| 68 | Q8×D5 | — | 12 | 1 | no | — |
| 69 | #709 | — | 6 | 3 | no | — |
| 70 | C5×C22⋊C4 | — | 2 | 3 | no | — |
| 71 | C5×C22⋊C4 | — | 2 | 6 | no | — |
| 72 | C42×C5 | #67 | 1 | 3 | yes | — |
| 73 | #687 | — | 12 | 3 | no | — |
| 74 | #686 | — | 12 | 3 | no | — |
| 75 | #681 | — | 6 | 3 | no | — |
| 76 | H(3,𝔽4) | D15 | 1 | 1 | yes | characteristic, complemented |
| 77 | #454 | — | 15 | 3 | no | — |
| 78 | #355 | — | 15 | 3 | no | — |
| 79 | C22×C3×C5 | — | 4 | 6 | no | — |
| 80 | C22×C3×C5 | — | 4 | 3 | no | — |
| 81 | C22×C3×C5 | — | 8 | 4 | no | — |
| 82 | C22×C3×C5 | #120 | 1 | 1 | yes | characteristic |
| 83 | C2×D15 | — | 4 | 1 | no | — |
| 84 | C3×C4×C5 | — | 2 | 3 | no | — |
| 85 | C3×C4×C5 | — | 4 | 3 | no | — |
| 86 | Dic15 | — | 4 | 1 | no | — |
| 87 | C24×C3 | — | 2 | 1 | no | — |
| 88 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 89 | C2×C3×Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 90 | C2×C3×D4 | — | 2 | 3 | no | — |
| 91 | C2×C3×D4 | C22⋊D5 | 1 | 3 | yes | — |
| 92 | #239 | — | 20 | 1 | no | — |
| 93 | D3×Q8 | — | 20 | 1 | no | — |
| 94 | #235 | — | 10 | 3 | no | — |
| 95 | C3×C22⋊C4 | — | 2 | 3 | no | — |
| 96 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 97 | C3×C42 | C22⋊D5 | 1 | 3 | yes | — |
| 98 | #214 | — | 20 | 3 | no | — |
| 99 | #213 | — | 20 | 3 | no | — |
| 100 | #208 | — | 10 | 3 | no | — |
| 101 | C23×C5 | — | 2 | 3 | no | — |
| 102 | C23×C5 | — | 8 | 2 | no | — |
| 103 | C23×C5 | — | 8 | 1 | no | — |
| 104 | C5×Q8 | — | 4 | 1 | no | — |
| 105 | C5×Q8 | — | 8 | 1 | no | — |
| 106 | C5×D4 | — | 4 | 3 | no | — |
| 107 | C5×D4 | — | 8 | 3 | no | — |
| 108 | C2×C4×C5 | — | 2 | 3 | no | — |
| 109 | C2×C4×C5 | C2×#67 | 1 | 3 | yes | — |
| 110 | C22⋊D5 | — | 12 | 1 | no | — |
| 111 | D20 | — | 6 | 3 | no | — |
| 112 | C4×D5 | — | 12 | 3 | no | — |
| 113 | Dic10 | — | 6 | 3 | no | — |
| 114 | #168 | — | 12 | 3 | no | — |
| 115 | 2−1+4 | — | 15 | 1 | no | — |
| 116 | AGL(1,ℤ/8) | — | 30 | 3 | no | — |
| 117 | #124 | — | 2 | 3 | no | — |
| 118 | #124 | C2×D15 | 1 | 3 | yes | — |
| 119 | #120 | — | 2 | 3 | no | — |
| 120 | #104 | — | 30 | 3 | no | — |
| 121 | #101 | — | 15 | 3 | no | — |
| 122 | C2×C3×C5 | — | 2 | 1 | no | — |
| 123 | C2×C3×C5 | — | 8 | 3 | no | — |
| 124 | C2×C3×C5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 125 | D15 | — | 8 | 1 | no | — |
| 126 | C23×C3 | — | 2 | 3 | no | — |
| 127 | C23×C3 | — | 8 | 2 | no | — |
| 128 | C23×C3 | — | 8 | 1 | no | — |
| 129 | C3×Q8 | — | 4 | 1 | no | — |
| 130 | C3×Q8 | — | 8 | 1 | no | — |
| 131 | C3×D4 | — | 4 | 3 | no | — |
| 132 | C3×D4 | — | 8 | 3 | no | — |
| 133 | C2×C3×C4 | — | 2 | 3 | no | — |
| 134 | C2×C3×C4 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 135 | #67 | — | 20 | 1 | no | — |
| 136 | D12 | — | 10 | 3 | no | — |
| 137 | C4×D3 | — | 20 | 3 | no | — |
| 138 | Dic6 | — | 10 | 3 | no | — |
| 139 | C3⋊C8 | — | 20 | 3 | no | — |
| 140 | C22×C5 | — | 4 | 6 | no | — |
| 141 | C22×C5 | — | 4 | 3 | no | — |
| 142 | C22×C5 | — | 8 | 4 | no | — |
| 143 | C22×C5 | #953 | 1 | 1 | yes | characteristic |
| 144 | C2×D5 | — | 12 | 1 | no | — |
| 145 | C4×C5 | — | 2 | 3 | no | — |
| 146 | C4×C5 | — | 4 | 3 | no | — |
| 147 | Dic5 | — | 12 | 1 | no | — |
| 148 | C24 | — | 2 | 1 | no | — |
| 149 | #41 | — | 30 | 3 | no | — |
| 150 | #41 | — | 60 | 1 | no | — |
| 151 | C2×Q8 | — | 2 | 1 | no | — |
| 152 | C2×Q8 | — | 60 | 1 | no | — |
| 153 | C2×Q8 | C22×D15 | 1 | 1 | yes | characteristic |
| 154 | C2×D4 | — | 2 | 3 | no | — |
| 155 | C2×D4 | #1220 | 1 | 3 | yes | — |
| 156 | SD16 | — | 60 | 3 | no | — |
| 157 | D8 | — | 60 | 3 | no | — |
| 158 | M16 | — | 30 | 3 | no | — |
| 159 | C22⋊C4 | — | 2 | 3 | no | — |
| 160 | C22⋊C4 | — | 2 | 6 | no | — |
| 161 | C42 | #1220 | 1 | 3 | yes | — |
| 162 | C3×C5 | #2202 | 1 | 1 | yes | characteristic, complemented, Hall |
| 163 | C22×C3 | — | 4 | 6 | no | — |
| 164 | C22×C3 | — | 4 | 3 | no | — |
| 165 | C22×C3 | — | 8 | 4 | no | — |
| 166 | C22×C3 | #4099 | 1 | 1 | yes | characteristic |
| 167 | C2×D3 | — | 20 | 1 | no | — |
| 168 | C3×C4 | — | 2 | 3 | no | — |
| 169 | C3×C4 | — | 4 | 3 | no | — |
| 170 | Dic3 | — | 20 | 1 | no | — |
| 171 | C2×C5 | — | 2 | 1 | no | — |
| 172 | C2×C5 | — | 8 | 3 | no | — |
| 173 | C2×C5 | #5244 | 1 | 1 | yes | characteristic |
| 174 | D5 | — | 24 | 1 | no | — |
| 175 | C23 | — | 2 | 3 | no | — |
| 176 | C23 | — | 8 | 2 | no | — |
| 177 | C23 | — | 8 | 1 | no | — |
| 178 | Q8 | — | 4 | 1 | no | — |
| 179 | Q8 | — | 8 | 1 | no | — |
| 180 | Q8 | — | 30 | 3 | no | — |
| 181 | D4 | — | 4 | 3 | no | — |
| 182 | D4 | — | 8 | 3 | no | — |
| 183 | D4 | — | 30 | 3 | no | — |
| 184 | D4 | — | 60 | 1 | no | — |
| 185 | C2×C4 | — | 2 | 3 | no | — |
| 186 | C2×C4 | — | 60 | 3 | no | — |
| 187 | C2×C4 | C2×#1220 | 1 | 3 | yes | — |
| 188 | C8 | — | 60 | 3 | no | — |
| 189 | C2×C3 | — | 2 | 1 | no | — |
| 190 | C2×C3 | — | 8 | 3 | no | — |
| 191 | C2×C3 | #65533 | 1 | 1 | yes | characteristic |
| 192 | D3 | — | 40 | 1 | no | — |
| 193 | C5 | #71951 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 194 | C22 | — | 4 | 6 | no | — |
| 195 | C22 | — | 4 | 3 | no | — |
| 196 | C22 | — | 8 | 4 | no | — |
| 197 | C22 | — | 60 | 1 | no | — |
| 198 | C22 | #91842 | 1 | 1 | yes | characteristic |
| 199 | C4 | — | 2 | 3 | no | — |
| 200 | C4 | — | 4 | 3 | no | — |
| 201 | C4 | — | 60 | 1 | no | — |
| 202 | C3 | #10602744 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 203 | C2 | — | 2 | 1 | no | — |
| 204 | C2 | — | 8 | 3 | no | — |
| 205 | C2 | — | 120 | 1 | no | — |
| 206 | C2 | #11751345 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×H(3,𝔽4) | C2 | semidirect |
| 2. #11751163 | C2 | semidirect |
| 3. #11746363 | C2 | semidirect |
| 5. #92098 | C22 | semidirect |
| 10. #91120 | C22 | semidirect |
| 13. C5×H(3,𝔽4) | D3 | semidirect |
| 30. C3×H(3,𝔽4) | D5 | semidirect |
| 76. H(3,𝔽4) | D15 | semidirect |
| 162. C3×C5 | #2202 | semidirect |
| 193. C5 | #71951 | semidirect |
| 202. C3 | #10602744 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×H(3,𝔽4) | 2 |
| #11751163 | 2 |
| #11746363 | 2 |
| #10602744 | 3 |
| #71951 | 5 |