Finite Group DatabasebetaFinite Group DatabasebetaOrder 960.
399 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 25 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 | C22×S5 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | D4×A5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C22⋊S5 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C4⋊S5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C4×S5 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×A5 | C22 | 1 | 2 | yes | complemented |
| 7 | C2×S5 | — | 2 | 4 | no | — |
| 8 | C2×S5 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×S5 | C22 | 1 | 2 | yes | complemented |
| 10 | C4×A5 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | A5⋊C4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | D4×S4 | — | 5 | 1 | no | maximal, Hall |
| 13 | D4×AGL(1,𝔽5) | — | 6 | 1 | no | maximal |
| 14 | C2×A5 | — | 2 | 2 | no | — |
| 15 | C2×A5 | C23 | 1 | 1 | yes | characteristic |
| 16 | S5 | — | 2 | 2 | no | — |
| 17 | S5 | D4 | 1 | 2 | yes | direct product factor |
| 18 | C22×S4 | — | 5 | 2 | no | — |
| 19 | C2×D3×D4 | — | 10 | 1 | no | maximal |
| 20 | D4×A4 | — | 5 | 1 | no | — |
| 21 | GL(2,ℤ/4) | — | 5 | 2 | no | — |
| 22 | C4⋊S4 | — | 5 | 1 | no | — |
| 23 | C4×S4 | — | 5 | 1 | no | — |
| 24 | C22×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 25 | D4×D5 | — | 6 | 1 | no | — |
| 26 | C22⋊AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 27 | C4⋊AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 28 | C4×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 29 | D42 | — | 15 | 1 | no | Sylow |
| 30 | A5 | C2×D4 | 1 | 1 | yes | characteristic, complemented |
| 31 | C23×D3 | — | 10 | 2 | no | — |
| 32 | C22×A4 | — | 5 | 2 | no | — |
| 33 | C2×S4 | — | 5 | 1 | no | — |
| 34 | C2×S4 | — | 5 | 2 | no | — |
| 35 | C2×S4 | — | 10 | 4 | no | — |
| 36 | C2×C3×D4 | — | 10 | 1 | no | — |
| 37 | C2×#67 | — | 10 | 2 | no | — |
| 38 | D3×D4 | — | 10 | 1 | no | — |
| 39 | D3×D4 | — | 10 | 1 | no | — |
| 40 | D3×D4 | — | 10 | 2 | no | — |
| 41 | D3×D4 | — | 10 | 1 | no | — |
| 42 | D3×D4 | — | 10 | 1 | no | — |
| 43 | D3×D4 | — | 10 | 2 | no | — |
| 44 | C2×D12 | — | 10 | 1 | no | — |
| 45 | C2×C4×D3 | — | 10 | 1 | no | — |
| 46 | C4×A4 | — | 5 | 1 | no | — |
| 47 | SL(2,ℤ/4) | — | 5 | 1 | no | — |
| 48 | C22×D5 | — | 6 | 2 | no | — |
| 49 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 50 | C2×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 51 | C2×AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 52 | C2×AGL(1,𝔽5) | — | 12 | 4 | no | — |
| 53 | C5×D4 | — | 6 | 1 | no | — |
| 54 | C22⋊D5 | — | 6 | 2 | no | — |
| 55 | D20 | — | 6 | 1 | no | — |
| 56 | C4×D5 | — | 6 | 1 | no | — |
| 57 | C22×D4 | — | 5 | 1 | no | — |
| 58 | C22×D4 | — | 15 | 1 | no | — |
| 59 | C22×D4 | — | 15 | 2 | no | — |
| 60 | #127 | — | 15 | 1 | no | — |
| 61 | #121 | — | 15 | 1 | no | — |
| 62 | #121 | — | 15 | 2 | no | — |
| 63 | #121 | — | 15 | 1 | no | — |
| 64 | #120 | — | 15 | 2 | no | — |
| 65 | #120 | — | 15 | 2 | no | — |
| 66 | C4×D4 | — | 15 | 1 | no | — |
| 67 | C4×D4 | — | 15 | 1 | no | — |
| 68 | C23×C3 | — | 10 | 2 | no | — |
| 69 | C22×D3 | — | 10 | 2 | no | — |
| 70 | C22×D3 | — | 10 | 1 | no | — |
| 71 | C22×D3 | — | 10 | 2 | no | — |
| 72 | C22×D3 | — | 10 | 2 | no | — |
| 73 | C22×D3 | — | 10 | 2 | no | — |
| 74 | C22×D3 | — | 10 | 2 | no | — |
| 75 | C22×D3 | — | 20 | 4 | no | — |
| 76 | C22×D3 | — | 20 | 4 | no | — |
| 77 | C2×A4 | — | 5 | 1 | no | — |
| 78 | C2×A4 | — | 10 | 2 | no | — |
| 79 | S4 | — | 5 | 2 | no | — |
| 80 | S4 | — | 10 | 2 | no | — |
| 81 | C3×D4 | — | 10 | 2 | no | — |
| 82 | C3×D4 | — | 10 | 1 | no | — |
| 83 | C3×D4 | — | 10 | 1 | no | — |
| 84 | C2×C3×C4 | — | 10 | 1 | no | — |
| 85 | #67 | — | 10 | 2 | no | — |
| 86 | #67 | — | 10 | 2 | no | — |
| 87 | #67 | — | 10 | 2 | no | — |
| 88 | #67 | — | 10 | 2 | no | — |
| 89 | C2×Dic3 | — | 10 | 1 | no | — |
| 90 | D12 | — | 10 | 1 | no | — |
| 91 | D12 | — | 10 | 1 | no | — |
| 92 | D12 | — | 10 | 2 | no | — |
| 93 | C4×D3 | — | 10 | 1 | no | — |
| 94 | C4×D3 | — | 10 | 1 | no | — |
| 95 | C4×D3 | — | 10 | 1 | no | — |
| 96 | C4×D3 | — | 10 | 1 | no | — |
| 97 | C22×C5 | — | 6 | 2 | no | — |
| 98 | C2×D5 | — | 6 | 1 | no | — |
| 99 | C2×D5 | — | 6 | 2 | no | — |
| 100 | C2×D5 | — | 12 | 2 | no | — |
| 101 | C2×D5 | — | 12 | 2 | no | — |
| 102 | AGL(1,𝔽5) | — | 6 | 2 | no | — |
| 103 | AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 104 | AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 105 | C4×C5 | — | 6 | 1 | no | — |
| 106 | Dic5 | — | 6 | 1 | no | — |
| 107 | C24 | — | 5 | 2 | no | — |
| 108 | C24 | — | 15 | 2 | no | — |
| 109 | C2×D4 | — | 10 | 1 | no | — |
| 110 | C2×D4 | — | 15 | 2 | no | — |
| 111 | C2×D4 | — | 15 | 1 | no | — |
| 112 | C2×D4 | — | 15 | 1 | no | — |
| 113 | C2×D4 | — | 15 | 1 | no | — |
| 114 | C2×D4 | — | 15 | 2 | no | — |
| 115 | C2×D4 | — | 15 | 1 | no | — |
| 116 | C2×D4 | — | 15 | 2 | no | — |
| 117 | C2×D4 | — | 15 | 2 | no | — |
| 118 | C2×D4 | — | 15 | 2 | no | — |
| 119 | C2×D4 | — | 15 | 2 | no | — |
| 120 | C2×D4 | — | 30 | 1 | no | — |
| 121 | C2×D4 | — | 30 | 2 | no | — |
| 122 | C2×D4 | — | 30 | 4 | no | — |
| 123 | C2×D4 | — | 30 | 4 | no | — |
| 124 | C22×C4 | — | 5 | 1 | no | — |
| 125 | C22×C4 | — | 15 | 1 | no | — |
| 126 | C22×C4 | — | 15 | 2 | no | — |
| 127 | C4⋊C4 | — | 15 | 1 | no | — |
| 128 | C4⋊C4 | — | 15 | 1 | no | — |
| 129 | C22⋊C4 | — | 15 | 1 | no | — |
| 130 | C22⋊C4 | — | 15 | 1 | no | — |
| 131 | C22⋊C4 | — | 15 | 2 | no | — |
| 132 | C22⋊C4 | — | 15 | 2 | no | — |
| 133 | C22⋊C4 | — | 15 | 2 | no | — |
| 134 | C42 | — | 15 | 1 | no | — |
| 135 | C22×C3 | — | 10 | 1 | no | — |
| 136 | C22×C3 | — | 10 | 2 | no | — |
| 137 | C22×C3 | — | 10 | 2 | no | — |
| 138 | C22×C3 | — | 20 | 4 | no | — |
| 139 | C2×D3 | — | 10 | 1 | no | — |
| 140 | C2×D3 | — | 10 | 2 | no | — |
| 141 | C2×D3 | — | 10 | 1 | no | — |
| 142 | C2×D3 | — | 10 | 2 | no | — |
| 143 | C2×D3 | — | 10 | 2 | no | — |
| 144 | C2×D3 | — | 10 | 2 | no | — |
| 145 | C2×D3 | — | 20 | 4 | no | — |
| 146 | C2×D3 | — | 20 | 2 | no | — |
| 147 | C2×D3 | — | 20 | 2 | no | — |
| 148 | C2×D3 | — | 20 | 2 | no | — |
| 149 | C2×D3 | — | 20 | 2 | no | — |
| 150 | C2×D3 | — | 20 | 4 | no | — |
| 151 | C2×D3 | — | 20 | 4 | no | — |
| 152 | A4 | — | 5 | 1 | no | — |
| 153 | C3×C4 | — | 10 | 1 | no | — |
| 154 | C3×C4 | — | 10 | 1 | no | — |
| 155 | Dic3 | — | 10 | 1 | no | — |
| 156 | Dic3 | — | 10 | 1 | no | — |
| 157 | C2×C5 | — | 6 | 1 | no | — |
| 158 | C2×C5 | — | 12 | 2 | no | — |
| 159 | D5 | — | 6 | 1 | no | — |
| 160 | D5 | — | 6 | 1 | no | — |
| 161 | D5 | — | 12 | 2 | no | — |
| 162 | C23 | — | 5 | 1 | no | — |
| 163 | C23 | — | 10 | 2 | no | — |
| 164 | C23 | — | 10 | 2 | no | — |
| 165 | C23 | — | 15 | 1 | no | — |
| 166 | C23 | — | 15 | 2 | no | — |
| 167 | C23 | — | 15 | 2 | no | — |
| 168 | C23 | — | 15 | 2 | no | — |
| 169 | C23 | — | 30 | 2 | no | — |
| 170 | C23 | — | 30 | 4 | no | — |
| 171 | C23 | — | 30 | 2 | no | — |
| 172 | C23 | — | 60 | 2 | no | — |
| 173 | D4 | — | 10 | 2 | no | — |
| 174 | D4 | — | 10 | 1 | no | — |
| 175 | D4 | — | 15 | 2 | no | — |
| 176 | D4 | — | 15 | 2 | no | — |
| 177 | D4 | — | 15 | 2 | no | — |
| 178 | D4 | — | 15 | 1 | no | — |
| 179 | D4 | — | 30 | 2 | no | — |
| 180 | D4 | — | 30 | 1 | no | — |
| 181 | D4 | — | 30 | 1 | no | — |
| 182 | D4 | — | 30 | 4 | no | — |
| 183 | D4 | — | 30 | 2 | no | — |
| 184 | D4 | — | 30 | 4 | no | — |
| 185 | D4 | — | 30 | 2 | no | — |
| 186 | D4 | — | 60 | 2 | no | — |
| 187 | D4 | S5 | 1 | 1 | yes | characteristic, direct product factor |
| 188 | C2×C4 | — | 10 | 1 | no | — |
| 189 | C2×C4 | — | 15 | 1 | no | — |
| 190 | C2×C4 | — | 15 | 1 | no | — |
| 191 | C2×C4 | — | 15 | 1 | no | — |
| 192 | C2×C4 | — | 15 | 1 | no | — |
| 193 | C2×C4 | — | 15 | 1 | no | — |
| 194 | C2×C4 | — | 15 | 2 | no | — |
| 195 | C2×C4 | — | 30 | 1 | no | — |
| 196 | C2×C4 | — | 30 | 4 | no | — |
| 197 | C2×C3 | — | 10 | 2 | no | — |
| 198 | C2×C3 | — | 10 | 1 | no | — |
| 199 | C2×C3 | — | 20 | 2 | no | — |
| 200 | C2×C3 | — | 20 | 2 | no | — |
| 201 | D3 | — | 10 | 2 | no | — |
| 202 | D3 | — | 10 | 1 | no | — |
| 203 | D3 | — | 10 | 1 | no | — |
| 204 | D3 | — | 20 | 2 | no | — |
| 205 | D3 | — | 20 | 2 | no | — |
| 206 | C5 | — | 6 | 1 | no | Sylow |
| 207 | C22 | — | 5 | 1 | no | — |
| 208 | C22 | — | 10 | 1 | no | — |
| 209 | C22 | — | 10 | 2 | no | — |
| 210 | C22 | — | 15 | 2 | no | — |
| 211 | C22 | — | 15 | 1 | no | — |
| 212 | C22 | — | 15 | 1 | no | — |
| 213 | C22 | — | 15 | 2 | no | — |
| 214 | C22 | — | 20 | 4 | no | — |
| 215 | C22 | — | 30 | 2 | no | — |
| 216 | C22 | — | 30 | 2 | no | — |
| 217 | C22 | — | 30 | 1 | no | — |
| 218 | C22 | — | 30 | 2 | no | — |
| 219 | C22 | — | 30 | 2 | no | — |
| 220 | C22 | — | 30 | 2 | no | — |
| 221 | C22 | — | 30 | 2 | no | — |
| 222 | C22 | — | 60 | 4 | no | — |
| 223 | C22 | C2×S5 | 1 | 2 | yes | complemented |
| 224 | C4 | — | 10 | 1 | no | — |
| 225 | C4 | — | 15 | 2 | no | — |
| 226 | C4 | — | 15 | 1 | no | — |
| 227 | C4 | — | 30 | 2 | no | — |
| 228 | C4 | — | 30 | 1 | no | — |
| 229 | C4 | C2×S5 | 1 | 1 | yes | characteristic, complemented |
| 230 | C3 | — | 10 | 1 | no | Sylow |
| 231 | C2 | — | 2 | 2 | no | — |
| 232 | C2 | — | 10 | 2 | no | — |
| 233 | C2 | — | 15 | 1 | no | — |
| 234 | C2 | — | 15 | 1 | no | — |
| 235 | C2 | — | 20 | 2 | no | — |
| 236 | C2 | — | 30 | 2 | no | — |
| 237 | C2 | C22×S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×S5 |
| commutator, socle, cosocle | C2×A5 | C23 |
| Fitting, hypercenter | D4 | S5 |
| hypocenter | A5 | C2×D4 |
| LCS | G ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×S5 | C2 | semidirect |
| 2. D4×A5 | C2 | semidirect |
| 3. C22⋊S5 | C2 | semidirect |
| 4. C4⋊S5 | C2 | semidirect |
| 5. C4×S5 | C2 | semidirect |
| 6. C22×A5 | C22 | semidirect |
| 9. C2×S5 | C22 | semidirect |
| 10. C4×A5 | C22 | semidirect |
| 11. A5⋊C4 | C22 | semidirect |
| 17. S5 | D4 | direct (N × H) |
| 30. A5 | C2×D4 | semidirect |
| 187. D4 | S5 | direct (N × H) |
| 223. C22 | C2×S5 | semidirect |
| 229. C4 | C2×S5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×S5 | 2 |
| D4×A5 | 2 |
| C22⋊S5 | 2 |
| C4⋊S5 | 2 |
| C4×S5 | 2 |
| D4×S4 | 5 |
| D4×AGL(1,𝔽5) | 6 |
| C2×D3×D4 | 10 |