Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4218 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 4218 conjugacy classes of subgroups — too many to draw. Only its 80 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⋊D3×C22⋊A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×D3×C22⋊A4 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C24×D32 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×D3×C22⋊A4 | — | 3 | 2 | no | maximal |
| 5 | C32×C22⋊A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | D32×A4 | — | 4 | 5 | no | maximal |
| 7 | C24×C3⋊D3 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C24×C3×D3 | C2×C3 | 1 | 2 | yes | complemented |
| 9 | C2×C3×C22⋊A4 | — | 3 | 2 | no | — |
| 10 | C23×D32 | — | 3 | 5 | no | — |
| 11 | C23×D32 | — | 3 | 5 | no | — |
| 12 | C23×D32 | — | 3 | 10 | no | — |
| 13 | D3×C22⋊A4 | — | 2 | 2 | no | — |
| 14 | D3×C22⋊A4 | — | 3 | 2 | no | — |
| 15 | D3×C22⋊A4 | — | 6 | 1 | no | — |
| 16 | D3×C22⋊A4 | D3 | 1 | 2 | yes | direct product factor |
| 17 | A4×C3⋊D3 | — | 4 | 5 | no | — |
| 18 | C3×D3×A4 | — | 4 | 10 | no | — |
| 19 | C25×D3 | — | 3 | 2 | no | — |
| 20 | C22×C22⋊A4 | — | 9 | 1 | no | — |
| 21 | C24×C32 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 22 | C23×C3⋊D3 | — | 3 | 5 | no | — |
| 23 | C23×C3⋊D3 | — | 3 | 5 | no | — |
| 24 | C23×C3×D3 | — | 3 | 10 | no | — |
| 25 | C23×C3×D3 | — | 3 | 10 | no | — |
| 26 | C3×C22⋊A4 | — | 2 | 2 | no | — |
| 27 | C3×C22⋊A4 | — | 2 | 1 | no | — |
| 28 | C3×C22⋊A4 | — | 4 | 1 | no | — |
| 29 | C3×C22⋊A4 | C2×D3 | 1 | 2 | yes | complemented |
| 30 | C22×D32 | — | 3 | 60 | no | — |
| 31 | C22×D32 | — | 3 | 10 | no | — |
| 32 | C22×D32 | — | 3 | 60 | no | — |
| 33 | C22×D32 | — | 3 | 30 | no | — |
| 34 | C22×D32 | — | 3 | 10 | no | — |
| 35 | C22×D32 | — | 3 | 5 | no | — |
| 36 | C22×D32 | — | 3 | 10 | no | — |
| 37 | C22×D32 | A4 | 1 | 5 | yes | complemented |
| 38 | C2×D3×A4 | — | 12 | 10 | no | — |
| 39 | C32×A4 | — | 4 | 5 | no | — |
| 40 | C3×D32 | — | 16 | 1 | no | — |
| 41 | C25×C3 | — | 3 | 2 | no | — |
| 42 | C24×D3 | — | 3 | 2 | no | — |
| 43 | C24×D3 | — | 6 | 1 | no | — |
| 44 | C24×D3 | — | 9 | 10 | no | — |
| 45 | C24×D3 | — | 9 | 10 | no | — |
| 46 | C24×D3 | — | 9 | 10 | no | — |
| 47 | C24×D3 | — | 9 | 10 | no | — |
| 48 | C24×D3 | C3×D3 | 1 | 2 | yes | complemented |
| 49 | C2×C22⋊A4 | — | 3 | 2 | no | — |
| 50 | C2×C22⋊A4 | — | 6 | 2 | no | — |
| 51 | C2×C22⋊A4 | — | 9 | 1 | no | — |
| 52 | C23×C32 | — | 3 | 5 | no | — |
| 53 | C22×C3⋊D3 | — | 3 | 30 | no | — |
| 54 | C22×C3⋊D3 | — | 3 | 10 | no | — |
| 55 | C22×C3⋊D3 | — | 3 | 5 | no | — |
| 56 | C22×C3⋊D3 | C2×A4 | 1 | 5 | yes | complemented |
| 57 | C22×C3×D3 | — | 3 | 60 | no | — |
| 58 | C22×C3×D3 | — | 3 | 20 | no | — |
| 59 | C22×C3×D3 | — | 3 | 10 | no | — |
| 60 | C22×C3×D3 | C2×A4 | 1 | 10 | yes | complemented |
| 61 | C2×C3×A4 | — | 12 | 10 | no | — |
| 62 | C2×D32 | — | 3 | 120 | no | — |
| 63 | C2×D32 | — | 3 | 30 | no | — |
| 64 | C2×D32 | — | 3 | 60 | no | — |
| 65 | C2×D32 | — | 3 | 30 | no | — |
| 66 | C2×D32 | — | 3 | 60 | no | — |
| 67 | C2×D32 | — | 3 | 5 | no | — |
| 68 | C2×D32 | — | 3 | 5 | no | — |
| 69 | C2×D32 | — | 3 | 10 | no | — |
| 70 | D3×A4 | — | 4 | 10 | no | — |
| 71 | D3×A4 | — | 8 | 10 | no | — |
| 72 | D3×A4 | — | 12 | 10 | no | — |
| 73 | D3×A4 | — | 24 | 5 | no | — |
| 74 | C26 | — | 9 | 1 | no | Sylow |
| 75 | C3×C3⋊D3 | — | 16 | 1 | no | — |
| 76 | C32×D3 | — | 16 | 2 | no | — |
| 77 | C24×C3 | — | 2 | 1 | no | — |
| 78 | C24×C3 | — | 9 | 10 | no | — |
| 79 | C24×C3 | — | 9 | 10 | no | — |
| 80 | C24×C3 | C2×C3×D3 | 1 | 2 | yes | complemented |
| 81 | C23×D3 | — | 3 | 10 | no | — |
| 82 | C23×D3 | — | 3 | 10 | no | — |
| 83 | C23×D3 | — | 3 | 10 | no | — |
| 84 | C23×D3 | — | 9 | 10 | no | — |
| 85 | C23×D3 | — | 9 | 10 | no | — |
| 86 | C23×D3 | — | 9 | 10 | no | — |
| 87 | C23×D3 | — | 9 | 20 | no | — |
| 88 | C23×D3 | — | 9 | 120 | no | — |
| 89 | C23×D3 | — | 9 | 60 | no | — |
| 90 | C23×D3 | — | 9 | 60 | no | — |
| 91 | C23×D3 | — | 9 | 60 | no | — |
| 92 | C23×D3 | — | 9 | 20 | no | — |
| 93 | C23×D3 | — | 9 | 10 | no | — |
| 94 | C23×D3 | — | 9 | 10 | no | — |
| 95 | C23×D3 | — | 18 | 5 | no | — |
| 96 | C23×D3 | — | 18 | 5 | no | — |
| 97 | C22⋊A4 | — | 2 | 2 | no | — |
| 98 | C22⋊A4 | — | 4 | 1 | no | — |
| 99 | C22⋊A4 | D32 | 1 | 1 | yes | characteristic, direct product factor |
| 100 | C22×A4 | — | 36 | 5 | no | — |
| 101 | C22×C32 | — | 3 | 10 | no | — |
| 102 | C22×C32 | C22×A4 | 1 | 5 | yes | complemented |
| 103 | C2×C3⋊D3 | — | 3 | 30 | no | — |
| 104 | C2×C3⋊D3 | — | 3 | 5 | no | — |
| 105 | C2×C3⋊D3 | — | 3 | 5 | no | — |
| 106 | C2×C3×D3 | — | 3 | 60 | no | — |
| 107 | C2×C3×D3 | — | 3 | 10 | no | — |
| 108 | C2×C3×D3 | — | 3 | 10 | no | — |
| 109 | C2×C3×D3 | — | 48 | 2 | no | — |
| 110 | C3×A4 | — | 4 | 10 | no | — |
| 111 | C3×A4 | — | 8 | 5 | no | — |
| 112 | C3×A4 | — | 8 | 10 | no | — |
| 113 | C3×A4 | — | 16 | 5 | no | — |
| 114 | D32 | — | 3 | 60 | no | — |
| 115 | D32 | — | 3 | 10 | no | — |
| 116 | D32 | — | 3 | 5 | no | — |
| 117 | D32 | — | 3 | 10 | no | — |
| 118 | D32 | C22⋊A4 | 1 | 1 | yes | characteristic, direct product factor |
| 119 | C25 | — | 3 | 2 | no | — |
| 120 | C25 | — | 9 | 1 | no | — |
| 121 | C25 | — | 27 | 5 | no | — |
| 122 | C25 | — | 27 | 10 | no | — |
| 123 | C25 | — | 27 | 5 | no | — |
| 124 | C33 | — | 16 | 1 | no | Sylow |
| 125 | C23×C3 | — | 3 | 10 | no | — |
| 126 | C23×C3 | — | 3 | 10 | no | — |
| 127 | C23×C3 | — | 6 | 5 | no | — |
| 128 | C23×C3 | — | 9 | 20 | no | — |
| 129 | C23×C3 | — | 9 | 60 | no | — |
| 130 | C23×C3 | — | 9 | 10 | no | — |
| 131 | C22×D3 | — | 3 | 60 | no | — |
| 132 | C22×D3 | — | 3 | 20 | no | — |
| 133 | C22×D3 | — | 3 | 10 | no | — |
| 134 | C22×D3 | — | 3 | 10 | no | — |
| 135 | C22×D3 | — | 6 | 5 | no | — |
| 136 | C22×D3 | — | 9 | 10 | no | — |
| 137 | C22×D3 | — | 9 | 60 | no | — |
| 138 | C22×D3 | — | 9 | 60 | no | — |
| 139 | C22×D3 | — | 9 | 60 | no | — |
| 140 | C22×D3 | — | 9 | 240 | no | — |
| 141 | C22×D3 | — | 9 | 60 | no | — |
| 142 | C22×D3 | — | 9 | 60 | no | — |
| 143 | C22×D3 | — | 9 | 60 | no | — |
| 144 | C22×D3 | — | 9 | 10 | no | — |
| 145 | C22×D3 | — | 9 | 10 | no | — |
| 146 | C22×D3 | — | 9 | 10 | no | — |
| 147 | C22×D3 | — | 9 | 20 | no | — |
| 148 | C22×D3 | — | 9 | 60 | no | — |
| 149 | C22×D3 | — | 9 | 10 | no | — |
| 150 | C22×D3 | — | 18 | 5 | no | — |
| 151 | C22×D3 | — | 18 | 10 | no | — |
| 152 | C22×D3 | — | 18 | 30 | no | — |
| 153 | C22×D3 | D3×A4 | 1 | 10 | yes | complemented |
| 154 | C2×A4 | — | 12 | 10 | no | — |
| 155 | C2×A4 | — | 24 | 10 | no | — |
| 156 | C2×A4 | — | 36 | 5 | no | — |
| 157 | C2×C32 | — | 3 | 5 | no | — |
| 158 | C2×C32 | — | 48 | 2 | no | — |
| 159 | C3⋊D3 | — | 3 | 5 | no | — |
| 160 | C3⋊D3 | C2×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 161 | C3×D3 | — | 3 | 10 | no | — |
| 162 | C3×D3 | — | 16 | 2 | no | — |
| 163 | C3×D3 | — | 32 | 2 | no | — |
| 164 | C3×D3 | — | 48 | 2 | no | — |
| 165 | C3×D3 | — | 96 | 1 | no | — |
| 166 | C3×D3 | C2×C22⋊A4 | 1 | 2 | yes | complemented |
| 167 | C24 | — | 9 | 10 | no | — |
| 168 | C24 | — | 9 | 5 | no | — |
| 169 | C24 | — | 9 | 10 | no | — |
| 170 | C24 | — | 27 | 5 | no | — |
| 171 | C24 | — | 27 | 10 | no | — |
| 172 | C24 | — | 27 | 5 | no | — |
| 173 | C24 | — | 27 | 5 | no | — |
| 174 | C24 | — | 27 | 10 | no | — |
| 175 | C24 | — | 27 | 60 | no | — |
| 176 | C24 | — | 27 | 30 | no | — |
| 177 | C24 | — | 27 | 60 | no | — |
| 178 | C24 | — | 27 | 10 | no | — |
| 179 | C24 | C3×D32 | 1 | 1 | yes | characteristic, complemented |
| 180 | C22×C3 | — | 2 | 5 | no | — |
| 181 | C22×C3 | — | 3 | 20 | no | — |
| 182 | C22×C3 | — | 6 | 10 | no | — |
| 183 | C22×C3 | — | 9 | 10 | no | — |
| 184 | C22×C3 | — | 9 | 10 | no | — |
| 185 | C22×C3 | — | 9 | 60 | no | — |
| 186 | C22×C3 | — | 144 | 1 | no | — |
| 187 | C22×C3 | C2×D3×A4 | 1 | 10 | yes | complemented |
| 188 | C2×D3 | — | 3 | 60 | no | — |
| 189 | C2×D3 | — | 3 | 10 | no | — |
| 190 | C2×D3 | — | 3 | 2 | no | — |
| 191 | C2×D3 | — | 3 | 10 | no | — |
| 192 | C2×D3 | — | 9 | 10 | no | — |
| 193 | C2×D3 | — | 9 | 10 | no | — |
| 194 | C2×D3 | — | 9 | 10 | no | — |
| 195 | C2×D3 | — | 9 | 120 | no | — |
| 196 | C2×D3 | — | 9 | 20 | no | — |
| 197 | C2×D3 | — | 9 | 60 | no | — |
| 198 | C2×D3 | — | 9 | 10 | no | — |
| 199 | C2×D3 | — | 9 | 10 | no | — |
| 200 | C2×D3 | — | 18 | 5 | no | — |
| 201 | C2×D3 | — | 18 | 30 | no | — |
| 202 | C2×D3 | — | 18 | 5 | no | — |
| 203 | A4 | — | 4 | 5 | no | — |
| 204 | A4 | — | 8 | 10 | no | — |
| 205 | A4 | — | 16 | 5 | no | — |
| 206 | C32 | — | 16 | 2 | no | — |
| 207 | C32 | — | 32 | 1 | no | — |
| 208 | C32 | — | 32 | 2 | no | — |
| 209 | C32 | — | 64 | 1 | no | — |
| 210 | C32 | C22×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 211 | C23 | — | 3 | 5 | no | — |
| 212 | C23 | — | 3 | 10 | no | — |
| 213 | C23 | — | 9 | 20 | no | — |
| 214 | C23 | — | 9 | 60 | no | — |
| 215 | C23 | — | 9 | 10 | no | — |
| 216 | C23 | — | 9 | 5 | no | — |
| 217 | C23 | — | 27 | 10 | no | — |
| 218 | C23 | — | 27 | 30 | no | — |
| 219 | C23 | — | 27 | 5 | no | — |
| 220 | C23 | — | 27 | 10 | no | — |
| 221 | C23 | — | 27 | 5 | no | — |
| 222 | C23 | — | 27 | 60 | no | — |
| 223 | C23 | — | 27 | 60 | no | — |
| 224 | C23 | — | 27 | 30 | no | — |
| 225 | C23 | — | 27 | 30 | no | — |
| 226 | C23 | — | 27 | 120 | no | — |
| 227 | C23 | — | 27 | 5 | no | — |
| 228 | C2×C3 | — | 3 | 10 | no | — |
| 229 | C2×C3 | — | 3 | 2 | no | — |
| 230 | C2×C3 | — | 6 | 5 | no | — |
| 231 | C2×C3 | — | 9 | 10 | no | — |
| 232 | C2×C3 | — | 48 | 2 | no | — |
| 233 | C2×C3 | — | 96 | 2 | no | — |
| 234 | C2×C3 | — | 144 | 1 | no | — |
| 235 | D3 | — | 3 | 10 | no | — |
| 236 | D3 | — | 3 | 2 | no | — |
| 237 | D3 | — | 6 | 1 | no | — |
| 238 | D3 | — | 9 | 10 | no | — |
| 239 | D3 | — | 18 | 5 | no | — |
| 240 | D3 | D3×C22⋊A4 | 1 | 2 | yes | direct product factor |
| 241 | C22 | — | 3 | 10 | no | — |
| 242 | C22 | — | 9 | 10 | no | — |
| 243 | C22 | — | 9 | 1 | no | — |
| 244 | C22 | — | 9 | 10 | no | — |
| 245 | C22 | — | 9 | 60 | no | — |
| 246 | C22 | — | 27 | 5 | no | — |
| 247 | C22 | — | 27 | 10 | no | — |
| 248 | C22 | — | 27 | 5 | no | — |
| 249 | C22 | — | 27 | 30 | no | — |
| 250 | C22 | — | 27 | 5 | no | — |
| 251 | C22 | — | 27 | 10 | no | — |
| 252 | C22 | — | 27 | 60 | no | — |
| 253 | C22 | D32×A4 | 1 | 5 | yes | complemented |
| 254 | C3 | — | 2 | 1 | no | — |
| 255 | C3 | — | 16 | 1 | no | — |
| 256 | C3 | — | 32 | 2 | no | — |
| 257 | C3 | — | 64 | 1 | no | — |
| 258 | C3 | C2×D3×C22⋊A4 | 1 | 2 | yes | complemented |
| 259 | C2 | — | 3 | 5 | no | — |
| 260 | C2 | — | 3 | 2 | no | — |
| 261 | C2 | — | 9 | 10 | no | — |
| 262 | C2 | — | 9 | 1 | no | — |
| 263 | C2 | — | 27 | 5 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C24×C32 | C22×C3 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊D3×C22⋊A4 | C2 | semidirect |
| 2. C3×D3×C22⋊A4 | C2 | semidirect |
| 3. C24×D32 | C3 | semidirect |
| 5. C32×C22⋊A4 | C22 | semidirect |
| 7. C24×C3⋊D3 | C2×C3 | semidirect |
| 8. C24×C3×D3 | C2×C3 | semidirect |
| 16. D3×C22⋊A4 | D3 | direct (N × H) |
| 21. C24×C32 | C22×C3 | semidirect |
| 29. C3×C22⋊A4 | C2×D3 | semidirect |
| 37. C22×D32 | A4 | semidirect |
| 48. C24×D3 | C3×D3 | semidirect |
| 56. C22×C3⋊D3 | C2×A4 | semidirect |
| 60. C22×C3×D3 | C2×A4 | semidirect |
| 80. C24×C3 | C2×C3×D3 | semidirect |
| 99. C22⋊A4 | D32 | direct (N × H) |
| 102. C22×C32 | C22×A4 | semidirect |
| 118. D32 | C22⋊A4 | direct (N × H) |
| 153. C22×D3 | D3×A4 | semidirect |
| 160. C3⋊D3 | C2×C22⋊A4 | semidirect |
| 166. C3×D3 | C2×C22⋊A4 | semidirect |
| 179. C24 | C3×D32 | semidirect |
| 187. C22×C3 | C2×D3×A4 | semidirect |
| 210. C32 | C22×C22⋊A4 | semidirect |
| 240. D3 | D3×C22⋊A4 | direct (N × H) |
| 253. C22 | D32×A4 | semidirect |
| 258. C3 | C2×D3×C22⋊A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×C22⋊A4 | 2 |
| C3×D3×C22⋊A4 | 2 |
| C24×D32 | 3 |
| C2×D3×C22⋊A4 | 3 |
| D32×A4 | 4 |