Finite Group DatabasebetaFinite Group Databasebeta← C2×(C3⋊D3)⋊(C22⋊A4) #13180099
Order 1728.
5853 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 5853 conjugacy classes of subgroups — too many to draw. Only its 70 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×C32⋊(C22⋊A4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | (C3⋊D3)⋊(C22⋊A4) | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C25×C3⋊D3 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×D3×C22⋊A4 | — | 3 | 1 | no | maximal |
| 5 | C32⋊(C22⋊A4) | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×(C3⋊D3)⋊A4 | — | 4 | 5 | no | maximal |
| 7 | C25×C32 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C24×C3⋊D3 | — | 3 | 5 | no | — |
| 9 | C24×C3⋊D3 | — | 3 | 5 | no | — |
| 10 | C24×C3⋊D3 | — | 3 | 10 | no | — |
| 11 | C24×C3⋊D3 | C2×C3 | 1 | 2 | yes | complemented |
| 12 | C2×C3×C22⋊A4 | — | 2 | 1 | no | — |
| 13 | C2×C3×C22⋊A4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | D3×C22⋊A4 | — | 3 | 2 | no | — |
| 15 | C2×C22⋊H(3,𝔽3) | — | 4 | 5 | no | — |
| 16 | (C3⋊D3)⋊A4 | — | 4 | 10 | no | — |
| 17 | C25×D3 | — | 3 | 1 | no | — |
| 18 | C25×D3 | — | 9 | 1 | no | — |
| 19 | C22×C22⋊A4 | — | 9 | 1 | no | — |
| 20 | C24×C32 | — | 3 | 5 | no | — |
| 21 | C24×C32 | — | 3 | 5 | no | — |
| 22 | C24×C32 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 23 | C23×C3⋊D3 | — | 3 | 60 | no | — |
| 24 | C23×C3⋊D3 | — | 3 | 60 | no | — |
| 25 | C23×C3⋊D3 | — | 3 | 30 | no | — |
| 26 | C23×C3⋊D3 | — | 3 | 10 | no | — |
| 27 | C23×C3⋊D3 | — | 3 | 10 | no | — |
| 28 | C23×C3⋊D3 | — | 3 | 5 | no | — |
| 29 | C23×C3⋊D3 | — | 3 | 10 | no | — |
| 30 | C23×C3⋊D3 | — | 3 | 10 | no | — |
| 31 | C23×C3⋊D3 | — | 3 | 10 | no | — |
| 32 | C23×C3⋊D3 | A4 | 1 | 5 | yes | complemented |
| 33 | C3×C22⋊A4 | — | 2 | 1 | no | — |
| 34 | C3×C22⋊A4 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 35 | C2×D3×A4 | — | 12 | 5 | no | — |
| 36 | C2×(C3⋊D3)⋊C3 | — | 16 | 1 | no | — |
| 37 | C22⋊H(3,𝔽3) | — | 4 | 5 | no | — |
| 38 | C25×C3 | — | 3 | 1 | no | — |
| 39 | C25×C3 | C3×D3 | 1 | 1 | yes | characteristic |
| 40 | C24×D3 | — | 3 | 2 | no | — |
| 41 | C24×D3 | — | 9 | 2 | no | — |
| 42 | C24×D3 | — | 9 | 5 | no | — |
| 43 | C24×D3 | — | 9 | 15 | no | — |
| 44 | C24×D3 | — | 9 | 30 | no | — |
| 45 | C24×D3 | — | 9 | 15 | no | — |
| 46 | C24×D3 | — | 9 | 10 | no | — |
| 47 | C24×D3 | — | 9 | 5 | no | — |
| 48 | C2×C22⋊A4 | — | 3 | 1 | no | — |
| 49 | C2×C22⋊A4 | — | 6 | 1 | no | — |
| 50 | C2×C22⋊A4 | — | 9 | 2 | no | — |
| 51 | C23×C32 | — | 3 | 30 | no | — |
| 52 | C23×C32 | — | 3 | 10 | no | — |
| 53 | C23×C32 | — | 3 | 5 | no | — |
| 54 | C23×C32 | — | 3 | 5 | no | — |
| 55 | C23×C32 | C2×A4 | 1 | 5 | yes | complemented |
| 56 | C22×C3⋊D3 | — | 3 | 30 | no | — |
| 57 | C22×C3⋊D3 | — | 3 | 120 | no | — |
| 58 | C22×C3⋊D3 | — | 3 | 60 | no | — |
| 59 | C22×C3⋊D3 | — | 3 | 60 | no | — |
| 60 | C22×C3⋊D3 | — | 3 | 30 | no | — |
| 61 | C22×C3⋊D3 | — | 3 | 60 | no | — |
| 62 | C22×C3⋊D3 | — | 3 | 20 | no | — |
| 63 | C22×C3⋊D3 | — | 3 | 10 | no | — |
| 64 | C22×C3⋊D3 | — | 3 | 5 | no | — |
| 65 | C22×C3⋊D3 | — | 3 | 5 | no | — |
| 66 | C22×C3⋊D3 | — | 3 | 10 | no | — |
| 67 | C22×C3⋊D3 | C2×A4 | 1 | 10 | yes | complemented |
| 68 | C2×C3×A4 | — | 4 | 5 | no | — |
| 69 | C2×C3×A4 | — | 8 | 5 | no | — |
| 70 | D3×A4 | — | 12 | 10 | no | — |
| 71 | C26 | — | 9 | 1 | no | Sylow |
| 72 | C2×H(3,𝔽3) | — | 16 | 1 | no | — |
| 73 | (C3⋊D3)⋊C3 | — | 16 | 2 | no | — |
| 74 | C24×C3 | — | 3 | 5 | no | — |
| 75 | C24×C3 | — | 3 | 15 | no | — |
| 76 | C24×C3 | — | 3 | 15 | no | — |
| 77 | C24×C3 | — | 3 | 5 | no | — |
| 78 | C24×C3 | — | 3 | 1 | no | — |
| 79 | C24×C3 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 80 | C23×D3 | — | 3 | 5 | no | — |
| 81 | C23×D3 | — | 9 | 5 | no | — |
| 82 | C23×D3 | — | 9 | 10 | no | — |
| 83 | C23×D3 | — | 9 | 10 | no | — |
| 84 | C23×D3 | — | 9 | 30 | no | — |
| 85 | C23×D3 | — | 9 | 30 | no | — |
| 86 | C23×D3 | — | 9 | 30 | no | — |
| 87 | C23×D3 | — | 9 | 10 | no | — |
| 88 | C23×D3 | — | 9 | 30 | no | — |
| 89 | C23×D3 | — | 9 | 180 | no | — |
| 90 | C23×D3 | — | 9 | 90 | no | — |
| 91 | C23×D3 | — | 9 | 180 | no | — |
| 92 | C23×D3 | — | 9 | 30 | no | — |
| 93 | C23×D3 | — | 9 | 60 | no | — |
| 94 | C23×D3 | — | 9 | 60 | no | — |
| 95 | C23×D3 | — | 9 | 30 | no | — |
| 96 | C23×D3 | — | 9 | 10 | no | — |
| 97 | C23×D3 | — | 9 | 15 | no | — |
| 98 | C23×D3 | — | 9 | 5 | no | — |
| 99 | C23×D3 | — | 9 | 10 | no | — |
| 100 | C22⋊A4 | — | 3 | 1 | no | — |
| 101 | C22⋊A4 | — | 6 | 1 | no | — |
| 102 | C22×A4 | — | 36 | 5 | no | — |
| 103 | C22×C32 | — | 3 | 5 | no | — |
| 104 | C22×C32 | — | 3 | 30 | no | — |
| 105 | C22×C32 | — | 3 | 10 | no | — |
| 106 | C22×C32 | — | 3 | 5 | no | — |
| 107 | C22×C32 | C22×A4 | 1 | 5 | yes | complemented |
| 108 | C2×C3⋊D3 | — | 3 | 60 | no | — |
| 109 | C2×C3⋊D3 | — | 3 | 10 | no | — |
| 110 | C2×C3⋊D3 | — | 3 | 60 | no | — |
| 111 | C2×C3⋊D3 | — | 3 | 5 | no | — |
| 112 | C2×C3⋊D3 | — | 3 | 10 | no | — |
| 113 | C2×C3⋊D3 | — | 3 | 10 | no | — |
| 114 | C2×C3⋊D3 | — | 3 | 10 | no | — |
| 115 | C2×C3⋊D3 | C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 116 | C2×C3×D3 | — | 48 | 1 | no | — |
| 117 | C3×A4 | — | 4 | 5 | no | — |
| 118 | C3×A4 | — | 8 | 5 | no | — |
| 119 | C25 | — | 9 | 2 | no | — |
| 120 | C25 | — | 27 | 5 | no | — |
| 121 | C25 | — | 27 | 10 | no | — |
| 122 | C25 | — | 27 | 5 | no | — |
| 123 | C25 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 124 | H(3,𝔽3) | — | 16 | 1 | no | Sylow |
| 125 | C23×C3 | — | 3 | 30 | no | — |
| 126 | C23×C3 | — | 3 | 90 | no | — |
| 127 | C23×C3 | — | 3 | 30 | no | — |
| 128 | C23×C3 | — | 3 | 10 | no | — |
| 129 | C23×C3 | — | 3 | 15 | no | — |
| 130 | C23×C3 | — | 3 | 15 | no | — |
| 131 | C23×C3 | — | 3 | 5 | no | — |
| 132 | C23×C3 | — | 3 | 5 | no | — |
| 133 | C23×C3 | — | 3 | 5 | no | — |
| 134 | C23×C3 | D3×A4 | 1 | 5 | yes | — |
| 135 | C22×D3 | — | 3 | 10 | no | — |
| 136 | C22×D3 | — | 9 | 10 | no | — |
| 137 | C22×D3 | — | 9 | 5 | no | — |
| 138 | C22×D3 | — | 9 | 15 | no | — |
| 139 | C22×D3 | — | 9 | 30 | no | — |
| 140 | C22×D3 | — | 9 | 15 | no | — |
| 141 | C22×D3 | — | 9 | 20 | no | — |
| 142 | C22×D3 | — | 9 | 60 | no | — |
| 143 | C22×D3 | — | 9 | 30 | no | — |
| 144 | C22×D3 | — | 9 | 90 | no | — |
| 145 | C22×D3 | — | 9 | 60 | no | — |
| 146 | C22×D3 | — | 9 | 180 | no | — |
| 147 | C22×D3 | — | 9 | 180 | no | — |
| 148 | C22×D3 | — | 9 | 180 | no | — |
| 149 | C22×D3 | — | 9 | 360 | no | — |
| 150 | C22×D3 | — | 9 | 120 | no | — |
| 151 | C22×D3 | — | 9 | 90 | no | — |
| 152 | C22×D3 | — | 9 | 60 | no | — |
| 153 | C22×D3 | — | 9 | 60 | no | — |
| 154 | C22×D3 | — | 9 | 30 | no | — |
| 155 | C22×D3 | — | 9 | 30 | no | — |
| 156 | C22×D3 | — | 9 | 10 | no | — |
| 157 | C22×D3 | — | 9 | 10 | no | — |
| 158 | C22×D3 | — | 9 | 5 | no | — |
| 159 | C2×A4 | — | 12 | 5 | no | — |
| 160 | C2×A4 | — | 24 | 5 | no | — |
| 161 | C2×A4 | — | 36 | 10 | no | — |
| 162 | C2×C32 | — | 3 | 5 | no | — |
| 163 | C2×C32 | — | 3 | 5 | no | — |
| 164 | C2×C32 | — | 16 | 1 | no | — |
| 165 | C2×C32 | — | 32 | 1 | no | — |
| 166 | C2×C32 | C2×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 167 | C3⋊D3 | — | 3 | 10 | no | — |
| 168 | C3⋊D3 | C2×C22⋊A4 | 1 | 2 | yes | complemented |
| 169 | C3×D3 | — | 48 | 2 | no | — |
| 170 | C24 | — | 3 | 5 | no | — |
| 171 | C24 | — | 3 | 5 | no | — |
| 172 | C24 | — | 9 | 5 | no | — |
| 173 | C24 | — | 27 | 10 | no | — |
| 174 | C24 | — | 27 | 10 | no | — |
| 175 | C24 | — | 27 | 10 | no | — |
| 176 | C24 | — | 27 | 10 | no | — |
| 177 | C24 | — | 27 | 60 | no | — |
| 178 | C24 | — | 27 | 30 | no | — |
| 179 | C24 | — | 27 | 60 | no | — |
| 180 | C24 | — | 27 | 10 | no | — |
| 181 | C24 | — | 27 | 5 | no | — |
| 182 | C24 | C2×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 183 | C22×C3 | — | 3 | 5 | no | — |
| 184 | C22×C3 | — | 3 | 15 | no | — |
| 185 | C22×C3 | — | 3 | 30 | no | — |
| 186 | C22×C3 | — | 3 | 90 | no | — |
| 187 | C22×C3 | — | 3 | 10 | no | — |
| 188 | C22×C3 | — | 3 | 30 | no | — |
| 189 | C22×C3 | — | 3 | 15 | no | — |
| 190 | C22×C3 | — | 3 | 5 | no | — |
| 191 | C22×C3 | — | 3 | 5 | no | — |
| 192 | C22×C3 | — | 144 | 1 | no | — |
| 193 | C22×C3 | C2×D3×A4 | 1 | 5 | yes | — |
| 194 | C2×D3 | — | 3 | 1 | no | — |
| 195 | C2×D3 | — | 9 | 1 | no | — |
| 196 | C2×D3 | — | 9 | 10 | no | — |
| 197 | C2×D3 | — | 9 | 10 | no | — |
| 198 | C2×D3 | — | 9 | 30 | no | — |
| 199 | C2×D3 | — | 9 | 30 | no | — |
| 200 | C2×D3 | — | 9 | 30 | no | — |
| 201 | C2×D3 | — | 9 | 180 | no | — |
| 202 | C2×D3 | — | 9 | 30 | no | — |
| 203 | C2×D3 | — | 9 | 180 | no | — |
| 204 | C2×D3 | — | 9 | 60 | no | — |
| 205 | C2×D3 | — | 9 | 60 | no | — |
| 206 | C2×D3 | — | 9 | 10 | no | — |
| 207 | C2×D3 | — | 9 | 15 | no | — |
| 208 | C2×D3 | — | 9 | 5 | no | — |
| 209 | C2×D3 | — | 9 | 10 | no | — |
| 210 | A4 | — | 12 | 5 | no | — |
| 211 | A4 | — | 24 | 5 | no | — |
| 212 | C32 | — | 16 | 1 | no | — |
| 213 | C32 | — | 32 | 1 | no | — |
| 214 | C32 | C22×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 215 | C23 | — | 3 | 30 | no | — |
| 216 | C23 | — | 3 | 10 | no | — |
| 217 | C23 | — | 3 | 5 | no | — |
| 218 | C23 | — | 3 | 5 | no | — |
| 219 | C23 | — | 9 | 10 | no | — |
| 220 | C23 | — | 27 | 5 | no | — |
| 221 | C23 | — | 27 | 10 | no | — |
| 222 | C23 | — | 27 | 5 | no | — |
| 223 | C23 | — | 27 | 30 | no | — |
| 224 | C23 | — | 27 | 20 | no | — |
| 225 | C23 | — | 27 | 60 | no | — |
| 226 | C23 | — | 27 | 60 | no | — |
| 227 | C23 | — | 27 | 60 | no | — |
| 228 | C23 | — | 27 | 120 | no | — |
| 229 | C23 | — | 27 | 30 | no | — |
| 230 | C23 | — | 27 | 10 | no | — |
| 231 | C23 | (C3⋊D3)⋊A4 | 1 | 5 | yes | complemented |
| 232 | C2×C3 | — | 3 | 15 | no | — |
| 233 | C2×C3 | — | 3 | 15 | no | — |
| 234 | C2×C3 | — | 3 | 5 | no | — |
| 235 | C2×C3 | — | 3 | 5 | no | — |
| 236 | C2×C3 | — | 3 | 1 | no | — |
| 237 | C2×C3 | — | 48 | 1 | no | — |
| 238 | C2×C3 | — | 96 | 1 | no | — |
| 239 | C2×C3 | — | 144 | 2 | no | — |
| 240 | C2×C3 | D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 241 | D3 | — | 3 | 2 | no | — |
| 242 | D3 | — | 9 | 2 | no | — |
| 243 | D3 | — | 9 | 30 | no | — |
| 244 | D3 | — | 9 | 10 | no | — |
| 245 | C22 | — | 3 | 5 | no | — |
| 246 | C22 | — | 3 | 10 | no | — |
| 247 | C22 | — | 3 | 30 | no | — |
| 248 | C22 | — | 3 | 5 | no | — |
| 249 | C22 | — | 9 | 1 | no | — |
| 250 | C22 | — | 27 | 10 | no | — |
| 251 | C22 | — | 27 | 10 | no | — |
| 252 | C22 | — | 27 | 10 | no | — |
| 253 | C22 | — | 27 | 60 | no | — |
| 254 | C22 | — | 27 | 10 | no | — |
| 255 | C22 | — | 27 | 60 | no | — |
| 256 | C22 | — | 27 | 5 | no | — |
| 257 | C22 | C2×(C3⋊D3)⋊A4 | 1 | 5 | yes | complemented |
| 258 | C3 | — | 3 | 1 | no | — |
| 259 | C3 | — | 48 | 1 | no | — |
| 260 | C3 | — | 96 | 1 | no | — |
| 261 | C3 | C2×D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 262 | C2 | — | 3 | 5 | no | — |
| 263 | C2 | — | 3 | 5 | no | — |
| 264 | C2 | — | 9 | 2 | no | — |
| 265 | C2 | — | 27 | 10 | no | — |
| 266 | C2 | (C3⋊D3)⋊(C22⋊A4) | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊D3)⋊(C22⋊A4) |
| commutator, cosocle, hypocenter | C24×C32 | C22×C3 |
| Frattini | C3 | C2×D3×C22⋊A4 |
| Fitting | C25×C32 | C2×C3 |
| socle | C25×C3 | C3×D3 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C32⋊(C22⋊A4) | C2 | semidirect |
| 2. (C3⋊D3)⋊(C22⋊A4) | C2 | direct (N × H) |
| 3. C25×C3⋊D3 | C3 | semidirect |
| 5. C32⋊(C22⋊A4) | C22 | semidirect |
| 7. C25×C32 | C2×C3 | semidirect |
| 11. C24×C3⋊D3 | C2×C3 | semidirect |
| 13. C2×C3×C22⋊A4 | D3 | semidirect |
| 22. C24×C32 | C22×C3 | semidirect |
| 32. C23×C3⋊D3 | A4 | semidirect |
| 34. C3×C22⋊A4 | C2×D3 | semidirect |
| 55. C23×C32 | C2×A4 | semidirect |
| 67. C22×C3⋊D3 | C2×A4 | semidirect |
| 107. C22×C32 | C22×A4 | semidirect |
| 115. C2×C3⋊D3 | C22⋊A4 | semidirect |
| 123. C25 | (C3⋊D3)⋊C3 | semidirect |
| 166. C2×C32 | C2×C22⋊A4 | semidirect |
| 168. C3⋊D3 | C2×C22⋊A4 | semidirect |
| 182. C24 | C2×(C3⋊D3)⋊C3 | semidirect |
| 214. C32 | C22×C22⋊A4 | semidirect |
| 231. C23 | (C3⋊D3)⋊A4 | semidirect |
| 257. C22 | C2×(C3⋊D3)⋊A4 | semidirect |
| 266. C2 | (C3⋊D3)⋊(C22⋊A4) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32⋊(C22⋊A4) | 2 |
| (C3⋊D3)⋊(C22⋊A4) | 2 |
| C25×C3⋊D3 | 3 |
| C2×D3×C22⋊A4 | 3 |
| C2×(C3⋊D3)⋊A4 | 4 |