Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
381 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 44 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×C3⋊(C3⋊S4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3×C3⋊S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×Dic3×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4⋊(C3⋊S4) | — | 3 | 1 | no | maximal |
| 5 | #64207 | — | 3 | 3 | no | maximal |
| 6 | #63978 | — | 3 | 1 | no | maximal |
| 7 | C2×C32×A4 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3⋊(C3⋊S4) | — | 2 | 1 | no | — |
| 9 | C3×C3⋊S4 | — | 2 | 1 | no | — |
| 10 | #6292 | — | 4 | 1 | no | maximal |
| 11 | C2×C3⋊S4 | — | 3 | 1 | no | — |
| 12 | C2×C3⋊S4 | — | 3 | 3 | no | — |
| 13 | C2×C3⋊S4 | — | 6 | 1 | no | — |
| 14 | C2×C3⋊S4 | — | 6 | 3 | no | — |
| 15 | C2×C3⋊S4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×C3×S4 | — | 3 | 3 | no | — |
| 17 | C2×#638 | — | 3 | 1 | no | — |
| 18 | C2×C3×#67 | — | 3 | 1 | no | — |
| 19 | C22×C3×Dic3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | C3×C4×A4 | — | 3 | 1 | no | — |
| 21 | C2×#626 | — | 3 | 1 | no | — |
| 22 | Dic3×A4 | D3 | 1 | 3 | yes | complemented |
| 23 | Dic3⋊Dic3 | — | 3 | 1 | no | — |
| 24 | #3722 | — | 3 | 1 | no | — |
| 25 | (C2×D3)⋊Dic3 | — | 3 | 1 | no | — |
| 26 | C2×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 27 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 28 | C32×A4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 29 | C32×Dic3 | — | 4 | 1 | no | — |
| 30 | C4⋊S4 | — | 9 | 3 | no | — |
| 31 | #939 | — | 9 | 1 | no | — |
| 32 | #930 | — | 9 | 1 | no | — |
| 33 | C23×C32 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 35 | C22×C3×D3 | — | 3 | 1 | no | — |
| 36 | C2×C3×A4 | — | 2 | 1 | no | — |
| 37 | C2×C3×A4 | — | 2 | 3 | no | — |
| 38 | C2×C3×A4 | C2×D3 | 1 | 3 | yes | — |
| 39 | C2×C3×A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 40 | C3⋊S4 | — | 2 | 1 | no | — |
| 41 | C3⋊S4 | — | 6 | 6 | no | — |
| 42 | C3⋊S4 | — | 6 | 2 | no | — |
| 43 | C3⋊S4 | — | 6 | 3 | no | — |
| 44 | C3⋊S4 | — | 6 | 1 | no | — |
| 45 | C3×S4 | — | 6 | 3 | no | — |
| 46 | #638 | — | 6 | 1 | no | — |
| 47 | #638 | — | 6 | 1 | no | — |
| 48 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 49 | C4⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 50 | C3×#67 | — | 6 | 1 | no | — |
| 51 | C3×#67 | — | 6 | 1 | no | — |
| 52 | C2×C3×Dic3 | — | 3 | 1 | no | — |
| 53 | C2×C3×Dic3 | — | 3 | 1 | no | — |
| 54 | C2×C3×Dic3 | — | 3 | 1 | no | — |
| 55 | #626 | — | 6 | 1 | no | — |
| 56 | #626 | — | 6 | 1 | no | — |
| 57 | #626 | — | 12 | 3 | no | — |
| 58 | C2×C33 | — | 4 | 1 | no | — |
| 59 | C3⋊(C3⋊D3) | — | 8 | 1 | no | — |
| 60 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 61 | C2×S4 | — | 3 | 3 | no | — |
| 62 | C2×S4 | — | 9 | 3 | no | — |
| 63 | C2×S4 | — | 18 | 3 | no | — |
| 64 | C2×C3×D4 | — | 9 | 1 | no | — |
| 65 | C22×C3×C4 | — | 3 | 1 | no | — |
| 66 | C2×#67 | — | 3 | 1 | no | — |
| 67 | C2×#67 | — | 9 | 1 | no | — |
| 68 | C2×#67 | — | 9 | 1 | no | — |
| 69 | C2×#67 | — | 9 | 1 | no | — |
| 70 | C2×#67 | — | 18 | 1 | no | — |
| 71 | C22×Dic3 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 72 | C2×D12 | — | 9 | 1 | no | — |
| 73 | C4×A4 | — | 3 | 3 | no | — |
| 74 | #217 | — | 9 | 1 | no | — |
| 75 | (C2×D3)⋊C4 | — | 9 | 1 | no | — |
| 76 | (C2×D3)⋊C4 | — | 9 | 1 | no | — |
| 77 | (C2×D3)⋊C4 | — | 9 | 1 | no | — |
| 78 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 79 | Dic3⋊C4 | — | 9 | 1 | no | — |
| 80 | C22×C32 | — | 3 | 1 | no | — |
| 81 | C22×C32 | — | 3 | 1 | no | — |
| 82 | C22×C32 | D12 | 1 | 1 | yes | characteristic, complemented |
| 83 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 84 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 85 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 86 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 87 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 88 | C2×C3⋊D3 | — | 12 | 3 | no | — |
| 89 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 90 | C2×C3⋊D3 | — | 24 | 3 | no | — |
| 91 | C2×C3×D3 | — | 6 | 1 | no | — |
| 92 | C2×C3×D3 | — | 6 | 1 | no | — |
| 93 | C2×C3×D3 | — | 6 | 1 | no | — |
| 94 | C2×C3×D3 | — | 12 | 3 | no | — |
| 95 | C3×A4 | — | 2 | 1 | no | — |
| 96 | C3×A4 | — | 2 | 3 | no | — |
| 97 | C3×A4 | D12 | 1 | 3 | yes | complemented |
| 98 | C3×A4 | #67 | 1 | 1 | yes | characteristic, complemented |
| 99 | C32×C4 | — | 12 | 1 | no | — |
| 100 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 101 | C3×Dic3 | — | 3 | 1 | no | — |
| 102 | C3×Dic3 | — | 4 | 3 | no | — |
| 103 | C3×Dic3 | — | 6 | 1 | no | — |
| 104 | C3×Dic3 | S4 | 1 | 1 | yes | characteristic, complemented |
| 105 | #121 | — | 27 | 1 | no | Sylow |
| 106 | C33 | — | 4 | 1 | no | Sylow |
| 107 | C23×C3 | — | 2 | 1 | no | — |
| 108 | C23×C3 | — | 9 | 1 | no | — |
| 109 | C23×C3 | D32 | 1 | 1 | yes | characteristic |
| 110 | C23×C3 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 111 | C22×D3 | — | 3 | 1 | no | — |
| 112 | C22×D3 | — | 9 | 1 | no | — |
| 113 | C22×D3 | — | 9 | 1 | no | — |
| 114 | C22×D3 | — | 18 | 1 | no | — |
| 115 | C2×A4 | — | 2 | 3 | no | — |
| 116 | C2×A4 | D32 | 1 | 3 | yes | — |
| 117 | S4 | — | 6 | 3 | no | — |
| 118 | S4 | — | 18 | 6 | no | — |
| 119 | S4 | — | 18 | 3 | no | — |
| 120 | C3×D4 | — | 18 | 1 | no | — |
| 121 | C3×D4 | — | 18 | 1 | no | — |
| 122 | C2×C3×C4 | — | 9 | 1 | no | — |
| 123 | C2×C3×C4 | — | 9 | 1 | no | — |
| 124 | C2×C3×C4 | — | 9 | 1 | no | — |
| 125 | #67 | — | 6 | 1 | no | — |
| 126 | #67 | — | 6 | 1 | no | — |
| 127 | #67 | — | 18 | 1 | no | — |
| 128 | #67 | — | 18 | 2 | no | — |
| 129 | #67 | — | 18 | 1 | no | — |
| 130 | #67 | — | 18 | 1 | no | — |
| 131 | #67 | — | 18 | 2 | no | — |
| 132 | #67 | — | 18 | 1 | no | — |
| 133 | #67 | — | 18 | 1 | no | — |
| 134 | #67 | — | 18 | 1 | no | — |
| 135 | C2×Dic3 | — | 3 | 1 | no | — |
| 136 | C2×Dic3 | — | 3 | 1 | no | — |
| 137 | C2×Dic3 | — | 3 | 1 | no | — |
| 138 | C2×Dic3 | — | 9 | 1 | no | — |
| 139 | C2×Dic3 | — | 9 | 1 | no | — |
| 140 | C2×Dic3 | — | 18 | 1 | no | — |
| 141 | D12 | — | 18 | 1 | no | — |
| 142 | D12 | — | 18 | 1 | no | — |
| 143 | D12 | — | 36 | 3 | no | — |
| 144 | C2×C32 | — | 3 | 1 | no | — |
| 145 | C2×C32 | — | 3 | 1 | no | — |
| 146 | C2×C32 | — | 4 | 1 | no | — |
| 147 | C2×C32 | — | 4 | 3 | no | — |
| 148 | C2×C32 | — | 8 | 3 | no | — |
| 149 | C2×C32 | — | 8 | 1 | no | — |
| 150 | C2×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 151 | C3⋊D3 | — | 8 | 1 | no | — |
| 152 | C3⋊D3 | — | 12 | 1 | no | — |
| 153 | C3⋊D3 | — | 24 | 3 | no | — |
| 154 | C3⋊D3 | — | 24 | 1 | no | — |
| 155 | C3⋊D3 | — | 24 | 6 | no | — |
| 156 | C3⋊D3 | — | 24 | 2 | no | — |
| 157 | C3×D3 | — | 12 | 1 | no | — |
| 158 | C3×D3 | — | 24 | 3 | no | — |
| 159 | C2×D4 | — | 9 | 1 | no | — |
| 160 | C2×D4 | — | 27 | 1 | no | — |
| 161 | C2×D4 | — | 27 | 1 | no | — |
| 162 | C22×C4 | — | 3 | 1 | no | — |
| 163 | C4⋊C4 | — | 27 | 1 | no | — |
| 164 | C22⋊C4 | — | 27 | 1 | no | — |
| 165 | C22⋊C4 | — | 27 | 1 | no | — |
| 166 | C22×C3 | — | 2 | 1 | no | — |
| 167 | C22×C3 | — | 3 | 1 | no | — |
| 168 | C22×C3 | — | 3 | 1 | no | — |
| 169 | C22×C3 | — | 3 | 1 | no | — |
| 170 | C22×C3 | — | 3 | 1 | no | — |
| 171 | C22×C3 | — | 6 | 1 | no | — |
| 172 | C22×C3 | — | 6 | 1 | no | — |
| 173 | C22×C3 | — | 18 | 1 | no | — |
| 174 | C22×C3 | — | 18 | 1 | no | — |
| 175 | C22×C3 | — | 18 | 1 | no | — |
| 176 | C22×C3 | #626 | 1 | 1 | yes | characteristic, complemented |
| 177 | C22×C3 | C4⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 178 | C2×D3 | — | 6 | 1 | no | — |
| 179 | C2×D3 | — | 6 | 1 | no | — |
| 180 | C2×D3 | — | 6 | 1 | no | — |
| 181 | C2×D3 | — | 12 | 3 | no | — |
| 182 | C2×D3 | — | 18 | 1 | no | — |
| 183 | C2×D3 | — | 18 | 1 | no | — |
| 184 | C2×D3 | — | 18 | 1 | no | — |
| 185 | C2×D3 | — | 18 | 1 | no | — |
| 186 | C2×D3 | — | 18 | 2 | no | — |
| 187 | C2×D3 | — | 18 | 1 | no | — |
| 188 | C2×D3 | — | 18 | 1 | no | — |
| 189 | C2×D3 | — | 36 | 1 | no | — |
| 190 | C2×D3 | — | 36 | 1 | no | — |
| 191 | C2×D3 | — | 36 | 3 | no | — |
| 192 | C2×D3 | — | 72 | 3 | no | — |
| 193 | A4 | — | 2 | 3 | no | — |
| 194 | A4 | #626 | 1 | 3 | yes | complemented |
| 195 | C3×C4 | — | 3 | 1 | no | — |
| 196 | C3×C4 | — | 9 | 1 | no | — |
| 197 | C3×C4 | — | 12 | 3 | no | — |
| 198 | C3×C4 | — | 18 | 1 | no | — |
| 199 | Dic3 | — | 3 | 1 | no | — |
| 200 | Dic3 | — | 6 | 1 | no | — |
| 201 | Dic3 | — | 18 | 1 | no | — |
| 202 | Dic3 | — | 18 | 1 | no | — |
| 203 | Dic3 | — | 18 | 2 | no | — |
| 204 | Dic3 | C3⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 205 | C32 | — | 4 | 1 | no | — |
| 206 | C32 | — | 4 | 3 | no | — |
| 207 | C32 | — | 8 | 3 | no | — |
| 208 | C32 | — | 8 | 1 | no | — |
| 209 | C32 | C4⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 210 | C23 | — | 9 | 1 | no | — |
| 211 | C23 | — | 27 | 1 | no | — |
| 212 | C23 | D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 213 | D4 | — | 18 | 1 | no | — |
| 214 | D4 | — | 18 | 1 | no | — |
| 215 | D4 | — | 54 | 1 | no | — |
| 216 | D4 | — | 54 | 1 | no | — |
| 217 | D4 | — | 54 | 1 | no | — |
| 218 | D4 | — | 54 | 1 | no | — |
| 219 | C2×C4 | — | 9 | 1 | no | — |
| 220 | C2×C4 | — | 9 | 1 | no | — |
| 221 | C2×C4 | — | 9 | 1 | no | — |
| 222 | C2×C4 | — | 27 | 1 | no | — |
| 223 | C2×C3 | — | 2 | 1 | no | — |
| 224 | C2×C3 | — | 3 | 1 | no | — |
| 225 | C2×C3 | — | 3 | 1 | no | — |
| 226 | C2×C3 | — | 3 | 1 | no | — |
| 227 | C2×C3 | — | 3 | 1 | no | — |
| 228 | C2×C3 | — | 4 | 3 | no | — |
| 229 | C2×C3 | — | 6 | 1 | no | — |
| 230 | C2×C3 | — | 6 | 1 | no | — |
| 231 | C2×C3 | — | 8 | 3 | no | — |
| 232 | C2×C3 | — | 36 | 1 | no | — |
| 233 | C2×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 234 | C2×C3 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 235 | D3 | — | 12 | 1 | no | — |
| 236 | D3 | — | 24 | 3 | no | — |
| 237 | D3 | — | 36 | 2 | no | — |
| 238 | D3 | — | 36 | 1 | no | — |
| 239 | D3 | — | 36 | 1 | no | — |
| 240 | D3 | — | 72 | 3 | no | — |
| 241 | D3 | — | 72 | 6 | no | — |
| 242 | C22 | — | 3 | 1 | no | — |
| 243 | C22 | — | 3 | 1 | no | — |
| 244 | C22 | — | 18 | 1 | no | — |
| 245 | C22 | — | 18 | 1 | no | — |
| 246 | C22 | — | 18 | 1 | no | — |
| 247 | C22 | — | 54 | 1 | no | — |
| 248 | C22 | — | 54 | 1 | no | — |
| 249 | C22 | — | 54 | 1 | no | — |
| 250 | C22 | #6292 | 1 | 1 | yes | characteristic, complemented |
| 251 | C4 | — | 3 | 1 | no | — |
| 252 | C4 | — | 9 | 1 | no | — |
| 253 | C4 | — | 18 | 1 | no | — |
| 254 | C4 | — | 54 | 1 | no | — |
| 255 | C3 | — | 2 | 1 | no | — |
| 256 | C3 | — | 4 | 3 | no | — |
| 257 | C3 | — | 8 | 3 | no | — |
| 258 | C3 | #64207 | 1 | 1 | yes | characteristic, complemented |
| 259 | C3 | C4⋊(C3⋊S4) | 1 | 1 | yes | characteristic, complemented |
| 260 | C2 | — | 3 | 1 | no | — |
| 261 | C2 | — | 3 | 1 | no | — |
| 262 | C2 | — | 36 | 1 | no | — |
| 263 | C2 | — | 108 | 1 | no | — |
| 264 | C2 | D3×C3⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊S4 |
| commutator, cosocle | C2×C32×A4 | C22 |
| Fitting, socle | C23×C32 | C2×D3 |
| hypocenter | C32×A4 | D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3⋊(C3⋊S4) | C2 | semidirect |
| 2. C2×C3×C3⋊S4 | C2 | semidirect |
| 3. C3×Dic3×A4 | C2 | semidirect |
| 15. C2×C3⋊S4 | D3 | semidirect |
| 19. C22×C3×Dic3 | D3 | semidirect |
| 22. Dic3×A4 | D3 | semidirect |
| 28. C32×A4 | D4 | semidirect |
| 71. C22×Dic3 | C3⋊D3 | semidirect |
| 82. C22×C32 | D12 | semidirect |
| 97. C3×A4 | D12 | semidirect |
| 98. C3×A4 | #67 | semidirect |
| 104. C3×Dic3 | S4 | semidirect |
| 176. C22×C3 | #626 | semidirect |
| 177. C22×C3 | C4⋊(C3⋊D3) | semidirect |
| 194. A4 | #626 | semidirect |
| 204. Dic3 | C3⋊S4 | semidirect |
| 209. C32 | C4⋊S4 | semidirect |
| 250. C22 | #6292 | semidirect |
| 258. C3 | #64207 | semidirect |
| 259. C3 | C4⋊(C3⋊S4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊(C3⋊S4) | 2 |
| C2×C3×C3⋊S4 | 2 |
| C3×Dic3×A4 | 2 |
| C4⋊(C3⋊S4) | 3 |
| #64207 | 3 |
| #63978 | 3 |
| #6292 | 4 |