Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
458 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 74 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×#6296 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#6291 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #88804 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×(C2×D3)⋊Dic3 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C3×Dic32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #63953 | — | 3 | 1 | no | maximal |
| 7 | #63852 | — | 3 | 2 | no | maximal |
| 8 | C22×C3⋊(C3⋊D3) | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C32×D3 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×C3⋊Dic3 | C22 | 1 | 2 | yes | — |
| 11 | C2×C3×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C32×Dic3 | C22 | 1 | 2 | yes | — |
| 13 | #6296 | — | 2 | 2 | no | — |
| 14 | #6291 | — | 2 | 2 | no | — |
| 15 | C2×#638 | — | 3 | 1 | no | — |
| 16 | C2×#627 | — | 3 | 2 | no | — |
| 17 | C2×#627 | — | 3 | 1 | no | — |
| 18 | C2×#626 | — | 3 | 2 | no | — |
| 19 | C2×#626 | — | 6 | 1 | no | — |
| 20 | #3752 | — | 3 | 2 | no | — |
| 21 | C3×#217 | — | 3 | 2 | no | — |
| 22 | C3×(C2×D3)⋊C4 | — | 3 | 2 | no | — |
| 23 | C3×C4×Dic3 | — | 3 | 2 | no | — |
| 24 | #3722 | — | 3 | 2 | no | — |
| 25 | #3722 | — | 3 | 2 | no | — |
| 26 | #3722 | — | 6 | 2 | no | — |
| 27 | (C2×D3)⋊Dic3 | D3 | 1 | 2 | yes | complemented |
| 28 | Dic32 | D3 | 1 | 1 | yes | characteristic, complemented |
| 29 | C22×C33 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×C3⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 31 | C2×C3⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 32 | C2×C32×D3 | — | 2 | 1 | no | — |
| 33 | C2×C32×D3 | — | 2 | 2 | no | — |
| 34 | C3×C3⋊Dic3 | — | 2 | 2 | no | — |
| 35 | C3×C3⋊Dic3 | D4 | 1 | 2 | yes | — |
| 36 | C32×Dic3 | — | 2 | 2 | no | — |
| 37 | #885 | — | 9 | 2 | no | — |
| 38 | #875 | — | 9 | 1 | no | — |
| 39 | C23×C32 | — | 3 | 1 | no | — |
| 40 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 41 | C22×C3⋊D3 | — | 3 | 2 | no | — |
| 42 | C22×C3⋊D3 | — | 6 | 2 | no | — |
| 43 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 44 | C22×C3⋊D3 | — | 12 | 1 | no | — |
| 45 | C22×C3×D3 | — | 2 | 1 | no | — |
| 46 | C22×C3×D3 | C2×D3 | 1 | 2 | yes | — |
| 47 | C2×C32×C4 | — | 3 | 2 | no | — |
| 48 | #638 | — | 6 | 2 | no | — |
| 49 | C2×C3⋊Dic3 | C2×D3 | 1 | 2 | yes | — |
| 50 | C2×C3⋊Dic3 | C2×D3 | 1 | 1 | yes | characteristic |
| 51 | C2×C3×Dic3 | — | 2 | 2 | no | — |
| 52 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 53 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 54 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 55 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 56 | C2×C3×Dic3 | — | 6 | 2 | no | — |
| 57 | C2×C3×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 58 | C2×C3×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 59 | #627 | — | 6 | 4 | no | — |
| 60 | #627 | — | 6 | 2 | no | — |
| 61 | #626 | — | 6 | 4 | no | — |
| 62 | #626 | — | 6 | 4 | no | — |
| 63 | C2×C33 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C2×C33 | #41 | 1 | 2 | yes | — |
| 65 | C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 66 | C32×D3 | — | 4 | 1 | no | — |
| 67 | C2×#67 | — | 9 | 2 | no | — |
| 68 | C2×#67 | — | 18 | 1 | no | — |
| 69 | C2×D12 | — | 9 | 1 | no | — |
| 70 | C2×Dic6 | — | 9 | 1 | no | — |
| 71 | C2×Dic6 | — | 9 | 2 | no | — |
| 72 | C3×C22⋊C4 | — | 9 | 2 | no | — |
| 73 | C3×C42 | — | 9 | 1 | no | — |
| 74 | #217 | — | 3 | 2 | no | — |
| 75 | (C2×D3)⋊C4 | — | 3 | 2 | no | — |
| 76 | (C2×D3)⋊C4 | — | 9 | 2 | no | — |
| 77 | (C2×D3)⋊C4 | — | 9 | 2 | no | — |
| 78 | (C2×D3)⋊C4 | — | 9 | 2 | no | — |
| 79 | (C2×D3)⋊C4 | — | 18 | 2 | no | — |
| 80 | C4×Dic3 | — | 3 | 2 | no | — |
| 81 | C22×C32 | — | 2 | 2 | no | — |
| 82 | C22×C32 | — | 2 | 1 | no | — |
| 83 | C22×C32 | — | 4 | 1 | no | — |
| 84 | C22×C32 | — | 6 | 1 | no | — |
| 85 | C22×C32 | — | 6 | 2 | no | — |
| 86 | C22×C32 | C22×D3 | 1 | 1 | yes | characteristic |
| 87 | C22×C32 | C22×D3 | 1 | 2 | yes | — |
| 88 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 89 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 90 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 91 | C2×C3⋊D3 | — | 6 | 4 | no | — |
| 92 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 93 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 94 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 95 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 96 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 97 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 98 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 99 | C2×C3⋊D3 | — | 12 | 4 | no | — |
| 100 | C2×C3×D3 | — | 2 | 2 | no | — |
| 101 | C2×C3×D3 | — | 2 | 2 | no | — |
| 102 | C2×C3×D3 | — | 2 | 2 | no | — |
| 103 | C2×C3×D3 | — | 2 | 2 | no | — |
| 104 | C2×C3×D3 | — | 4 | 2 | no | — |
| 105 | C32×C4 | — | 6 | 2 | no | — |
| 106 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 107 | C3⋊Dic3 | D12 | 1 | 2 | yes | — |
| 108 | C3×Dic3 | — | 2 | 2 | no | — |
| 109 | C3×Dic3 | — | 2 | 2 | no | — |
| 110 | C3×Dic3 | — | 3 | 4 | no | — |
| 111 | C3×Dic3 | — | 4 | 2 | no | — |
| 112 | C3×Dic3 | — | 6 | 2 | no | — |
| 113 | C3×Dic3 | — | 6 | 2 | no | — |
| 114 | C3×Dic3 | — | 6 | 2 | no | — |
| 115 | C3×Dic3 | — | 12 | 2 | no | — |
| 116 | #124 | — | 27 | 1 | no | Sylow |
| 117 | C33 | #124 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 118 | C23×C3 | — | 3 | 2 | no | — |
| 119 | C23×C3 | — | 6 | 1 | no | — |
| 120 | C22×D3 | — | 9 | 1 | no | — |
| 121 | C22×D3 | — | 9 | 2 | no | — |
| 122 | C22×D3 | — | 18 | 2 | no | — |
| 123 | C22×D3 | — | 18 | 1 | no | — |
| 124 | C22×D3 | — | 36 | 1 | no | — |
| 125 | C22×D3 | D32 | 1 | 1 | yes | characteristic |
| 126 | C2×C3×C4 | — | 3 | 2 | no | — |
| 127 | C2×C3×C4 | — | 3 | 2 | no | — |
| 128 | C2×C3×C4 | — | 6 | 2 | no | — |
| 129 | C2×C3×C4 | — | 9 | 1 | no | — |
| 130 | C2×C3×C4 | — | 9 | 2 | no | — |
| 131 | #67 | — | 18 | 4 | no | — |
| 132 | #67 | — | 18 | 4 | no | — |
| 133 | C2×Dic3 | — | 3 | 2 | no | — |
| 134 | C2×Dic3 | — | 3 | 2 | no | — |
| 135 | C2×Dic3 | — | 3 | 2 | no | — |
| 136 | C2×Dic3 | — | 6 | 1 | no | — |
| 137 | C2×Dic3 | — | 6 | 2 | no | — |
| 138 | C2×Dic3 | D32 | 1 | 2 | yes | — |
| 139 | D12 | — | 18 | 2 | no | — |
| 140 | Dic6 | — | 18 | 4 | no | — |
| 141 | Dic6 | — | 18 | 2 | no | — |
| 142 | C2×C32 | — | 2 | 2 | no | — |
| 143 | C2×C32 | — | 2 | 2 | no | — |
| 144 | C2×C32 | — | 2 | 2 | no | — |
| 145 | C2×C32 | — | 2 | 2 | no | — |
| 146 | C2×C32 | — | 2 | 1 | no | — |
| 147 | C2×C32 | — | 4 | 1 | no | — |
| 148 | C2×C32 | — | 4 | 2 | no | — |
| 149 | C2×C32 | — | 12 | 1 | no | — |
| 150 | C2×C32 | C2×D12 | 1 | 1 | yes | characteristic |
| 151 | C2×C32 | #235 | 1 | 2 | yes | — |
| 152 | C2×C32 | #235 | 1 | 2 | yes | — |
| 153 | C2×C32 | D3×D4 | 1 | 2 | yes | — |
| 154 | C2×C32 | #237 | 1 | 2 | yes | — |
| 155 | C3⋊D3 | — | 12 | 4 | no | — |
| 156 | C3⋊D3 | — | 12 | 2 | no | — |
| 157 | C3⋊D3 | — | 12 | 4 | no | — |
| 158 | C3⋊D3 | — | 12 | 1 | no | — |
| 159 | C3⋊D3 | — | 12 | 2 | no | — |
| 160 | C3×D3 | — | 4 | 2 | no | — |
| 161 | C3×D3 | — | 4 | 2 | no | — |
| 162 | C2×Q8 | — | 27 | 1 | no | — |
| 163 | C2×D4 | — | 27 | 1 | no | — |
| 164 | C22⋊C4 | — | 9 | 2 | no | — |
| 165 | C22⋊C4 | — | 27 | 2 | no | — |
| 166 | C42 | — | 9 | 1 | no | — |
| 167 | C22×C3 | — | 2 | 2 | no | — |
| 168 | C22×C3 | — | 2 | 1 | no | — |
| 169 | C22×C3 | — | 4 | 1 | no | — |
| 170 | C22×C3 | — | 6 | 2 | no | — |
| 171 | C22×C3 | — | 6 | 2 | no | — |
| 172 | C22×C3 | — | 6 | 2 | no | — |
| 173 | C22×C3 | — | 6 | 2 | no | — |
| 174 | C22×C3 | — | 12 | 2 | no | — |
| 175 | C22×C3 | C2×D32 | 1 | 2 | yes | — |
| 176 | C22×C3 | C2×D32 | 1 | 1 | yes | characteristic |
| 177 | C2×D3 | — | 2 | 2 | no | — |
| 178 | C2×D3 | — | 2 | 1 | no | — |
| 179 | C2×D3 | — | 18 | 2 | no | — |
| 180 | C2×D3 | — | 18 | 4 | no | — |
| 181 | C2×D3 | — | 18 | 2 | no | — |
| 182 | C2×D3 | — | 18 | 2 | no | — |
| 183 | C2×D3 | — | 18 | 2 | no | — |
| 184 | C2×D3 | — | 18 | 2 | no | — |
| 185 | C2×D3 | — | 18 | 1 | no | — |
| 186 | C2×D3 | — | 36 | 2 | no | — |
| 187 | C2×D3 | — | 36 | 2 | no | — |
| 188 | C2×D3 | — | 36 | 2 | no | — |
| 189 | C2×D3 | — | 36 | 2 | no | — |
| 190 | C2×D3 | — | 36 | 4 | no | — |
| 191 | C3×C4 | — | 6 | 2 | no | — |
| 192 | C3×C4 | — | 6 | 2 | no | — |
| 193 | C3×C4 | — | 9 | 2 | no | — |
| 194 | C3×C4 | — | 12 | 2 | no | — |
| 195 | C3×C4 | — | 18 | 2 | no | — |
| 196 | Dic3 | — | 2 | 2 | no | — |
| 197 | Dic3 | — | 3 | 4 | no | — |
| 198 | Dic3 | — | 6 | 2 | no | — |
| 199 | Dic3 | — | 6 | 2 | no | — |
| 200 | Dic3 | — | 6 | 2 | no | — |
| 201 | Dic3 | — | 12 | 2 | no | — |
| 202 | C32 | — | 2 | 2 | no | — |
| 203 | C32 | — | 2 | 1 | no | — |
| 204 | C32 | — | 4 | 1 | no | — |
| 205 | C32 | #875 | 1 | 1 | yes | characteristic, complemented |
| 206 | C32 | #885 | 1 | 2 | yes | complemented |
| 207 | C23 | — | 3 | 1 | no | — |
| 208 | C23 | — | 27 | 1 | no | — |
| 209 | Q8 | — | 54 | 2 | no | — |
| 210 | D4 | — | 54 | 2 | no | — |
| 211 | C2×C4 | — | 3 | 2 | no | — |
| 212 | C2×C4 | — | 9 | 1 | no | — |
| 213 | C2×C4 | — | 9 | 2 | no | — |
| 214 | C2×C3 | — | 2 | 1 | no | — |
| 215 | C2×C3 | — | 2 | 2 | no | — |
| 216 | C2×C3 | — | 2 | 2 | no | — |
| 217 | C2×C3 | — | 2 | 2 | no | — |
| 218 | C2×C3 | — | 2 | 2 | no | — |
| 219 | C2×C3 | — | 4 | 1 | no | — |
| 220 | C2×C3 | — | 4 | 2 | no | — |
| 221 | C2×C3 | — | 12 | 2 | no | — |
| 222 | C2×C3 | — | 12 | 2 | no | — |
| 223 | C2×C3 | #3798 | 1 | 2 | yes | — |
| 224 | C2×C3 | D3×D12 | 1 | 2 | yes | — |
| 225 | C2×C3 | #3804 | 1 | 2 | yes | — |
| 226 | C2×C3 | #3804 | 1 | 2 | yes | — |
| 227 | C2×C3 | #3811 | 1 | 1 | yes | characteristic |
| 228 | D3 | — | 4 | 1 | no | — |
| 229 | D3 | — | 36 | 2 | no | — |
| 230 | D3 | — | 36 | 4 | no | — |
| 231 | D3 | — | 36 | 4 | no | — |
| 232 | D3 | — | 36 | 2 | no | — |
| 233 | D3 | — | 36 | 1 | no | — |
| 234 | C22 | — | 6 | 1 | no | — |
| 235 | C22 | — | 6 | 2 | no | — |
| 236 | C22 | — | 54 | 1 | no | — |
| 237 | C22 | — | 54 | 2 | no | — |
| 238 | C22 | D33 | 1 | 1 | yes | characteristic |
| 239 | C4 | — | 6 | 2 | no | — |
| 240 | C4 | — | 9 | 2 | no | — |
| 241 | C4 | — | 18 | 2 | no | — |
| 242 | C3 | — | 2 | 1 | no | — |
| 243 | C3 | — | 2 | 2 | no | — |
| 244 | C3 | — | 4 | 1 | no | — |
| 245 | C3 | #63852 | 1 | 2 | yes | complemented |
| 246 | C3 | #63953 | 1 | 1 | yes | characteristic, complemented |
| 247 | C2 | — | 12 | 1 | no | — |
| 248 | C2 | — | 108 | 1 | no | — |
| 249 | C2 | #88955 | 1 | 1 | yes | characteristic |
| 250 | C2 | #88959 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D33 |
| commutator, Fitting, socle, cosocle | C22×C33 | C23 |
| prime radical | C2×#6291 | C2 |
| hypocenter | C33 | #124 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6296 | C2 | semidirect |
| 3. #88804 | C2 | semidirect |
| 4. C3×(C2×D3)⋊Dic3 | C2 | semidirect |
| 5. C3×Dic32 | C2 | semidirect |
| 27. (C2×D3)⋊Dic3 | D3 | semidirect |
| 28. Dic32 | D3 | semidirect |
| 117. C33 | #124 | semidirect |
| 205. C32 | #875 | semidirect |
| 206. C32 | #885 | semidirect |
| 245. C3 | #63852 | semidirect |
| 246. C3 | #63953 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6296 | 2 |
| C2×#6291 | 2 |
| #88804 | 2 |
| C3×(C2×D3)⋊Dic3 | 2 |
| C3×Dic32 | 2 |
| #63953 | 3 |
| #63852 | 3 |