Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
549 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 114 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 | Q8×#272 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #88722 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Q8×(C3⋊D3)⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #88650 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #88649 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #88648 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | ((C3⋊D3)⋊C3)⋊Q8 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #64316 | — | 3 | 1 | no | maximal |
| 9 | #64315 | — | 3 | 1 | no | maximal |
| 10 | Q8×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 11 | C4×#272 | C22 | 1 | 3 | yes | — |
| 12 | #6229 | C22 | 1 | 3 | yes | — |
| 13 | C4⋊((C3⋊D3)⋊C3) | C22 | 1 | 3 | yes | — |
| 14 | C4×(C3⋊D3)⋊C3 | C22 | 1 | 3 | yes | — |
| 15 | C4×(C3⋊D3)⋊C3 | C22 | 1 | 3 | yes | — |
| 16 | #634⋊C3 | C22 | 1 | 3 | yes | — |
| 17 | #6200 | C22 | 1 | 3 | yes | — |
| 18 | #6200 | C22 | 1 | 1 | yes | characteristic |
| 19 | #6198 | C22 | 1 | 3 | yes | — |
| 20 | ((C3⋊D3)⋊C3)⋊C4 | C22 | 1 | 6 | yes | — |
| 21 | #6196 | C22 | 1 | 3 | yes | — |
| 22 | #3832 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | Q8×C3⋊D3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C3×#239 | — | 3 | 1 | no | — |
| 25 | C3×D3×Q8 | — | 3 | 1 | no | — |
| 26 | C3×D3×Q8 | — | 3 | 1 | no | — |
| 27 | C3×D3×Q8 | — | 3 | 1 | no | — |
| 28 | C3×D3×Q8 | — | 6 | 1 | no | — |
| 29 | #3799 | — | 3 | 3 | no | — |
| 30 | #3799 | — | 3 | 3 | no | — |
| 31 | #3798 | — | 3 | 3 | no | — |
| 32 | #3798 | — | 3 | 3 | no | — |
| 33 | #3797 | — | 3 | 3 | no | — |
| 34 | #3796 | — | 3 | 3 | no | — |
| 35 | #3795 | — | 3 | 3 | no | — |
| 36 | D3×Dic6 | — | 3 | 3 | no | — |
| 37 | C2×#272 | C23 | 1 | 1 | yes | characteristic |
| 38 | C2×(C3⋊D3)⋊C3 | C23 | 1 | 3 | yes | — |
| 39 | C2×(C3⋊D3)⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 40 | C4×H(3,𝔽3) | C23 | 1 | 3 | yes | — |
| 41 | #1084 | C23 | 1 | 3 | yes | — |
| 42 | (C3⋊Dic3)⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 43 | (C3⋊Dic3)⋊C3 | C23 | 1 | 3 | yes | — |
| 44 | #1010 | — | 9 | 1 | no | — |
| 45 | #1007 | — | 9 | 1 | no | — |
| 46 | #1007 | — | 9 | 1 | no | — |
| 47 | C32×Q8 | — | 2 | 1 | no | — |
| 48 | C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 49 | C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 50 | C4⋊(C3⋊D3) | C2×D3 | 1 | 3 | yes | — |
| 51 | C4×C3⋊D3 | C2×D3 | 1 | 3 | yes | — |
| 52 | C4×C3⋊D3 | C2×D3 | 1 | 3 | yes | — |
| 53 | #634 | C2×D3 | 1 | 3 | yes | — |
| 54 | C3×D12 | — | 3 | 3 | no | — |
| 55 | C3×C4×D3 | — | 3 | 3 | no | — |
| 56 | C3×C4×D3 | — | 3 | 3 | no | — |
| 57 | C3×C4×D3 | — | 3 | 3 | no | — |
| 58 | C3×C4×D3 | — | 3 | 3 | no | — |
| 59 | C3×C4×D3 | — | 6 | 3 | no | — |
| 60 | C3×Dic6 | — | 3 | 3 | no | — |
| 61 | C3×Dic6 | — | 3 | 3 | no | — |
| 62 | C3×Dic6 | — | 3 | 3 | no | — |
| 63 | C3×Dic6 | — | 6 | 3 | no | — |
| 64 | #627 | — | 3 | 3 | no | — |
| 65 | #627 | — | 3 | 3 | no | — |
| 66 | #626 | — | 3 | 3 | no | — |
| 67 | #626 | — | 3 | 3 | no | — |
| 68 | #626 | — | 3 | 1 | no | — |
| 69 | #626 | — | 3 | 3 | no | — |
| 70 | #625 | — | 3 | 3 | no | — |
| 71 | #625 | — | 3 | 1 | no | — |
| 72 | #624 | — | 3 | 6 | no | — |
| 73 | D3×Dic3 | — | 3 | 6 | no | — |
| 74 | C2×H(3,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 75 | #272 | — | 2 | 1 | no | — |
| 76 | (C3⋊D3)⋊C3 | — | 2 | 3 | no | — |
| 77 | (C3⋊D3)⋊C3 | — | 2 | 1 | no | — |
| 78 | C3×#41 | — | 9 | 1 | no | — |
| 79 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 80 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 81 | #239 | — | 3 | 1 | no | — |
| 82 | #239 | — | 9 | 1 | no | — |
| 83 | #239 | — | 9 | 3 | no | — |
| 84 | #239 | — | 9 | 3 | no | — |
| 85 | D3×Q8 | — | 3 | 1 | no | — |
| 86 | D3×Q8 | — | 3 | 1 | no | — |
| 87 | D3×Q8 | — | 3 | 1 | no | — |
| 88 | D3×Q8 | — | 6 | 1 | no | — |
| 89 | D3×Q8 | — | 9 | 1 | no | — |
| 90 | D3×Q8 | — | 9 | 3 | no | — |
| 91 | D3×Q8 | — | 9 | 3 | no | — |
| 92 | #237 | — | 9 | 3 | no | — |
| 93 | #237 | — | 9 | 3 | no | — |
| 94 | #235 | — | 9 | 3 | no | — |
| 95 | #235 | — | 9 | 3 | no | — |
| 96 | #235 | — | 9 | 3 | no | — |
| 97 | #235 | — | 9 | 3 | no | — |
| 98 | #235 | — | 9 | 3 | no | — |
| 99 | C2×Dic6 | — | 9 | 3 | no | — |
| 100 | C2×C3⋊D3 | C22×D3 | 1 | 3 | yes | — |
| 101 | C2×C3⋊D3 | C22×D3 | 1 | 1 | yes | characteristic |
| 102 | C2×C3×D3 | — | 3 | 3 | no | — |
| 103 | C2×C3×D3 | — | 3 | 1 | no | — |
| 104 | C2×C3×D3 | — | 3 | 1 | no | — |
| 105 | C2×C3×D3 | — | 3 | 1 | no | — |
| 106 | C2×C3×D3 | — | 6 | 1 | no | — |
| 107 | C32×C4 | — | 2 | 3 | no | — |
| 108 | C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 109 | C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 110 | C3⋊Dic3 | C22×D3 | 1 | 3 | yes | — |
| 111 | C3⋊Dic3 | C22×D3 | 1 | 1 | yes | characteristic |
| 112 | C3×Dic3 | — | 3 | 3 | no | — |
| 113 | C3×Dic3 | — | 3 | 1 | no | — |
| 114 | C3×Dic3 | — | 3 | 3 | no | — |
| 115 | C3×Dic3 | — | 3 | 3 | no | — |
| 116 | C3×Dic3 | — | 6 | 3 | no | — |
| 117 | 2−1+4 | — | 27 | 1 | no | Sylow |
| 118 | H(3,𝔽3) | 2−1+4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 119 | C3×Q8 | — | 3 | 1 | no | — |
| 120 | C3×Q8 | — | 3 | 1 | no | — |
| 121 | C3×Q8 | — | 6 | 1 | no | — |
| 122 | C3×Q8 | — | 9 | 3 | no | — |
| 123 | C3×Q8 | — | 9 | 3 | no | — |
| 124 | C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 125 | C3×D4 | — | 9 | 3 | no | — |
| 126 | C2×C3×C4 | — | 9 | 3 | no | — |
| 127 | C2×C3×C4 | — | 9 | 3 | no | — |
| 128 | C2×C3×C4 | — | 9 | 3 | no | — |
| 129 | #67 | — | 9 | 3 | no | — |
| 130 | #67 | — | 9 | 1 | no | — |
| 131 | #67 | — | 9 | 3 | no | — |
| 132 | #67 | — | 9 | 3 | no | — |
| 133 | #67 | — | 9 | 1 | no | — |
| 134 | #67 | — | 9 | 3 | no | — |
| 135 | C2×Dic3 | — | 9 | 6 | no | — |
| 136 | D12 | — | 3 | 3 | no | — |
| 137 | D12 | — | 9 | 3 | no | — |
| 138 | D12 | — | 9 | 3 | no | — |
| 139 | D12 | — | 9 | 1 | no | — |
| 140 | D12 | — | 9 | 3 | no | — |
| 141 | C4×D3 | — | 3 | 3 | no | — |
| 142 | C4×D3 | — | 3 | 3 | no | — |
| 143 | C4×D3 | — | 3 | 3 | no | — |
| 144 | C4×D3 | — | 3 | 3 | no | — |
| 145 | C4×D3 | — | 6 | 3 | no | — |
| 146 | C4×D3 | — | 9 | 3 | no | — |
| 147 | C4×D3 | — | 9 | 3 | no | — |
| 148 | C4×D3 | — | 9 | 6 | no | — |
| 149 | C4×D3 | — | 9 | 6 | no | — |
| 150 | Dic6 | — | 3 | 3 | no | — |
| 151 | Dic6 | — | 3 | 3 | no | — |
| 152 | Dic6 | — | 3 | 3 | no | — |
| 153 | Dic6 | — | 6 | 3 | no | — |
| 154 | Dic6 | — | 9 | 3 | no | — |
| 155 | Dic6 | — | 9 | 3 | no | — |
| 156 | Dic6 | — | 9 | 3 | no | — |
| 157 | Dic6 | — | 9 | 3 | no | — |
| 158 | C2×C32 | — | 2 | 1 | no | — |
| 159 | C2×C32 | C23×D3 | 1 | 1 | yes | characteristic |
| 160 | C2×C32 | C23×D3 | 1 | 1 | yes | characteristic |
| 161 | C3⋊D3 | — | 2 | 1 | no | — |
| 162 | C3⋊D3 | — | 2 | 3 | no | — |
| 163 | C3×D3 | — | 6 | 2 | no | — |
| 164 | C3×D3 | — | 6 | 3 | no | — |
| 165 | C3×D3 | — | 6 | 1 | no | — |
| 166 | C3×D3 | — | 6 | 1 | no | — |
| 167 | C3×D3 | — | 6 | 1 | no | — |
| 168 | #41 | — | 9 | 1 | no | — |
| 169 | #41 | — | 27 | 3 | no | — |
| 170 | #41 | — | 27 | 3 | no | — |
| 171 | #41 | — | 27 | 3 | no | — |
| 172 | C2×Q8 | — | 9 | 1 | no | — |
| 173 | C2×Q8 | — | 9 | 1 | no | — |
| 174 | C2×Q8 | — | 27 | 3 | no | — |
| 175 | C22×C3 | — | 9 | 3 | no | — |
| 176 | C22×C3 | — | 9 | 1 | no | — |
| 177 | C22×C3 | — | 9 | 1 | no | — |
| 178 | C2×D3 | — | 3 | 1 | no | — |
| 179 | C2×D3 | — | 3 | 1 | no | — |
| 180 | C2×D3 | — | 3 | 1 | no | — |
| 181 | C2×D3 | — | 3 | 3 | no | — |
| 182 | C2×D3 | — | 6 | 1 | no | — |
| 183 | C2×D3 | — | 9 | 3 | no | — |
| 184 | C2×D3 | — | 9 | 1 | no | — |
| 185 | C3×C4 | — | 3 | 3 | no | — |
| 186 | C3×C4 | — | 3 | 3 | no | — |
| 187 | C3×C4 | — | 6 | 3 | no | — |
| 188 | C3×C4 | — | 9 | 1 | no | — |
| 189 | C3×C4 | — | 9 | 3 | no | — |
| 190 | C3×C4 | — | 9 | 3 | no | — |
| 191 | C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 192 | Dic3 | — | 3 | 3 | no | — |
| 193 | Dic3 | — | 3 | 3 | no | — |
| 194 | Dic3 | — | 3 | 3 | no | — |
| 195 | Dic3 | — | 3 | 1 | no | — |
| 196 | Dic3 | — | 6 | 3 | no | — |
| 197 | Dic3 | — | 9 | 1 | no | — |
| 198 | Dic3 | — | 9 | 3 | no | — |
| 199 | C32 | — | 2 | 1 | no | — |
| 200 | C32 | #1007 | 1 | 1 | yes | characteristic, complemented |
| 201 | C32 | #1010 | 1 | 1 | yes | characteristic, complemented |
| 202 | Q8 | — | 9 | 3 | no | — |
| 203 | Q8 | — | 9 | 3 | no | — |
| 204 | Q8 | — | 27 | 3 | no | — |
| 205 | Q8 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 206 | D4 | — | 9 | 3 | no | — |
| 207 | D4 | — | 27 | 1 | no | — |
| 208 | D4 | — | 27 | 3 | no | — |
| 209 | D4 | — | 27 | 3 | no | — |
| 210 | C2×C4 | — | 9 | 3 | no | — |
| 211 | C2×C4 | — | 9 | 3 | no | — |
| 212 | C2×C4 | — | 9 | 3 | no | — |
| 213 | C2×C4 | — | 27 | 6 | no | — |
| 214 | C2×C3 | — | 3 | 1 | no | — |
| 215 | C2×C3 | — | 3 | 1 | no | — |
| 216 | C2×C3 | — | 6 | 1 | no | — |
| 217 | C2×C3 | — | 18 | 1 | no | — |
| 218 | C2×C3 | — | 18 | 3 | no | — |
| 219 | C2×C3 | — | 18 | 1 | no | — |
| 220 | C2×C3 | C22×D32 | 1 | 1 | yes | characteristic |
| 221 | D3 | — | 6 | 1 | no | — |
| 222 | D3 | — | 6 | 1 | no | — |
| 223 | D3 | — | 6 | 2 | no | — |
| 224 | D3 | — | 6 | 1 | no | — |
| 225 | D3 | — | 6 | 3 | no | — |
| 226 | D3 | — | 18 | 3 | no | — |
| 227 | D3 | — | 18 | 1 | no | — |
| 228 | C22 | — | 9 | 1 | no | — |
| 229 | C22 | — | 9 | 1 | no | — |
| 230 | C22 | — | 9 | 3 | no | — |
| 231 | C4 | — | 9 | 1 | no | — |
| 232 | C4 | — | 9 | 3 | no | — |
| 233 | C4 | — | 9 | 3 | no | — |
| 234 | C4 | C2×((C3⋊D3)⋊C3)⋊C2 | 1 | 3 | yes | — |
| 235 | C3 | — | 3 | 1 | no | — |
| 236 | C3 | — | 3 | 1 | no | — |
| 237 | C3 | — | 6 | 1 | no | — |
| 238 | C3 | #64315 | 1 | 1 | yes | characteristic |
| 239 | C2 | — | 18 | 1 | no | — |
| 240 | C2 | — | 18 | 1 | no | — |
| 241 | C2 | — | 18 | 3 | no | — |
| 242 | C2 | C22×((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×#272 | C2 | semidirect |
| 2. #88722 | C2 | semidirect |
| 3. Q8×(C3⋊D3)⋊C3 | C2 | semidirect |
| 4. #88650 | C2 | semidirect |
| 5. #88649 | C2 | semidirect |
| 6. #88648 | C2 | semidirect |
| 7. ((C3⋊D3)⋊C3)⋊Q8 | C2 | semidirect |
| 22. #3832 | D3 | semidirect |
| 23. Q8×C3⋊D3 | D3 | semidirect |
| 118. H(3,𝔽3) | 2−1+4 | semidirect |
| 200. C32 | #1007 | semidirect |
| 201. C32 | #1010 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#272 | 2 |
| #88722 | 2 |
| Q8×(C3⋊D3)⋊C3 | 2 |
| #88650 | 2 |
| #88649 | 2 |
| #88648 | 2 |
| ((C3⋊D3)⋊C3)⋊Q8 | 2 |
| #64316 | 3 |
| #64315 | 3 |