Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
1421 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 1421 conjugacy classes of subgroups — too many to draw. Only its 153 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 | C23×Q8⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#63689 | C2 | 1 | 12 | yes | maximal, direct product factor |
| 3 | C22×#3689 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×#3688 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C22×#942 | — | 3 | 1 | no | maximal |
| 6 | C22×#611 | — | 4 | 1 | no | maximal |
| 7 | C22×Q8⋊C9 | C22 | 1 | 1 | yes | characteristic |
| 8 | C22×Q8⋊C9 | C22 | 1 | 6 | yes | complemented |
| 9 | #63689 | C22 | 1 | 16 | yes | direct product factor |
| 10 | C2×#3689 | C22 | 1 | 6 | yes | complemented |
| 11 | C2×#3688 | C22 | 1 | 6 | yes | complemented |
| 12 | C23×C3×Q8 | D3 | 1 | 1 | yes | characteristic |
| 13 | C22×#235 | — | 3 | 1 | no | — |
| 14 | C22×#216 | — | 3 | 1 | no | — |
| 15 | C22×#216 | — | 3 | 1 | no | — |
| 16 | C2×#942 | — | 3 | 12 | no | — |
| 17 | C2×#942 | — | 3 | 12 | no | — |
| 18 | C22×#215 | — | 3 | 1 | no | — |
| 19 | C22×#215 | — | 3 | 1 | no | — |
| 20 | C22×#208 | — | 3 | 1 | no | — |
| 21 | C24×C9 | — | 4 | 1 | no | — |
| 22 | C23×D9 | — | 4 | 1 | no | — |
| 23 | C2×#611 | — | 4 | 12 | no | — |
| 24 | C22×Dic9 | — | 4 | 1 | no | — |
| 25 | C2×Q8⋊C9 | C23 | 1 | 3 | yes | — |
| 26 | C2×Q8⋊C9 | C23 | 1 | 4 | yes | complemented |
| 27 | #3689 | C23 | 1 | 4 | yes | complemented |
| 28 | #3688 | C23 | 1 | 4 | yes | complemented |
| 29 | C22×#137 | — | 9 | 1 | no | Sylow |
| 30 | C22×C3×Q8 | — | 3 | 1 | no | — |
| 31 | C22×C3×Q8 | — | 3 | 6 | no | — |
| 32 | C22×C3×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | C22×C3×Q8 | C2×D3 | 1 | 6 | yes | — |
| 34 | C23×C3×C4 | — | 3 | 1 | no | — |
| 35 | C22×#67 | — | 6 | 1 | no | — |
| 36 | C2×#235 | — | 3 | 12 | no | — |
| 37 | C2×#235 | — | 6 | 6 | no | — |
| 38 | C22×D12 | — | 3 | 1 | no | — |
| 39 | C22×C4×D3 | — | 6 | 1 | no | — |
| 40 | C22×Dic6 | — | 3 | 1 | no | — |
| 41 | C2×#216 | — | 3 | 6 | no | — |
| 42 | C2×#216 | — | 3 | 12 | no | — |
| 43 | C2×#216 | — | 3 | 6 | no | — |
| 44 | #942 | — | 3 | 48 | no | — |
| 45 | #942 | — | 3 | 16 | no | — |
| 46 | C2×#215 | — | 3 | 12 | no | — |
| 47 | C2×#215 | — | 3 | 6 | no | — |
| 48 | C2×#215 | — | 3 | 6 | no | — |
| 49 | C2×#208 | — | 3 | 12 | no | — |
| 50 | C22×C3⋊C8 | — | 3 | 1 | no | — |
| 51 | C22×C3⋊C8 | — | 3 | 1 | no | — |
| 52 | C23×C9 | — | 4 | 1 | no | — |
| 53 | C23×C9 | — | 4 | 6 | no | — |
| 54 | C23×C9 | — | 8 | 4 | no | — |
| 55 | C22×D9 | — | 4 | 6 | no | — |
| 56 | C22×D9 | — | 8 | 4 | no | — |
| 57 | #611 | — | 4 | 16 | no | — |
| 58 | C2×Dic9 | — | 4 | 6 | no | — |
| 59 | Q8⋊C9 | C24 | 1 | 1 | yes | characteristic |
| 60 | C22×#41 | — | 9 | 1 | no | — |
| 61 | C23×Q8 | D9 | 1 | 1 | yes | characteristic, complemented |
| 62 | C2×#137 | — | 9 | 12 | no | — |
| 63 | C2×#137 | — | 9 | 12 | no | — |
| 64 | C22×Q16 | — | 9 | 1 | no | — |
| 65 | C22×Q16 | — | 9 | 1 | no | — |
| 66 | C22×SD16 | — | 9 | 1 | no | — |
| 67 | C22×SD16 | — | 9 | 1 | no | — |
| 68 | C22×M16 | — | 9 | 1 | no | — |
| 69 | C24×C3 | S4 | 1 | 1 | yes | characteristic |
| 70 | C23×D3 | — | 6 | 1 | no | — |
| 71 | C2×C3×Q8 | — | 3 | 3 | no | — |
| 72 | C2×C3×Q8 | — | 3 | 6 | no | — |
| 73 | C2×C3×Q8 | — | 3 | 12 | no | — |
| 74 | C2×C3×Q8 | — | 6 | 3 | no | — |
| 75 | C2×C3×Q8 | — | 6 | 4 | no | — |
| 76 | C2×C3×Q8 | C22×D3 | 1 | 3 | yes | — |
| 77 | C2×C3×Q8 | C22×D3 | 1 | 4 | yes | — |
| 78 | C22×C3×C4 | — | 3 | 1 | no | — |
| 79 | C22×C3×C4 | — | 3 | 6 | no | — |
| 80 | C22×C3×C4 | — | 3 | 1 | no | — |
| 81 | C22×C3×C4 | — | 3 | 6 | no | — |
| 82 | C2×#67 | — | 6 | 12 | no | — |
| 83 | C22×Dic3 | — | 6 | 1 | no | — |
| 84 | #235 | — | 3 | 16 | no | — |
| 85 | #235 | — | 6 | 24 | no | — |
| 86 | C2×D12 | — | 3 | 6 | no | — |
| 87 | C2×D12 | — | 6 | 3 | no | — |
| 88 | C2×C4×D3 | — | 6 | 12 | no | — |
| 89 | C2×Dic6 | — | 3 | 6 | no | — |
| 90 | C2×Dic6 | — | 6 | 3 | no | — |
| 91 | #216 | — | 3 | 12 | no | — |
| 92 | #216 | — | 3 | 16 | no | — |
| 93 | #216 | — | 3 | 4 | no | — |
| 94 | #215 | — | 3 | 16 | no | — |
| 95 | #215 | — | 3 | 12 | no | — |
| 96 | #215 | — | 3 | 4 | no | — |
| 97 | #208 | — | 3 | 16 | no | — |
| 98 | C2×C3⋊C8 | — | 3 | 6 | no | — |
| 99 | C2×C3⋊C8 | — | 3 | 6 | no | — |
| 100 | C22×C9 | — | 4 | 4 | no | — |
| 101 | C22×C9 | — | 4 | 3 | no | — |
| 102 | C22×C9 | — | 4 | 4 | no | — |
| 103 | C22×C9 | — | 8 | 12 | no | — |
| 104 | C2×D9 | — | 4 | 4 | no | — |
| 105 | C2×D9 | — | 8 | 12 | no | — |
| 106 | Dic9 | — | 4 | 4 | no | — |
| 107 | C2×#41 | — | 9 | 12 | no | — |
| 108 | C2×#41 | — | 18 | 6 | no | — |
| 109 | C22×Q8 | — | 3 | 1 | no | — |
| 110 | C22×Q8 | — | 3 | 6 | no | — |
| 111 | C22×Q8 | — | 9 | 1 | no | — |
| 112 | C22×Q8 | C2×D9 | 1 | 1 | yes | characteristic |
| 113 | C22×Q8 | C2×D9 | 1 | 6 | yes | complemented |
| 114 | C22×D4 | — | 9 | 1 | no | — |
| 115 | C22×D4 | — | 18 | 1 | no | — |
| 116 | C23×C4 | — | 3 | 1 | no | — |
| 117 | C23×C4 | — | 18 | 1 | no | — |
| 118 | #137 | — | 9 | 48 | no | — |
| 119 | #137 | — | 9 | 16 | no | — |
| 120 | C2×Q16 | — | 9 | 6 | no | — |
| 121 | C2×Q16 | — | 9 | 12 | no | — |
| 122 | C2×Q16 | — | 9 | 6 | no | — |
| 123 | C2×SD16 | — | 9 | 6 | no | — |
| 124 | C2×SD16 | — | 9 | 12 | no | — |
| 125 | C2×SD16 | — | 9 | 6 | no | — |
| 126 | C2×M16 | — | 9 | 12 | no | — |
| 127 | C22×C8 | — | 9 | 1 | no | — |
| 128 | C22×C8 | — | 9 | 1 | no | — |
| 129 | C23×C3 | — | 2 | 4 | no | — |
| 130 | C23×C3 | C2×S4 | 1 | 6 | yes | — |
| 131 | C23×C3 | C2×S4 | 1 | 1 | yes | characteristic |
| 132 | C22×D3 | — | 6 | 6 | no | — |
| 133 | C22×D3 | — | 12 | 4 | no | — |
| 134 | C3×Q8 | — | 3 | 3 | no | — |
| 135 | C3×Q8 | — | 3 | 4 | no | — |
| 136 | C3×Q8 | — | 6 | 1 | no | — |
| 137 | C3×Q8 | — | 6 | 6 | no | — |
| 138 | C3×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 139 | C2×C3×C4 | — | 3 | 3 | no | — |
| 140 | C2×C3×C4 | — | 3 | 4 | no | — |
| 141 | C2×C3×C4 | — | 3 | 3 | no | — |
| 142 | C2×C3×C4 | — | 3 | 6 | no | — |
| 143 | C2×C3×C4 | — | 3 | 12 | no | — |
| 144 | #67 | — | 6 | 16 | no | — |
| 145 | C2×Dic3 | — | 6 | 6 | no | — |
| 146 | D12 | — | 3 | 4 | no | — |
| 147 | D12 | — | 6 | 6 | no | — |
| 148 | C4×D3 | — | 6 | 16 | no | — |
| 149 | Dic6 | — | 3 | 4 | no | — |
| 150 | Dic6 | — | 6 | 6 | no | — |
| 151 | C3⋊C8 | — | 3 | 4 | no | — |
| 152 | C3⋊C8 | — | 3 | 4 | no | — |
| 153 | C2×C9 | — | 4 | 6 | no | — |
| 154 | C2×C9 | — | 4 | 1 | no | — |
| 155 | C2×C9 | — | 8 | 4 | no | — |
| 156 | D9 | — | 8 | 4 | no | — |
| 157 | C24 | — | 18 | 1 | no | — |
| 158 | C24 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 159 | #41 | — | 9 | 16 | no | — |
| 160 | #41 | — | 18 | 24 | no | — |
| 161 | C2×Q8 | — | 3 | 3 | no | — |
| 162 | C2×Q8 | — | 3 | 6 | no | — |
| 163 | C2×Q8 | — | 3 | 12 | no | — |
| 164 | C2×Q8 | — | 6 | 3 | no | — |
| 165 | C2×Q8 | — | 6 | 4 | no | — |
| 166 | C2×Q8 | — | 9 | 6 | no | — |
| 167 | C2×Q8 | — | 18 | 3 | no | — |
| 168 | C2×Q8 | C22×D9 | 1 | 3 | yes | — |
| 169 | C2×Q8 | C22×D9 | 1 | 4 | yes | complemented |
| 170 | C2×D4 | — | 9 | 6 | no | — |
| 171 | C2×D4 | — | 18 | 3 | no | — |
| 172 | C2×D4 | — | 18 | 12 | no | — |
| 173 | C22×C4 | — | 3 | 6 | no | — |
| 174 | C22×C4 | — | 3 | 6 | no | — |
| 175 | C22×C4 | — | 3 | 1 | no | — |
| 176 | C22×C4 | — | 3 | 1 | no | — |
| 177 | C22×C4 | — | 18 | 12 | no | — |
| 178 | C22×C4 | — | 18 | 1 | no | — |
| 179 | Q16 | — | 9 | 12 | no | — |
| 180 | Q16 | — | 9 | 16 | no | — |
| 181 | Q16 | — | 9 | 4 | no | — |
| 182 | SD16 | — | 9 | 12 | no | — |
| 183 | SD16 | — | 9 | 16 | no | — |
| 184 | SD16 | — | 9 | 4 | no | — |
| 185 | M16 | — | 9 | 16 | no | — |
| 186 | C2×C8 | — | 9 | 6 | no | — |
| 187 | C2×C8 | — | 9 | 6 | no | — |
| 188 | C22×C3 | — | 2 | 12 | no | — |
| 189 | C22×C3 | #983 | 1 | 4 | yes | — |
| 190 | C22×C3 | C22×S4 | 1 | 4 | yes | — |
| 191 | C22×C3 | C22×S4 | 1 | 3 | yes | — |
| 192 | C2×D3 | — | 6 | 4 | no | — |
| 193 | C2×D3 | — | 12 | 12 | no | — |
| 194 | C3×C4 | — | 3 | 1 | no | — |
| 195 | C3×C4 | — | 3 | 3 | no | — |
| 196 | C3×C4 | — | 3 | 4 | no | — |
| 197 | Dic3 | — | 6 | 4 | no | — |
| 198 | C9 | — | 4 | 1 | no | Sylow |
| 199 | C23 | — | 2 | 4 | no | — |
| 200 | C23 | — | 18 | 6 | no | — |
| 201 | C23 | — | 36 | 4 | no | — |
| 202 | C23 | C2×(C22⋊C9)⋊C2 | 1 | 6 | yes | — |
| 203 | C23 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 204 | Q8 | — | 3 | 3 | no | — |
| 205 | Q8 | — | 3 | 4 | no | — |
| 206 | Q8 | — | 6 | 6 | no | — |
| 207 | Q8 | — | 6 | 1 | no | — |
| 208 | Q8 | — | 9 | 4 | no | — |
| 209 | Q8 | — | 18 | 6 | no | — |
| 210 | Q8 | C23×D9 | 1 | 1 | yes | characteristic |
| 211 | D4 | — | 9 | 4 | no | — |
| 212 | D4 | — | 18 | 6 | no | — |
| 213 | D4 | — | 18 | 16 | no | — |
| 214 | C2×C4 | — | 3 | 3 | no | — |
| 215 | C2×C4 | — | 3 | 4 | no | — |
| 216 | C2×C4 | — | 3 | 6 | no | — |
| 217 | C2×C4 | — | 3 | 3 | no | — |
| 218 | C2×C4 | — | 3 | 12 | no | — |
| 219 | C2×C4 | — | 18 | 16 | no | — |
| 220 | C2×C4 | — | 18 | 6 | no | — |
| 221 | C8 | — | 9 | 4 | no | — |
| 222 | C8 | — | 9 | 4 | no | — |
| 223 | C2×C3 | — | 2 | 4 | no | — |
| 224 | C2×C3 | C2×#983 | 1 | 6 | yes | — |
| 225 | C2×C3 | C23×S4 | 1 | 1 | yes | characteristic |
| 226 | D3 | — | 12 | 4 | no | — |
| 227 | C22 | — | 2 | 12 | no | — |
| 228 | C22 | — | 18 | 4 | no | — |
| 229 | C22 | — | 36 | 12 | no | — |
| 230 | C22 | #63689 | 1 | 4 | yes | direct product factor |
| 231 | C22 | C22×(C22⋊C9)⋊C2 | 1 | 4 | yes | — |
| 232 | C22 | C22×(C22⋊C9)⋊C2 | 1 | 3 | yes | — |
| 233 | C4 | — | 3 | 1 | no | — |
| 234 | C4 | — | 3 | 4 | no | — |
| 235 | C4 | — | 3 | 3 | no | — |
| 236 | C4 | — | 18 | 4 | no | — |
| 237 | C3 | C22×#983 | 1 | 1 | yes | characteristic |
| 238 | C2 | — | 2 | 4 | no | — |
| 239 | C2 | — | 36 | 4 | no | — |
| 240 | C2 | C2×#63689 | 1 | 6 | yes | direct product factor |
| 241 | C2 | C23×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C24 |
| Frattini | C2×C3 | C23×S4 |
| Fitting | C23×C3×Q8 | D3 |
| socle | C23×C3 | C2×S4 |
| hypercenter | C24 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8⋊C9 | C2 | semidirect |
| 2. C2×#63689 | C2 | direct (N × H) |
| 3. C22×#3689 | C2 | semidirect |
| 4. C22×#3688 | C2 | semidirect |
| 8. C22×Q8⋊C9 | C22 | semidirect |
| 9. #63689 | C22 | direct (N × H) |
| 10. C2×#3689 | C22 | semidirect |
| 11. C2×#3688 | C22 | semidirect |
| 26. C2×Q8⋊C9 | C23 | semidirect |
| 27. #3689 | C23 | semidirect |
| 28. #3688 | C23 | semidirect |
| 61. C23×Q8 | D9 | semidirect |
| 113. C22×Q8 | C2×D9 | semidirect |
| 169. C2×Q8 | C22×D9 | semidirect |
| 230. C22 | #63689 | direct (N × H) |
| 240. C2 | C2×#63689 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8⋊C9 | 2 |
| C2×#63689 | 2 |
| C22×#3689 | 2 |
| C22×#3688 | 2 |
| C22×#942 | 3 |
| C22×#611 | 4 |