Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
646 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 83 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 | C9×#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D9×#41⋊C3 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11722412 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | D3×#994 | — | 3 | 1 | no | maximal |
| 5 | D9×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C3×Q8×D9 | — | 4 | 1 | no | maximal |
| 7 | C9×#41⋊C3 | C22 | 1 | 3 | yes | complemented |
| 8 | D9×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | #88615 | C22 | 1 | 3 | yes | complemented |
| 10 | C3×#994 | D3 | 1 | 1 | yes | characteristic |
| 11 | D3×#41⋊C3 | — | 3 | 3 | no | — |
| 12 | C3⋊#994 | — | 3 | 3 | no | — |
| 13 | C9×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #63715 | — | 3 | 1 | no | — |
| 15 | #63715 | C2×C3 | 1 | 3 | yes | complemented |
| 16 | #41×D9 | — | 3 | 1 | no | — |
| 17 | #41×D9 | C2×C3 | 1 | 3 | yes | — |
| 18 | #63710 | — | 3 | 3 | no | — |
| 19 | C2×D4×D9 | — | 3 | 3 | no | — |
| 20 | #63700 | — | 2 | 1 | no | — |
| 21 | C3×Q8×C9 | — | 4 | 1 | no | — |
| 22 | C3×C4×D9 | — | 4 | 3 | no | — |
| 23 | C3×Dic18 | — | 4 | 3 | no | — |
| 24 | C9×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic, complemented |
| 25 | D3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 26 | C2×#994 | — | 9 | 1 | no | — |
| 27 | C3×D3×Q8 | — | 12 | 1 | no | — |
| 28 | C3×#41⋊C3 | C2×D3 | 1 | 3 | yes | — |
| 29 | D3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 30 | C3⋊(#41⋊C3) | — | 3 | 3 | no | — |
| 31 | C23×D9 | — | 3 | 2 | no | — |
| 32 | C9×#41 | — | 3 | 1 | no | — |
| 33 | C9×#41 | C22×C3 | 1 | 3 | yes | — |
| 34 | C2×D4×C9 | — | 3 | 3 | no | — |
| 35 | C2×#611 | — | 3 | 6 | no | — |
| 36 | #3701 | — | 3 | 1 | no | — |
| 37 | #3701 | C22×C3 | 1 | 3 | yes | — |
| 38 | Q8×D9 | A4 | 1 | 1 | yes | characteristic |
| 39 | Q8×D9 | C22×C3 | 1 | 1 | yes | characteristic |
| 40 | #3699 | — | 3 | 3 | no | — |
| 41 | #3699 | — | 3 | 3 | no | — |
| 42 | D4×D9 | — | 3 | 6 | no | — |
| 43 | D4×D9 | — | 3 | 3 | no | — |
| 44 | D4×D9 | — | 3 | 6 | no | — |
| 45 | D4×D9 | — | 3 | 3 | no | — |
| 46 | #3697 | — | 3 | 3 | no | — |
| 47 | #3697 | — | 3 | 3 | no | — |
| 48 | C2×D36 | — | 3 | 3 | no | — |
| 49 | C2×C4×D9 | — | 3 | 3 | no | — |
| 50 | #41⋊C9 | — | 2 | 3 | no | — |
| 51 | C2×C3×D9 | — | 4 | 1 | no | — |
| 52 | C3×C4×C9 | — | 4 | 3 | no | — |
| 53 | C3×Dic9 | — | 4 | 3 | no | — |
| 54 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 55 | #1009 | — | 3 | 3 | no | — |
| 56 | #1009 | — | 9 | 1 | no | — |
| 57 | D3×#41 | — | 3 | 3 | no | — |
| 58 | D3×#41 | — | 9 | 1 | no | — |
| 59 | #1004 | — | 9 | 3 | no | — |
| 60 | C2×D3×D4 | — | 9 | 3 | no | — |
| 61 | #994 | — | 2 | 1 | no | — |
| 62 | #994 | — | 9 | 3 | no | — |
| 63 | #994 | D9 | 1 | 1 | yes | characteristic, direct product factor |
| 64 | C2×#41⋊C3 | — | 9 | 3 | no | — |
| 65 | C32×Q8 | — | 4 | 1 | no | — |
| 66 | C3×C4×D3 | — | 12 | 3 | no | — |
| 67 | C3×Dic6 | — | 12 | 3 | no | — |
| 68 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 69 | C23×C9 | — | 3 | 2 | no | — |
| 70 | C22×D9 | — | 3 | 6 | no | — |
| 71 | C22×D9 | — | 3 | 3 | no | — |
| 72 | C22×D9 | — | 12 | 4 | no | — |
| 73 | Q8×C9 | — | 8 | 1 | no | — |
| 74 | Q8×C9 | C2×A4 | 1 | 1 | yes | characteristic |
| 75 | Q8×C9 | C23×C3 | 1 | 1 | yes | characteristic |
| 76 | D4×C9 | — | 3 | 3 | no | — |
| 77 | D4×C9 | — | 3 | 3 | no | — |
| 78 | C2×C4×C9 | — | 3 | 3 | no | — |
| 79 | #611 | — | 3 | 6 | no | — |
| 80 | #611 | — | 3 | 6 | no | — |
| 81 | C2×Dic9 | — | 3 | 3 | no | — |
| 82 | D36 | — | 3 | 3 | no | — |
| 83 | D36 | — | 3 | 3 | no | — |
| 84 | C4×D9 | — | 3 | 1 | no | — |
| 85 | C4×D9 | — | 3 | 3 | no | — |
| 86 | C4×D9 | — | 3 | 3 | no | — |
| 87 | C4×D9 | C2×A4 | 1 | 3 | yes | — |
| 88 | Dic18 | — | 3 | 1 | no | — |
| 89 | Dic18 | C2×A4 | 1 | 3 | yes | — |
| 90 | Q8⋊C9 | — | 2 | 1 | no | — |
| 91 | C2×H(5,𝔽2) | — | 9 | 1 | no | Sylow |
| 92 | C2×C3×C9 | — | 4 | 1 | no | — |
| 93 | C3×D9 | — | 4 | 2 | no | — |
| 94 | C23×D3 | — | 9 | 2 | no | — |
| 95 | C3×#41 | — | 3 | 1 | no | — |
| 96 | C3×#41 | C2×C3×D3 | 1 | 3 | yes | — |
| 97 | C2×C3×Q8 | — | 36 | 1 | no | — |
| 98 | C2×C3×D4 | — | 3 | 3 | no | — |
| 99 | C2×#67 | — | 9 | 6 | no | — |
| 100 | #239 | — | 3 | 3 | no | — |
| 101 | #239 | — | 9 | 1 | no | — |
| 102 | D3×Q8 | — | 3 | 1 | no | — |
| 103 | D3×Q8 | — | 3 | 1 | no | — |
| 104 | #237 | — | 9 | 3 | no | — |
| 105 | #237 | — | 9 | 3 | no | — |
| 106 | D3×D4 | — | 9 | 3 | no | — |
| 107 | D3×D4 | — | 9 | 3 | no | — |
| 108 | D3×D4 | — | 9 | 6 | no | — |
| 109 | D3×D4 | — | 9 | 6 | no | — |
| 110 | #235 | — | 9 | 3 | no | — |
| 111 | #235 | — | 9 | 3 | no | — |
| 112 | C2×D12 | — | 9 | 3 | no | — |
| 113 | C2×C4×D3 | — | 9 | 3 | no | — |
| 114 | #41⋊C3 | — | 2 | 3 | no | — |
| 115 | #41⋊C3 | — | 9 | 3 | no | — |
| 116 | #41⋊C3 | C2×D9 | 1 | 3 | yes | complemented |
| 117 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 118 | C2×C3×D3 | — | 12 | 1 | no | — |
| 119 | C32×C4 | — | 4 | 3 | no | — |
| 120 | C3×Dic3 | — | 12 | 3 | no | — |
| 121 | C22×C9 | — | 3 | 3 | no | — |
| 122 | C22×C9 | — | 12 | 2 | no | — |
| 123 | C2×D9 | — | 3 | 3 | no | — |
| 124 | C2×D9 | — | 6 | 6 | no | — |
| 125 | C2×D9 | — | 12 | 6 | no | — |
| 126 | C2×D9 | C22×A4 | 1 | 1 | yes | characteristic |
| 127 | C4×C9 | — | 3 | 1 | no | — |
| 128 | C4×C9 | — | 8 | 3 | no | — |
| 129 | C4×C9 | C22×A4 | 1 | 3 | yes | — |
| 130 | Dic9 | — | 3 | 1 | no | — |
| 131 | Dic9 | C22×A4 | 1 | 3 | yes | — |
| 132 | H(5,𝔽2) | — | 9 | 3 | no | — |
| 133 | H(5,𝔽2) | — | 27 | 3 | no | — |
| 134 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 135 | H(5,𝔽2) | C3×D9 | 1 | 1 | yes | characteristic, complemented |
| 136 | C2×#41 | — | 9 | 3 | no | — |
| 137 | C2×#41 | — | 27 | 1 | no | — |
| 138 | C22×D4 | — | 27 | 3 | no | — |
| 139 | C3×C9 | — | 4 | 1 | no | Sylow |
| 140 | C23×C3 | — | 3 | 2 | no | — |
| 141 | C22×D3 | — | 9 | 3 | no | — |
| 142 | C22×D3 | — | 9 | 6 | no | — |
| 143 | C22×D3 | — | 36 | 4 | no | — |
| 144 | C3×Q8 | — | 4 | 1 | no | — |
| 145 | C3×Q8 | — | 8 | 1 | no | — |
| 146 | C3×Q8 | — | 36 | 3 | no | — |
| 147 | C3×Q8 | D3×A4 | 1 | 1 | yes | characteristic |
| 148 | C3×Q8 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 149 | C3×D4 | — | 3 | 3 | no | — |
| 150 | C3×D4 | — | 3 | 3 | no | — |
| 151 | C2×C3×C4 | — | 3 | 3 | no | — |
| 152 | C2×C3×C4 | — | 36 | 3 | no | — |
| 153 | #67 | — | 9 | 6 | no | — |
| 154 | #67 | — | 9 | 6 | no | — |
| 155 | C2×Dic3 | — | 9 | 3 | no | — |
| 156 | D12 | — | 9 | 3 | no | — |
| 157 | D12 | — | 9 | 3 | no | — |
| 158 | C4×D3 | — | 3 | 3 | no | — |
| 159 | C4×D3 | — | 9 | 3 | no | — |
| 160 | C4×D3 | — | 9 | 1 | no | — |
| 161 | C4×D3 | — | 9 | 3 | no | — |
| 162 | Dic6 | — | 3 | 3 | no | — |
| 163 | Dic6 | — | 9 | 1 | no | — |
| 164 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 165 | SL(2,𝔽3) | C22×D9 | 1 | 1 | yes | characteristic, complemented |
| 166 | C2×C32 | — | 4 | 1 | no | — |
| 167 | C3×D3 | — | 12 | 2 | no | — |
| 168 | C2×C9 | — | 6 | 3 | no | — |
| 169 | C2×C9 | — | 8 | 1 | no | — |
| 170 | C2×C9 | C23×A4 | 1 | 1 | yes | characteristic |
| 171 | D9 | — | 6 | 3 | no | — |
| 172 | D9 | #994 | 1 | 2 | yes | direct product factor |
| 173 | C24 | — | 27 | 2 | no | — |
| 174 | #41 | — | 3 | 1 | no | — |
| 175 | #41 | — | 9 | 3 | no | — |
| 176 | #41 | — | 27 | 3 | no | — |
| 177 | #41 | — | 27 | 3 | no | — |
| 178 | #41 | — | 27 | 3 | no | — |
| 179 | #41 | — | 27 | 1 | no | — |
| 180 | #41 | — | 27 | 3 | no | — |
| 181 | #41 | C2×C3×D9 | 1 | 3 | yes | — |
| 182 | C2×Q8 | — | 9 | 1 | no | — |
| 183 | C2×Q8 | — | 9 | 1 | no | — |
| 184 | C2×D4 | — | 3 | 3 | no | — |
| 185 | C2×D4 | — | 27 | 3 | no | — |
| 186 | C2×D4 | — | 27 | 3 | no | — |
| 187 | C2×D4 | — | 27 | 3 | no | — |
| 188 | C2×D4 | — | 27 | 6 | no | — |
| 189 | C2×D4 | — | 27 | 6 | no | — |
| 190 | C2×D4 | — | 27 | 6 | no | — |
| 191 | C22×C4 | — | 27 | 3 | no | — |
| 192 | C22×C3 | — | 3 | 3 | no | — |
| 193 | C22×C3 | — | 12 | 2 | no | — |
| 194 | C22×C3 | — | 36 | 1 | no | — |
| 195 | C2×D3 | — | 3 | 1 | no | — |
| 196 | C2×D3 | — | 9 | 3 | no | — |
| 197 | C2×D3 | — | 18 | 6 | no | — |
| 198 | C2×D3 | — | 36 | 6 | no | — |
| 199 | C3×C4 | — | 3 | 1 | no | — |
| 200 | C3×C4 | — | 4 | 3 | no | — |
| 201 | C3×C4 | — | 8 | 3 | no | — |
| 202 | C3×C4 | — | 36 | 3 | no | — |
| 203 | C3×C4 | C2×D3×A4 | 1 | 3 | yes | — |
| 204 | Dic3 | — | 3 | 3 | no | — |
| 205 | Dic3 | — | 9 | 1 | no | — |
| 206 | C32 | — | 4 | 1 | no | — |
| 207 | C9 | — | 8 | 1 | no | — |
| 208 | C9 | C2×#994 | 1 | 1 | yes | characteristic, complemented |
| 209 | C23 | — | 3 | 2 | no | — |
| 210 | C23 | — | 27 | 3 | no | — |
| 211 | C23 | — | 27 | 6 | no | — |
| 212 | C23 | — | 108 | 4 | no | — |
| 213 | Q8 | — | 9 | 3 | no | — |
| 214 | Q8 | — | 27 | 1 | no | — |
| 215 | Q8 | A4×D9 | 1 | 1 | yes | characteristic |
| 216 | Q8 | C22×C3×D9 | 1 | 1 | yes | characteristic |
| 217 | D4 | — | 3 | 3 | no | — |
| 218 | D4 | — | 3 | 3 | no | — |
| 219 | D4 | — | 27 | 3 | no | — |
| 220 | D4 | — | 27 | 6 | no | — |
| 221 | D4 | — | 27 | 6 | no | — |
| 222 | D4 | — | 27 | 3 | no | — |
| 223 | C2×C4 | — | 3 | 3 | no | — |
| 224 | C2×C4 | — | 9 | 3 | no | — |
| 225 | C2×C4 | — | 27 | 3 | no | — |
| 226 | C2×C4 | — | 27 | 1 | no | — |
| 227 | C2×C4 | — | 27 | 3 | no | — |
| 228 | C2×C4 | — | 27 | 3 | no | — |
| 229 | C2×C3 | — | 4 | 1 | no | — |
| 230 | C2×C3 | — | 6 | 3 | no | — |
| 231 | C2×C3 | — | 8 | 1 | no | — |
| 232 | C2×C3 | — | 36 | 2 | no | — |
| 233 | C2×C3 | C22×D3×A4 | 1 | 1 | yes | characteristic |
| 234 | D3 | — | 3 | 2 | no | — |
| 235 | D3 | — | 18 | 3 | no | — |
| 236 | C22 | — | 3 | 3 | no | — |
| 237 | C22 | — | 9 | 1 | no | — |
| 238 | C22 | — | 12 | 2 | no | — |
| 239 | C22 | — | 27 | 3 | no | — |
| 240 | C22 | — | 54 | 6 | no | — |
| 241 | C22 | — | 108 | 6 | no | — |
| 242 | C4 | — | 3 | 1 | no | — |
| 243 | C4 | — | 9 | 3 | no | — |
| 244 | C4 | — | 27 | 1 | no | — |
| 245 | C4 | C2×A4×D9 | 1 | 3 | yes | — |
| 246 | C3 | — | 4 | 1 | no | — |
| 247 | C3 | — | 8 | 1 | no | — |
| 248 | C3 | D3×#994 | 1 | 1 | yes | characteristic |
| 249 | C2 | — | 6 | 3 | no | — |
| 250 | C2 | — | 9 | 2 | no | — |
| 251 | C2 | — | 54 | 3 | no | — |
| 252 | C2 | C22×A4×D9 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×A4×D9 |
| commutator, cosocle, hypocenter | Q8×C9 | C23×C3 |
| Frattini, socle | C2×C3 | C22×D3×A4 |
| Fitting | C9×H(5,𝔽2) | C2×C3 |
| hypercenter | Q8 | A4×D9 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#994 | C2 | semidirect |
| 2. D9×#41⋊C3 | C2 | semidirect |
| 3. #11722412 | C2 | semidirect |
| 5. D9×H(5,𝔽2) | C3 | semidirect |
| 7. C9×#41⋊C3 | C22 | semidirect |
| 8. D9×SL(2,𝔽3) | C22 | semidirect |
| 9. #88615 | C22 | semidirect |
| 13. C9×H(5,𝔽2) | C2×C3 | semidirect |
| 15. #63715 | C2×C3 | semidirect |
| 24. C9×SL(2,𝔽3) | C23 | semidirect |
| 63. #994 | D9 | direct (N × H) |
| 116. #41⋊C3 | C2×D9 | semidirect |
| 135. H(5,𝔽2) | C3×D9 | semidirect |
| 165. SL(2,𝔽3) | C22×D9 | semidirect |
| 172. D9 | #994 | direct (N × H) |
| 208. C9 | C2×#994 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#994 | 2 |
| D9×#41⋊C3 | 2 |
| #11722412 | 2 |
| D3×#994 | 3 |
| D9×H(5,𝔽2) | 3 |
| C3×Q8×D9 | 4 |