Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
293 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 39 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 | D3×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #64265 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#984 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D3×#226 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #64196 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#984 | — | 3 | 1 | no | maximal |
| 7 | D3×#137 | — | 3 | 1 | no | maximal |
| 8 | C3×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | D3×Dic6 | — | 4 | 1 | no | maximal |
| 10 | D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 11 | C3⋊(#41⋊C3) | C22 | 1 | 1 | yes | characteristic |
| 12 | C3⋊#226 | C22 | 1 | 2 | yes | complemented |
| 13 | C3×#226 | C22 | 1 | 2 | yes | complemented |
| 14 | D3×#41 | D3 | 1 | 1 | yes | characteristic |
| 15 | C2×D3×Q8 | — | 3 | 1 | no | — |
| 16 | C2×#41⋊C3 | — | 3 | 1 | no | — |
| 17 | #984 | — | 3 | 1 | no | — |
| 18 | #984 | — | 3 | 2 | no | — |
| 19 | #984 | — | 6 | 1 | no | — |
| 20 | #984 | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 21 | C2×#226 | — | 3 | 2 | no | — |
| 22 | C3×#137 | — | 3 | 1 | no | — |
| 23 | #951 | — | 3 | 1 | no | — |
| 24 | #942 | — | 3 | 1 | no | — |
| 25 | #918 | — | 3 | 2 | no | — |
| 26 | D3×Q16 | — | 3 | 2 | no | — |
| 27 | #915 | — | 3 | 2 | no | — |
| 28 | D3×SD16 | — | 3 | 2 | no | — |
| 29 | #909 | — | 3 | 1 | no | — |
| 30 | D3×M16 | — | 3 | 1 | no | — |
| 31 | #634 | — | 4 | 1 | no | — |
| 32 | C3×C4×D3 | — | 4 | 1 | no | — |
| 33 | C3×Dic6 | — | 4 | 1 | no | — |
| 34 | C3×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 35 | #627 | — | 4 | 2 | no | — |
| 36 | D3×Dic3 | — | 4 | 2 | no | — |
| 37 | C2×#137 | — | 9 | 1 | no | Sylow |
| 38 | C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 39 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 40 | #239 | C2×D3 | 1 | 1 | yes | characteristic |
| 41 | D3×Q8 | — | 3 | 2 | no | — |
| 42 | D3×Q8 | — | 3 | 2 | no | — |
| 43 | D3×Q8 | — | 6 | 1 | no | — |
| 44 | D3×Q8 | — | 6 | 1 | no | — |
| 45 | D3×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 46 | #237 | — | 3 | 1 | no | — |
| 47 | D3×D4 | — | 3 | 1 | no | — |
| 48 | #235 | — | 3 | 1 | no | — |
| 49 | C2×C4×D3 | — | 3 | 1 | no | — |
| 50 | C2×C4×D3 | — | 6 | 1 | no | — |
| 51 | C2×Dic6 | — | 3 | 1 | no | — |
| 52 | C2×Dic6 | — | 6 | 1 | no | — |
| 53 | C2×Dic6 | — | 12 | 1 | no | — |
| 54 | #41⋊C3 | — | 2 | 1 | no | — |
| 55 | #41⋊C3 | — | 3 | 1 | no | — |
| 56 | #41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | C2×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 58 | #226 | — | 3 | 2 | no | — |
| 59 | #226 | — | 6 | 2 | no | — |
| 60 | #226 | C2×D3 | 1 | 2 | yes | complemented |
| 61 | C3×Q16 | — | 3 | 2 | no | — |
| 62 | C3×SD16 | — | 3 | 2 | no | — |
| 63 | C3×M16 | — | 3 | 1 | no | — |
| 64 | #216 | — | 3 | 2 | no | — |
| 65 | #216 | — | 3 | 2 | no | — |
| 66 | #215 | — | 3 | 2 | no | — |
| 67 | #214 | — | 3 | 2 | no | — |
| 68 | #208 | — | 3 | 1 | no | — |
| 69 | Dic12 | — | 3 | 2 | no | — |
| 70 | #204 | — | 3 | 2 | no | — |
| 71 | ΓL(1,𝔽25) | — | 3 | 2 | no | — |
| 72 | D3×C8 | — | 3 | 2 | no | — |
| 73 | C2×C3×D3 | — | 4 | 1 | no | — |
| 74 | C32×C4 | — | 4 | 1 | no | — |
| 75 | C3⋊Dic3 | — | 4 | 2 | no | — |
| 76 | C3×Dic3 | — | 4 | 2 | no | — |
| 77 | C3×Dic3 | — | 4 | 1 | no | — |
| 78 | C2×#41 | — | 3 | 1 | no | — |
| 79 | C22×Q8 | — | 9 | 1 | no | — |
| 80 | #137 | — | 3 | 1 | no | — |
| 81 | #137 | — | 9 | 1 | no | — |
| 82 | #137 | — | 9 | 2 | no | — |
| 83 | #137 | — | 9 | 2 | no | — |
| 84 | #137 | — | 9 | 1 | no | — |
| 85 | #137 | — | 9 | 1 | no | — |
| 86 | C2×Q16 | — | 9 | 2 | no | — |
| 87 | C2×SD16 | — | 9 | 2 | no | — |
| 88 | C2×M16 | — | 9 | 1 | no | — |
| 89 | C22×D3 | — | 3 | 1 | no | — |
| 90 | C3×Q8 | — | 3 | 2 | no | — |
| 91 | C3×Q8 | — | 6 | 1 | no | — |
| 92 | C3×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | C3×D4 | — | 3 | 1 | no | — |
| 94 | C2×C3×C4 | — | 3 | 1 | no | — |
| 95 | C2×C3×C4 | — | 6 | 1 | no | — |
| 96 | C2×C3×C4 | — | 12 | 1 | no | — |
| 97 | #67 | — | 6 | 1 | no | — |
| 98 | C2×Dic3 | — | 3 | 1 | no | — |
| 99 | C2×Dic3 | — | 6 | 1 | no | — |
| 100 | C2×Dic3 | — | 12 | 2 | no | — |
| 101 | D12 | — | 3 | 1 | no | — |
| 102 | C4×D3 | — | 3 | 1 | no | — |
| 103 | C4×D3 | — | 3 | 1 | no | — |
| 104 | C4×D3 | — | 3 | 1 | no | — |
| 105 | C4×D3 | — | 6 | 2 | no | — |
| 106 | C4×D3 | — | 6 | 2 | no | — |
| 107 | C4×D3 | S4 | 1 | 1 | yes | characteristic |
| 108 | Dic6 | — | 3 | 1 | no | — |
| 109 | Dic6 | — | 3 | 2 | no | — |
| 110 | Dic6 | — | 4 | 1 | no | — |
| 111 | Dic6 | — | 6 | 2 | no | — |
| 112 | Dic6 | — | 6 | 1 | no | — |
| 113 | Dic6 | — | 6 | 2 | no | — |
| 114 | Dic6 | — | 12 | 2 | no | — |
| 115 | Dic6 | — | 12 | 1 | no | — |
| 116 | Dic6 | — | 24 | 1 | no | — |
| 117 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 118 | SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 119 | C3×C8 | — | 3 | 2 | no | — |
| 120 | C3⋊C8 | — | 3 | 2 | no | — |
| 121 | C2×C32 | — | 4 | 1 | no | — |
| 122 | C3×D3 | — | 4 | 2 | no | — |
| 123 | #41 | — | 3 | 1 | no | — |
| 124 | #41 | — | 9 | 1 | no | — |
| 125 | #41 | — | 9 | 1 | no | — |
| 126 | #41 | D32 | 1 | 1 | yes | characteristic |
| 127 | C2×Q8 | — | 3 | 1 | no | — |
| 128 | C2×Q8 | — | 3 | 1 | no | — |
| 129 | C2×Q8 | — | 9 | 1 | no | — |
| 130 | C2×Q8 | — | 9 | 2 | no | — |
| 131 | C2×Q8 | — | 9 | 2 | no | — |
| 132 | C2×Q8 | — | 18 | 1 | no | — |
| 133 | C2×Q8 | — | 18 | 1 | no | — |
| 134 | C2×Q8 | — | 18 | 1 | no | — |
| 135 | C2×D4 | — | 9 | 1 | no | — |
| 136 | C22×C4 | — | 9 | 1 | no | — |
| 137 | C22×C4 | — | 18 | 1 | no | — |
| 138 | Q16 | — | 3 | 2 | no | — |
| 139 | Q16 | — | 9 | 2 | no | — |
| 140 | Q16 | — | 9 | 2 | no | — |
| 141 | Q16 | — | 9 | 2 | no | — |
| 142 | SD16 | — | 3 | 2 | no | — |
| 143 | SD16 | — | 9 | 2 | no | — |
| 144 | SD16 | — | 9 | 2 | no | — |
| 145 | SD16 | — | 9 | 2 | no | — |
| 146 | M16 | — | 3 | 1 | no | — |
| 147 | M16 | — | 9 | 2 | no | — |
| 148 | M16 | — | 9 | 1 | no | — |
| 149 | C2×C8 | — | 9 | 2 | no | — |
| 150 | C22×C3 | — | 3 | 1 | no | — |
| 151 | C22×C3 | — | 12 | 1 | no | — |
| 152 | C2×D3 | — | 3 | 1 | no | — |
| 153 | C2×D3 | — | 6 | 2 | no | — |
| 154 | C2×D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 155 | C3×C4 | — | 3 | 1 | no | — |
| 156 | C3×C4 | — | 4 | 1 | no | — |
| 157 | C3×C4 | — | 6 | 2 | no | — |
| 158 | C3×C4 | — | 8 | 1 | no | — |
| 159 | C3×C4 | — | 12 | 1 | no | — |
| 160 | C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 161 | Dic3 | — | 3 | 1 | no | — |
| 162 | Dic3 | — | 4 | 2 | no | — |
| 163 | Dic3 | — | 6 | 2 | no | — |
| 164 | Dic3 | — | 12 | 2 | no | — |
| 165 | Dic3 | — | 24 | 2 | no | — |
| 166 | Dic3 | C2×S4 | 1 | 1 | yes | characteristic |
| 167 | C32 | — | 4 | 1 | no | Sylow |
| 168 | C23 | — | 9 | 1 | no | — |
| 169 | Q8 | — | 3 | 2 | no | — |
| 170 | Q8 | — | 6 | 1 | no | — |
| 171 | Q8 | — | 9 | 1 | no | — |
| 172 | Q8 | — | 9 | 2 | no | — |
| 173 | Q8 | — | 18 | 2 | no | — |
| 174 | Q8 | — | 18 | 2 | no | — |
| 175 | Q8 | — | 18 | 1 | no | — |
| 176 | Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 177 | D4 | — | 3 | 1 | no | — |
| 178 | D4 | — | 9 | 1 | no | — |
| 179 | D4 | — | 18 | 1 | no | — |
| 180 | C2×C4 | — | 3 | 1 | no | — |
| 181 | C2×C4 | — | 3 | 1 | no | — |
| 182 | C2×C4 | — | 6 | 1 | no | — |
| 183 | C2×C4 | — | 9 | 1 | no | — |
| 184 | C2×C4 | — | 9 | 1 | no | — |
| 185 | C2×C4 | — | 9 | 1 | no | — |
| 186 | C2×C4 | — | 9 | 1 | no | — |
| 187 | C2×C4 | — | 18 | 2 | no | — |
| 188 | C2×C4 | — | 18 | 2 | no | — |
| 189 | C2×C4 | — | 18 | 1 | no | — |
| 190 | C8 | — | 3 | 2 | no | — |
| 191 | C8 | — | 9 | 2 | no | — |
| 192 | C2×C3 | — | 4 | 1 | no | — |
| 193 | C2×C3 | — | 6 | 1 | no | — |
| 194 | C2×C3 | — | 8 | 1 | no | — |
| 195 | C2×C3 | — | 12 | 2 | no | — |
| 196 | C2×C3 | C22×S4 | 1 | 1 | yes | characteristic |
| 197 | D3 | — | 6 | 1 | no | — |
| 198 | D3 | #984 | 1 | 2 | yes | direct product factor |
| 199 | C22 | — | 3 | 1 | no | — |
| 200 | C22 | — | 3 | 1 | no | — |
| 201 | C22 | — | 9 | 1 | no | — |
| 202 | C22 | — | 18 | 2 | no | — |
| 203 | C4 | — | 3 | 1 | no | — |
| 204 | C4 | — | 3 | 1 | no | — |
| 205 | C4 | — | 6 | 2 | no | — |
| 206 | C4 | — | 9 | 1 | no | — |
| 207 | C4 | — | 18 | 2 | no | — |
| 208 | C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 209 | C3 | — | 4 | 1 | no | — |
| 210 | C3 | — | 8 | 1 | no | — |
| 211 | C3 | C2×#984 | 1 | 1 | yes | characteristic, complemented |
| 212 | C2 | — | 3 | 2 | no | — |
| 213 | C2 | — | 6 | 1 | no | — |
| 214 | C2 | — | 18 | 1 | no | — |
| 215 | C2 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C23 |
| Fitting | C3×#41 | C2×D3 |
| socle | C2×C3 | C22×S4 |
| prime radical | D3×#41⋊C3 | C2 |
| hypercenter | C4 | D3×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #64265 | C2 | semidirect |
| 3. C3×#984 | C2 | semidirect |
| 4. D3×#226 | C2 | semidirect |
| 5. #64196 | C2 | semidirect |
| 12. C3⋊#226 | C22 | semidirect |
| 13. C3×#226 | C22 | semidirect |
| 20. #984 | D3 | direct (N × H) |
| 60. #226 | C2×D3 | semidirect |
| 198. D3 | #984 | direct (N × H) |
| 211. C3 | C2×#984 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×#41⋊C3 | 2 |
| #64265 | 2 |
| C3×#984 | 2 |
| D3×#226 | 2 |
| #64196 | 2 |
| C2×#984 | 3 |
| D3×#137 | 3 |
| D3×Dic6 | 4 |