Finite Group DatabasebetaFinite Group DatabasebetaOrder 2880.
384 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 34 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 | C3×C5×#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#5935 | — | 3 | 3 | no | maximal |
| 3 | C5×#5241 | — | 3 | 1 | no | maximal |
| 4 | C2×C5×C3⋊S4 | — | 4 | 1 | no | maximal |
| 5 | C5×C3⋊GL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 6 | C3⋊#5935 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 7 | C3×C5×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C5×#997 | D3 | 1 | 3 | yes | complemented |
| 9 | C5×#949 | — | 3 | 2 | no | — |
| 10 | C5×#940 | — | 3 | 1 | no | — |
| 11 | C5×#837 | — | 3 | 2 | no | — |
| 12 | C5×#833 | — | 3 | 1 | no | — |
| 13 | C2×C3×C5×A4 | — | 4 | 1 | no | — |
| 14 | C2×C3×C5×A4 | — | 8 | 1 | no | — |
| 15 | C5×C3⋊S4 | — | 4 | 2 | no | — |
| 16 | C3×C5×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 17 | C5×#453 | — | 9 | 1 | no | Hall |
| 18 | C3×#997 | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×C5×S4 | — | 12 | 3 | no | — |
| 20 | C3×C5×#41 | — | 3 | 2 | no | — |
| 21 | C2×C3×C5×D4 | — | 3 | 1 | no | — |
| 22 | C2×C3×C5×D4 | — | 6 | 1 | no | — |
| 23 | C2×C5×#67 | — | 6 | 1 | no | — |
| 24 | C2×C5×#67 | — | 6 | 1 | no | — |
| 25 | C2×C5×D12 | — | 3 | 1 | no | — |
| 26 | C5×GL(2,𝔽3) | — | 12 | 6 | no | — |
| 27 | C5×#215 | — | 6 | 2 | no | — |
| 28 | C5×#213 | — | 6 | 2 | no | — |
| 29 | C4×C5×Dic3 | — | 3 | 1 | no | — |
| 30 | C5×#208 | — | 3 | 2 | no | — |
| 31 | #5935 | — | 3 | 3 | no | — |
| 32 | #5241 | — | 3 | 1 | no | — |
| 33 | C2×C5×C3⋊D3 | — | 16 | 1 | no | — |
| 34 | C3×C5×A4 | — | 4 | 1 | no | — |
| 35 | C3×C5×A4 | — | 8 | 1 | no | — |
| 36 | C5×H(5,𝔽2) | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 37 | C5×AGL(1,ℤ/8) | — | 9 | 2 | no | — |
| 38 | C5×#127 | — | 9 | 1 | no | — |
| 39 | C5×#104 | — | 9 | 2 | no | — |
| 40 | C5×#100 | — | 9 | 1 | no | — |
| 41 | C2×C3⋊S4 | — | 4 | 1 | no | — |
| 42 | C3⋊GL(2,𝔽3) | — | 4 | 2 | no | — |
| 43 | C23×C3×C5 | — | 2 | 1 | no | — |
| 44 | C23×C3×C5 | — | 3 | 1 | no | — |
| 45 | C23×C3×C5 | S4 | 1 | 1 | yes | characteristic |
| 46 | C22×C5×D3 | — | 6 | 1 | no | — |
| 47 | C2×C5×A4 | — | 4 | 3 | no | — |
| 48 | C2×C5×A4 | — | 8 | 3 | no | — |
| 49 | C5×S4 | — | 12 | 6 | no | — |
| 50 | C3×C5×Q8 | S4 | 1 | 2 | yes | complemented |
| 51 | C3×C5×D4 | — | 3 | 2 | no | — |
| 52 | C3×C5×D4 | — | 6 | 2 | no | — |
| 53 | C2×C3×C4×C5 | — | 3 | 1 | no | — |
| 54 | C2×C3×C4×C5 | — | 6 | 1 | no | — |
| 55 | C5×#67 | — | 12 | 2 | no | — |
| 56 | C5×#67 | — | 12 | 2 | no | — |
| 57 | C2×C5×Dic3 | — | 6 | 1 | no | — |
| 58 | C5×D12 | — | 6 | 2 | no | — |
| 59 | C5×SL(2,𝔽3) | — | 4 | 6 | no | — |
| 60 | C5×C3⋊C8 | — | 6 | 2 | no | — |
| 61 | C3×H(5,𝔽2) | C5×D3 | 1 | 1 | yes | characteristic, complemented |
| 62 | #997 | C5×D3 | 1 | 3 | yes | complemented |
| 63 | #949 | — | 3 | 2 | no | — |
| 64 | #940 | — | 3 | 1 | no | — |
| 65 | #837 | — | 3 | 2 | no | — |
| 66 | #833 | — | 3 | 1 | no | — |
| 67 | C2×C32×C5 | — | 16 | 1 | no | — |
| 68 | C5×C3⋊D3 | — | 16 | 2 | no | — |
| 69 | C5×#41 | — | 3 | 2 | no | — |
| 70 | C2×C5×D4 | — | 3 | 1 | no | — |
| 71 | C2×C5×D4 | — | 6 | 1 | no | — |
| 72 | C2×C5×D4 | — | 9 | 1 | no | — |
| 73 | C2×C5×D4 | — | 18 | 1 | no | — |
| 74 | C2×C5×D4 | — | 18 | 1 | no | — |
| 75 | C5×SD16 | — | 18 | 2 | no | — |
| 76 | C5×D8 | — | 18 | 2 | no | — |
| 77 | C5×M16 | — | 9 | 2 | no | — |
| 78 | C42×C5 | — | 9 | 1 | no | — |
| 79 | C2×C3×A4 | — | 4 | 1 | no | — |
| 80 | C2×C3×A4 | — | 8 | 1 | no | — |
| 81 | C3⋊S4 | — | 4 | 2 | no | — |
| 82 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 83 | #453 | — | 9 | 1 | no | Sylow |
| 84 | C22×C3×C5 | — | 3 | 1 | no | — |
| 85 | C22×C3×C5 | — | 4 | 1 | no | — |
| 86 | C22×C3×C5 | — | 6 | 1 | no | — |
| 87 | C22×C3×C5 | — | 8 | 1 | no | — |
| 88 | C22×C3×C5 | — | 12 | 1 | no | — |
| 89 | C2×C5×D3 | — | 12 | 1 | no | — |
| 90 | C2×C5×D3 | — | 12 | 1 | no | — |
| 91 | C2×C5×D3 | — | 12 | 1 | no | — |
| 92 | C2×C5×D3 | — | 48 | 3 | no | — |
| 93 | C5×A4 | — | 4 | 3 | no | — |
| 94 | C5×A4 | — | 8 | 3 | no | — |
| 95 | C3×C4×C5 | — | 3 | 2 | no | — |
| 96 | C5×Dic3 | — | 6 | 2 | no | — |
| 97 | C2×S4 | — | 12 | 3 | no | — |
| 98 | C3×#41 | — | 3 | 2 | no | — |
| 99 | C2×C3×D4 | — | 3 | 1 | no | — |
| 100 | C2×C3×D4 | — | 6 | 1 | no | — |
| 101 | C2×#67 | — | 6 | 1 | no | — |
| 102 | C2×#67 | — | 6 | 1 | no | — |
| 103 | C2×D12 | — | 3 | 1 | no | — |
| 104 | GL(2,𝔽3) | — | 12 | 6 | no | — |
| 105 | #215 | — | 6 | 2 | no | — |
| 106 | #213 | — | 6 | 2 | no | — |
| 107 | C4×Dic3 | — | 3 | 1 | no | — |
| 108 | #208 | — | 3 | 2 | no | — |
| 109 | C32×C5 | — | 16 | 1 | no | Hall |
| 110 | C23×C5 | — | 2 | 1 | no | — |
| 111 | C23×C5 | — | 3 | 1 | no | — |
| 112 | C23×C5 | — | 18 | 1 | no | — |
| 113 | C23×C5 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 114 | C5×Q8 | C3⋊S4 | 1 | 2 | yes | complemented |
| 115 | C5×D4 | — | 3 | 2 | no | — |
| 116 | C5×D4 | — | 6 | 2 | no | — |
| 117 | C5×D4 | — | 18 | 2 | no | — |
| 118 | C5×D4 | — | 36 | 2 | no | — |
| 119 | C5×D4 | — | 36 | 2 | no | — |
| 120 | C2×C4×C5 | — | 3 | 1 | no | — |
| 121 | C2×C4×C5 | — | 6 | 1 | no | — |
| 122 | C2×C4×C5 | — | 18 | 1 | no | — |
| 123 | C5×C8 | — | 18 | 2 | no | — |
| 124 | C2×C3⋊D3 | — | 16 | 1 | no | — |
| 125 | C3×A4 | — | 4 | 1 | no | — |
| 126 | C3×A4 | — | 8 | 1 | no | — |
| 127 | H(5,𝔽2) | C5×C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 128 | AGL(1,ℤ/8) | — | 9 | 2 | no | — |
| 129 | #127 | — | 9 | 1 | no | — |
| 130 | #104 | — | 9 | 2 | no | — |
| 131 | #100 | — | 9 | 1 | no | — |
| 132 | C2×C3×C5 | — | 6 | 1 | no | — |
| 133 | C2×C3×C5 | — | 12 | 1 | no | — |
| 134 | C2×C3×C5 | — | 16 | 3 | no | — |
| 135 | C2×C3×C5 | #1020 | 1 | 1 | yes | characteristic |
| 136 | C5×D3 | — | 24 | 1 | no | — |
| 137 | C5×D3 | — | 48 | 6 | no | — |
| 138 | C23×C3 | — | 2 | 1 | no | — |
| 139 | C23×C3 | — | 3 | 1 | no | — |
| 140 | C23×C3 | C5×S4 | 1 | 1 | yes | characteristic |
| 141 | C22×D3 | — | 6 | 1 | no | — |
| 142 | C2×A4 | — | 4 | 3 | no | — |
| 143 | C2×A4 | — | 8 | 3 | no | — |
| 144 | S4 | — | 12 | 6 | no | — |
| 145 | C3×Q8 | C5×S4 | 1 | 2 | yes | complemented |
| 146 | C3×D4 | — | 3 | 2 | no | — |
| 147 | C3×D4 | — | 6 | 2 | no | — |
| 148 | C2×C3×C4 | — | 3 | 1 | no | — |
| 149 | C2×C3×C4 | — | 6 | 1 | no | — |
| 150 | #67 | — | 12 | 2 | no | — |
| 151 | #67 | — | 12 | 2 | no | — |
| 152 | C2×Dic3 | — | 6 | 1 | no | — |
| 153 | D12 | — | 6 | 2 | no | — |
| 154 | SL(2,𝔽3) | — | 4 | 6 | no | — |
| 155 | C3⋊C8 | — | 6 | 2 | no | — |
| 156 | C22×C5 | — | 3 | 1 | no | — |
| 157 | C22×C5 | — | 4 | 1 | no | — |
| 158 | C22×C5 | — | 6 | 1 | no | — |
| 159 | C22×C5 | — | 8 | 1 | no | — |
| 160 | C22×C5 | — | 12 | 1 | no | — |
| 161 | C22×C5 | — | 36 | 1 | no | — |
| 162 | C22×C5 | — | 36 | 1 | no | — |
| 163 | C22×C5 | — | 36 | 1 | no | — |
| 164 | C4×C5 | — | 3 | 2 | no | — |
| 165 | C4×C5 | — | 18 | 2 | no | — |
| 166 | C2×C32 | — | 16 | 1 | no | — |
| 167 | C3⋊D3 | — | 16 | 2 | no | — |
| 168 | #41 | — | 3 | 2 | no | — |
| 169 | C2×D4 | — | 3 | 1 | no | — |
| 170 | C2×D4 | — | 6 | 1 | no | — |
| 171 | C2×D4 | — | 9 | 1 | no | — |
| 172 | C2×D4 | — | 18 | 1 | no | — |
| 173 | C2×D4 | — | 18 | 1 | no | — |
| 174 | SD16 | — | 18 | 2 | no | — |
| 175 | D8 | — | 18 | 2 | no | — |
| 176 | M16 | — | 9 | 2 | no | — |
| 177 | C42 | — | 9 | 1 | no | — |
| 178 | C3×C5 | — | 16 | 3 | no | — |
| 179 | C3×C5 | #5935 | 1 | 1 | yes | characteristic, complemented |
| 180 | C22×C3 | — | 3 | 1 | no | — |
| 181 | C22×C3 | — | 4 | 1 | no | — |
| 182 | C22×C3 | — | 6 | 1 | no | — |
| 183 | C22×C3 | — | 8 | 1 | no | — |
| 184 | C22×C3 | — | 12 | 1 | no | — |
| 185 | C2×D3 | — | 12 | 1 | no | — |
| 186 | C2×D3 | — | 12 | 1 | no | — |
| 187 | C2×D3 | — | 12 | 1 | no | — |
| 188 | C2×D3 | — | 48 | 3 | no | — |
| 189 | A4 | — | 4 | 3 | no | — |
| 190 | A4 | — | 8 | 3 | no | — |
| 191 | C3×C4 | — | 3 | 2 | no | — |
| 192 | Dic3 | — | 6 | 2 | no | — |
| 193 | C2×C5 | — | 6 | 1 | no | — |
| 194 | C2×C5 | — | 12 | 1 | no | — |
| 195 | C2×C5 | — | 72 | 1 | no | — |
| 196 | C2×C5 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 197 | C32 | — | 16 | 1 | no | Sylow |
| 198 | C23 | — | 2 | 1 | no | — |
| 199 | C23 | — | 3 | 1 | no | — |
| 200 | C23 | — | 18 | 1 | no | — |
| 201 | C23 | C5×C3⋊S4 | 1 | 1 | yes | characteristic |
| 202 | Q8 | C5×C3⋊S4 | 1 | 2 | yes | complemented |
| 203 | D4 | — | 3 | 2 | no | — |
| 204 | D4 | — | 6 | 2 | no | — |
| 205 | D4 | — | 18 | 2 | no | — |
| 206 | D4 | — | 36 | 2 | no | — |
| 207 | D4 | — | 36 | 2 | no | — |
| 208 | C2×C4 | — | 3 | 1 | no | — |
| 209 | C2×C4 | — | 6 | 1 | no | — |
| 210 | C2×C4 | — | 18 | 1 | no | — |
| 211 | C8 | — | 18 | 2 | no | — |
| 212 | C2×C3 | — | 6 | 1 | no | — |
| 213 | C2×C3 | — | 12 | 1 | no | — |
| 214 | C2×C3 | — | 16 | 3 | no | — |
| 215 | C2×C3 | C5×#1020 | 1 | 1 | yes | characteristic |
| 216 | D3 | — | 24 | 1 | no | — |
| 217 | D3 | — | 48 | 6 | no | — |
| 218 | C5 | C3⋊#5935 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 219 | C22 | — | 3 | 1 | no | — |
| 220 | C22 | — | 4 | 1 | no | — |
| 221 | C22 | — | 6 | 1 | no | — |
| 222 | C22 | — | 8 | 1 | no | — |
| 223 | C22 | — | 12 | 1 | no | — |
| 224 | C22 | — | 36 | 1 | no | — |
| 225 | C22 | — | 36 | 1 | no | — |
| 226 | C22 | — | 36 | 1 | no | — |
| 227 | C4 | — | 3 | 2 | no | — |
| 228 | C4 | — | 18 | 2 | no | — |
| 229 | C3 | — | 16 | 3 | no | — |
| 230 | C3 | C5×#5935 | 1 | 1 | yes | characteristic, complemented |
| 231 | C2 | — | 6 | 1 | no | — |
| 232 | C2 | — | 12 | 1 | no | — |
| 233 | C2 | — | 72 | 1 | no | — |
| 234 | C2 | C5×C3⋊#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×#997 | C2×C5 |
| Frattini | C2 | C5×C3⋊#1020 |
| Fitting | C3×C5×H(5,𝔽2) | D3 |
| socle | C2×C3×C5 | #1020 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×#997 | C2 | semidirect |
| 6. C3⋊#5935 | C5 | direct (N × H) |
| 7. C3×C5×H(5,𝔽2) | D3 | semidirect |
| 8. C5×#997 | D3 | semidirect |
| 18. C3×#997 | C2×C5 | semidirect |
| 36. C5×H(5,𝔽2) | C3⋊D3 | semidirect |
| 50. C3×C5×Q8 | S4 | semidirect |
| 61. C3×H(5,𝔽2) | C5×D3 | semidirect |
| 62. #997 | C5×D3 | semidirect |
| 114. C5×Q8 | C3⋊S4 | semidirect |
| 127. H(5,𝔽2) | C5×C3⋊D3 | semidirect |
| 145. C3×Q8 | C5×S4 | semidirect |
| 179. C3×C5 | #5935 | semidirect |
| 202. Q8 | C5×C3⋊S4 | semidirect |
| 218. C5 | C3⋊#5935 | direct (N × H) |
| 230. C3 | C5×#5935 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×#997 | 2 |
| C5×#5935 | 3 |
| C5×#5241 | 3 |
| C2×C5×C3⋊S4 | 4 |
| C5×C3⋊GL(2,𝔽3) | 4 |
| C3⋊#5935 | 5 |