Finite Group DatabasebetaFinite Group DatabasebetaOrder 960.
338 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 338 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 | D5×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #91956 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D5×GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #91894 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | D5×AGL(1,ℤ/8) | — | 3 | 1 | no | maximal, Hall |
| 7 | C5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | D5×D12 | — | 4 | 1 | no | maximal |
| 9 | D5×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C5⋊(#41⋊C3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C5⋊GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | C5×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 13 | C2×#986 | — | 5 | 1 | no | maximal, Hall |
| 14 | D5×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×D4×D5 | — | 3 | 1 | no | — |
| 16 | C5×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 17 | #4095 | — | 3 | 1 | no | — |
| 18 | #4078 | — | 3 | 1 | no | — |
| 19 | #4060 | — | 3 | 2 | no | — |
| 20 | D5×SD16 | — | 3 | 2 | no | — |
| 21 | #4057 | — | 3 | 2 | no | — |
| 22 | D5×D8 | — | 3 | 2 | no | — |
| 23 | #4054 | — | 3 | 1 | no | — |
| 24 | D5×M16 | — | 3 | 1 | no | — |
| 25 | C2×D3×D5 | — | 4 | 2 | no | — |
| 26 | D60 | — | 4 | 1 | no | — |
| 27 | C5×D12 | — | 4 | 1 | no | — |
| 28 | C3×C4×D5 | — | 4 | 1 | no | — |
| 29 | C5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic, complemented |
| 30 | #1203 | — | 4 | 2 | no | — |
| 31 | C2×#41⋊C3 | — | 5 | 1 | no | — |
| 32 | #986 | — | 5 | 1 | no | — |
| 33 | #986 | — | 5 | 2 | no | — |
| 34 | #986 | D5 | 1 | 1 | yes | characteristic, direct product factor |
| 35 | C2×GL(2,𝔽3) | — | 5 | 2 | no | — |
| 36 | C23×D5 | — | 6 | 1 | no | — |
| 37 | C5×#41 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 38 | C2×C5×D4 | — | 3 | 1 | no | — |
| 39 | C2×C22⋊D5 | — | 6 | 1 | no | — |
| 40 | #713 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 41 | Q8×D5 | C2×D3 | 1 | 1 | yes | characteristic |
| 42 | #711 | — | 3 | 1 | no | — |
| 43 | D4×D5 | — | 3 | 1 | no | — |
| 44 | D4×D5 | — | 3 | 2 | no | — |
| 45 | D4×D5 | — | 3 | 2 | no | — |
| 46 | D4×D5 | — | 6 | 1 | no | — |
| 47 | D4×D5 | — | 6 | 1 | no | — |
| 48 | #709 | — | 3 | 1 | no | — |
| 49 | C2×D20 | — | 3 | 1 | no | — |
| 50 | C2×C4×D5 | — | 3 | 1 | no | — |
| 51 | C5×SD16 | — | 3 | 2 | no | — |
| 52 | C5×D8 | — | 3 | 2 | no | — |
| 53 | C5×M16 | — | 3 | 1 | no | — |
| 54 | #688 | — | 3 | 2 | no | — |
| 55 | #687 | — | 3 | 2 | no | — |
| 56 | #686 | — | 3 | 2 | no | — |
| 57 | #686 | — | 3 | 2 | no | — |
| 58 | #681 | — | 3 | 1 | no | — |
| 59 | D40 | — | 3 | 2 | no | — |
| 60 | #677 | — | 3 | 2 | no | — |
| 61 | #676 | — | 3 | 2 | no | — |
| 62 | C8×D5 | — | 3 | 2 | no | — |
| 63 | C2×AGL(1,ℤ/8) | — | 15 | 1 | no | Sylow |
| 64 | C2×D15 | — | 4 | 2 | no | — |
| 65 | C2×C5×D3 | — | 4 | 2 | no | — |
| 66 | C2×C3×D5 | — | 4 | 1 | no | — |
| 67 | D3×D5 | — | 8 | 4 | no | — |
| 68 | C3×C4×C5 | — | 4 | 1 | no | — |
| 69 | C3×Dic5 | — | 4 | 1 | no | — |
| 70 | C2×D12 | — | 20 | 1 | no | — |
| 71 | #41⋊C3 | — | 5 | 1 | no | — |
| 72 | #41⋊C3 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 73 | C2×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 74 | GL(2,𝔽3) | — | 5 | 2 | no | — |
| 75 | GL(2,𝔽3) | C2×D5 | 1 | 2 | yes | complemented |
| 76 | C23×C5 | — | 6 | 1 | no | — |
| 77 | C22×D5 | — | 3 | 1 | no | — |
| 78 | C22×D5 | — | 6 | 2 | no | — |
| 79 | C22×D5 | — | 6 | 1 | no | — |
| 80 | C22×D5 | — | 6 | 2 | no | — |
| 81 | C22×D5 | — | 12 | 4 | no | — |
| 82 | C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 83 | C5×D4 | — | 3 | 1 | no | — |
| 84 | C5×D4 | — | 3 | 2 | no | — |
| 85 | C5×D4 | — | 6 | 1 | no | — |
| 86 | C2×C4×C5 | — | 3 | 1 | no | — |
| 87 | C22⋊D5 | — | 6 | 1 | no | — |
| 88 | C22⋊D5 | — | 6 | 2 | no | — |
| 89 | C22⋊D5 | — | 6 | 2 | no | — |
| 90 | C2×Dic5 | — | 3 | 1 | no | — |
| 91 | D20 | — | 3 | 1 | no | — |
| 92 | D20 | — | 3 | 2 | no | — |
| 93 | D20 | — | 6 | 1 | no | — |
| 94 | C4×D5 | — | 3 | 1 | no | — |
| 95 | C4×D5 | — | 3 | 1 | no | — |
| 96 | C4×D5 | — | 3 | 1 | no | — |
| 97 | C4×D5 | S4 | 1 | 1 | yes | characteristic |
| 98 | Dic10 | — | 3 | 1 | no | — |
| 99 | C5×C8 | — | 3 | 2 | no | — |
| 100 | #168 | — | 3 | 2 | no | — |
| 101 | C2×#41 | — | 5 | 1 | no | — |
| 102 | C22×D4 | — | 15 | 1 | no | — |
| 103 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 104 | AGL(1,ℤ/8) | — | 15 | 1 | no | — |
| 105 | AGL(1,ℤ/8) | — | 15 | 2 | no | — |
| 106 | AGL(1,ℤ/8) | — | 15 | 2 | no | — |
| 107 | AGL(1,ℤ/8) | — | 15 | 1 | no | — |
| 108 | AGL(1,ℤ/8) | — | 15 | 1 | no | — |
| 109 | C2×SD16 | — | 15 | 2 | no | — |
| 110 | C2×D8 | — | 15 | 2 | no | — |
| 111 | C2×M16 | — | 15 | 1 | no | — |
| 112 | C2×C3×C5 | — | 4 | 1 | no | — |
| 113 | D15 | — | 8 | 2 | no | — |
| 114 | C3×D5 | — | 4 | 2 | no | — |
| 115 | C5×D3 | — | 8 | 2 | no | — |
| 116 | C22×D3 | — | 20 | 2 | no | — |
| 117 | C2×C3×C4 | — | 20 | 1 | no | — |
| 118 | D12 | — | 4 | 1 | no | — |
| 119 | D12 | — | 20 | 1 | no | — |
| 120 | D12 | — | 20 | 2 | no | — |
| 121 | SL(2,𝔽3) | C22×D5 | 1 | 1 | yes | characteristic, complemented |
| 122 | C22×C5 | — | 3 | 1 | no | — |
| 123 | C22×C5 | — | 6 | 2 | no | — |
| 124 | C22×C5 | — | 12 | 2 | no | — |
| 125 | C2×D5 | — | 3 | 1 | no | — |
| 126 | C2×D5 | — | 6 | 2 | no | — |
| 127 | C2×D5 | — | 6 | 2 | no | — |
| 128 | C2×D5 | — | 12 | 4 | no | — |
| 129 | C2×D5 | — | 12 | 2 | no | — |
| 130 | C2×D5 | — | 12 | 4 | no | — |
| 131 | C2×D5 | C2×S4 | 1 | 1 | yes | characteristic |
| 132 | C4×C5 | — | 3 | 1 | no | — |
| 133 | C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 134 | Dic5 | — | 3 | 1 | no | — |
| 135 | Dic5 | C2×S4 | 1 | 1 | yes | characteristic |
| 136 | C24 | — | 30 | 1 | no | — |
| 137 | #41 | — | 5 | 1 | no | — |
| 138 | #41 | — | 15 | 1 | no | — |
| 139 | #41 | — | 15 | 1 | no | — |
| 140 | #41 | D3×D5 | 1 | 1 | yes | characteristic, complemented |
| 141 | C2×Q8 | — | 5 | 1 | no | — |
| 142 | C2×D4 | — | 3 | 1 | no | — |
| 143 | C2×D4 | — | 15 | 1 | no | — |
| 144 | C2×D4 | — | 15 | 1 | no | — |
| 145 | C2×D4 | — | 15 | 2 | no | — |
| 146 | C2×D4 | — | 15 | 2 | no | — |
| 147 | C2×D4 | — | 30 | 1 | no | — |
| 148 | C2×D4 | — | 30 | 1 | no | — |
| 149 | C2×D4 | — | 30 | 1 | no | — |
| 150 | C22×C4 | — | 15 | 1 | no | — |
| 151 | SD16 | — | 3 | 2 | no | — |
| 152 | SD16 | — | 15 | 2 | no | — |
| 153 | SD16 | — | 15 | 2 | no | — |
| 154 | SD16 | — | 15 | 2 | no | — |
| 155 | D8 | — | 3 | 2 | no | — |
| 156 | D8 | — | 15 | 2 | no | — |
| 157 | D8 | — | 15 | 2 | no | — |
| 158 | D8 | — | 15 | 2 | no | — |
| 159 | M16 | — | 3 | 1 | no | — |
| 160 | M16 | — | 15 | 2 | no | — |
| 161 | M16 | — | 15 | 1 | no | — |
| 162 | C2×C8 | — | 15 | 2 | no | — |
| 163 | C3×C5 | — | 4 | 1 | no | Hall |
| 164 | C22×C3 | — | 20 | 1 | no | — |
| 165 | C2×D3 | — | 4 | 2 | no | — |
| 166 | C2×D3 | — | 20 | 2 | no | — |
| 167 | C2×D3 | — | 40 | 4 | no | — |
| 168 | C3×C4 | — | 4 | 1 | no | — |
| 169 | C3×C4 | — | 20 | 1 | no | — |
| 170 | C2×C5 | — | 6 | 1 | no | — |
| 171 | C2×C5 | — | 12 | 2 | no | — |
| 172 | C2×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 173 | D5 | — | 6 | 1 | no | — |
| 174 | D5 | — | 12 | 2 | no | — |
| 175 | D5 | #986 | 1 | 2 | yes | direct product factor |
| 176 | C23 | — | 6 | 1 | no | — |
| 177 | C23 | — | 15 | 1 | no | — |
| 178 | C23 | — | 30 | 1 | no | — |
| 179 | C23 | — | 30 | 2 | no | — |
| 180 | C23 | — | 30 | 2 | no | — |
| 181 | C23 | — | 60 | 4 | no | — |
| 182 | Q8 | — | 15 | 1 | no | — |
| 183 | Q8 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 184 | D4 | — | 3 | 1 | no | — |
| 185 | D4 | — | 3 | 2 | no | — |
| 186 | D4 | — | 6 | 1 | no | — |
| 187 | D4 | — | 15 | 1 | no | — |
| 188 | D4 | — | 15 | 2 | no | — |
| 189 | D4 | — | 30 | 2 | no | — |
| 190 | D4 | — | 30 | 1 | no | — |
| 191 | D4 | — | 30 | 2 | no | — |
| 192 | D4 | — | 30 | 1 | no | — |
| 193 | C2×C4 | — | 3 | 1 | no | — |
| 194 | C2×C4 | — | 5 | 1 | no | — |
| 195 | C2×C4 | — | 15 | 1 | no | — |
| 196 | C2×C4 | — | 15 | 1 | no | — |
| 197 | C2×C4 | — | 15 | 1 | no | — |
| 198 | C2×C4 | — | 15 | 1 | no | — |
| 199 | C8 | — | 3 | 2 | no | — |
| 200 | C8 | — | 15 | 2 | no | — |
| 201 | C2×C3 | — | 4 | 1 | no | — |
| 202 | C2×C3 | — | 20 | 2 | no | — |
| 203 | D3 | — | 8 | 2 | no | — |
| 204 | D3 | — | 40 | 2 | no | — |
| 205 | C5 | C2×#986 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 206 | C22 | — | 3 | 1 | no | — |
| 207 | C22 | — | 5 | 1 | no | — |
| 208 | C22 | — | 6 | 2 | no | — |
| 209 | C22 | — | 12 | 2 | no | — |
| 210 | C22 | — | 15 | 1 | no | — |
| 211 | C22 | — | 30 | 2 | no | — |
| 212 | C22 | — | 30 | 2 | no | — |
| 213 | C22 | — | 60 | 4 | no | — |
| 214 | C22 | — | 60 | 2 | no | — |
| 215 | C22 | — | 60 | 4 | no | — |
| 216 | C4 | — | 3 | 1 | no | — |
| 217 | C4 | — | 5 | 1 | no | — |
| 218 | C4 | — | 15 | 1 | no | — |
| 219 | C4 | D5×S4 | 1 | 1 | yes | characteristic |
| 220 | C3 | — | 4 | 1 | no | Sylow |
| 221 | C2 | — | 5 | 2 | no | — |
| 222 | C2 | — | 6 | 1 | no | — |
| 223 | C2 | — | 12 | 2 | no | — |
| 224 | C2 | — | 30 | 1 | no | — |
| 225 | C2 | — | 60 | 2 | no | — |
| 226 | C2 | C2×D5×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×S4 |
| commutator, cosocle, hypocenter | C5×SL(2,𝔽3) | C23 |
| Fitting | C5×#41 | C2×D3 |
| socle | C2×C5 | C22×S4 |
| hypercenter | C4 | D5×S4 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D5×#41⋊C3 | C2 | semidirect |
| 2. #91956 | C2 | semidirect |
| 3. C5×#986 | C2 | semidirect |
| 4. D5×GL(2,𝔽3) | C2 | semidirect |
| 5. #91894 | C2 | semidirect |
| 7. C5×#41⋊C3 | C22 | semidirect |
| 9. D5×SL(2,𝔽3) | C22 | semidirect |
| 10. C5⋊(#41⋊C3) | C22 | semidirect |
| 11. C5⋊GL(2,𝔽3) | C22 | semidirect |
| 12. C5×GL(2,𝔽3) | C22 | semidirect |
| 14. D5×#41 | D3 | semidirect |
| 29. C5×SL(2,𝔽3) | C23 | semidirect |
| 34. #986 | D5 | direct (N × H) |
| 37. C5×#41 | C2×D3 | semidirect |
| 40. #713 | C2×D3 | semidirect |
| 72. #41⋊C3 | C2×D5 | semidirect |
| 75. GL(2,𝔽3) | C2×D5 | semidirect |
| 121. SL(2,𝔽3) | C22×D5 | semidirect |
| 140. #41 | D3×D5 | semidirect |
| 175. D5 | #986 | direct (N × H) |
| 205. C5 | C2×#986 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×#41⋊C3 | 2 |
| #91956 | 2 |
| C5×#986 | 2 |
| D5×GL(2,𝔽3) | 2 |
| #91894 | 2 |
| D5×AGL(1,ℤ/8) | 3 |
| D5×D12 | 4 |
| C2×#986 | 5 |