Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
527 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 128 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 | #64297 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #64116 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#909 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×#3718 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #63822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #63811 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #63809 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #63802 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #5779 | — | 3 | 2 | no | maximal |
| 10 | #3828 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#235 | C22 | 1 | 2 | yes | — |
| 12 | C2×C3×Dic6 | C22 | 1 | 2 | yes | — |
| 13 | #3804 | — | 2 | 4 | no | — |
| 14 | #3799 | — | 2 | 2 | no | — |
| 15 | #3797 | C22 | 1 | 4 | yes | — |
| 16 | #3796 | C22 | 1 | 4 | yes | — |
| 17 | D3×Dic6 | — | 2 | 2 | no | — |
| 18 | C32×M16 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C3⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 20 | #3743 | C22 | 1 | 2 | yes | — |
| 21 | C8×C3⋊D3 | C22 | 1 | 2 | yes | — |
| 22 | C3×Dic12 | C22 | 1 | 4 | yes | — |
| 23 | C3×#204 | C22 | 1 | 4 | yes | — |
| 24 | #3718 | C22 | 1 | 1 | yes | characteristic |
| 25 | #3718 | C22 | 1 | 2 | yes | — |
| 26 | #3718 | C22 | 1 | 2 | yes | — |
| 27 | #3715 | C22 | 1 | 2 | yes | — |
| 28 | #3713 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1010 | — | 3 | 2 | no | — |
| 30 | #1007 | — | 3 | 2 | no | — |
| 31 | C3×#137 | — | 3 | 2 | no | — |
| 32 | #950 | — | 3 | 2 | no | — |
| 33 | C2×#216 | — | 3 | 2 | no | — |
| 34 | #918 | — | 3 | 4 | no | — |
| 35 | D3×Q16 | — | 3 | 4 | no | — |
| 36 | #916 | — | 3 | 4 | no | — |
| 37 | #915 | — | 3 | 4 | no | — |
| 38 | #909 | D3 | 1 | 2 | yes | complemented |
| 39 | #907 | — | 3 | 2 | no | — |
| 40 | #907 | — | 6 | 1 | no | — |
| 41 | C2×C32×C4 | C23 | 1 | 1 | yes | characteristic |
| 42 | #638 | D4 | 1 | 2 | yes | — |
| 43 | C4⋊(C3⋊D3) | D4 | 1 | 1 | yes | characteristic |
| 44 | C4×C3⋊D3 | C23 | 1 | 2 | yes | — |
| 45 | #634 | D4 | 1 | 1 | yes | characteristic |
| 46 | C3×#67 | — | 2 | 2 | no | — |
| 47 | C2×C3×Dic3 | — | 2 | 2 | no | — |
| 48 | C3×D12 | C23 | 1 | 2 | yes | — |
| 49 | C3×C4×D3 | — | 2 | 2 | no | — |
| 50 | C3×Dic6 | — | 2 | 2 | no | — |
| 51 | C3×Dic6 | C23 | 1 | 2 | yes | — |
| 52 | C3×Dic6 | C23 | 1 | 4 | yes | — |
| 53 | #627 | — | 2 | 4 | no | — |
| 54 | #626 | — | 2 | 4 | no | — |
| 55 | #624 | — | 2 | 4 | no | — |
| 56 | D3×Dic3 | — | 2 | 4 | no | — |
| 57 | C32×C8 | C23 | 1 | 2 | yes | — |
| 58 | C3⋊(C3⋊C8) | C23 | 1 | 1 | yes | characteristic |
| 59 | C3⋊(C3⋊C8) | C23 | 1 | 1 | yes | characteristic |
| 60 | #578 | — | 9 | 1 | no | Sylow |
| 61 | C3×#41 | — | 3 | 2 | no | — |
| 62 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 63 | #239 | — | 3 | 4 | no | — |
| 64 | #239 | — | 6 | 2 | no | — |
| 65 | D3×Q8 | — | 3 | 4 | no | — |
| 66 | D3×Q8 | — | 3 | 4 | no | — |
| 67 | D3×Q8 | — | 6 | 2 | no | — |
| 68 | #237 | — | 3 | 4 | no | — |
| 69 | #237 | — | 6 | 4 | no | — |
| 70 | #235 | — | 3 | 2 | no | — |
| 71 | #235 | — | 6 | 1 | no | — |
| 72 | #235 | — | 6 | 2 | no | — |
| 73 | #235 | — | 6 | 4 | no | — |
| 74 | #235 | C2×D3 | 1 | 2 | yes | — |
| 75 | C2×Dic6 | — | 6 | 2 | no | — |
| 76 | C2×Dic6 | C2×D3 | 1 | 2 | yes | — |
| 77 | C3×Q16 | — | 3 | 4 | no | — |
| 78 | C3×SD16 | — | 3 | 4 | no | — |
| 79 | C3×M16 | — | 2 | 1 | no | — |
| 80 | C3×M16 | C2×D3 | 1 | 2 | yes | — |
| 81 | #216 | — | 3 | 4 | no | — |
| 82 | #216 | — | 3 | 4 | no | — |
| 83 | #216 | — | 3 | 2 | no | — |
| 84 | #215 | — | 3 | 2 | no | — |
| 85 | #214 | — | 3 | 2 | no | — |
| 86 | #213 | — | 3 | 2 | no | — |
| 87 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 88 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 89 | Dic12 | C2×D3 | 1 | 4 | yes | — |
| 90 | #204 | C2×D3 | 1 | 4 | yes | — |
| 91 | ΓL(1,𝔽25) | — | 3 | 4 | no | — |
| 92 | ΓL(1,𝔽25) | — | 6 | 2 | no | — |
| 93 | D3×C8 | — | 3 | 4 | no | — |
| 94 | D3×C8 | — | 6 | 2 | no | — |
| 95 | C22×C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C3⋊D3 | C2×D4 | 1 | 2 | yes | — |
| 97 | C2×C3×D3 | — | 2 | 2 | no | — |
| 98 | C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C32×C4 | C24 | 1 | 1 | yes | characteristic |
| 100 | C3⋊Dic3 | C2×D4 | 1 | 2 | yes | — |
| 101 | C3×Dic3 | — | 2 | 4 | no | — |
| 102 | C3×Dic3 | — | 2 | 2 | no | — |
| 103 | 2−1+4 | — | 9 | 2 | no | — |
| 104 | #137 | — | 3 | 2 | no | — |
| 105 | #137 | — | 9 | 4 | no | — |
| 106 | #135 | — | 9 | 2 | no | — |
| 107 | #135 | — | 9 | 1 | no | — |
| 108 | C2×Q16 | — | 9 | 2 | no | — |
| 109 | C2×Q16 | — | 9 | 1 | no | — |
| 110 | #131 | — | 9 | 1 | no | — |
| 111 | C3×Q8 | — | 3 | 2 | no | — |
| 112 | C3×Q8 | — | 3 | 4 | no | — |
| 113 | C3×Q8 | — | 6 | 2 | no | — |
| 114 | C3×D4 | — | 3 | 2 | no | — |
| 115 | C3×D4 | — | 6 | 2 | no | — |
| 116 | C2×C3×C4 | — | 2 | 1 | no | — |
| 117 | C2×C3×C4 | — | 6 | 2 | no | — |
| 118 | C2×C3×C4 | — | 6 | 2 | no | — |
| 119 | C2×C3×C4 | C22×D3 | 1 | 2 | yes | — |
| 120 | #67 | — | 2 | 2 | no | — |
| 121 | #67 | — | 3 | 4 | no | — |
| 122 | #67 | — | 6 | 2 | no | — |
| 123 | #67 | — | 6 | 4 | no | — |
| 124 | C2×Dic3 | — | 2 | 2 | no | — |
| 125 | C2×Dic3 | — | 6 | 4 | no | — |
| 126 | D12 | — | 3 | 2 | no | — |
| 127 | D12 | — | 6 | 1 | no | — |
| 128 | D12 | — | 6 | 4 | no | — |
| 129 | D12 | C22×D3 | 1 | 2 | yes | — |
| 130 | C4×D3 | — | 2 | 2 | no | — |
| 131 | C4×D3 | — | 3 | 4 | no | — |
| 132 | C4×D3 | — | 6 | 2 | no | — |
| 133 | C4×D3 | — | 6 | 4 | no | — |
| 134 | C4×D3 | — | 6 | 4 | no | — |
| 135 | C4×D3 | — | 6 | 4 | no | — |
| 136 | Dic6 | — | 2 | 2 | no | — |
| 137 | Dic6 | — | 3 | 2 | no | — |
| 138 | Dic6 | — | 6 | 1 | no | — |
| 139 | Dic6 | — | 6 | 4 | no | — |
| 140 | Dic6 | — | 6 | 4 | no | — |
| 141 | Dic6 | C22×D3 | 1 | 4 | yes | — |
| 142 | Dic6 | C22×D3 | 1 | 2 | yes | — |
| 143 | C3×C8 | — | 2 | 2 | no | — |
| 144 | C3×C8 | C22×D3 | 1 | 4 | yes | — |
| 145 | C3⋊C8 | — | 3 | 2 | no | — |
| 146 | C3⋊C8 | — | 3 | 2 | no | — |
| 147 | C3⋊C8 | — | 6 | 1 | no | — |
| 148 | C3⋊C8 | — | 6 | 1 | no | — |
| 149 | C2×C32 | — | 2 | 1 | no | — |
| 150 | C2×C32 | C22×D4 | 1 | 1 | yes | characteristic |
| 151 | C3⋊D3 | — | 2 | 2 | no | — |
| 152 | C3×D3 | — | 4 | 2 | no | — |
| 153 | #41 | — | 3 | 2 | no | — |
| 154 | #41 | — | 9 | 1 | no | — |
| 155 | #41 | — | 9 | 4 | no | — |
| 156 | #41 | — | 18 | 4 | no | — |
| 157 | #41 | — | 18 | 2 | no | — |
| 158 | C2×Q8 | — | 3 | 2 | no | — |
| 159 | C2×Q8 | — | 9 | 4 | no | — |
| 160 | C2×Q8 | — | 18 | 2 | no | — |
| 161 | Q16 | — | 3 | 4 | no | — |
| 162 | Q16 | — | 9 | 2 | no | — |
| 163 | Q16 | — | 9 | 1 | no | — |
| 164 | Q16 | — | 9 | 2 | no | — |
| 165 | SD16 | — | 3 | 4 | no | — |
| 166 | SD16 | — | 9 | 2 | no | — |
| 167 | D8 | — | 9 | 1 | no | — |
| 168 | M16 | — | 9 | 2 | no | — |
| 169 | M16 | D32 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | — | 9 | 1 | no | — |
| 171 | C2×C8 | — | 9 | 2 | no | — |
| 172 | C22×C3 | — | 2 | 1 | no | — |
| 173 | C22×C3 | — | 6 | 2 | no | — |
| 174 | C22×C3 | D3×D4 | 1 | 2 | yes | — |
| 175 | C2×D3 | — | 2 | 2 | no | — |
| 176 | C2×D3 | — | 3 | 4 | no | — |
| 177 | C2×D3 | — | 6 | 2 | no | — |
| 178 | C3×C4 | — | 2 | 1 | no | — |
| 179 | C3×C4 | — | 2 | 1 | no | — |
| 180 | C3×C4 | — | 6 | 2 | no | — |
| 181 | C3×C4 | — | 6 | 4 | no | — |
| 182 | C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 183 | C3×C4 | C23×D3 | 1 | 2 | yes | — |
| 184 | Dic3 | — | 2 | 4 | no | — |
| 185 | Dic3 | — | 2 | 2 | no | — |
| 186 | Dic3 | — | 3 | 4 | no | — |
| 187 | Dic3 | — | 6 | 2 | no | — |
| 188 | C32 | #578 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 189 | Q8 | — | 3 | 4 | no | — |
| 190 | Q8 | — | 3 | 2 | no | — |
| 191 | Q8 | — | 6 | 2 | no | — |
| 192 | Q8 | — | 9 | 1 | no | — |
| 193 | Q8 | — | 18 | 4 | no | — |
| 194 | D4 | — | 3 | 2 | no | — |
| 195 | D4 | — | 6 | 2 | no | — |
| 196 | D4 | — | 9 | 2 | no | — |
| 197 | D4 | — | 9 | 1 | no | — |
| 198 | D4 | — | 18 | 4 | no | — |
| 199 | C2×C4 | — | 6 | 2 | no | — |
| 200 | C2×C4 | — | 6 | 2 | no | — |
| 201 | C2×C4 | — | 9 | 2 | no | — |
| 202 | C2×C4 | — | 18 | 4 | no | — |
| 203 | C2×C4 | — | 18 | 4 | no | — |
| 204 | C2×C4 | C2×D32 | 1 | 1 | yes | characteristic |
| 205 | C8 | — | 9 | 1 | no | — |
| 206 | C8 | — | 9 | 1 | no | — |
| 207 | C8 | C2×D32 | 1 | 2 | yes | — |
| 208 | C2×C3 | — | 2 | 2 | no | — |
| 209 | C2×C3 | — | 2 | 1 | no | — |
| 210 | C2×C3 | — | 4 | 1 | no | — |
| 211 | C2×C3 | — | 12 | 2 | no | — |
| 212 | C2×C3 | C2×D3×D4 | 1 | 2 | yes | — |
| 213 | D3 | — | 4 | 2 | no | — |
| 214 | D3 | — | 6 | 4 | no | — |
| 215 | D3 | — | 12 | 2 | no | — |
| 216 | C22 | — | 6 | 2 | no | — |
| 217 | C22 | — | 9 | 2 | no | — |
| 218 | C22 | #3802 | 1 | 1 | yes | characteristic |
| 219 | C4 | — | 6 | 2 | no | — |
| 220 | C4 | — | 6 | 4 | no | — |
| 221 | C4 | — | 9 | 2 | no | — |
| 222 | C4 | #3802 | 1 | 1 | yes | characteristic |
| 223 | C4 | C22×D32 | 1 | 1 | yes | characteristic |
| 224 | C3 | — | 2 | 1 | no | — |
| 225 | C3 | #5779 | 1 | 2 | yes | complemented |
| 226 | C2 | — | 2 | 1 | no | — |
| 227 | C2 | — | 12 | 2 | no | — |
| 228 | C2 | — | 18 | 2 | no | — |
| 229 | C2 | C2×#3802 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#3802 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C32×M16 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | M16 | D32 |
| hypocenter | C32 | #578 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #64297 | C2 | semidirect |
| 2. #64116 | C2 | semidirect |
| 3. C3×#909 | C2 | semidirect |
| 4. C2×#3718 | C2 | semidirect |
| 5. #63822 | C2 | semidirect |
| 6. #63811 | C2 | semidirect |
| 7. #63809 | C2 | semidirect |
| 8. #63802 | C2 | semidirect |
| 38. #909 | D3 | semidirect |
| 188. C32 | #578 | semidirect |
| 225. C3 | #5779 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64297 | 2 |
| #64116 | 2 |
| C3×#909 | 2 |
| C2×#3718 | 2 |
| #63822 | 2 |
| #63811 | 2 |
| #63809 | 2 |
| #63802 | 2 |
| #5779 | 3 |