Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
804 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 97 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×#63700 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11722391 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11722390 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10596306 | — | 3 | 1 | no | maximal |
| 5 | #10593100 | — | 3 | 3 | no | maximal |
| 6 | #88746 | — | 4 | 1 | no | maximal |
| 7 | C3×#41⋊C9 | C22 | 1 | 3 | yes | complemented |
| 8 | C3⋊#3689 | C22 | 1 | 3 | yes | complemented |
| 9 | C3⋊#3688 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 11 | #64366 | — | 3 | 1 | no | — |
| 12 | #64159 | — | 3 | 3 | no | — |
| 13 | #64158 | — | 3 | 3 | no | — |
| 14 | #64157 | — | 3 | 1 | no | — |
| 15 | #64141 | — | 3 | 3 | no | — |
| 16 | C2×#3753 | — | 3 | 3 | no | — |
| 17 | #63700 | D3 | 1 | 3 | yes | complemented |
| 18 | #63692 | — | 3 | 9 | no | — |
| 19 | #63691 | — | 3 | 9 | no | — |
| 20 | C3×Q8×C9 | — | 4 | 1 | no | — |
| 21 | C4⋊(C3⋊D9) | — | 4 | 3 | no | — |
| 22 | C4×C3⋊D9 | — | 4 | 3 | no | — |
| 23 | C3×Q8⋊C9 | C23 | 1 | 1 | yes | characteristic |
| 24 | #5835 | — | 9 | 3 | no | — |
| 25 | #5835 | — | 9 | 1 | no | — |
| 26 | C32×#41 | — | 3 | 1 | no | — |
| 27 | C32×#41 | C2×D3 | 1 | 3 | yes | — |
| 28 | C2×C32×D4 | — | 3 | 3 | no | — |
| 29 | #3832 | — | 6 | 1 | no | — |
| 30 | D4×C3⋊D3 | — | 6 | 3 | no | — |
| 31 | #3828 | — | 3 | 3 | no | — |
| 32 | C2×C4⋊(C3⋊D3) | — | 3 | 3 | no | — |
| 33 | #3756 | — | 3 | 1 | no | — |
| 34 | #3755 | — | 3 | 3 | no | — |
| 35 | #3754 | — | 3 | 3 | no | — |
| 36 | #3753 | — | 3 | 3 | no | — |
| 37 | #3753 | — | 3 | 6 | no | — |
| 38 | #3748 | — | 3 | 3 | no | — |
| 39 | C2×C3⋊(C3⋊C8) | — | 3 | 3 | no | — |
| 40 | #3701 | — | 12 | 3 | no | — |
| 41 | #41⋊C9 | C2×D3 | 1 | 9 | yes | complemented |
| 42 | #3689 | — | 3 | 9 | no | — |
| 43 | #3688 | — | 3 | 3 | no | — |
| 44 | C2×C3⋊D9 | — | 4 | 3 | no | — |
| 45 | C3×C4×C9 | — | 4 | 3 | no | — |
| 46 | C3⋊Dic9 | — | 4 | 1 | no | — |
| 47 | C3×H(5,𝔽2) | D9 | 1 | 3 | yes | complemented |
| 48 | C3×H(5,𝔽2) | C3⋊D3 | 1 | 1 | yes | characteristic |
| 49 | #1009 | — | 9 | 3 | no | — |
| 50 | #1009 | — | 9 | 1 | no | — |
| 51 | #950 | — | 9 | 9 | no | — |
| 52 | #950 | — | 9 | 3 | no | — |
| 53 | #949 | — | 9 | 9 | no | — |
| 54 | #949 | — | 9 | 3 | no | — |
| 55 | #948 | — | 9 | 3 | no | — |
| 56 | #948 | — | 9 | 1 | no | — |
| 57 | #932 | — | 9 | 9 | no | — |
| 58 | #932 | — | 9 | 3 | no | — |
| 59 | C2×#213 | — | 9 | 9 | no | — |
| 60 | C2×#213 | — | 9 | 3 | no | — |
| 61 | C23×C32 | — | 6 | 1 | no | — |
| 62 | C22×C3⋊D3 | — | 6 | 3 | no | — |
| 63 | C32×Q8 | S4 | 1 | 1 | yes | characteristic |
| 64 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 65 | C32×D4 | — | 3 | 3 | no | — |
| 66 | C32×D4 | — | 3 | 3 | no | — |
| 67 | C2×C32×C4 | — | 3 | 3 | no | — |
| 68 | #638 | — | 6 | 3 | no | — |
| 69 | C4⋊(C3⋊D3) | — | 3 | 3 | no | — |
| 70 | C4⋊(C3⋊D3) | — | 6 | 3 | no | — |
| 71 | C4×C3⋊D3 | — | 6 | 3 | no | — |
| 72 | #634 | — | 3 | 1 | no | — |
| 73 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 74 | C3⋊(C3⋊C8) | — | 3 | 3 | no | — |
| 75 | Q8×C9 | — | 4 | 3 | no | — |
| 76 | D36 | — | 12 | 9 | no | — |
| 77 | C4×D9 | — | 12 | 9 | no | — |
| 78 | Q8⋊C9 | C22×D3 | 1 | 3 | yes | — |
| 79 | #576 | — | 27 | 1 | no | Sylow |
| 80 | C2×C3×C9 | — | 4 | 1 | no | — |
| 81 | C3⋊D9 | — | 8 | 3 | no | — |
| 82 | C3×#41 | — | 3 | 3 | no | — |
| 83 | C3×#41 | — | 3 | 1 | no | — |
| 84 | C3×#41 | C2×D9 | 1 | 9 | yes | — |
| 85 | C3×#41 | C2×C3⋊D3 | 1 | 3 | yes | — |
| 86 | C2×C3×D4 | — | 3 | 9 | no | — |
| 87 | C2×C3×D4 | — | 3 | 3 | no | — |
| 88 | #239 | — | 18 | 3 | no | — |
| 89 | #239 | — | 18 | 1 | no | — |
| 90 | D3×D4 | — | 18 | 3 | no | — |
| 91 | D3×D4 | — | 18 | 9 | no | — |
| 92 | #235 | — | 9 | 3 | no | — |
| 93 | #235 | — | 9 | 9 | no | — |
| 94 | C2×D12 | — | 9 | 3 | no | — |
| 95 | C2×D12 | — | 9 | 9 | no | — |
| 96 | #216 | — | 9 | 3 | no | — |
| 97 | #216 | — | 9 | 1 | no | — |
| 98 | #215 | — | 9 | 9 | no | — |
| 99 | #215 | — | 9 | 3 | no | — |
| 100 | #214 | — | 9 | 9 | no | — |
| 101 | #214 | — | 9 | 3 | no | — |
| 102 | #213 | — | 9 | 9 | no | — |
| 103 | #213 | — | 9 | 3 | no | — |
| 104 | #213 | — | 9 | 18 | no | — |
| 105 | #213 | — | 9 | 6 | no | — |
| 106 | #208 | — | 9 | 3 | no | — |
| 107 | #208 | — | 9 | 9 | no | — |
| 108 | C2×C3⋊C8 | — | 9 | 3 | no | — |
| 109 | C2×C3⋊C8 | — | 9 | 9 | no | — |
| 110 | C22×C32 | — | 3 | 3 | no | — |
| 111 | C22×C32 | — | 24 | 1 | no | — |
| 112 | C2×C3⋊D3 | — | 6 | 3 | no | — |
| 113 | C2×C3⋊D3 | — | 24 | 3 | no | — |
| 114 | C32×C4 | — | 3 | 1 | no | — |
| 115 | C32×C4 | C2×S4 | 1 | 3 | yes | — |
| 116 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 117 | C2×D9 | — | 12 | 9 | no | — |
| 118 | C4×C9 | — | 4 | 9 | no | — |
| 119 | Dic9 | — | 12 | 3 | no | — |
| 120 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 121 | H(5,𝔽2) | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 122 | AGL(1,ℤ/8) | — | 27 | 3 | no | — |
| 123 | AGL(1,ℤ/8) | — | 27 | 3 | no | — |
| 124 | #135 | — | 27 | 3 | no | — |
| 125 | C2×D8 | — | 27 | 3 | no | — |
| 126 | #131 | — | 27 | 1 | no | — |
| 127 | C3×C9 | — | 4 | 1 | no | Sylow |
| 128 | C23×C3 | — | 6 | 1 | no | — |
| 129 | C23×C3 | — | 6 | 3 | no | — |
| 130 | C22×D3 | — | 18 | 3 | no | — |
| 131 | C22×D3 | — | 18 | 9 | no | — |
| 132 | C3×Q8 | (C22⋊C9)⋊C2 | 1 | 3 | yes | — |
| 133 | C3×Q8 | C22×D9 | 1 | 3 | yes | — |
| 134 | C3×Q8 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 135 | C3×Q8 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 136 | C3×D4 | — | 3 | 9 | no | — |
| 137 | C3×D4 | — | 3 | 3 | no | — |
| 138 | C3×D4 | — | 3 | 3 | no | — |
| 139 | C3×D4 | — | 3 | 9 | no | — |
| 140 | C2×C3×C4 | — | 3 | 3 | no | — |
| 141 | C2×C3×C4 | — | 3 | 9 | no | — |
| 142 | #67 | — | 18 | 3 | no | — |
| 143 | #67 | — | 18 | 9 | no | — |
| 144 | D12 | — | 9 | 3 | no | — |
| 145 | D12 | — | 9 | 9 | no | — |
| 146 | D12 | — | 18 | 3 | no | — |
| 147 | D12 | — | 18 | 9 | no | — |
| 148 | C4×D3 | — | 18 | 3 | no | — |
| 149 | C4×D3 | — | 18 | 9 | no | — |
| 150 | Dic6 | — | 9 | 1 | no | — |
| 151 | Dic6 | — | 9 | 3 | no | — |
| 152 | C3⋊C8 | — | 9 | 1 | no | — |
| 153 | C3⋊C8 | — | 9 | 3 | no | — |
| 154 | C3⋊C8 | — | 9 | 3 | no | — |
| 155 | C3⋊C8 | — | 9 | 9 | no | — |
| 156 | C2×C32 | — | 6 | 3 | no | — |
| 157 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 158 | C3⋊D3 | — | 12 | 3 | no | — |
| 159 | C2×C9 | — | 4 | 3 | no | — |
| 160 | D9 | — | 24 | 9 | no | — |
| 161 | #41 | — | 3 | 1 | no | — |
| 162 | #41 | — | 27 | 3 | no | — |
| 163 | #41 | — | 54 | 1 | no | — |
| 164 | #41 | C2×C3⋊D9 | 1 | 3 | yes | — |
| 165 | C2×D4 | — | 3 | 3 | no | — |
| 166 | C2×D4 | — | 27 | 3 | no | — |
| 167 | C2×D4 | — | 54 | 3 | no | — |
| 168 | Q16 | — | 27 | 1 | no | — |
| 169 | SD16 | — | 27 | 3 | no | — |
| 170 | SD16 | — | 27 | 3 | no | — |
| 171 | D8 | — | 27 | 3 | no | — |
| 172 | D8 | — | 27 | 6 | no | — |
| 173 | M16 | — | 27 | 3 | no | — |
| 174 | C2×C8 | — | 27 | 3 | no | — |
| 175 | C22×C3 | — | 3 | 3 | no | — |
| 176 | C22×C3 | — | 3 | 9 | no | — |
| 177 | C22×C3 | — | 24 | 3 | no | — |
| 178 | C22×C3 | — | 24 | 1 | no | — |
| 179 | C2×D3 | — | 18 | 3 | no | — |
| 180 | C2×D3 | — | 18 | 9 | no | — |
| 181 | C2×D3 | — | 72 | 9 | no | — |
| 182 | C2×D3 | — | 72 | 3 | no | — |
| 183 | C3×C4 | — | 3 | 1 | no | — |
| 184 | C3×C4 | — | 3 | 3 | no | — |
| 185 | C3×C4 | C2×(C22⋊C9)⋊C2 | 1 | 9 | yes | — |
| 186 | C3×C4 | C2×C3⋊S4 | 1 | 3 | yes | — |
| 187 | Dic3 | — | 18 | 1 | no | — |
| 188 | Dic3 | — | 18 | 3 | no | — |
| 189 | C32 | #5925 | 1 | 1 | yes | characteristic |
| 190 | C9 | — | 4 | 3 | no | — |
| 191 | C23 | — | 6 | 1 | no | — |
| 192 | C23 | — | 54 | 3 | no | — |
| 193 | Q8 | — | 27 | 1 | no | — |
| 194 | Q8 | C3⋊((C22⋊C9)⋊C2) | 1 | 1 | yes | characteristic |
| 195 | Q8 | C22×C3⋊D9 | 1 | 1 | yes | characteristic |
| 196 | D4 | — | 3 | 3 | no | — |
| 197 | D4 | — | 3 | 3 | no | — |
| 198 | D4 | — | 27 | 3 | no | — |
| 199 | D4 | — | 54 | 3 | no | — |
| 200 | D4 | — | 54 | 3 | no | — |
| 201 | C2×C4 | — | 3 | 3 | no | — |
| 202 | C2×C4 | — | 54 | 3 | no | — |
| 203 | C8 | — | 27 | 1 | no | — |
| 204 | C8 | — | 27 | 3 | no | — |
| 205 | C2×C3 | — | 6 | 9 | no | — |
| 206 | C2×C3 | — | 6 | 3 | no | — |
| 207 | C2×C3 | C22×(C22⋊C9)⋊C2 | 1 | 3 | yes | — |
| 208 | C2×C3 | C22×C3⋊S4 | 1 | 1 | yes | characteristic |
| 209 | D3 | — | 36 | 9 | no | — |
| 210 | D3 | — | 36 | 3 | no | — |
| 211 | C22 | — | 3 | 3 | no | — |
| 212 | C22 | — | 24 | 1 | no | — |
| 213 | C22 | — | 54 | 3 | no | — |
| 214 | C22 | — | 216 | 3 | no | — |
| 215 | C4 | — | 3 | 1 | no | — |
| 216 | C4 | — | 54 | 1 | no | — |
| 217 | C4 | C2×C3⋊((C22⋊C9)⋊C2) | 1 | 3 | yes | — |
| 218 | C3 | #10593100 | 1 | 3 | yes | complemented |
| 219 | C3 | #10596588 | 1 | 1 | yes | characteristic |
| 220 | C2 | — | 6 | 3 | no | — |
| 221 | C2 | — | 108 | 3 | no | — |
| 222 | C2 | C22×C3⋊((C22⋊C9)⋊C2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | Q8 | C3⋊((C22⋊C9)⋊C2) |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63700 | C2 | semidirect |
| 2. #11722391 | C2 | semidirect |
| 3. #11722390 | C2 | semidirect |
| 7. C3×#41⋊C9 | C22 | semidirect |
| 8. C3⋊#3689 | C22 | semidirect |
| 9. C3⋊#3688 | C22 | semidirect |
| 17. #63700 | D3 | semidirect |
| 41. #41⋊C9 | C2×D3 | semidirect |
| 47. C3×H(5,𝔽2) | D9 | semidirect |
| 121. H(5,𝔽2) | C3⋊D9 | semidirect |
| 218. C3 | #10593100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63700 | 2 |
| #11722391 | 2 |
| #11722390 | 2 |
| #10596306 | 3 |
| #10593100 | 3 |
| #88746 | 4 |