Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1090 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 199 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×#64153 | C2 | 1 | 3 | yes | maximal |
| 2 | #11723666 | C2 | 1 | 6 | yes | maximal |
| 3 | #11723665 | C2 | 1 | 6 | yes | maximal |
| 4 | #10595421 | — | 3 | 3 | no | maximal |
| 5 | C2×C33×Q8 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×C3×#634 | C22 | 1 | 3 | yes | — |
| 7 | C2×#6296 | C22 | 1 | 2 | yes | — |
| 8 | C2×#6296 | C22 | 1 | 2 | yes | — |
| 9 | C3×#3750 | C22 | 1 | 12 | yes | — |
| 10 | C3×C4×C3⋊Dic3 | C22 | 1 | 3 | yes | — |
| 11 | #88815 | C22 | 1 | 6 | yes | — |
| 12 | #88813 | C22 | 1 | 6 | yes | — |
| 13 | #64153 | D3 | 1 | 3 | yes | — |
| 14 | C3×#944 | — | 3 | 6 | no | — |
| 15 | C3×#944 | — | 6 | 3 | no | — |
| 16 | #63840 | — | 3 | 6 | no | — |
| 17 | #63839 | — | 3 | 12 | no | — |
| 18 | #63838 | — | 3 | 6 | no | — |
| 19 | #63837 | — | 3 | 12 | no | — |
| 20 | C33×Q8 | — | 4 | 1 | no | — |
| 21 | C2×C33×C4 | C23 | 1 | 3 | yes | — |
| 22 | C2×C3×C3⋊Dic3 | C23 | 1 | 6 | yes | — |
| 23 | C2×C3×C3⋊Dic3 | C23 | 1 | 6 | yes | — |
| 24 | C3×#634 | — | 4 | 3 | no | — |
| 25 | #6296 | — | 4 | 2 | no | — |
| 26 | #6296 | Q8 | 1 | 8 | yes | complemented |
| 27 | #5722 | — | 9 | 3 | no | — |
| 28 | C2×C32×Q8 | — | 2 | 3 | no | — |
| 29 | C2×C32×Q8 | — | 4 | 1 | no | — |
| 30 | C2×C32×Q8 | C2×D3 | 1 | 3 | yes | — |
| 31 | C2×#634 | C2×D3 | 1 | 3 | yes | — |
| 32 | C2×C3×Dic6 | — | 3 | 6 | no | — |
| 33 | C2×C3×Dic6 | — | 6 | 3 | no | — |
| 34 | C2×#627 | — | 3 | 6 | no | — |
| 35 | C2×#627 | — | 3 | 6 | no | — |
| 36 | #3750 | C2×D3 | 1 | 12 | yes | — |
| 37 | C4×C3⋊Dic3 | C2×D3 | 1 | 3 | yes | — |
| 38 | C3×Dic3⋊C4 | — | 3 | 12 | no | — |
| 39 | C3×Dic3⋊C4 | — | 3 | 12 | no | — |
| 40 | C3×Dic3⋊C4 | — | 6 | 12 | no | — |
| 41 | C3×C4×Dic3 | — | 3 | 6 | no | — |
| 42 | C3×C4×Dic3 | — | 6 | 3 | no | — |
| 43 | #3724 | — | 3 | 6 | no | — |
| 44 | Dic3⋊Dic3 | — | 3 | 12 | no | — |
| 45 | Dic32 | — | 3 | 6 | no | — |
| 46 | Dic32 | — | 3 | 12 | no | — |
| 47 | C22×C33 | C24 | 1 | 1 | yes | characteristic |
| 48 | C33×C4 | — | 2 | 3 | no | — |
| 49 | C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 50 | C3×C3⋊Dic3 | C2×Q8 | 1 | 12 | yes | — |
| 51 | C3×C4⋊Q8 | — | 9 | 3 | no | — |
| 52 | #944 | — | 3 | 6 | no | — |
| 53 | #944 | — | 6 | 3 | no | — |
| 54 | #944 | — | 9 | 6 | no | — |
| 55 | #888 | — | 9 | 12 | no | — |
| 56 | #887 | — | 9 | 12 | no | — |
| 57 | C4×Dic6 | — | 9 | 6 | no | — |
| 58 | C32×Q8 | — | 4 | 3 | no | — |
| 59 | C32×Q8 | — | 4 | 6 | no | — |
| 60 | C32×Q8 | — | 4 | 4 | no | — |
| 61 | C2×C32×C4 | — | 2 | 3 | no | — |
| 62 | C2×C32×C4 | — | 2 | 6 | no | — |
| 63 | C2×C32×C4 | — | 4 | 3 | no | — |
| 64 | C2×C32×C4 | C22×D3 | 1 | 6 | yes | — |
| 65 | C2×C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 66 | C2×C3⋊Dic3 | C22×D3 | 1 | 6 | yes | — |
| 67 | C2×C3⋊Dic3 | C22×D3 | 1 | 6 | yes | — |
| 68 | #634 | — | 4 | 3 | no | — |
| 69 | C2×C3×Dic3 | — | 3 | 12 | no | — |
| 70 | C2×C3×Dic3 | — | 3 | 12 | no | — |
| 71 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 72 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 73 | C3×Dic6 | — | 12 | 6 | no | — |
| 74 | C3×Dic6 | — | 12 | 6 | no | — |
| 75 | #627 | — | 3 | 24 | no | — |
| 76 | #627 | — | 12 | 6 | no | — |
| 77 | Q82 | — | 27 | 1 | no | Sylow |
| 78 | C2×C33 | C22×Q8 | 1 | 2 | yes | — |
| 79 | C2×C33 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 80 | C2×C3×Q8 | — | 2 | 3 | no | — |
| 81 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 82 | C2×C3×Q8 | — | 9 | 3 | no | — |
| 83 | C2×C3×Q8 | D32 | 1 | 3 | yes | — |
| 84 | C2×Dic6 | — | 3 | 6 | no | — |
| 85 | C2×Dic6 | — | 6 | 3 | no | — |
| 86 | C2×Dic6 | — | 9 | 6 | no | — |
| 87 | C2×Dic6 | — | 9 | 6 | no | — |
| 88 | C3×C4⋊C4 | — | 9 | 12 | no | — |
| 89 | C3×C42 | — | 9 | 3 | no | — |
| 90 | C4⋊Dic3 | — | 9 | 6 | no | — |
| 91 | Dic3⋊C4 | — | 3 | 12 | no | — |
| 92 | Dic3⋊C4 | — | 3 | 12 | no | — |
| 93 | Dic3⋊C4 | — | 6 | 12 | no | — |
| 94 | Dic3⋊C4 | — | 9 | 12 | no | — |
| 95 | C4×Dic3 | — | 3 | 6 | no | — |
| 96 | C4×Dic3 | — | 6 | 3 | no | — |
| 97 | C4×Dic3 | — | 9 | 12 | no | — |
| 98 | C4×Dic3 | — | 9 | 6 | no | — |
| 99 | C22×C32 | — | 2 | 3 | no | — |
| 100 | C22×C32 | — | 4 | 1 | no | — |
| 101 | C22×C32 | C23×D3 | 1 | 3 | yes | — |
| 102 | C32×C4 | — | 2 | 3 | no | — |
| 103 | C32×C4 | — | 2 | 6 | no | — |
| 104 | C32×C4 | — | 2 | 6 | no | — |
| 105 | C32×C4 | — | 4 | 6 | no | — |
| 106 | C32×C4 | — | 4 | 6 | no | — |
| 107 | C3⋊Dic3 | — | 2 | 6 | no | — |
| 108 | C3⋊Dic3 | D3×Q8 | 1 | 12 | yes | — |
| 109 | C3×Dic3 | — | 3 | 24 | no | — |
| 110 | C3×Dic3 | — | 6 | 12 | no | — |
| 111 | C3×Dic3 | — | 6 | 12 | no | — |
| 112 | C3×Dic3 | — | 12 | 6 | no | — |
| 113 | C4⋊Q8 | — | 9 | 3 | no | — |
| 114 | C4⋊Q8 | — | 27 | 6 | no | — |
| 115 | C4×Q8 | — | 27 | 6 | no | — |
| 116 | C33 | Q82 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 117 | C3×Q8 | — | 4 | 3 | no | — |
| 118 | C3×Q8 | — | 4 | 6 | no | — |
| 119 | C3×Q8 | — | 4 | 4 | no | — |
| 120 | C3×Q8 | — | 36 | 3 | no | — |
| 121 | C2×C3×C4 | — | 2 | 3 | no | — |
| 122 | C2×C3×C4 | — | 2 | 6 | no | — |
| 123 | C2×C3×C4 | — | 4 | 3 | no | — |
| 124 | C2×C3×C4 | — | 9 | 6 | no | — |
| 125 | C2×C3×C4 | — | 9 | 6 | no | — |
| 126 | C2×C3×C4 | C2×D32 | 1 | 6 | yes | — |
| 127 | C2×C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 128 | C2×Dic3 | — | 3 | 12 | no | — |
| 129 | C2×Dic3 | — | 3 | 12 | no | — |
| 130 | C2×Dic3 | — | 6 | 6 | no | — |
| 131 | C2×Dic3 | — | 6 | 6 | no | — |
| 132 | Dic6 | — | 9 | 24 | no | — |
| 133 | Dic6 | — | 12 | 6 | no | — |
| 134 | Dic6 | — | 12 | 6 | no | — |
| 135 | Dic6 | — | 36 | 6 | no | — |
| 136 | C2×C32 | — | 2 | 6 | no | — |
| 137 | C2×C32 | — | 2 | 3 | no | — |
| 138 | C2×C32 | — | 4 | 2 | no | — |
| 139 | C2×C32 | — | 4 | 1 | no | — |
| 140 | C2×C32 | #1004 | 1 | 3 | yes | — |
| 141 | C2×C32 | C2×D3×Q8 | 1 | 6 | yes | — |
| 142 | C2×Q8 | — | 9 | 3 | no | — |
| 143 | C2×Q8 | — | 27 | 2 | no | — |
| 144 | C2×Q8 | — | 27 | 2 | no | — |
| 145 | C2×Q8 | #1113 | 1 | 1 | yes | characteristic |
| 146 | C4⋊C4 | — | 9 | 12 | no | — |
| 147 | C4⋊C4 | — | 27 | 6 | no | — |
| 148 | C42 | — | 9 | 3 | no | — |
| 149 | C42 | — | 27 | 6 | no | — |
| 150 | C22×C3 | — | 2 | 3 | no | — |
| 151 | C22×C3 | — | 4 | 1 | no | — |
| 152 | C22×C3 | C22×D32 | 1 | 3 | yes | — |
| 153 | C3×C4 | — | 2 | 6 | no | — |
| 154 | C3×C4 | — | 2 | 6 | no | — |
| 155 | C3×C4 | — | 2 | 3 | no | — |
| 156 | C3×C4 | — | 4 | 6 | no | — |
| 157 | C3×C4 | — | 4 | 6 | no | — |
| 158 | C3×C4 | — | 9 | 12 | no | — |
| 159 | C3×C4 | — | 18 | 6 | no | — |
| 160 | Dic3 | — | 3 | 24 | no | — |
| 161 | Dic3 | — | 6 | 12 | no | — |
| 162 | Dic3 | — | 6 | 12 | no | — |
| 163 | Dic3 | — | 12 | 6 | no | — |
| 164 | C32 | — | 2 | 3 | no | — |
| 165 | C32 | — | 4 | 1 | no | — |
| 166 | C32 | #5722 | 1 | 3 | yes | complemented |
| 167 | Q8 | — | 4 | 1 | no | — |
| 168 | Q8 | — | 27 | 8 | no | — |
| 169 | Q8 | — | 36 | 3 | no | — |
| 170 | Q8 | — | 108 | 2 | no | — |
| 171 | C2×C4 | — | 9 | 6 | no | — |
| 172 | C2×C4 | — | 9 | 6 | no | — |
| 173 | C2×C4 | C2×#1113 | 1 | 3 | yes | — |
| 174 | C2×C3 | — | 2 | 6 | no | — |
| 175 | C2×C3 | — | 2 | 3 | no | — |
| 176 | C2×C3 | — | 4 | 2 | no | — |
| 177 | C2×C3 | — | 4 | 1 | no | — |
| 178 | C2×C3 | D32×Q8 | 1 | 6 | yes | — |
| 179 | C2×C3 | #64330 | 1 | 3 | yes | — |
| 180 | C22 | C22×#1113 | 1 | 1 | yes | characteristic |
| 181 | C4 | — | 2 | 3 | no | — |
| 182 | C4 | — | 9 | 12 | no | — |
| 183 | C4 | — | 18 | 6 | no | — |
| 184 | C3 | — | 2 | 3 | no | — |
| 185 | C3 | — | 4 | 1 | no | — |
| 186 | C3 | #10595421 | 1 | 3 | yes | complemented |
| 187 | C2 | Q8×#1113 | 1 | 2 | yes | — |
| 188 | C2 | #11724633 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#1113 |
| commutator, socle, prime radical, cosocle | C22×C33 | C24 |
| Fitting | C2×C33×Q8 | C22 |
| hypercenter | C2×Q8 | #1113 |
| hypocenter | C33 | Q82 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 26. #6296 | Q8 | semidirect |
| 116. C33 | Q82 | semidirect |
| 166. C32 | #5722 | semidirect |
| 186. C3 | #10595421 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#64153 | 2 |
| #11723666 | 2 |
| #11723665 | 2 |
| #10595421 | 3 |