Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
346 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 49 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 | C22×#625 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#3792 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #63957 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #63739 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C23×C3×D3 | — | 2 | 1 | no | — |
| 6 | C22×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#625 | — | 2 | 1 | no | — |
| 8 | C2×#625 | — | 2 | 2 | no | — |
| 9 | C2×#625 | C4 | 1 | 4 | yes | complemented |
| 10 | C2×#625 | C22 | 1 | 2 | yes | — |
| 11 | #3792 | — | 2 | 2 | no | — |
| 12 | C2×#622 | C22 | 1 | 2 | yes | complemented |
| 13 | C3×#217 | — | 2 | 1 | no | — |
| 14 | #3724 | C22 | 1 | 2 | yes | complemented |
| 15 | Dic32 | — | 2 | 1 | no | — |
| 16 | C22×#67 | — | 6 | 1 | no | — |
| 17 | #877 | — | 6 | 1 | no | — |
| 18 | C23×C32 | D4 | 1 | 1 | yes | characteristic |
| 19 | C22×C3×D3 | — | 2 | 4 | no | — |
| 20 | C22×C3×D3 | — | 2 | 2 | no | — |
| 21 | C22×C3×D3 | — | 4 | 4 | no | — |
| 22 | C2×C3⋊Dic3 | C2×C4 | 1 | 2 | yes | — |
| 23 | C2×C3⋊Dic3 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×C3⋊Dic3 | D4 | 1 | 2 | yes | — |
| 25 | C2×C3⋊Dic3 | C23 | 1 | 1 | yes | characteristic |
| 26 | C2×C3×Dic3 | — | 2 | 2 | no | — |
| 27 | #625 | — | 2 | 4 | no | — |
| 28 | #625 | — | 2 | 2 | no | — |
| 29 | #625 | C2×C4 | 1 | 4 | yes | — |
| 30 | #622 | — | 2 | 2 | no | — |
| 31 | #418 | — | 9 | 1 | no | maximal, Sylow |
| 32 | C24×C3 | — | 6 | 1 | no | — |
| 33 | C23×D3 | — | 2 | 1 | no | — |
| 34 | C2×#67 | — | 6 | 4 | no | — |
| 35 | C2×#67 | — | 6 | 2 | no | — |
| 36 | C2×#67 | — | 6 | 2 | no | — |
| 37 | C2×#67 | — | 6 | 4 | no | — |
| 38 | C22×Dic3 | — | 6 | 1 | no | — |
| 39 | C22×Dic3 | — | 6 | 1 | no | — |
| 40 | C3×C22⋊C4 | — | 6 | 1 | no | — |
| 41 | #217 | — | 2 | 1 | no | — |
| 42 | Dic3⋊C4 | — | 6 | 2 | no | — |
| 43 | C4×Dic3 | — | 6 | 2 | no | — |
| 44 | C22×C32 | — | 2 | 2 | no | — |
| 45 | C22×C32 | C22⋊C4 | 1 | 2 | yes | — |
| 46 | C22×C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 47 | C2×C3×D3 | — | 2 | 4 | no | — |
| 48 | C2×C3×D3 | — | 4 | 8 | no | — |
| 49 | C2×C3×D3 | — | 4 | 4 | no | — |
| 50 | C3⋊Dic3 | C22⋊C4 | 1 | 2 | yes | — |
| 51 | C3⋊Dic3 | C22×C4 | 1 | 1 | yes | characteristic |
| 52 | C3⋊Dic3 | C2×D4 | 1 | 1 | yes | characteristic |
| 53 | C3×Dic3 | — | 4 | 2 | no | — |
| 54 | C22×D4 | — | 9 | 1 | no | — |
| 55 | C2×M16 | — | 9 | 1 | no | — |
| 56 | #117 | — | 9 | 1 | no | — |
| 57 | #102 | — | 9 | 4 | no | — |
| 58 | C23×C3 | — | 2 | 1 | no | — |
| 59 | C23×C3 | — | 2 | 1 | no | — |
| 60 | C23×C3 | — | 6 | 2 | no | — |
| 61 | C23×C3 | — | 6 | 4 | no | — |
| 62 | C23×C3 | — | 12 | 4 | no | — |
| 63 | C22×D3 | — | 2 | 2 | no | — |
| 64 | C22×D3 | — | 2 | 4 | no | — |
| 65 | C22×D3 | — | 4 | 4 | no | — |
| 66 | C2×C3×C4 | — | 6 | 2 | no | — |
| 67 | #67 | — | 6 | 4 | no | — |
| 68 | #67 | — | 6 | 8 | no | — |
| 69 | #67 | — | 6 | 4 | no | — |
| 70 | C2×Dic3 | — | 2 | 2 | no | — |
| 71 | C2×Dic3 | — | 6 | 2 | no | — |
| 72 | C2×Dic3 | — | 6 | 2 | no | — |
| 73 | C2×Dic3 | — | 6 | 1 | no | — |
| 74 | C2×Dic3 | — | 6 | 2 | no | — |
| 75 | C2×Dic3 | — | 6 | 1 | no | — |
| 76 | C2×Dic3 | — | 6 | 2 | no | — |
| 77 | C2×Dic3 | — | 6 | 1 | no | — |
| 78 | C2×Dic3 | — | 6 | 1 | no | — |
| 79 | C2×C32 | — | 2 | 2 | no | — |
| 80 | C2×C32 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 81 | C2×C32 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 82 | C3×D3 | — | 4 | 4 | no | — |
| 83 | C24 | — | 6 | 1 | no | — |
| 84 | C2×D4 | — | 9 | 2 | no | — |
| 85 | C2×D4 | — | 9 | 4 | no | — |
| 86 | C2×D4 | — | 18 | 1 | no | — |
| 87 | C2×D4 | — | 18 | 2 | no | — |
| 88 | C22×C4 | — | 9 | 1 | no | — |
| 89 | M16 | — | 18 | 2 | no | — |
| 90 | C2×C8 | — | 9 | 2 | no | — |
| 91 | C4⋊C4 | — | 9 | 2 | no | — |
| 92 | C22⋊C4 | — | 6 | 1 | no | — |
| 93 | C42 | — | 18 | 1 | no | — |
| 94 | C22×C3 | — | 2 | 4 | no | — |
| 95 | C22×C3 | — | 2 | 1 | no | — |
| 96 | C22×C3 | — | 2 | 1 | no | — |
| 97 | C22×C3 | — | 2 | 2 | no | — |
| 98 | C22×C3 | — | 2 | 2 | no | — |
| 99 | C22×C3 | — | 4 | 2 | no | — |
| 100 | C22×C3 | — | 6 | 4 | no | — |
| 101 | C22×C3 | — | 12 | 8 | no | — |
| 102 | C22×C3 | — | 12 | 4 | no | — |
| 103 | C2×D3 | — | 2 | 4 | no | — |
| 104 | C2×D3 | — | 4 | 8 | no | — |
| 105 | C2×D3 | — | 4 | 4 | no | — |
| 106 | C3×C4 | — | 12 | 2 | no | — |
| 107 | Dic3 | — | 4 | 2 | no | — |
| 108 | Dic3 | — | 6 | 2 | no | — |
| 109 | Dic3 | — | 6 | 1 | no | — |
| 110 | Dic3 | — | 6 | 1 | no | — |
| 111 | Dic3 | — | 6 | 2 | no | — |
| 112 | Dic3 | — | 6 | 1 | no | — |
| 113 | Dic3 | — | 6 | 1 | no | — |
| 114 | C32 | #418 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 115 | C23 | — | 6 | 2 | no | — |
| 116 | C23 | — | 6 | 4 | no | — |
| 117 | C23 | — | 12 | 4 | no | — |
| 118 | C23 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 119 | D4 | — | 9 | 4 | no | — |
| 120 | D4 | — | 18 | 2 | no | — |
| 121 | D4 | — | 18 | 4 | no | — |
| 122 | C2×C4 | — | 6 | 2 | no | — |
| 123 | C2×C4 | — | 9 | 1 | no | — |
| 124 | C2×C4 | — | 9 | 1 | no | — |
| 125 | C2×C4 | — | 9 | 2 | no | — |
| 126 | C2×C4 | — | 9 | 2 | no | — |
| 127 | C8 | — | 18 | 2 | no | — |
| 128 | C2×C3 | — | 2 | 4 | no | — |
| 129 | C2×C3 | — | 2 | 1 | no | — |
| 130 | C2×C3 | — | 2 | 1 | no | — |
| 131 | C2×C3 | — | 2 | 2 | no | — |
| 132 | C2×C3 | — | 2 | 2 | no | — |
| 133 | C2×C3 | — | 4 | 2 | no | — |
| 134 | C2×C3 | — | 12 | 4 | no | — |
| 135 | D3 | — | 4 | 4 | no | — |
| 136 | C22 | — | 2 | 2 | no | — |
| 137 | C22 | — | 6 | 4 | no | — |
| 138 | C22 | — | 12 | 8 | no | — |
| 139 | C22 | — | 12 | 4 | no | — |
| 140 | C22 | D32⋊C4 | 1 | 2 | yes | — |
| 141 | C22 | C2×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 142 | C4 | — | 9 | 1 | no | — |
| 143 | C4 | — | 9 | 1 | no | — |
| 144 | C4 | — | 9 | 2 | no | — |
| 145 | C4 | — | 12 | 2 | no | — |
| 146 | C3 | — | 2 | 1 | no | — |
| 147 | C3 | — | 2 | 1 | no | — |
| 148 | C2 | — | 2 | 2 | no | — |
| 149 | C2 | — | 12 | 4 | no | — |
| 150 | C2 | C2×D32⋊C4 | 1 | 1 | yes | characteristic |
| 151 | C2 | #64236 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#3792 | C2 | semidirect |
| 3. #63957 | C2 | semidirect |
| 4. #63739 | C2 | semidirect |
| 9. C2×#625 | C4 | semidirect |
| 12. C2×#622 | C22 | semidirect |
| 14. #3724 | C22 | semidirect |
| 114. C32 | #418 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#625 | 2 |
| C2×#3792 | 2 |
| #63957 | 2 |
| #63739 | 2 |
| #418 | 9 |