Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
473 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 46 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 | Q8⋊(Q8⋊C9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#3483 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#41⋊C9 | — | 4 | 3 | no | maximal |
| 4 | C4×Q8⋊C9 | — | 4 | 2 | no | maximal |
| 5 | C3×Q82 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | C3×#555 | — | 3 | 3 | no | — |
| 7 | C3×#550 | — | 3 | 2 | no | — |
| 8 | C3×#532 | — | 3 | 3 | no | — |
| 9 | C3×C4×#41 | — | 3 | 2 | no | — |
| 10 | #41⋊C9 | — | 4 | 6 | no | — |
| 11 | C2×Q8⋊C9 | — | 4 | 2 | no | — |
| 12 | C2×Q8⋊C9 | — | 4 | 3 | no | — |
| 13 | #3483 | C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2×C3×#41 | — | 3 | 1 | no | — |
| 15 | C2×C3×#41 | — | 3 | 2 | no | — |
| 16 | C2×C3×#41 | A4 | 1 | 3 | yes | — |
| 17 | C3×C4⋊Q8 | — | 3 | 3 | no | — |
| 18 | C3×#127 | — | 3 | 3 | no | — |
| 19 | C3×#124 | — | 3 | 3 | no | — |
| 20 | C3×#124 | — | 3 | 3 | no | — |
| 21 | C3×#121 | — | 3 | 12 | no | — |
| 22 | C3×C4×Q8 | — | 3 | 2 | no | — |
| 23 | C3×C4×Q8 | A4 | 1 | 2 | yes | — |
| 24 | C3×C4×D4 | — | 3 | 6 | no | — |
| 25 | C3×C4×D4 | — | 3 | 6 | no | — |
| 26 | C3×#117 | — | 3 | 6 | no | — |
| 27 | C2×C3×C42 | — | 3 | 3 | no | — |
| 28 | C2×C4×C9 | — | 16 | 1 | no | — |
| 29 | Q8⋊C9 | — | 4 | 3 | no | — |
| 30 | Q8⋊C9 | — | 4 | 2 | no | — |
| 31 | Q82 | C2×C9 | 1 | 1 | yes | characteristic |
| 32 | #555 | — | 3 | 3 | no | — |
| 33 | #550 | — | 3 | 2 | no | — |
| 34 | #532 | — | 3 | 3 | no | — |
| 35 | C4×#41 | — | 3 | 2 | no | — |
| 36 | C3×#41 | — | 4 | 6 | no | — |
| 37 | C3×#41 | — | 6 | 6 | no | — |
| 38 | C3×#41 | — | 6 | 2 | no | — |
| 39 | C3×#41 | — | 12 | 2 | no | — |
| 40 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 41 | C2×C3×Q8 | C2×A4 | 1 | 2 | yes | — |
| 42 | C2×C3×Q8 | C2×A4 | 1 | 3 | yes | — |
| 43 | C2×C3×D4 | — | 3 | 6 | no | — |
| 44 | C2×C3×D4 | — | 3 | 3 | no | — |
| 45 | C2×C3×D4 | — | 3 | 3 | no | — |
| 46 | C22×C3×C4 | — | 3 | 6 | no | — |
| 47 | C22×C3×C4 | — | 3 | 3 | no | — |
| 48 | C3×C4⋊C4 | — | 3 | 2 | no | — |
| 49 | C3×C4⋊C4 | — | 3 | 6 | no | — |
| 50 | C3×C22⋊C4 | — | 3 | 12 | no | — |
| 51 | C3×C42 | — | 3 | 3 | no | — |
| 52 | C3×C42 | — | 3 | 3 | no | — |
| 53 | C3×C42 | — | 3 | 2 | no | — |
| 54 | C22×C9 | — | 16 | 1 | no | — |
| 55 | C4×C9 | — | 16 | 2 | no | — |
| 56 | C2×#41 | — | 3 | 1 | no | — |
| 57 | C2×#41 | — | 3 | 2 | no | — |
| 58 | C2×#41 | C22⋊C9 | 1 | 3 | yes | — |
| 59 | C4⋊Q8 | — | 3 | 3 | no | — |
| 60 | #127 | — | 3 | 3 | no | — |
| 61 | #124 | — | 3 | 3 | no | — |
| 62 | #124 | — | 3 | 3 | no | — |
| 63 | #121 | — | 3 | 12 | no | — |
| 64 | C4×Q8 | — | 3 | 2 | no | — |
| 65 | C4×Q8 | C22⋊C9 | 1 | 2 | yes | — |
| 66 | C4×D4 | — | 3 | 6 | no | — |
| 67 | C4×D4 | — | 3 | 6 | no | — |
| 68 | #117 | — | 3 | 6 | no | — |
| 69 | C2×C42 | — | 3 | 3 | no | — |
| 70 | C23×C3 | — | 3 | 3 | no | — |
| 71 | C3×Q8 | — | 3 | 2 | no | — |
| 72 | C3×Q8 | — | 4 | 3 | no | — |
| 73 | C3×Q8 | — | 12 | 1 | no | — |
| 74 | C3×Q8 | #41⋊C3 | 1 | 2 | yes | — |
| 75 | C3×D4 | — | 6 | 6 | no | — |
| 76 | C3×D4 | — | 6 | 6 | no | — |
| 77 | C3×D4 | — | 12 | 3 | no | — |
| 78 | C3×D4 | — | 12 | 3 | no | — |
| 79 | C2×C3×C4 | — | 3 | 2 | no | — |
| 80 | C2×C3×C4 | — | 3 | 2 | no | — |
| 81 | C2×C3×C4 | — | 3 | 3 | no | — |
| 82 | C2×C3×C4 | — | 6 | 6 | no | — |
| 83 | C2×C3×C4 | — | 6 | 6 | no | — |
| 84 | C2×C3×C4 | — | 6 | 6 | no | — |
| 85 | C2×C3×C4 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 86 | C2×C9 | — | 16 | 2 | no | — |
| 87 | C2×C9 | — | 16 | 1 | no | — |
| 88 | #41 | — | 4 | 6 | no | — |
| 89 | #41 | — | 6 | 6 | no | — |
| 90 | #41 | — | 6 | 2 | no | — |
| 91 | #41 | — | 12 | 2 | no | — |
| 92 | C2×Q8 | — | 3 | 1 | no | — |
| 93 | C2×Q8 | C2×C22⋊C9 | 1 | 2 | yes | — |
| 94 | C2×Q8 | C2×C22⋊C9 | 1 | 3 | yes | — |
| 95 | C2×D4 | — | 3 | 6 | no | — |
| 96 | C2×D4 | — | 3 | 3 | no | — |
| 97 | C2×D4 | — | 3 | 3 | no | — |
| 98 | C22×C4 | — | 3 | 6 | no | — |
| 99 | C22×C4 | — | 3 | 3 | no | — |
| 100 | C4⋊C4 | — | 3 | 2 | no | — |
| 101 | C4⋊C4 | — | 3 | 6 | no | — |
| 102 | C22⋊C4 | — | 3 | 12 | no | — |
| 103 | C42 | — | 3 | 3 | no | — |
| 104 | C42 | — | 3 | 3 | no | — |
| 105 | C42 | — | 3 | 2 | no | — |
| 106 | C22×C3 | — | 6 | 6 | no | — |
| 107 | C22×C3 | — | 6 | 3 | no | — |
| 108 | C22×C3 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 109 | C3×C4 | — | 3 | 2 | no | — |
| 110 | C3×C4 | — | 3 | 2 | no | — |
| 111 | C3×C4 | — | 6 | 2 | no | — |
| 112 | C3×C4 | — | 6 | 3 | no | — |
| 113 | C3×C4 | #997 | 1 | 2 | yes | — |
| 114 | C9 | — | 16 | 1 | no | Sylow |
| 115 | C23 | — | 3 | 3 | no | — |
| 116 | Q8 | — | 3 | 2 | no | — |
| 117 | Q8 | — | 4 | 3 | no | — |
| 118 | Q8 | — | 12 | 1 | no | — |
| 119 | Q8 | #41⋊C9 | 1 | 2 | yes | complemented |
| 120 | D4 | — | 6 | 6 | no | — |
| 121 | D4 | — | 6 | 6 | no | — |
| 122 | D4 | — | 12 | 3 | no | — |
| 123 | D4 | — | 12 | 3 | no | — |
| 124 | C2×C4 | — | 3 | 2 | no | — |
| 125 | C2×C4 | — | 3 | 2 | no | — |
| 126 | C2×C4 | — | 3 | 3 | no | — |
| 127 | C2×C4 | — | 6 | 6 | no | — |
| 128 | C2×C4 | — | 6 | 6 | no | — |
| 129 | C2×C4 | — | 6 | 6 | no | — |
| 130 | C2×C4 | C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 131 | C2×C3 | — | 12 | 3 | no | — |
| 132 | C2×C3 | #5948 | 1 | 2 | yes | — |
| 133 | C2×C3 | C2×#997 | 1 | 1 | yes | characteristic |
| 134 | C22 | — | 6 | 6 | no | — |
| 135 | C22 | — | 6 | 3 | no | — |
| 136 | C22 | C2×C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 137 | C4 | — | 3 | 2 | no | — |
| 138 | C4 | — | 3 | 2 | no | — |
| 139 | C4 | — | 6 | 2 | no | — |
| 140 | C4 | — | 6 | 3 | no | — |
| 141 | C4 | #63703 | 1 | 2 | yes | — |
| 142 | C3 | #85654 | 1 | 1 | yes | characteristic |
| 143 | C2 | — | 12 | 3 | no | — |
| 144 | C2 | #10593123 | 1 | 2 | yes | — |
| 145 | C2 | C2×#63703 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | C22⋊A4 |
| commutator, hypocenter | Q82 | C2×C9 |
| Frattini, socle | C22×C3 | C2×C22⋊A4 |
| Fitting, prime radical | C3×#3483 | C3 |
| cosocle | C3×Q82 | C2×C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2×C3×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8⋊(Q8⋊C9) | C2 | semidirect |
| 13. #3483 | C9 | semidirect |
| 119. Q8 | #41⋊C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8⋊(Q8⋊C9) | 2 |
| C3×#3483 | 3 |
| C2×#41⋊C9 | 4 |
| C4×Q8⋊C9 | 4 |