Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1700 conjugacy classes of subgroups; only the 746 normal subgroups are shown.
This group has 1700 conjugacy classes of subgroups — too many to draw. Only its 746 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 | #52307 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23347 | C2 | 1 | 9 | yes | maximal, complemented |
| 3 | #23312 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C8×#124 | C2 | 1 | 9 | yes | maximal, complemented |
| 5 | #21442 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C22×#131 | C22 | 1 | 2 | yes | — |
| 7 | C2×C8×Q8 | C22 | 1 | 3 | yes | — |
| 8 | C2×C8×D4 | C22 | 1 | 9 | yes | — |
| 9 | #2811 | C4 | 1 | 1 | yes | characteristic |
| 10 | #2785 | C4 | 1 | 6 | yes | — |
| 11 | #2733 | C4 | 1 | 9 | yes | — |
| 12 | #2388 | C22 | 1 | 6 | yes | — |
| 13 | C4×#124 | C22 | 1 | 9 | yes | — |
| 14 | #1984 | C22 | 1 | 9 | yes | — |
| 15 | #1954 | C22 | 1 | 9 | yes | — |
| 16 | #1923 | C22 | 1 | 36 | yes | — |
| 17 | #1870 | C22 | 1 | 18 | yes | — |
| 18 | #1851 | C22 | 1 | 18 | yes | — |
| 19 | #1831 | C22 | 1 | 9 | yes | — |
| 20 | #1780 | C22 | 1 | 6 | yes | — |
| 21 | C8×C22⋊C4 | C22 | 1 | 18 | yes | — |
| 22 | C42×C8 | C22 | 1 | 3 | yes | — |
| 23 | C22×#41 | C23 | 1 | 2 | yes | — |
| 24 | C2×#131 | — | 4 | 12 | no | — |
| 25 | C22×M16 | C23 | 1 | 6 | yes | — |
| 26 | C23×C8 | C23 | 1 | 6 | yes | — |
| 27 | C2×#124 | C2×C4 | 1 | 9 | yes | — |
| 28 | C2×C4×Q8 | C23 | 1 | 3 | yes | — |
| 29 | C2×C4×D4 | C23 | 1 | 9 | yes | — |
| 30 | C8×Q8 | — | 4 | 6 | no | — |
| 31 | C8×D4 | — | 4 | 18 | no | — |
| 32 | C2×C4⋊C8 | C23 | 1 | 9 | yes | — |
| 33 | C2×C4⋊C8 | C23 | 1 | 9 | yes | — |
| 34 | C2×C22⋊C8 | C23 | 1 | 18 | yes | — |
| 35 | C2×C22⋊C8 | C23 | 1 | 18 | yes | — |
| 36 | C2×C4×C8 | C23 | 1 | 9 | yes | — |
| 37 | C2×C4×C8 | C23 | 1 | 9 | yes | — |
| 38 | #395 | C2×C4 | 1 | 6 | yes | — |
| 39 | #394 | C2×C4 | 1 | 18 | yes | — |
| 40 | #393 | C2×C4 | 1 | 6 | yes | — |
| 41 | #392 | C2×C4 | 1 | 18 | yes | — |
| 42 | #391 | C2×C4 | 1 | 9 | yes | — |
| 43 | #390 | C2×C4 | 1 | 9 | yes | — |
| 44 | #388 | C2×C4 | 1 | 36 | yes | — |
| 45 | #382 | C2×C4 | 1 | 9 | yes | — |
| 46 | C4×C22⋊C4 | C23 | 1 | 18 | yes | — |
| 47 | C43 | C23 | 1 | 3 | yes | — |
| 48 | #336 | C23 | 1 | 36 | yes | — |
| 49 | C2×#41 | — | 4 | 6 | no | — |
| 50 | C2×#41 | — | 4 | 6 | no | — |
| 51 | C22×Q8 | C22×C4 | 1 | 2 | yes | — |
| 52 | C22×Q8 | C22×C4 | 1 | 3 | yes | — |
| 53 | C22×D4 | C22×C4 | 1 | 9 | yes | — |
| 54 | C22×D4 | C22×C4 | 1 | 6 | yes | — |
| 55 | C23×C4 | C24 | 1 | 6 | yes | — |
| 56 | #131 | — | 4 | 32 | no | — |
| 57 | C2×M16 | — | 4 | 18 | no | — |
| 58 | C22×C8 | — | 2 | 36 | no | — |
| 59 | C22×C8 | C24 | 1 | 6 | yes | — |
| 60 | C22×C8 | C24 | 1 | 9 | yes | — |
| 61 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 62 | #124 | — | 2 | 36 | no | — |
| 63 | C4×Q8 | — | 4 | 3 | no | — |
| 64 | C4×Q8 | — | 4 | 3 | no | — |
| 65 | C4×D4 | — | 4 | 9 | no | — |
| 66 | C4×D4 | — | 4 | 9 | no | — |
| 67 | C2×C4⋊C4 | C22×C4 | 1 | 9 | yes | — |
| 68 | C2×C4⋊C4 | C22×C4 | 1 | 9 | yes | — |
| 69 | C2×C22⋊C4 | C22×C4 | 1 | 36 | yes | — |
| 70 | C2×C22⋊C4 | C22×C4 | 1 | 18 | yes | — |
| 71 | C2×C42 | C22×C4 | 1 | 9 | yes | — |
| 72 | C2×C42 | C22×C4 | 1 | 3 | yes | — |
| 73 | C2×C42 | C24 | 1 | 9 | yes | — |
| 74 | C4⋊C8 | — | 4 | 9 | no | — |
| 75 | C4⋊C8 | — | 4 | 9 | no | — |
| 76 | C22⋊C8 | — | 4 | 18 | no | — |
| 77 | C22⋊C8 | — | 4 | 18 | no | — |
| 78 | C4×C8 | — | 2 | 18 | no | — |
| 79 | C4×C8 | — | 4 | 9 | no | — |
| 80 | #95 | C22×C4 | 1 | 18 | yes | — |
| 81 | #95 | C22×C4 | 1 | 18 | yes | — |
| 82 | C24 | C23×C4 | 1 | 6 | yes | — |
| 83 | #41 | — | 4 | 8 | no | — |
| 84 | #41 | — | 4 | 24 | no | — |
| 85 | C2×Q8 | — | 4 | 3 | no | — |
| 86 | C2×Q8 | — | 4 | 3 | no | — |
| 87 | C2×Q8 | — | 4 | 6 | no | — |
| 88 | C2×Q8 | #131 | 1 | 12 | yes | complemented |
| 89 | C2×D4 | — | 2 | 18 | no | — |
| 90 | C2×D4 | — | 4 | 9 | no | — |
| 91 | C2×D4 | — | 4 | 9 | no | — |
| 92 | C2×D4 | — | 4 | 18 | no | — |
| 93 | C22×C4 | — | 2 | 18 | no | — |
| 94 | C22×C4 | — | 2 | 18 | no | — |
| 95 | C22×C4 | C23×C4 | 1 | 6 | yes | — |
| 96 | C22×C4 | C23×C4 | 1 | 9 | yes | — |
| 97 | C22×C4 | C23×C4 | 1 | 9 | yes | — |
| 98 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 99 | M16 | — | 4 | 24 | no | — |
| 100 | C2×C8 | — | 2 | 9 | no | — |
| 101 | C2×C8 | — | 2 | 18 | no | — |
| 102 | C2×C8 | — | 2 | 18 | no | — |
| 103 | C2×C8 | — | 4 | 24 | no | — |
| 104 | C2×C8 | H(5,𝔽2) | 1 | 6 | yes | — |
| 105 | C4⋊C4 | — | 4 | 9 | no | — |
| 106 | C4⋊C4 | — | 4 | 9 | no | — |
| 107 | C22⋊C4 | — | 2 | 72 | no | — |
| 108 | C22⋊C4 | — | 4 | 18 | no | — |
| 109 | C42 | — | 2 | 18 | no | — |
| 110 | C42 | — | 2 | 9 | no | — |
| 111 | C42 | — | 2 | 9 | no | — |
| 112 | C42 | — | 4 | 3 | no | — |
| 113 | C23 | — | 2 | 18 | no | — |
| 114 | C23 | — | 2 | 18 | no | — |
| 115 | C23 | — | 2 | 6 | no | — |
| 116 | C23 | C24×C4 | 1 | 1 | yes | characteristic |
| 117 | Q8 | — | 4 | 12 | no | — |
| 118 | Q8 | — | 4 | 8 | no | — |
| 119 | D4 | — | 4 | 36 | no | — |
| 120 | D4 | — | 4 | 24 | no | — |
| 121 | C2×C4 | — | 2 | 9 | no | — |
| 122 | C2×C4 | — | 2 | 9 | no | — |
| 123 | C2×C4 | — | 2 | 9 | no | — |
| 124 | C2×C4 | — | 2 | 18 | no | — |
| 125 | C2×C4 | — | 2 | 18 | no | — |
| 126 | C2×C4 | — | 4 | 6 | no | — |
| 127 | C2×C4 | — | 4 | 18 | no | — |
| 128 | C2×C4 | C2×#131 | 1 | 18 | yes | — |
| 129 | C2×C4 | C2×H(5,𝔽2) | 1 | 3 | yes | — |
| 130 | C2×C4 | #585 | 1 | 3 | yes | — |
| 131 | C8 | — | 4 | 6 | no | — |
| 132 | C8 | — | 4 | 9 | no | — |
| 133 | C8 | H(3,𝔽4) | 1 | 4 | yes | — |
| 134 | C22 | — | 4 | 6 | no | — |
| 135 | C22 | — | 4 | 18 | no | — |
| 136 | C22 | — | 4 | 18 | no | — |
| 137 | C22 | C4×H(5,𝔽2) | 1 | 3 | yes | — |
| 138 | C22 | C22×#131 | 1 | 3 | yes | — |
| 139 | C22 | #3573 | 1 | 1 | yes | characteristic |
| 140 | C4 | — | 2 | 18 | no | — |
| 141 | C4 | — | 4 | 6 | no | — |
| 142 | C4 | — | 4 | 9 | no | — |
| 143 | C4 | C2×H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 144 | C4 | #3529 | 1 | 3 | yes | — |
| 145 | C2 | — | 8 | 6 | no | — |
| 146 | C2 | C4×H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 147 | C2 | #60304 | 1 | 3 | yes | — |
| 148 | C2 | #60308 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator, socle | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #52307 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23347 | C2 | semidirect |
| 3. #23312 | C2 | semidirect |
| 4. C8×#124 | C2 | semidirect |
| 5. #21442 | C2 | semidirect |
| 88. C2×Q8 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #52307 | 2 |
| #23347 | 2 |
| #23312 | 2 |
| C8×#124 | 2 |
| #21442 | 2 |