Finite Group DatabasebetaFinite Group DatabasebetaOrder 4096.
4467484379 conjugacy classes of subgroups; only the 533115 normal subgroups are shown.
This group has 4467484379 conjugacy classes of subgroups — too many to draw. Only its 533115 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 | C2×#72219197 | C2 | 1 | 62 | yes | maximal, complemented |
| 2 | C22×#14679 | C22 | 1 | 620 | yes | complemented |
| 3 | #72219197 | — | 2 | 992 | no | — |
| 4 | C211 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C210 | — | 2 | 992 | no | — |
| 6 | C210 | C4 | 1 | 32 | yes | complemented |
| 7 | C210 | C22 | 1 | 31 | yes | — |
| 8 | C29 | — | 2 | 317440 | no | — |
| 9 | C29 | — | 2 | 15872 | no | — |
| 10 | C29 | — | 2 | 14880 | no | — |
| 11 | C29 | C2×C4 | 1 | 496 | yes | — |
| 12 | C29 | D4 | 1 | 992 | yes | — |
| 13 | C29 | C23 | 1 | 155 | yes | — |
| 14 | C23×#379 | C23 | 1 | 1240 | yes | complemented |
| 15 | C2×#14679 | — | 2 | 14880 | no | — |
| 16 | C28 | — | 2 | 20316160 | no | — |
| 17 | C28 | — | 2 | 2222080 | no | — |
| 18 | C28 | — | 2 | 2539520 | no | — |
| 19 | C28 | — | 2 | 119040 | no | — |
| 20 | C28 | — | 2 | 238080 | no | — |
| 21 | C28 | — | 2 | 34720 | no | — |
| 22 | C28 | C22⋊C4 | 1 | 7936 | yes | complemented |
| 23 | C28 | C22×C4 | 1 | 1240 | yes | — |
| 24 | C28 | C2×D4 | 1 | 7440 | yes | — |
| 25 | C28 | C24 | 1 | 155 | yes | — |
| 26 | C24×C22⋊C4 | C24 | 1 | 496 | yes | complemented |
| 27 | C22×#379 | — | 2 | 34720 | no | — |
| 28 | #14679 | — | 4 | 39680 | no | — |
| 29 | C27 | — | 2 | 260046848 | no | — |
| 30 | C27 | — | 2 | 60948480 | no | — |
| 31 | C27 | — | 2 | 81264640 | no | — |
| 32 | C27 | — | 2 | 17776640 | no | — |
| 33 | C27 | — | 2 | 2222080 | no | — |
| 34 | C27 | — | 2 | 8888320 | no | — |
| 35 | C27 | — | 2 | 138880 | no | — |
| 36 | C27 | — | 2 | 833280 | no | — |
| 37 | C27 | — | 2 | 14880 | no | — |
| 38 | C27 | — | 2 | 952320 | no | — |
| 39 | C27 | C2×C22⋊C4 | 1 | 29760 | yes | — |
| 40 | C27 | #120 | 1 | 39680 | yes | — |
| 41 | C27 | C23×C4 | 1 | 620 | yes | — |
| 42 | C27 | C22×D4 | 1 | 8680 | yes | — |
| 43 | C27 | C25 | 1 | 31 | yes | — |
| 44 | C25×C4 | C25 | 1 | 32 | yes | complemented |
| 45 | C23×C22⋊C4 | — | 2 | 14880 | no | — |
| 46 | C2×#379 | — | 4 | 138880 | no | — |
| 47 | C26 | — | 2 | 536870912 | no | — |
| 48 | C26 | — | 2 | 520093696 | no | — |
| 49 | C26 | — | 2 | 260046848 | no | — |
| 50 | C26 | — | 2 | 121896960 | no | — |
| 51 | C26 | — | 2 | 20316160 | no | — |
| 52 | C26 | — | 2 | 243793920 | no | — |
| 53 | C26 | — | 2 | 26664960 | no | — |
| 54 | C26 | — | 2 | 35553280 | no | — |
| 55 | C26 | — | 2 | 317440 | no | — |
| 56 | C26 | — | 2 | 4444160 | no | — |
| 57 | C26 | — | 2 | 992 | no | — |
| 58 | C26 | — | 2 | 2222080 | no | — |
| 59 | C26 | — | 2 | 29760 | no | — |
| 60 | C26 | — | 2 | 416640 | no | — |
| 61 | C26 | — | 2 | 1666560 | no | — |
| 62 | C26 | #379 | 1 | 79360 | yes | complemented |
| 63 | C26 | C22×C22⋊C4 | 1 | 17360 | yes | — |
| 64 | C26 | C2×#120 | 1 | 69440 | yes | — |
| 65 | C26 | C24×C4 | 1 | 62 | yes | — |
| 66 | C26 | C23×D4 | 1 | 1860 | yes | — |
| 67 | C26 | C26 | 1 | 1 | yes | characteristic |
| 68 | C24×C4 | — | 2 | 992 | no | — |
| 69 | C22×C22⋊C4 | — | 4 | 69440 | no | — |
| 70 | #379 | — | 8 | 79360 | no | — |
| 71 | C25 | — | 2 | 536870912 | no | — |
| 72 | C25 | — | 2 | 260046848 | no | — |
| 73 | C25 | — | 2 | 520093696 | no | — |
| 74 | C25 | — | 2 | 243793920 | no | — |
| 75 | C25 | — | 2 | 121896960 | no | — |
| 76 | C25 | — | 2 | 20316160 | no | — |
| 77 | C25 | — | 2 | 317440 | no | — |
| 78 | C25 | — | 2 | 26664960 | no | — |
| 79 | C25 | — | 2 | 4444160 | no | — |
| 80 | C25 | — | 2 | 35553280 | no | — |
| 81 | C25 | — | 2 | 1666560 | no | — |
| 82 | C25 | — | 2 | 2222080 | no | — |
| 83 | C25 | — | 2 | 992 | no | — |
| 84 | C25 | — | 2 | 29760 | no | — |
| 85 | C25 | — | 2 | 416640 | no | — |
| 86 | C25 | C2×#379 | 1 | 69440 | yes | — |
| 87 | C25 | #2846 | 1 | 79360 | yes | — |
| 88 | C25 | C23×C22⋊C4 | 1 | 1860 | yes | — |
| 89 | C25 | C22×#120 | 1 | 17360 | yes | — |
| 90 | C25 | C25×C4 | 1 | 1 | yes | characteristic |
| 91 | C25 | C24×D4 | 1 | 62 | yes | — |
| 92 | C23×C4 | — | 4 | 4960 | no | — |
| 93 | C2×C22⋊C4 | — | 8 | 59520 | no | — |
| 94 | C24 | — | 2 | 260046848 | no | — |
| 95 | C24 | — | 2 | 60948480 | no | — |
| 96 | C24 | — | 2 | 81264640 | no | — |
| 97 | C24 | — | 2 | 17776640 | no | — |
| 98 | C24 | — | 2 | 8888320 | no | — |
| 99 | C24 | — | 2 | 2222080 | no | — |
| 100 | C24 | — | 2 | 14880 | no | — |
| 101 | C24 | — | 2 | 833280 | no | — |
| 102 | C24 | — | 2 | 138880 | no | — |
| 103 | C24 | — | 2 | 952320 | no | — |
| 104 | C24 | #14679 | 1 | 39680 | yes | complemented |
| 105 | C24 | C22×#379 | 1 | 8680 | yes | — |
| 106 | C24 | C2×#2846 | 1 | 29760 | yes | — |
| 107 | C24 | C24×C22⋊C4 | 1 | 31 | yes | — |
| 108 | C24 | C23×#120 | 1 | 620 | yes | — |
| 109 | C22×C4 | — | 8 | 4960 | no | — |
| 110 | C22⋊C4 | — | 16 | 7936 | no | — |
| 111 | C23 | — | 2 | 20316160 | no | — |
| 112 | C23 | — | 2 | 2222080 | no | — |
| 113 | C23 | — | 2 | 2539520 | no | — |
| 114 | C23 | — | 2 | 238080 | no | — |
| 115 | C23 | — | 2 | 119040 | no | — |
| 116 | C23 | — | 2 | 34720 | no | — |
| 117 | C23 | C2×#14679 | 1 | 7440 | yes | — |
| 118 | C23 | #6342371 | 1 | 7936 | yes | — |
| 119 | C23 | C23×#379 | 1 | 155 | yes | — |
| 120 | C23 | C22×#2846 | 1 | 1240 | yes | — |
| 121 | C2×C4 | — | 16 | 992 | no | — |
| 122 | C22 | — | 2 | 317440 | no | — |
| 123 | C22 | — | 2 | 14880 | no | — |
| 124 | C22 | — | 2 | 15872 | no | — |
| 125 | C22 | C2×#6342371 | 1 | 496 | yes | — |
| 126 | C22 | #72219197 | 1 | 992 | yes | complemented |
| 127 | C22 | C22×#14679 | 1 | 155 | yes | — |
| 128 | C4 | — | 32 | 32 | no | — |
| 129 | C2 | — | 2 | 992 | no | — |
| 130 | C2 | #72232847 | 1 | 32 | yes | — |
| 131 | C2 | C2×#72219197 | 1 | 31 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C26 | C26 |
| commutator | C25 | C25×C4 |
| prime radical | C211 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#72219197 | C2 | semidirect |
| 2. C22×#14679 | C22 | semidirect |
| 6. C210 | C4 | semidirect |
| 14. C23×#379 | C23 | semidirect |
| 22. C28 | C22⋊C4 | semidirect |
| 26. C24×C22⋊C4 | C24 | semidirect |
| 44. C25×C4 | C25 | semidirect |
| 62. C26 | #379 | semidirect |
| 104. C24 | #14679 | semidirect |
| 126. C22 | #72219197 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#72219197 | 2 |
| C211 | 2 |