Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
114 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #497 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #489 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #329 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×Q16 | — | 2 | 1 | no | — |
| 7 | C2×SD16 | — | 2 | 1 | no | — |
| 8 | C4⋊Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | #124 | — | 2 | 2 | no | — |
| 10 | #124 | — | 2 | 1 | no | — |
| 11 | #124 | C22 | 1 | 2 | yes | — |
| 12 | #124 | C22 | 1 | 1 | yes | characteristic |
| 13 | #120 | — | 2 | 2 | no | — |
| 14 | #120 | — | 2 | 1 | no | — |
| 15 | #103 | — | 2 | 2 | no | — |
| 16 | #102 | — | 2 | 2 | no | — |
| 17 | #97 | C22 | 1 | 2 | yes | complemented |
| 18 | #97 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C24 | — | 2 | 1 | no | — |
| 20 | C2×Q8 | — | 2 | 1 | no | — |
| 21 | C2×Q8 | D4 | 1 | 2 | yes | — |
| 22 | C2×Q8 | D4 | 1 | 1 | yes | characteristic |
| 23 | C2×D4 | — | 2 | 1 | no | — |
| 24 | C2×D4 | — | 2 | 2 | no | — |
| 25 | C2×D4 | D4 | 1 | 2 | yes | — |
| 26 | C2×D4 | D4 | 1 | 1 | yes | characteristic |
| 27 | Q16 | — | 4 | 2 | no | — |
| 28 | SD16 | — | 4 | 2 | no | — |
| 29 | C2×C8 | — | 2 | 1 | no | — |
| 30 | C2×C8 | — | 2 | 2 | no | — |
| 31 | C4⋊C4 | — | 2 | 2 | no | — |
| 32 | C22⋊C4 | — | 2 | 2 | no | — |
| 33 | C22⋊C4 | — | 2 | 1 | no | — |
| 34 | C22⋊C4 | — | 2 | 2 | no | — |
| 35 | C22⋊C4 | — | 2 | 2 | no | — |
| 36 | C22⋊C4 | — | 2 | 2 | no | — |
| 37 | C42 | — | 2 | 1 | no | — |
| 38 | C42 | C23 | 1 | 1 | yes | characteristic |
| 39 | C23 | — | 2 | 1 | no | — |
| 40 | C23 | — | 2 | 2 | no | — |
| 41 | C23 | — | 8 | 1 | no | — |
| 42 | C23 | — | 8 | 2 | no | — |
| 43 | Q8 | — | 4 | 1 | no | — |
| 44 | Q8 | — | 4 | 2 | no | — |
| 45 | Q8 | — | 4 | 1 | no | — |
| 46 | Q8 | — | 4 | 1 | no | — |
| 47 | D4 | — | 4 | 1 | no | — |
| 48 | D4 | — | 4 | 2 | no | — |
| 49 | D4 | — | 8 | 1 | no | — |
| 50 | D4 | — | 8 | 2 | no | — |
| 51 | C2×C4 | — | 2 | 1 | no | — |
| 52 | C2×C4 | — | 2 | 2 | no | — |
| 53 | C2×C4 | — | 4 | 1 | no | — |
| 54 | C2×C4 | C2×D4 | 1 | 2 | yes | — |
| 55 | C2×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 56 | C8 | — | 2 | 2 | no | — |
| 57 | C8 | — | 4 | 2 | no | — |
| 58 | C22 | — | 4 | 1 | no | — |
| 59 | C22 | — | 4 | 2 | no | — |
| 60 | C22 | — | 4 | 2 | no | — |
| 61 | C22 | — | 4 | 2 | no | — |
| 62 | C22 | — | 4 | 2 | no | — |
| 63 | C22 | — | 8 | 4 | no | — |
| 64 | C22 | #120 | 1 | 1 | yes | characteristic |
| 65 | C4 | — | 2 | 1 | no | — |
| 66 | C4 | — | 2 | 2 | no | — |
| 67 | C4 | — | 4 | 1 | no | — |
| 68 | C4 | — | 4 | 2 | no | — |
| 69 | C4 | — | 8 | 1 | no | — |
| 70 | C2 | — | 8 | 1 | no | — |
| 71 | C2 | — | 8 | 2 | no | — |
| 72 | C2 | #453 | 1 | 1 | yes | characteristic |
| 73 | C2 | #455 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #497 | C2 | semidirect |
| 3. #489 | C2 | semidirect |
| 4. #329 | C2 | semidirect |
| 5. #329 | C2 | semidirect |
| 8. C4⋊Q8 | C22 | semidirect |
| 17. #97 | C22 | semidirect |
| 18. #97 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽4) | 2 |
| #497 | 2 |
| #489 | 2 |
| #329 | 2 |