Finite Group Databasebeta

← #2202×C211 #41009822

Subgroups — #2202×C211 #41009822

Order 27008.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽4)×C211C211yesmaximal, characteristic, complemented
2#454×C211C213yesmaximal, complemented
3#355×C211C213yesmaximal, complemented
42−1+4×C211C2211yescharacteristic, complemented
5AGL(1,ℤ/8)×C211—23no—
6#124×C211—23no—
7#124×C211C2213yes—
8#120×C211—23no—
9#104×C211—23no—
10#101×C211C2213yescomplemented
11C24×C211—21no—
12#41×C211—23no—
13#41×C211—41no—
14C2×Q8×C211—21no—
15C2×Q8×C211—41no—
16C2×Q8×C211C2311yescharacteristic
17C2×D4×C211—23no—
18C2×D4×C211D413yes—
19SD16×C211—43no—
20D8×C211—43no—
21M16×C211—23no—
22C22⋊C4×C211—23no—
23C22⋊C4×C211—26no—
24C42×C211D413yes—
25C23×C211—23no—
26C23×C211—82no—
27C23×C211—81no—
28Q8×C211—23no—
29Q8×C211—41no—
30Q8×C211—81no—
31D4×C211—23no—
32D4×C211—43no—
33D4×C211—41no—
34D4×C211—83no—
35C2×C4×C211—23no—
36C2×C4×C211—43no—
37C2×C4×C211C2×D413yes—
38C8×C211—43no—
39C22×C211—46no—
40C22×C211—41no—
41C22×C211—43no—
42C22×C211—84no—
43C22×C211#12011yescharacteristic
44C4×C211—23no—
45C4×C211—43no—
46C4×C211—41no—
47C2×C211—21no—
48C2×C211—81no—
49C2×C211—83no—
50C2×C211B(4,𝔽2)11yescharacteristic
51C211#220211yescharacteristic, direct product factor, Sylow
52#2202C21111yesmaximal, characteristic, direct product factor, Sylow
53H(3,𝔽4)C2×C21111yescharacteristic, complemented
54#454C2×C21113yescomplemented
55#355C2×C21113yescomplemented
562−1+4C22×C21111yescharacteristic, complemented
57AGL(1,ℤ/8)—23no—
58#124—23no—
59#124C22×C21113yes—
60#120—23no—
61#104—23no—
62#101C22×C21113yescomplemented
63C24—21no—
64#41—23no—
65#41—41no—
66C2×Q8—21no—
67C2×Q8—41no—
68C2×Q8C23×C21111yescharacteristic
69C2×D4—23no—
70C2×D4D4×C21113yes—
71SD16—43no—
72D8—43no—
73M16—23no—
74C22⋊C4—23no—
75C22⋊C4—26no—
76C42D4×C21113yes—
77C23—23no—
78C23—82no—
79C23—81no—
80Q8—23no—
81Q8—41no—
82Q8—81no—
83D4—23no—
84D4—43no—
85D4—41no—
86D4—83no—
87C2×C4—23no—
88C2×C4—43no—
89C2×C4C2×D4×C21113yes—
90C8—43no—
91C22—46no—
92C22—41no—
93C22—43no—
94C22—84no—
95C22#120×C21111yescharacteristic
96C4—23no—
97C4—43no—
98C4—41no—
99C2—21no—
100C2—83no—
101C2—81no—
102C2B(4,𝔽2)×C21111yescharacteristic
subgroupG/H
center, socleC2×C211B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C211
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C22×C211 ◁ C2×Q8×C211 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. H(3,𝔽4)×C211C2semidirect
2. #454×C211C2semidirect
3. #355×C211C2semidirect
4. 2−1+4×C211C22semidirect
10. #101×C211C22semidirect
51. C211#2202direct (N × H)
52. #2202C211direct (N × H)
53. H(3,𝔽4)C2×C211semidirect
54. #454C2×C211semidirect
55. #355C2×C211semidirect
56. 2−1+4C22×C211semidirect
62. #101C22×C211semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
H(3,𝔽4)×C2112
#454×C2112
#355×C2112
#2202211