Finite Group Databasebeta

← #2029×C211 #41009649

Subgroups — #2029×C211 #41009649

Order 27008.

466 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 84 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
1#543×C211C211yesmaximal, characteristic, complemented
2#379×C211C213yesmaximal
3#342×C211C213yesmaximal, complemented
4C25×C211C2211yescharacteristic
5C22⋊Q8×C211—26no—
6C2×C22⋊C4×C211—26no—
7C2×C22⋊C4×C211C2213yes—
8C2×C22⋊C4×C211C2213yes—
9C24×C211—23no—
10C24×C211—23no—
11C24×C211—44no—
12C24×C211Q812yescomplemented
13C24×C211C2311yescharacteristic
14C2×Q8×C211—22no—
15C22×C4×C211—23no—
16C22×C4×C211D416yes—
17C4⋊C4×C211—26no—
18C22⋊C4×C211—26no—
19C22⋊C4×C211—26no—
20C22⋊C4×C211—412no—
21C23×C211—212no—
22C23×C211—26no—
23C23×C211—44no—
24C23×C211—42no—
25C23×C211—46no—
26C23×C211—412no—
27C23×C211—41no—
28C23×C211—43no—
29C23×C211C2×D413yes—
30C23×C211C2×Q811yescharacteristic
31C23×C211#4113yes—
32Q8×C211—82no—
33C2×C4×C211—26no—
34C2×C4×C211—46no—
35C2×C4×C211—46no—
36C2×C4×C211—43no—
37C22×C211—26no—
38C22×C211—212no—
39C22×C211—41no—
40C22×C211—43no—
41C22×C211—412no—
42C22×C211—46no—
43C22×C211—42no—
44C22×C211—44no—
45C22×C211#12011yescharacteristic
46C22×C211C22⋊Q813yes—
47C22×C211#12413yes—
48C4×C211—46no—
49C4×C211—83no—
50C2×C211—23no—
51C2×C211—23no—
52C2×C211—44no—
53C2×C211#39311yescharacteristic
54C2×C211B(4,𝔽2)12yes—
55C211#202911yescharacteristic, direct product factor, Sylow
56#2029C21111yesmaximal, characteristic, direct product factor, Sylow
57#543C2×C21111yescharacteristic, complemented
58#379C2×C21113yes—
59#342C2×C21113yescomplemented
60C25C22×C21111yescharacteristic
61C22⋊Q8—26no—
62C2×C22⋊C4—26no—
63C2×C22⋊C4C22×C21113yes—
64C2×C22⋊C4C22×C21113yes—
65C24—23no—
66C24—23no—
67C24—44no—
68C24Q8×C21112yescomplemented
69C24C23×C21111yescharacteristic
70C2×Q8—22no—
71C22×C4—23no—
72C22×C4D4×C21116yes—
73C4⋊C4—26no—
74C22⋊C4—26no—
75C22⋊C4—26no—
76C22⋊C4—412no—
77C23—212no—
78C23—26no—
79C23—44no—
80C23—42no—
81C23—46no—
82C23—412no—
83C23—41no—
84C23—43no—
85C23C2×D4×C21113yes—
86C23C2×Q8×C21111yescharacteristic
87C23#41×C21113yes—
88Q8—82no—
89C2×C4—26no—
90C2×C4—46no—
91C2×C4—46no—
92C2×C4—43no—
93C22—26no—
94C22—212no—
95C22—41no—
96C22—43no—
97C22—412no—
98C22—46no—
99C22—42no—
100C22—44no—
101C22#120×C21111yescharacteristic
102C22C22⋊Q8×C21113yes—
103C22#124×C21113yes—
104C4—46no—
105C4—83no—
106C2—23no—
107C2—23no—
108C2—44no—
109C2#393×C21111yescharacteristic
110C2B(4,𝔽2)×C21112yes—
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC24C23×C211
prime radicalC25×C211C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C24×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. #543×C211C2semidirect
3. #342×C211C2semidirect
12. C24×C211Q8semidirect
55. C211#2029direct (N × H)
56. #2029C211direct (N × H)
57. #543C2×C211semidirect
59. #342C2×C211semidirect
68. C24Q8×C211semidirect

Maximal subgroups

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

subgroupindex
#543×C2112
#379×C2112
#342×C2112
#2029211