Finite Group Databasebeta

← C22⋊C9×C22⋊A4 #13180064

Subgroups — C22⋊C9×C22⋊A4 #13180064

Order 1728.

942 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C22×C3×C22⋊A4C311yesmaximal, characteristic
2C22⋊(C22⋊(C22⋊C9))C312yesmaximal, complemented
3C24×C22⋊C9C311yesmaximal, characteristic, complemented
4C9×C22⋊A4—41nomaximal
5A4×C22⋊C9—45nomaximal
6C2×C3×C22⋊A4—31no—
7C23×C22⋊C9—35no—
8C26×C3C3211yescharacteristic
9C22×C22⋊A4C913yescomplemented
10C3×C22⋊A4A411yescharacteristic
11C22×C3×A4—45no—
12C24×C9—41no—
13C22⋊(C22⋊C9)—42no—
14C22⋊(C22⋊C9)—410no—
15C22⋊(C22⋊C9)—1210no—
16C22×C22⋊C9—310no—
17C22×C22⋊C9A415yescomplemented
18C3×C22⋊C9—161no—
19C9×A4—165no—
20C25×C3—35no—
21C25×C3—31no—
22C25×C3—95no—
23C2×C22⋊A4—33no—
24C2×C3×A4—125no—
25C23×C9—125no—
26C2×C22⋊C9—35no—
27C26C3×C911yescharacteristic, complemented, Sylow
28C24×C3—310no—
29C24×C3—310no—
30C24×C3—95no—
31C24×C3—95no—
32C24×C3—95no—
33C24×C3—930no—
34C24×C3—920no—
35C24×C3C3×A415yes—
36C24×C3C3×A411yescharacteristic
37C22⋊A4C22⋊C913yesdirect product factor
38C22×A4—415no—
39C22×C32—161no—
40C3×A4—45no—
41C22×C9—45no—
42C22×C9—1210no—
43C22⋊C9—1610no—
44C22⋊C9—162no—
45C22⋊C9—4810no—
46C22⋊C9C22⋊A411yescharacteristic, direct product factor
47C25—35no—
48C25—31no—
49C25—95no—
50C3×C9—641noSylow
51C23×C3—35no—
52C23×C3—35no—
53C23×C3—35no—
54C23×C3—95no—
55C23×C3—910no—
56C23×C3—940no—
57C23×C3—930no—
58C23×C3—930no—
59C23×C3—930no—
60C23×C3—95no—
61C2×A4—1215no—
62C2×C32—481no—
63C2×C9—125no—
64C24—310no—
65C24—310no—
66C24—95no—
67C24—95no—
68C24—95no—
69C24—930no—
70C24—920no—
71C24C9×A415yescomplemented
72C24C3×C22⋊C911yescharacteristic, complemented
73C22×C3—310no—
74C22×C3—310no—
75C22×C3—95no—
76C22×C3—95no—
77C22×C3—95no—
78C22×C3—930no—
79C22×C3—920no—
80C22×C3—163no—
81C22×C3A4215yes—
82C22×C3C3×C22⋊A411yescharacteristic
83A4—415no—
84C32—161no—
85C9—41no—
86C9—642no—
87C23—35no—
88C23—35no—
89C23—35no—
90C23—95no—
91C23—910no—
92C23—940no—
93C23—930no—
94C23—930no—
95C23—930no—
96C23—95no—
97C2×C3—35no—
98C2×C3—31no—
99C2×C3—95no—
100C2×C3—483no—
101C22—310no—
102C22—310no—
103C22—95no—
104C22—95no—
105C22—95no—
106C22—930no—
107C22—920no—
108C22A4×C22⋊C915yescomplemented
109C22C9×C22⋊A411yescharacteristic, complemented
110C3—163no—
111C3A4×C22⋊A411yescharacteristic
112C2—35no—
113C2—31no—
114C2—95no—
subgroupG/H
center, Frattini, hypercenterC3A4×C22⋊A4
commutator, hypocenterC26C3×C9
Fitting, socle, cosocleC26×C3C32
prime radicalC22×C3×C22⋊A4C3
LCSG ▷ C26
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
2. C22⋊(C22⋊(C22⋊C9))C3semidirect
3. C24×C22⋊C9C3semidirect
9. C22×C22⋊A4C9semidirect
17. C22×C22⋊C9A4semidirect
27. C26C3×C9semidirect
37. C22⋊A4C22⋊C9direct (N × H)
46. C22⋊C9C22⋊A4direct (N × H)
71. C24C9×A4semidirect
72. C24C3×C22⋊C9semidirect
108. C22A4×C22⋊C9semidirect
109. C22C9×C22⋊A4semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊A43
C22⋊(C22⋊(C22⋊C9))3
C24×C22⋊C93
C9×C22⋊A44
A4×C22⋊C94