Finite Group Databasebeta

← Dic3×C22⋊(C22⋊C9) #13164663

Subgroups — Dic3×C22⋊(C22⋊C9) #13164663

Order 1728.

1154 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×C3×C22⋊(C22⋊C9)C211yesmaximal, characteristic
2C24×C3×Dic3C311yesmaximal, characteristic
3C4×C22⋊(C22⋊C9)—31nomaximal
4C3×C22⋊(C22⋊C9)C411yescharacteristic, complemented
5Dic3×C22⋊C9—45nomaximal
6C25×C32C2×C311yescharacteristic
7C23×C3×Dic3—35no—
8C23×C3×Dic3—35no—
9C2×C22⋊(C22⋊C9)—21no—
10C2×C22⋊(C22⋊C9)D311yescharacteristic
11C2×C3×C22⋊C9—45no—
12C24×C3×C4—31no—
13C24×Dic3C911yescharacteristic, complemented
14C24×C32—35no—
15C24×C32—35no—
16C24×C32C3×C411yescharacteristic
17C22×C3×Dic3—330no—
18C22×C3×Dic3—310no—
19C22×C3×Dic3—35no—
20C22×C3×Dic3A415yes—
21C22⋊(C22⋊C9)—21no—
22C22⋊(C22⋊C9)Dic311yescharacteristic, direct product factor
23C4×C22⋊C9—125no—
24C3×C22⋊C9—45no—
25C9×Dic3—161no—
26C25×C3—21no—
27C25×C3C2×C911yescharacteristic
28C25×C3C3×D311yescharacteristic
29C23×C3×C4—95no—
30C23×C3×C4—95no—
31C23×Dic3—35no—
32C23×Dic3—35no—
33C23×C32—330no—
34C23×C32—310no—
35C23×C32—35no—
36C23×C32—35no—
37C23×C32C2×A415yes—
38C2×C3×Dic3—330no—
39C2×C3×Dic3—35no—
40C2×C3×Dic3—35no—
41C2×C22⋊C9—45no—
42C2×C22⋊C9—85no—
43C24×C4—31noSylow
44C2×C3×C9—161no—
45C24×C3—21no—
46C24×C3—35no—
47C24×C3—35no—
48C24×C3—35no—
49C24×C3—35no—
50C24×C3—65no—
51C24×C3—65no—
52C24×C3C4×C911yescharacteristic, complemented
53C24×C3C3×Dic311yescharacteristic
54C22×C3×C4—35no—
55C22×C3×C4—910no—
56C22×C3×C4—930no—
57C22×C3×C4—95no—
58C22×Dic3—330no—
59C22×Dic3—310no—
60C22×Dic3—35no—
61C22×Dic3C22⋊C915yescomplemented
62C22×C32—35no—
63C22×C32—330no—
64C22×C32—310no—
65C22×C32—35no—
66C22×C32C4×A415yes—
67C3×Dic3—35no—
68C3×Dic3C22⋊A411yescharacteristic
69C22⋊C9—45no—
70C22⋊C9—85no—
71C4×C9—481no—
72C25D3×C911yescharacteristic
73C23×C4—95no—
74C23×C4—95no—
75C3×C9—161noSylow
76C23×C3—25no—
77C23×C3—310no—
78C23×C3—310no—
79C23×C3—330no—
80C23×C3—330no—
81C23×C3—35no—
82C23×C3—35no—
83C23×C3—35no—
84C23×C3—35no—
85C23×C3—65no—
86C23×C3—65no—
87C23×C3—610no—
88C23×C3—630no—
89C23×C3C2×C22⋊C915yes—
90C23×C3D3×A415yes—
91C2×C3×C4—95no—
92C2×C3×C4—930no—
93C2×C3×C4—95no—
94C2×Dic3—330no—
95C2×Dic3—35no—
96C2×Dic3—35no—
97C2×C32—35no—
98C2×C32—35no—
99C2×C32C2×C22⋊A411yescharacteristic
100C2×C9—161no—
101C2×C9—321no—
102C24—35no—
103C24—35no—
104C24C9×Dic311yescharacteristic, complemented
105C22×C4—35no—
106C22×C4—910no—
107C22×C4—930no—
108C22×C4—95no—
109C22×C3—25no—
110C22×C3—35no—
111C22×C3—35no—
112C22×C3—310no—
113C22×C3—330no—
114C22×C3—310no—
115C22×C3—330no—
116C22×C3—35no—
117C22×C3—35no—
118C22×C3—65no—
119C22×C3—65no—
120C22×C3—610no—
121C22×C3—630no—
122C22×C3C4×C22⋊C915yescomplemented
123C22×C3Dic3×A415yes—
124C3×C4—31no—
125C3×C4—95no—
126Dic3—35no—
127Dic3C22⋊(C22⋊C9)11yescharacteristic, direct product factor
128C32C4×C22⋊A411yescharacteristic
129C9—161no—
130C9—321no—
131C23—310no—
132C23—330no—
133C23—35no—
134C23—35no—
135C23D3×C22⋊C915yes—
136C2×C4—95no—
137C2×C4—930no—
138C2×C4—95no—
139C2×C3—21no—
140C2×C3—35no—
141C2×C3—35no—
142C2×C3—35no—
143C2×C3—35no—
144C2×C3—65no—
145C2×C3—65no—
146C2×C3C2×C22⋊(C22⋊C9)11yescharacteristic
147C2×C3D3×C22⋊A411yescharacteristic
148C22—35no—
149C22—310no—
150C22—330no—
151C22—35no—
152C22Dic3×C22⋊C915yescomplemented
153C4—31no—
154C4—95no—
155C3—21no—
156C3C4×C22⋊(C22⋊C9)11yescharacteristic, complemented
157C3Dic3×C22⋊A411yescharacteristic
158C2—35no—
159C2—35no—
160C2D3×C22⋊(C22⋊C9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C3D3×C22⋊A4
commutator, hypocenterC24×C3C4×C9
Fitting, socle, prime radical, cosocleC25×C32C2×C3
LCSG ▷ C24×C3
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
4. C3×C22⋊(C22⋊C9)C4semidirect
13. C24×Dic3C9semidirect
22. C22⋊(C22⋊C9)Dic3direct (N × H)
52. C24×C3C4×C9semidirect
61. C22×Dic3C22⋊C9semidirect
104. C24C9×Dic3semidirect
122. C22×C3C4×C22⋊C9semidirect
127. Dic3C22⋊(C22⋊C9)direct (N × H)
152. C22Dic3×C22⋊C9semidirect
156. C3C4×C22⋊(C22⋊C9)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊(C22⋊C9)2
C24×C3×Dic33
C4×C22⋊(C22⋊C9)3
Dic3×C22⋊C94