Finite Group Databasebeta

← C22⋊A4×AGL(1,ℤ/9) #72220098

Subgroups — C22⋊A4×AGL(1,ℤ/9) #72220098

Order 2592.

1062 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 62 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
13−1+2×C22⋊A4C211yesmaximal, characteristic, complemented
2C3×D3×C22⋊A4—31nomaximal
3C24×AGL(1,ℤ/9)C311yesmaximal, characteristic, complemented
4D9×C22⋊A4C311yesmaximal, characteristic, complemented
5D9⋊(C22⋊A4)C312yesmaximal, complemented
6A4×AGL(1,ℤ/9)—45nomaximal
7C32×C22⋊A4D311yescharacteristic
8C24×3−1+2C2×C311yescharacteristic, complemented
9C23×AGL(1,ℤ/9)—35no—
10C23×AGL(1,ℤ/9)—35no—
11#88907—21no—
12C9⋊(C22⋊A4)—22no—
13C9⋊(C22⋊A4)C2×C312yescomplemented
14C9×C22⋊A4—21no—
15C9×C22⋊A4C2×C311yescharacteristic, complemented
16A4×3−1+2—45no—
17C24×C3×D3—31no—
18C2×C3×C22⋊A4—91no—
19D3×C22⋊A4—32no—
20D3×C22⋊A4—31no—
21C24×D9C3211yescharacteristic, complemented
22C3×D3×A4—125no—
23C23×3−1+2—35no—
24C22×AGL(1,ℤ/9)—330no—
25C22×AGL(1,ℤ/9)—310no—
26C22×AGL(1,ℤ/9)—35no—
27C22×AGL(1,ℤ/9)A415yescomplemented
28A4×D9—45no—
29D9⋊A4—410no—
30C3×AGL(1,ℤ/9)—161no—
31C24×C32C3×D311yescharacteristic
32C23×C3×D3—95no—
33C23×C3×D3—95no—
34C3×C22⋊A4—31no—
35C3×C22⋊A4—61no—
36C3×C22⋊A4C3×D311yescharacteristic
37C3×C22⋊A4C3×D312yes—
38C24×C9—21no—
39C24×C9C2×C3211yescharacteristic, complemented
40C23×D9—35no—
41C23×D9—35no—
42C22⋊(C22⋊C9)—22no—
43C22⋊(C22⋊C9)—21no—
44C32×A4—45no—
45C22×3−1+2—310no—
46C22×3−1+2C2×A415yescomplemented
47C2×AGL(1,ℤ/9)—330no—
48C2×AGL(1,ℤ/9)—35no—
49C2×AGL(1,ℤ/9)—35no—
50#1094—85no—
51C9⋊A4—410no—
52C9⋊A4—810no—
53C9×A4—45no—
54C9×A4—85no—
55C25×C3—91no—
56C24×D3—31no—
57C2×C22⋊A4—92no—
58C2×C22⋊A4—91no—
59C3×3−1+2—161noSylow
60C23×C32—35no—
61C22×C3×D3—35no—
62C22×C3×D3—910no—
63C22×C3×D3—930no—
64C22×C3×D3—95no—
65C2×C3×A4—365no—
66D3×A4—1210no—
67D3×A4—125no—
68C23×C9—35no—
69C23×C9—65no—
70C22×D9—330no—
71C22×D9—310no—
72C22×D9—35no—
73C22×D9C3×A415yescomplemented
74C32×D3—481no—
75C2×3−1+2—35no—
76AGL(1,ℤ/9)—35no—
77AGL(1,ℤ/9)—162no—
78AGL(1,ℤ/9)C22⋊A411yescharacteristic, direct product factor
79C3×D9—161no—
80C24×C3—31no—
81C24×C3—275no—
82C24×C3—275no—
83C24×C3C32×D311yescharacteristic
84C23×D3—95no—
85C23×D3—95no—
86C22⋊A4—21no—
87C22⋊A4—32no—
88C22⋊A4AGL(1,ℤ/9)11yescharacteristic, direct product factor
89C22×C32—310no—
90C22×C32D3×A415yes—
91C2×C3×D3—95no—
92C2×C3×D3—930no—
93C2×C3×D3—95no—
94C3×A4—45no—
95C3×A4—410no—
96C3×A4—125no—
97C3×A4—245no—
98C22×C9—25no—
99C22×C9—310no—
100C22×C9—610no—
101C22×C9C2×C3×A415yescomplemented
102C2×D9—330no—
103C2×D9—35no—
104C2×D9—35no—
105C22⋊C9—85no—
106C22⋊C9—810no—
107C25—91noSylow
108C33—161no—
1093−1+2—162no—
1103−1+2—322no—
1113−1+2—321no—
1123−1+2C2×C22⋊A411yescharacteristic, complemented
113C3×C9—161no—
114C3×C9—321no—
115C23×C3—35no—
116C23×C3—95no—
117C23×C3—95no—
118C23×C3—275no—
119C23×C3—2710no—
120C23×C3—2730no—
121C22×D3—35no—
122C22×D3—910no—
123C22×D3—930no—
124C22×D3—95no—
125C2×A4—365no—
126C2×A4—3610no—
127C2×C32—35no—
128C2×C32—1441no—
129C3×D3—31no—
130C3×D3—95no—
131C3×D3—481no—
132C3×D3—482no—
133C2×C9—35no—
134C2×C9—65no—
135D9—35no—
136D9C3×C22⋊A411yescharacteristic, complemented
137C24—275no—
138C24—275no—
139C24C3×AGL(1,ℤ/9)11yescharacteristic, complemented
140C22×C3—310no—
141C22×C3—35no—
142C22×C3—910no—
143C22×C3—275no—
144C22×C3—275no—
145C22×C3—2730no—
146C22×C3C3×D3×A415yes—
147C2×D3—95no—
148C2×D3—930no—
149C2×D3—95no—
150A4—45no—
151A4—85no—
152A4—1210no—
153C32—161no—
154C32—162no—
155C32—481no—
156C32—961no—
157C32D3×C22⋊A411yescharacteristic
158C9—21no—
159C9—321no—
160C9—322no—
161C9C2×C3×C22⋊A411yescharacteristic, complemented
162C23—35no—
163C23—95no—
164C23—2710no—
165C23—2730no—
166C23—275no—
167C2×C3—35no—
168C2×C3—95no—
169C2×C3—91no—
170C2×C3—275no—
171C2×C3—1441no—
172C2×C3—1442no—
173D3—31no—
174D3—95no—
175C22—310no—
176C22—275no—
177C22—275no—
178C22—2730no—
179C22A4×AGL(1,ℤ/9)15yescomplemented
180C3—31no—
181C3—161no—
182C3—321no—
183C3—482no—
184C3C3×D3×C22⋊A411yescharacteristic
185C2—35no—
186C2—91no—
187C2—275no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC24×C9C2×C32
FrattiniC3C3×D3×C22⋊A4
FittingC24×3−1+2C2×C3
socleC24×C3C32×D3
LCSG ▷ C24×C9
UCSC1

Complements

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

NH ≅ G/Ntype
1. 3−1+2×C22⋊A4C2semidirect
3. C24×AGL(1,ℤ/9)C3semidirect
4. D9×C22⋊A4C3semidirect
5. D9⋊(C22⋊A4)C3semidirect
8. C24×3−1+2C2×C3semidirect
13. C9⋊(C22⋊A4)C2×C3semidirect
15. C9×C22⋊A4C2×C3semidirect
21. C24×D9C32semidirect
27. C22×AGL(1,ℤ/9)A4semidirect
39. C24×C9C2×C32semidirect
46. C22×3−1+2C2×A4semidirect
73. C22×D9C3×A4semidirect
78. AGL(1,ℤ/9)C22⋊A4direct (N × H)
88. C22⋊A4AGL(1,ℤ/9)direct (N × H)
101. C22×C9C2×C3×A4semidirect
112. 3−1+2C2×C22⋊A4semidirect
136. D9C3×C22⋊A4semidirect
139. C24C3×AGL(1,ℤ/9)semidirect
161. C9C2×C3×C22⋊A4semidirect
179. C22A4×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊A42
C3×D3×C22⋊A43
C24×AGL(1,ℤ/9)3
D9×C22⋊A43
D9⋊(C22⋊A4)3
A4×AGL(1,ℤ/9)4