Finite Group Databasebeta

← C22×C4×C9×A4 #13164821

Subgroups — C22×C4×C9×A4 #13164821

Order 1728.

1167 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1167 conjugacy classes of subgroups — too many to draw. Only its 351 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
1C23×C9×A4C211yesmaximal, characteristic
2C2×C4×C9×A4C216yesmaximal, direct product factor
3C22×C3×C4×A4C311yesmaximal, characteristic
4C24×C4×C9C311yesmaximal, characteristic, complemented
5C22×C4×C22⋊C9C312yesmaximal, complemented
6C22×C9×A4C414yesdirect product factor
7C22×C9×A4C2213yes—
8C22×C3×C4×C9—41nomaximal
9C4×C9×A4C2214yesdirect product factor
10C23×C3×A4C2×C311yescharacteristic
11C2×C3×C4×A4C2×C316yes—
12C25×C9C2×C311yescharacteristic
13C23×C22⋊C9C2×C312yes—
14C23×C4×C9—31no—
15C23×C4×C9—31no—
16C23×C4×C9—36no—
17C23×C4×C9C2×C316yescomplemented
18C2×C4×C22⋊C9C2×C3112yescomplemented
19C23×C3×C9—41no—
20C2×C9×A4C2×C416yesdirect product factor
21C2×C9×A4C2311yescharacteristic
22C2×C3×C4×C9—46no—
23C24×C3×C4C3211yescharacteristic
24C22×C4×A4C913yesdirect product factor
25C22×C3×A4C3×C414yes—
26C22×C3×A4C22×C313yes—
27C22×C32×C4—41no—
28C3×C4×A4C22×C314yes—
29C24×C9—31no—
30C24×C9—34no—
31C24×C9—33no—
32C24×C9C3×C414yescomplemented
33C24×C9C22×C313yes—
34C22×C22⋊C9C3×C418yescomplemented
35C22×C22⋊C9C22×C316yes—
36C22×C4×C9—36no—
37C22×C4×C9—38no—
38C22×C4×C9—31no—
39C22×C4×C9—36no—
40C22×C4×C9—36no—
41C22×C4×C9—36no—
42C22×C4×C9—312no—
43C22×C4×C9—42no—
44C22×C4×C9A411yescharacteristic, direct product factor
45C22×C4×C9C22×C314yescomplemented
46C4×C22⋊C9C22×C318yescomplemented
47C22×C3×C9—44no—
48C22×C3×C9—43no—
49C9×A4C22×C411yescharacteristic, direct product factor
50C3×C4×C9—44no—
51C25×C3C2×C3211yescharacteristic
52C23×A4C2×C913yes—
53C23×C3×C4—31no—
54C23×C3×C4—31no—
55C23×C3×C4—36no—
56C23×C3×C4C2×C3216yes—
57C2×C4×A4C2×C9118yesdirect product factor
58C23×C32—41no—
59C2×C3×A4C2×C3×C416yes—
60C2×C3×A4C23×C311yescharacteristic
61C2×C32×C4—46no—
62C23×C9—33no—
63C23×C9—312no—
64C23×C9—36no—
65C23×C9—32no—
66C23×C9—312no—
67C23×C9—34no—
68C23×C9—34no—
69C23×C9—33no—
70C23×C9—33no—
71C23×C9—42no—
72C23×C9C2×C3×C416yescomplemented
73C23×C9C2×A411yescharacteristic
74C23×C9C23×C311yescharacteristic
75C2×C22⋊C9C2×C3×C4112yescomplemented
76C2×C22⋊C9C23×C312yes—
77C2×C4×C9—312no—
78C2×C4×C9—36no—
79C2×C4×C9—312no—
80C2×C4×C9—34no—
81C2×C4×C9—34no—
82C2×C4×C9—412no—
83C2×C4×C9C2×A416yesdirect product factor
84C24×C4C3×C911yescharacteristic, complemented, Sylow
85C2×C3×C9—46no—
86C2×C3×C9—41no—
87C24×C3—34no—
88C24×C3—33no—
89C24×C3—31no—
90C24×C3C32×C414yes—
91C24×C3C22×C3213yes—
92C22×A4C4×C9112yesdirect product factor
93C22×A4C22×C919yes—
94C22×C3×C4—36no—
95C22×C3×C4—38no—
96C22×C3×C4—31no—
97C22×C3×C4—36no—
98C22×C3×C4—36no—
99C22×C3×C4—36no—
100C22×C3×C4—312no—
101C22×C3×C4—43no—
102C22×C3×C4C3×A411yescharacteristic
103C22×C3×C4C22×C3214yes—
104C4×A4C22×C9112yesdirect product factor
105C22×C32—43no—
106C22×C32—44no—
107C3×A4C22×C3×C411yescharacteristic
108C32×C4—44no—
109C22×C9—36no—
110C22×C9—31no—
111C22×C9—36no—
112C22×C9—38no—
113C22×C9—312no—
114C22×C9—36no—
115C22×C9—36no—
116C22×C9—33no—
117C22×C9—31no—
118C22×C9—46no—
119C22×C9—48no—
120C22×C9C4×A414yesdirect product factor
121C22×C9C22×C3×C411yescharacteristic, complemented
122C22×C9C22×A413yes—
123C22⋊C9C22×C3×C412yescomplemented
124C4×C9—34no—
125C4×C9—48no—
126C4×C9C22×A414yesdirect product factor
127C25C2×C3×C911yescharacteristic
128C23×C4—31no—
129C23×C4—31no—
130C23×C4—36no—
131C23×C4C2×C3×C916yescomplemented
132C3×C9—41noSylow
133C23×C3—33no—
134C23×C3—312no—
135C23×C3—36no—
136C23×C3—32no—
137C23×C3—312no—
138C23×C3—33no—
139C23×C3—33no—
140C23×C3—34no—
141C23×C3—34no—
142C23×C3—43no—
143C23×C3C2×C32×C416yes—
144C23×C3C2×C3×A411yescharacteristic
145C23×C3C23×C3211yescharacteristic
146C2×A4C2×C4×C9118yesdirect product factor
147C2×A4C23×C913yes—
148C2×C3×C4—312no—
149C2×C3×C4—36no—
150C2×C3×C4—312no—
151C2×C3×C4—34no—
152C2×C3×C4—34no—
153C2×C3×C4—418no—
154C2×C3×C4C2×C3×A416yes—
155C2×C32—46no—
156C2×C32—41no—
157C2×C9—36no—
158C2×C9—31no—
159C2×C9—31no—
160C2×C9—412no—
161C2×C9—42no—
162C2×C9C2×C4×A416yesdirect product factor
163C2×C9C23×A411yescharacteristic
164C24—33no—
165C24—34no—
166C24—31no—
167C24C3×C4×C914yescomplemented
168C24C22×C3×C913yes—
169C22×C4—36no—
170C22×C4—38no—
171C22×C4—31no—
172C22×C4—36no—
173C22×C4—36no—
174C22×C4—36no—
175C22×C4—312no—
176C22×C4C9×A411yescharacteristic, direct product factor
177C22×C4C22×C3×C914yescomplemented
178C22×C3—36no—
179C22×C3—31no—
180C22×C3—36no—
181C22×C3—38no—
182C22×C3—36no—
183C22×C3—312no—
184C22×C3—36no—
185C22×C3—31no—
186C22×C3—33no—
187C22×C3—49no—
188C22×C3—412no—
189C22×C3C3×C4×A414yes—
190C22×C3C22×C32×C411yescharacteristic
191C22×C3C22×C3×A413yes—
192A4C22×C4×C913yesdirect product factor
193C3×C4—34no—
194C3×C4—412no—
195C3×C4C22×C3×A414yes—
196C32—41no—
197C9—42no—
198C9C22×C4×A411yescharacteristic, direct product factor
199C23—33no—
200C23—312no—
201C23—36no—
202C23—32no—
203C23—312no—
204C23—33no—
205C23—33no—
206C23—34no—
207C23—34no—
208C23C2×C3×C4×C916yescomplemented
209C23C2×C9×A411yescharacteristic
210C23C23×C3×C911yescharacteristic
211C2×C4—312no—
212C2×C4—36no—
213C2×C4—34no—
214C2×C4—312no—
215C2×C4—34no—
216C2×C4C2×C9×A416yesdirect product factor
217C2×C3—31no—
218C2×C3—36no—
219C2×C3—31no—
220C2×C3—43no—
221C2×C3—418no—
222C2×C3C2×C3×C4×A416yes—
223C2×C3C23×C3×A411yescharacteristic
224C22—36no—
225C22—31no—
226C22—36no—
227C22—38no—
228C22—31no—
229C22—33no—
230C22—36no—
231C22—312no—
232C22—36no—
233C22C4×C9×A414yesdirect product factor
234C22C22×C3×C4×C911yescharacteristic, complemented
235C22C22×C9×A413yes—
236C4—34no—
237C4C22×C9×A414yesdirect product factor
238C3—43no—
239C3C22×C3×C4×A411yescharacteristic
240C2—31no—
241C2—36no—
242C2—31no—
243C2C2×C4×C9×A416yesdirect product factor
244C2C23×C9×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C4×C9A4
commutator, hypocenterC22C22×C3×C4×C9
FrattiniC2×C3C23×C3×A4
FittingC24×C4×C9C3
socleC25×C3C2×C32
prime radicalC23×C3×A4C2×C3
cosocleC23×C3C23×C32
LCSG ▷ C22
UCSC1 ◁ C22×C4×C9

Complements

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

NH ≅ G/Ntype
2. C2×C4×C9×A4C2direct (N × H)
4. C24×C4×C9C3semidirect
5. C22×C4×C22⋊C9C3semidirect
6. C22×C9×A4C4direct (N × H)
9. C4×C9×A4C22direct (N × H)
17. C23×C4×C9C2×C3semidirect
18. C2×C4×C22⋊C9C2×C3semidirect
20. C2×C9×A4C2×C4direct (N × H)
24. C22×C4×A4C9direct (N × H)
32. C24×C9C3×C4semidirect
34. C22×C22⋊C9C3×C4semidirect
44. C22×C4×C9A4direct (N × H)
45. C22×C4×C9C22×C3semidirect
46. C4×C22⋊C9C22×C3semidirect
49. C9×A4C22×C4direct (N × H)
57. C2×C4×A4C2×C9direct (N × H)
72. C23×C9C2×C3×C4semidirect
75. C2×C22⋊C9C2×C3×C4semidirect
83. C2×C4×C9C2×A4direct (N × H)
84. C24×C4C3×C9semidirect
92. C22×A4C4×C9direct (N × H)
104. C4×A4C22×C9direct (N × H)
120. C22×C9C4×A4direct (N × H)
121. C22×C9C22×C3×C4semidirect
123. C22⋊C9C22×C3×C4semidirect
126. C4×C9C22×A4direct (N × H)
131. C23×C4C2×C3×C9semidirect
146. C2×A4C2×C4×C9direct (N × H)
162. C2×C9C2×C4×A4direct (N × H)
167. C24C3×C4×C9semidirect
176. C22×C4C9×A4direct (N × H)
177. C22×C4C22×C3×C9semidirect
192. A4C22×C4×C9direct (N × H)
198. C9C22×C4×A4direct (N × H)
208. C23C2×C3×C4×C9semidirect
216. C2×C4C2×C9×A4direct (N × H)
233. C22C4×C9×A4direct (N × H)
234. C22C22×C3×C4×C9semidirect
237. C4C22×C9×A4direct (N × H)
243. C2C2×C4×C9×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C9×A42
C2×C4×C9×A42
C22×C3×C4×A43
C24×C4×C93
C22×C4×C22⋊C93
C22×C3×C4×C94