Finite Group Databasebeta

← C4⋊(Dic9⋊A4) #13147155

Subgroups — C4⋊(Dic9⋊A4) #13147155

Order 1728.

355 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 73 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×C4×C9⋊A4C211yesmaximal, characteristic
2C2×Dic9⋊A4C212yesmaximal
3A4×C4⋊Dic3—31nomaximal
4C22×C4⋊Dic9C311yesmaximal, characteristic, complemented
5C22×C9⋊A4C2211yescharacteristic
6C4×C9⋊A4C412yescomplemented
7Dic9⋊A4—22no—
8C4⋊(3−1+2⋊C4)—41nomaximal
9C2×C3×C4×A4D311yescharacteristic
10C2×Dic3×A4—32no—
11C23×C4×C9C2×C311yescharacteristic
12C23×Dic9C2×C312yes—
13C2×C4×C22⋊C9—21no—
14C2×C4⋊Dic9—31no—
15C2×C4⋊Dic9—31no—
16C2×C4⋊Dic9—32no—
17C2×C9⋊A4C2×C411yescharacteristic
18C2×C9⋊A4D411yescharacteristic
19C2×C9⋊A4Q811yescharacteristic
20C2×C4×3−1+2—41no—
21C2×3−1+2⋊C4—42no—
22C22×C4⋊Dic3—31no—
23A4×C4⋊C4—91no—
24C22×C3×A4C2×D311yescharacteristic
25C3×C4×A4Dic312yes—
26Dic3×A4—62no—
27C24×C9C22×C311yescharacteristic
28C22×C22⋊C9—21no—
29C3×C4⋊Dic3—121no—
30C22×C4×C9—31no—
31C22×C4×C9—32no—
32C22×C4×C9—31no—
33C22×C4×C9C3×C412yescomplemented
34C22×Dic9—22no—
35C22×Dic9—32no—
36C22×Dic9—32no—
37C22×Dic9—62no—
38C4×C22⋊C9—22no—
39C4⋊Dic9—32no—
40C4⋊Dic9—31no—
41C4⋊Dic9—32no—
42C4⋊Dic9A411yescharacteristic, complemented
43C22×3−1+2—41no—
44C9⋊A4C4⋊C411yescharacteristic, complemented
45C4×3−1+2—42no—
463−1+2⋊C4—82no—
47C23×C3×C4C3×D311yescharacteristic
48C23×Dic3—32no—
49C2×C4×A4—31no—
50C2×C4×A4—92no—
51C2×C4⋊Dic3—91no—
52C2×C4⋊Dic3—92no—
53C2×C4⋊Dic3—91no—
54C2×C3×A4Dic611yescharacteristic
55C2×C3×A4D1211yescharacteristic
56C2×C3×A4C2×Dic311yescharacteristic
57C2×C32×C4—41no—
58C2×C3×Dic3—122no—
59C23×C9—31no—
60C23×C9—31no—
61C23×C9—31no—
62C23×C9—31no—
63C23×C9C2×C3×C411yescharacteristic
64C23×C9C3×D411yescharacteristic
65C23×C9C3×Q811yescharacteristic
66C2×C22⋊C9—21no—
67C2×C22⋊C9—21no—
68C2×C22⋊C9—21no—
69C2×C4×C9—32no—
70C2×C4×C9—31no—
71C2×C4×C9—32no—
72C2×C4×C9—32no—
73C2×C4×C9—32no—
74C2×C4×C9—81no—
75C2×C4×C9C2×A411yescharacteristic
76C2×Dic9—32no—
77C2×Dic9—62no—
78C2×Dic9—62no—
79C2×Dic9—62no—
80C2×Dic9—62no—
81C2×Dic9C2×A412yes—
82C22×C4⋊C4—91noSylow
83C2×3−1+2—41no—
84C2×3−1+2—41no—
85C2×3−1+2—41no—
86C24×C3C2×C3×D311yescharacteristic
87C22×A4—31no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4—32no—
91C22×C3×C4C3×Dic312yes—
92C22×Dic3—62no—
93C22×Dic3—92no—
94C22×Dic3—92no—
95C22×Dic3—182no—
96C4×A4—32no—
97C4×A4—182no—
98C3×C4⋊C4—361no—
99C4⋊Dic3—31no—
100C4⋊Dic3—92no—
101C4⋊Dic3—92no—
102C4⋊Dic3—91no—
103C22×C32—41no—
104C3×A4C4⋊Dic311yescharacteristic
105C32×C4—42no—
106C3×Dic3—242no—
107C22×C9—31no—
108C22×C9—31no—
109C22×C9—31no—
110C22×C9—32no—
111C22×C9—31no—
112C22×C9—31no—
113C22×C9—31no—
114C22×C9—31no—
115C22×C9—31no—
116C22×C9—31no—
117C22×C9—81no—
118C22×C9C3×C4⋊C411yescharacteristic, complemented
119C22×C9C22×A411yescharacteristic
120C22⋊C9—21no—
121C4×C9—32no—
122C4×C9—82no—
123C4×C9C4×A412yescomplemented
124Dic9—22no—
125Dic9—62no—
126C23×C4—92no—
127C23×C4AGL(1,ℤ/9)11yescharacteristic
128C2×C4⋊C4—271no—
129C2×C4⋊C4—272no—
130C2×C4⋊C4—271no—
1313−1+2—41noSylow
132C23×C3—31no—
133C23×C3—31no—
134C23×C3—31no—
135C23×C3—31no—
136C23×C3C3×Dic611yescharacteristic
137C23×C3C3×D1211yescharacteristic
138C23×C3C2×C3×Dic311yescharacteristic
139C2×A4—31no—
140C2×A4—31no—
141C2×A4—31no—
142C2×C3×C4—32no—
143C2×C3×C4—31no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4—121no—
148C2×C3×C4—362no—
149C2×C3×C4D3×A411yescharacteristic
150C2×Dic3—32no—
151C2×Dic3—92no—
152C2×Dic3—182no—
153C2×Dic3—182no—
154C2×Dic3—182no—
155C2×Dic3—182no—
156C2×C32—41no—
157C2×C32—41no—
158C2×C32—41no—
159C2×C9—31no—
160C2×C9—31no—
161C2×C9—31no—
162C2×C9—31no—
163C2×C9—81no—
164C2×C9—81no—
165C2×C9—81no—
166C2×C9C2×C4×A411yescharacteristic
167C2×C9D4×A411yescharacteristic
168C2×C9Q8×A411yescharacteristic
169C24C2×AGL(1,ℤ/9)11yescharacteristic
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—182no—
174C22×C4—272no—
175C22×C4—272no—
176C22×C4—542no—
177C22×C43−1+2⋊C412yescomplemented
178C4⋊C4—91no—
179C4⋊C4—272no—
180C4⋊C4—272no—
181C4⋊C4—271no—
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—32no—
186C22×C3—31no—
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—31no—
190C22×C3—31no—
191C22×C3—31no—
192C22×C3—121no—
193C22×C3C3×C4⋊Dic311yescharacteristic
194C22×C3C2×D3×A411yescharacteristic
195A4—31no—
196C3×C4—32no—
197C3×C4—122no—
198C3×C4—722no—
199C3×C4Dic3×A412yes—
200Dic3—62no—
201Dic3—182no—
202C32—41no—
203C9—81no—
204C9A4×C4⋊C411yescharacteristic, complemented
205C23—31no—
206C23—31no—
207C23—31no—
208C23—31no—
209C233−1+2⋊Q811yescharacteristic
210C23C4⋊AGL(1,ℤ/9)11yescharacteristic
211C23C2×3−1+2⋊C411yescharacteristic
212C2×C4—32no—
213C2×C4—31no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—32no—
217C2×C4—92no—
218C2×C4—272no—
219C2×C4—542no—
220C2×C4—542no—
221C2×C4—542no—
222C2×C4—542no—
223C2×C4D9⋊A411yescharacteristic
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—31no—
228C2×C3—121no—
229C2×C3—121no—
230C2×C3—121no—
231C2×C3A4×Dic611yescharacteristic
232C2×C3A4×D1211yescharacteristic
233C2×C3C2×Dic3×A411yescharacteristic
234C22—31no—
235C22—31no—
236C22—31no—
237C22—32no—
238C22—31no—
239C22—31no—
240C22—31no—
241C22—31no—
242C22—31no—
243C22—31no—
244C22C4⋊(3−1+2⋊C4)11yescharacteristic, complemented
245C22C2×D9⋊A411yescharacteristic
246C4—32no—
247C4—182no—
248C4—542no—
249C4Dic9⋊A412yescomplemented
250C3—121no—
251C3A4×C4⋊Dic311yescharacteristic
252C2—31no—
253C2—31no—
254C2—31no—
255C2—31no—
256C2Dic18⋊A411yescharacteristic
257C2C4⋊(D9⋊A4)11yescharacteristic
258C2C2×Dic9⋊A411yescharacteristic
subgroupG/H
centerC22C2×D9⋊A4
commutatorC23×C9C2×C3×C4
FrattiniC22×C3C2×D3×A4
FittingC23×C4×C9C2×C3
socleC24×C3C2×C3×D3
prime radicalC22×C3×A4C2×D3
cosocleC24×C9C22×C3
hypercenterC2×C4D9⋊A4
hypocenterC22×C9C3×C4⋊C4
LCSG ▷ C23×C9 ▷ C22×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊Dic9C3semidirect
6. C4×C9⋊A4C4semidirect
33. C22×C4×C9C3×C4semidirect
42. C4⋊Dic9A4semidirect
44. C9⋊A4C4⋊C4semidirect
118. C22×C9C3×C4⋊C4semidirect
123. C4×C9C4×A4semidirect
177. C22×C43−1+2⋊C4semidirect
204. C9A4×C4⋊C4semidirect
244. C22C4⋊(3−1+2⋊C4)semidirect
249. C4Dic9⋊A4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C9⋊A42
C2×Dic9⋊A42
A4×C4⋊Dic33
C22×C4⋊Dic93
C4⋊(3−1+2⋊C4)4