Finite Group Databasebeta

← C22×C3⋊(((C22⋊C9)⋊C2)⋊C3) #72244567

Subgroups — C22×C3⋊(((C22⋊C9)⋊C2)⋊C3) #72244567

Order 2592.

1686 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1686 conjugacy classes of subgroups — too many to draw. Only its 142 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×#1094C211yesmaximal, characteristic, complemented
2C2×C3⋊(((C22⋊C9)⋊C2)⋊C3)C216yesmaximal, direct product factor
3C22×C3×#638—31nomaximal
4C22×C3⋊((C22⋊C9)⋊C2)C311yesmaximal, characteristic, complemented
5C22×((C22⋊C9)⋊C2)⋊C3—33nomaximal
6C22×C3⋊AGL(1,ℤ/9)—41nomaximal
7C2×C3×#1094C2213yescomplemented
8C3⋊(((C22⋊C9)⋊C2)⋊C3)C2214yesdirect product factor
9C24×C33D311yescharacteristic
10C23×C3×C3⋊D3—31no—
11C2×C3×#638—36no—
12C2×C3×#638—36no—
13C22×C3×C3⋊Dic3—31no—
14C22×#1094—23no—
15C22×#1094D313yescomplemented
16C22×C3×C22⋊C9—21no—
17C22×C3×C22⋊C9C2×C311yescharacteristic, complemented
18C2×C3⋊((C22⋊C9)⋊C2)C2×C316yescomplemented
19C2×((C22⋊C9)⋊C2)⋊C3—318no—
20C22×C3×3−1+2—41no—
21C2×C3⋊AGL(1,ℤ/9)—46no—
22C3×#1094C2311yescharacteristic, complemented
23C22×#638—31no—
24C22×C3×#67—93no—
25C22×C3×#67—91no—
26C22×(C22⋊C9)⋊C2—33no—
27C23×C33—31no—
28C23×C33—33no—
29C23×C33C2×D313yes—
30C22×C3×C3⋊D3—36no—
31C22×C3×C3⋊D3—63no—
32C22×C3×C3⋊D3—61no—
33C3×#638—312no—
34C3×#638—34no—
35C2×C3×C3⋊Dic3—36no—
36C22×C3⋊D9—41no—
37C22×AGL(1,ℤ/9)—123no—
38C2×#1094—29no—
39C2×#1094C2×D319yescomplemented
40C2×C3×C22⋊C9—23no—
41C2×C3×C22⋊C9C22×C313yescomplemented
42C3⋊((C22⋊C9)⋊C2)C22×C314yescomplemented
43((C22⋊C9)⋊C2)⋊C3—312no—
44C2×C3×3−1+2—43no—
45C3⋊AGL(1,ℤ/9)—44no—
46C24×C32—21no—
47C24×C32—33no—
48C24×C32C3×D311yescharacteristic
49C24×C32C3⋊D311yescharacteristic
50C23×C3⋊D3—31no—
51C23×C3×D3—93no—
52C23×C3×D3—91no—
53C2×#638—36no—
54C2×#638—36no—
55C22×C3⋊Dic3—31no—
56C2×C3×#67—96no—
57C2×C3×#67—918no—
58C2×C3×#67—918no—
59C2×C3×#67—96no—
60C22×C3×Dic3—93no—
61C22×C3×Dic3—91no—
62C22×C22⋊C9—23no—
63C22×C22⋊C9C3×D313yescomplemented
64C2×(C22⋊C9)⋊C2—318no—
65C22×C33—33no—
66C22×C33—33no—
67C22×C33—33no—
68C22×C33—61no—
69C22×C33S411yescharacteristic
70C22×C33C22×D311yescharacteristic
71C2×C3×C3⋊D3—34no—
72C2×C3×C3⋊D3—66no—
73C2×C3×C3⋊D3—66no—
74C3×C3⋊Dic3—34no—
75C22×3−1+2—43no—
76C22×3−1+2—83no—
77C22×C3×C9—41no—
78C22×C3×C9—81no—
79C2×C3⋊D9—46no—
80C2×AGL(1,ℤ/9)—1218no—
81#1094—23no—
82#1094C22×D313yescomplemented
83C3×C22⋊C9—21no—
84C3×C22⋊C9C23×C311yescharacteristic, complemented
85C22×C3×D4—271no—
86C22×#67—93no—
87C22×#67—91no—
88C3×3−1+2—41noSylow
89C23×C32—23no—
90C23×C32—31no—
91C23×C32—33no—
92C23×C32—33no—
93C23×C32—31no—
94C23×C32—33no—
95C23×C32—39no—
96C23×C32—39no—
97C23×C32—61no—
98C23×C32—63no—
99C23×C32—69no—
100C23×C32—63no—
101C23×C32C2×C3×D313yes—
102C23×C32C2×C3⋊D313yes—
103C22×C3⋊D3—36no—
104C22×C3⋊D3—63no—
105C22×C3⋊D3—61no—
106C22×C3×D3—918no—
107C22×C3×D3—96no—
108C22×C3×D3—189no—
109C22×C3×D3—183no—
110C22×C3×D3—183no—
111C22×C3×D3—181no—
112#638—312no—
113#638—34no—
114C2×C3⋊Dic3—36no—
115C3×#67—912no—
116C3×#67—936no—
117C3×#67—912no—
118C3×#67—94no—
119C2×C3×Dic3—918no—
120C2×C3×Dic3—96no—
121C22×D9—123no—
122C2×C22⋊C9—29no—
123C2×C22⋊C9C2×C3×D319yescomplemented
124(C22⋊C9)⋊C2—312no—
125C2×C33—31no—
126C2×C33—33no—
127C2×C33C2×S413yes—
128C3×C3⋊D3—64no—
129C2×3−1+2—49no—
130C2×3−1+2—89no—
131C2×C3×C9—43no—
132C2×C3×C9—83no—
133C3⋊D9—44no—
134AGL(1,ℤ/9)—1212no—
135C24×C3—31no—
136C24×C3—61no—
137C24×C3—271no—
138C24×C3AGL(1,ℤ/9)13yescomplemented
139C24×C3C3×C3⋊D311yescharacteristic
140C23×D3—93no—
141C23×D3—91no—
142C2×C3×D4—276no—
143C2×C3×D4—276no—
144C22×C3×C4—271no—
145C2×#67—918no—
146C2×#67—96no—
147C2×#67—918no—
148C2×#67—96no—
149C22×Dic3—93no—
150C22×Dic3—91no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—33no—
154C22×C32—33no—
155C22×C32—33no—
156C22×C32—33no—
157C22×C32—39no—
158C22×C32—39no—
159C22×C32—33no—
160C22×C32—33no—
161C22×C32—39no—
162C22×C32—33no—
163C22×C32—33no—
164C22×C32—63no—
165C22×C32—63no—
166C22×C32—69no—
167C22×C32—69no—
168C22×C32—69no—
169C22×C32—63no—
170C22×C32—61no—
171C22×C32—61no—
172C22×C32—69no—
173C22×C32—62no—
174C22×C32C3×S411yescharacteristic
175C22×C32C3⋊S411yescharacteristic
176C22×C32C22×C3×D311yescharacteristic
177C22×C32C22×C3⋊D311yescharacteristic
178C2×C3⋊D3—34no—
179C2×C3⋊D3—66no—
180C2×C3⋊D3—66no—
181C2×C3×D3—912no—
182C2×C3×D3—94no—
183C2×C3×D3—1818no—
184C2×C3×D3—186no—
185C2×C3×D3—1818no—
186C2×C3×D3—186no—
187C3⋊Dic3—34no—
188C3×Dic3—912no—
189C3×Dic3—94no—
190C22×C9—43no—
191C22×C9—83no—
192C2×D9—1218no—
193C22⋊C9—23no—
194C22⋊C9C22×C3×D313yescomplemented
195C22×D4—271noSylow
196C33C22×S411yescharacteristic
1973−1+2—43no—
1983−1+2—83no—
199C3×C9—41no—
200C3×C9—81no—
201C23×C3—31no—
202C23×C3—33no—
203C23×C3—31no—
204C23×C3—39no—
205C23×C3—33no—
206C23×C3—33no—
207C23×C3—33no—
208C23×C3—61no—
209C23×C3—63no—
210C23×C3—63no—
211C23×C3—69no—
212C23×C3—63no—
213C23×C3—276no—
214C23×C3—543no—
215C23×C3—541no—
216C23×C3C2×AGL(1,ℤ/9)19yescomplemented
217C23×C3C2×C3×C3⋊D313yes—
218C22×D3—96no—
219C22×D3—918no—
220C22×D3—183no—
221C22×D3—189no—
222C22×D3—183no—
223C22×D3—181no—
224C3×D4—2712no—
225C3×D4—274no—
226C2×C3×C4—276no—
227#67—936no—
228#67—912no—
229#67—912no—
230#67—94no—
231C2×Dic3—96no—
232C2×Dic3—918no—
233C2×C32—23no—
234C2×C32—31no—
235C2×C32—33no—
236C2×C32—33no—
237C2×C32—39no—
238C2×C32—31no—
239C2×C32—33no—
240C2×C32—39no—
241C2×C32—61no—
242C2×C32—63no—
243C2×C32—63no—
244C2×C32—69no—
245C2×C32C2×C3×S413yes—
246C2×C32C2×C3⋊S413yes—
247C3⋊D3—64no—
248C3×D3—1812no—
249C3×D3—184no—
250C2×C9—49no—
251C2×C9—89no—
252D9—1212no—
253C24—271no—
254C24C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
255C2×D4—276no—
256C2×D4—276no—
257C22×C4—271no—
258C22×C3—39no—
259C22×C3—33no—
260C22×C3—33no—
261C22×C3—33no—
262C22×C3—33no—
263C22×C3—33no—
264C22×C3—39no—
265C22×C3—39no—
266C22×C3—33no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—63no—
270C22×C3—63no—
271C22×C3—63no—
272C22×C3—69no—
273C22×C3—69no—
274C22×C3—69no—
275C22×C3—63no—
276C22×C3—61no—
277C22×C3—63no—
278C22×C3—66no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—274no—
282C22×C3—546no—
283C22×C3—546no—
284C22×C3((C22⋊C9)⋊C2)⋊C313yescomplemented
285C22×C3C22×AGL(1,ℤ/9)13yescomplemented
286C22×C3C3×C3⋊S411yescharacteristic
287C22×C3C22×C3×C3⋊D311yescharacteristic
288C2×D3—912no—
289C2×D3—94no—
290C2×D3—1818no—
291C2×D3—186no—
292C2×D3—186no—
293C2×D3—1818no—
294C3×C4—274no—
295Dic3—912no—
296Dic3—94no—
297C32—21no—
298C32—33no—
299C32C22×C3×S411yescharacteristic
300C32C22×C3⋊S411yescharacteristic
301C9—43no—
302C9—83no—
303C23—31no—
304C23—33no—
305C23—276no—
306C23—543no—
307C23—541no—
308C23C2×C3⋊AGL(1,ℤ/9)13yescomplemented
309D4—2712no—
310D4—274no—
311C2×C4—276no—
312C2×C3—31no—
313C2×C3—33no—
314C2×C3—33no—
315C2×C3—39no—
316C2×C3—31no—
317C2×C3—33no—
318C2×C3—33no—
319C2×C3—61no—
320C2×C3—63no—
321C2×C3—63no—
322C2×C3—69no—
323C2×C3—63no—
324C2×C3—544no—
325C2×C3C2×((C22⋊C9)⋊C2)⋊C319yescomplemented
326C2×C3C2×C3×C3⋊S413yes—
327D3—1812no—
328D3—184no—
329C22—33no—
330C22—33no—
331C22—33no—
332C22—61no—
333C22—274no—
334C22—546no—
335C22—546no—
336C22C3⋊(((C22⋊C9)⋊C2)⋊C3)11yescharacteristic, direct product factor
337C22C22×C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
338C4—274no—
339C3—31no—
340C3—61no—
341C3C22×((C22⋊C9)⋊C2)⋊C313yescomplemented
342C3C22×C3×C3⋊S411yescharacteristic
343C2—31no—
344C2—33no—
345C2—544no—
346C2C2×C3⋊(((C22⋊C9)⋊C2)⋊C3)13yesdirect product factor
subgroupG/H
center, hypercenterC22C3⋊(((C22⋊C9)⋊C2)⋊C3)
commutator, cosocle, hypocenterC3×C22⋊C9C23×C3
FrattiniC3C22×C3×C3⋊S4
FittingC24×C33D3
socleC24×C32C3×D3
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×#1094C2semidirect
2. C2×C3⋊(((C22⋊C9)⋊C2)⋊C3)C2direct (N × H)
4. C22×C3⋊((C22⋊C9)⋊C2)C3semidirect
7. C2×C3×#1094C22semidirect
8. C3⋊(((C22⋊C9)⋊C2)⋊C3)C22direct (N × H)
15. C22×#1094D3semidirect
17. C22×C3×C22⋊C9C2×C3semidirect
18. C2×C3⋊((C22⋊C9)⋊C2)C2×C3semidirect
22. C3×#1094C23semidirect
39. C2×#1094C2×D3semidirect
41. C2×C3×C22⋊C9C22×C3semidirect
42. C3⋊((C22⋊C9)⋊C2)C22×C3semidirect
63. C22×C22⋊C9C3×D3semidirect
82. #1094C22×D3semidirect
84. C3×C22⋊C9C23×C3semidirect
123. C2×C22⋊C9C2×C3×D3semidirect
138. C24×C3AGL(1,ℤ/9)semidirect
194. C22⋊C9C22×C3×D3semidirect
216. C23×C3C2×AGL(1,ℤ/9)semidirect
254. C24C3⋊AGL(1,ℤ/9)semidirect
284. C22×C3((C22⋊C9)⋊C2)⋊C3semidirect
285. C22×C3C22×AGL(1,ℤ/9)semidirect
308. C23C2×C3⋊AGL(1,ℤ/9)semidirect
325. C2×C3C2×((C22⋊C9)⋊C2)⋊C3semidirect
336. C22C3⋊(((C22⋊C9)⋊C2)⋊C3)direct (N × H)
337. C22C22×C3⋊AGL(1,ℤ/9)semidirect
341. C3C22×((C22⋊C9)⋊C2)⋊C3semidirect
346. C2C2×C3⋊(((C22⋊C9)⋊C2)⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#10942
C2×C3⋊(((C22⋊C9)⋊C2)⋊C3)2
C22×C3×#6383
C22×C3⋊((C22⋊C9)⋊C2)3
C22×((C22⋊C9)⋊C2)⋊C33
C22×C3⋊AGL(1,ℤ/9)4