Finite Group Databasebeta

← C9×#5052 #13142867

Subgroups — C9×#5052 #13142867

Order 1728.

404 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic
2C2×C9×#216C211yesmaximal, characteristic, complemented
3C9×#924C211yesmaximal, characteristic
4C2×C9×#208C211yesmaximal, characteristic
5C9×#835C211yesmaximal, characteristic, complemented
6C9×#810C211yesmaximal, characteristic, complemented
7C9×#808C211yesmaximal, characteristic, complemented
8C3×#5052C311yesmaximal, characteristic
9C9×#470—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×C4⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C22⋊C4—21no—
16C9×#217—21no—
17C9×#216—22no—
18C9×C4⋊Dic3C2211yescharacteristic
19C9×Dic3⋊C4—21no—
20C9×#208—22no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23C32×C22⋊Q8C2×C311yescharacteristic
24C2×C3×#216C2×C311yescharacteristic
25C3×#924C2×C311yescharacteristic
26C2×C3×#208C2×C311yescharacteristic
27C3×#835C2×C311yescharacteristic
28C3×#810C2×C311yescharacteristic
29C3×#808C2×C311yescharacteristic
30C2×C9×Q16—31no—
31C2×C9×M16—31no—
32C9×C22⋊Q8—21no—
33C9×C22⋊Q8—31no—
34C9×C22⋊Q8D311yescharacteristic
35C9×#106—31no—
36C9×#103—31no—
37C9×#103—31no—
38C23×C3×C9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9D411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C2×C9×Dic3—21no—
48C2×C9×Dic3—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C9×C3⋊C8—21no—
52C9×C3⋊C8D412yes—
53C3×#470—31no—
54#5052C911yescharacteristic, direct product factor
55C2×C32×Q8C22×C311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C2×C3×Dic6C22×C311yescharacteristic
58C32×C4⋊C4—21no—
59C32×C4⋊C4C22×C311yescharacteristic
60C32×C22⋊C4—21no—
61C3×#217—21no—
62C3×#216—22no—
63C3×C4⋊Dic3C22×C311yescharacteristic
64C3×Dic3⋊C4—21no—
65C3×#208—22no—
66C2×C3×C3⋊C8C22×C311yescharacteristic
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C2×Q8×C9—21no—
69C2×Q8×C9—31no—
70C2×Q8×C9C2×D311yescharacteristic
71C22×C4×C9—21no—
72C22×C4×C9C2×D311yescharacteristic
73C9×Q16—62no—
74C9×M16—62no—
75C2×C8×C9—31no—
76C2×C8×C9—31no—
77C9×C4⋊C4—22no—
78C9×C4⋊C4—21no—
79C9×C4⋊C4—21no—
80C9×C4⋊C4—31no—
81C9×C4⋊C4—61no—
82C9×C4⋊C4C2×D311yescharacteristic
83C9×C22⋊C4—22no—
84C9×C22⋊C4—21no—
85C9×C22⋊C4—61no—
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9—21no—
89C22×C3×C9C2×D411yescharacteristic
90C3×C4×C9—21no—
91C3×C4×C9—41no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C9×Dic3—41no—
96C9×Dic3—41no—
97C2×C3×Q16—31no—
98C2×C3×M16—31no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—31no—
101C3×C22⋊Q8C2×C911yescharacteristic
102C3×C22⋊Q8C3×D311yescharacteristic
103C2×#216C2×C911yescharacteristic, complemented
104#924C2×C911yescharacteristic
105C2×#208C2×C911yescharacteristic
106C3×#106—31no—
107C3×#103—31no—
108C3×#103—31no—
109#835C2×C911yescharacteristic, complemented
110#810C2×C911yescharacteristic, complemented
111#808C2×C911yescharacteristic, complemented
112C23×C32C3×D411yescharacteristic
113C32×Q8—21no—
114C32×Q8—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4C3×D411yescharacteristic
120C2×C32×C4C23×C311yescharacteristic
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C23×C9—21no—
126C23×C9#6711yescharacteristic
127C3×C3⋊C8—21no—
128C3×C3⋊C8C3×D412yes—
129Q8×C9—21no—
130Q8×C9—21no—
131Q8×C9—41no—
132Q8×C9—41no—
133Q8×C9—61no—
134Q8×C9—61no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—22no—
138C2×C4×C9—22no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—41no—
144C2×C4×C9—41no—
145C2×C4×C9—61no—
146C2×C4×C9—61no—
147C2×C4×C9#6711yescharacteristic
148C2×C4×C9C22×D311yescharacteristic
149C8×C9—32no—
150C8×C9—61no—
151#470—31noSylow
152C2×C3×C9—41no—
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9#13711yescharacteristic
155C2×C3×C9#13711yescharacteristic
156C2×C3×Q8—21no—
157C2×C3×Q8—31no—
158C2×C3×Q8C22×C911yescharacteristic
159C2×C3×Q8C2×C3×D311yescharacteristic
160C22×C3×C4—21no—
161C22×C3×C4C22×C911yescharacteristic
162C22×C3×C4C2×C3×D311yescharacteristic
163C2×Dic6C22×C911yescharacteristic
164C3×Q16—62no—
165C3×M16—62no—
166C2×C3×C8—31no—
167C2×C3×C8—31no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4C22×C911yescharacteristic
175C3×C4⋊C4C2×C3×D311yescharacteristic
176C3×C22⋊C4—22no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—61no—
180#217—21no—
181#216—22no—
182C4⋊Dic3C22×C911yescharacteristic
183Dic3⋊C4—21no—
184#208—22no—
185C2×C3⋊C8C22×C911yescharacteristic
186C2×C3⋊C8C22×C911yescharacteristic
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32C2×C3×D411yescharacteristic
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4C2×C3×D411yescharacteristic
195C32×C4C3×#4111yescharacteristic
196C3×Dic3—41no—
197C3×Dic3—41no—
198C22×C9—22no—
199C22×C9—21no—
200C22×C9—21no—
201C22×C9—21no—
202C22×C9—21no—
203C22×C9—41no—
204C22×C9—41no—
205C22×C9C2×#6711yescharacteristic
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—41no—
210C4×C9—42no—
211C4×C9—42no—
212C4×C9—41no—
213C4×C9—41no—
214C4×C9—121no—
215C4×C9—121no—
216C4×C9D3×D411yescharacteristic
217C4×C9#23711yescharacteristic
218C2×Q16—31no—
219C2×M16—31no—
220C22⋊Q8—31no—
221C22⋊Q8D3×C911yescharacteristic
222#106—31no—
223#103—31no—
224#103—31no—
225C3×C9#47011yescharacteristic, complemented, Sylow
226C23×C3—21no—
227C23×C3D4×C911yescharacteristic
228C23×C3C3×#6711yescharacteristic
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—61no—
236C3×Q8—61no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4D4×C911yescharacteristic
254C2×C3×C4C23×C911yescharacteristic
255C2×C3×C4C3×#6711yescharacteristic
256C2×C3×C4C22×C3×D311yescharacteristic
257C2×Dic3—21no—
258C2×Dic3—21no—
259Dic6—21no—
260Dic6—21no—
261C3×C8—32no—
262C3×C8—61no—
263C3⋊C8—21no—
264C3⋊C8D4×C912yes—
265C2×C32—41no—
266C2×C32C3×#12111yescharacteristic
267C2×C32C3×#13711yescharacteristic
268C2×C32C3×#13711yescharacteristic
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—42no—
273C2×C9—41no—
274C2×C9#93911yescharacteristic
275C2×C9#94211yescharacteristic
276C2×C9#95111yescharacteristic
277C2×Q8—31no—
278C2×Q8C2×D3×C911yescharacteristic
279C22×C4C2×D3×C911yescharacteristic
280Q16—62no—
281M16—62no—
282C2×C8—31no—
283C2×C8—31no—
284C4⋊C4—21no—
285C4⋊C4—31no—
286C4⋊C4—61no—
287C4⋊C4C2×D3×C911yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—61no—
290C22×C3—22no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C2×D4×C911yescharacteristic
301C22×C3C2×C3×#6711yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—121no—
314C3×C4—121no—
315C3×C4C2×D4×C911yescharacteristic
316C3×C4C9×#4111yescharacteristic
317C3×C4C3×D3×D411yescharacteristic
318C3×C4C3×#23711yescharacteristic
319Dic3—41no—
320Dic3—41no—
321C32C3×#47011yescharacteristic
322C9—21no—
323C9#505211yescharacteristic, direct product factor
324C23C9×#6711yescharacteristic
325Q8—21no—
326Q8—21no—
327Q8—61no—
328Q8—61no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4C9×#6711yescharacteristic
336C2×C4C22×D3×C911yescharacteristic
337C8—32no—
338C8—61no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—42no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3C9×#12111yescharacteristic
346C2×C3C9×#13711yescharacteristic
347C2×C3C9×#13711yescharacteristic
348C2×C3C3×#93911yescharacteristic
349C2×C3C3×#94211yescharacteristic
350C2×C3C3×#95111yescharacteristic
351C22—21no—
352C22—21no—
353C22—21no—
354C22C2×C9×#6711yescharacteristic
355C4—21no—
356C4—41no—
357C4—41no—
358C4—121no—
359C4—121no—
360C4D3×D4×C911yescharacteristic
361C4C9×#23711yescharacteristic
362C3—21no—
363C3C9×#47011yescharacteristic, complemented
364C3C3×#505211yescharacteristic
365C2—41no—
366C2C9×#93911yescharacteristic
367C2C9×#94211yescharacteristic
368C2C9×#95111yescharacteristic
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×C3×D4
prime radicalC23×C32C3×D4
cosocleC2×C32×C4C23×C3
hypercenterC9×C22⋊Q8D3
hypocenterC3C9×#470
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×C4×C9 ◁ C9×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C9×#216C2semidirect
5. C9×#835C2semidirect
6. C9×#810C2semidirect
7. C9×#808C2semidirect
54. #5052C9direct (N × H)
103. C2×#216C2×C9semidirect
109. #835C2×C9semidirect
110. #810C2×C9semidirect
111. #808C2×C9semidirect
225. C3×C9#470semidirect
323. C9#5052direct (N × H)
363. C3C9×#470semidirect

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C2×C9×#2162
C9×#9242
C2×C9×#2082
C9×#8352
C9×#8102
C9×#8082
C3×#50523
C9×#4703