Finite Group Databasebeta

← C9×#5048 #13142868

Subgroups — C9×#5048 #13142868

Order 1728.

446 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic
4C22×C9×C3⋊C8C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic, complemented
7C9×#808C211yesmaximal, characteristic, complemented
8C3×#5048C311yesmaximal, characteristic
9C9×#465—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×C9×#67—21no—
13C2×C9×D12C2211yescharacteristic
14C3×C9×C4⋊C4—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C22⋊C4—21no—
17C9×#215—22no—
18C9×(C2×D3)⋊C4—21no—
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C9×C3⋊C8—22no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23C32×C22⋊Q8C2×C311yescharacteristic
24C2×C3×#215C2×C311yescharacteristic
25C3×#930C2×C311yescharacteristic
26C22×C3×C3⋊C8C2×C311yescharacteristic
27C3×#835C2×C311yescharacteristic
28C3×#809C2×C311yescharacteristic
29C3×#808C2×C311yescharacteristic
30C2×C9×SD16—31no—
31C22×C8×C9—31no—
32C9×C22⋊Q8—21no—
33C9×C22⋊Q8D311yescharacteristic, complemented
34C9×#121—31no—
35C9×#106—31no—
36C9×#103—31no—
37C9×#102—31no—
38C23×C3×C9D411yescharacteristic
39C22×D3×C9—21no—
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9D411yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48C9×#67—42no—
49C2×C9×Dic3—21no—
50C9×D12—21no—
51C9×D12—21no—
52C9×C3⋊C8—21no—
53C9×C3⋊C8D412yescomplemented
54C3×#465—31no—
55#5048C911yescharacteristic, direct product factor
56C2×C32×Q8C22×C311yescharacteristic
57C22×C32×C4C22×C311yescharacteristic
58C2×C3×#67—21no—
59C2×C3×D12C22×C311yescharacteristic
60C32×C4⋊C4—21no—
61C32×C4⋊C4C22×C311yescharacteristic
62C32×C22⋊C4—21no—
63C3×#215—22no—
64C3×(C2×D3)⋊C4—21no—
65C3×C4⋊Dic3C22×C311yescharacteristic
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C2×C3×C3⋊C8C22×C311yescharacteristic
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—31no—
72C2×D4×C9—61no—
73C22×C4×C9—21no—
74C22×C4×C9C2×D311yescharacteristic
75C9×SD16—62no—
76C2×C8×C9—31no—
77C2×C8×C9—31no—
78C2×C8×C9—62no—
79C9×C4⋊C4—22no—
80C9×C4⋊C4—21no—
81C9×C4⋊C4—21no—
82C9×C4⋊C4—31no—
83C9×C4⋊C4C2×D311yescharacteristic
84C9×C22⋊C4—22no—
85C9×C22⋊C4—21no—
86C9×C22⋊C4—61no—
87C22×C3×C9—21no—
88C22×C3×C9—21no—
89C22×C3×C9SD1612yescomplemented
90C22×C3×C9C2×D411yescharacteristic
91C2×D3×C9—41no—
92C2×D3×C9—41no—
93C2×D3×C9—41no—
94C3×C4×C9—21no—
95C3×C4×C9—41no—
96C3×C4×C9—41no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9#4111yescharacteristic
99C9×Dic3—41no—
100C2×C3×SD16—31no—
101C22×C3×C8—31no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8C2×C911yescharacteristic, complemented
104C3×C22⋊Q8C3×D311yescharacteristic
105C3×#121—31no—
106C2×#215C2×C911yescharacteristic, complemented
107#930C2×C911yescharacteristic
108C22×C3⋊C8C2×C911yescharacteristic, complemented
109C3×#106—31no—
110C3×#103—31no—
111C3×#102—31no—
112#835C2×C911yescharacteristic, complemented
113#809C2×C911yescharacteristic, complemented
114#808C2×C911yescharacteristic, complemented
115C23×C32C3×D411yescharacteristic
116C22×C3×D3—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4C3×D411yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C3×#67—42no—
126C2×C3×Dic3—21no—
127C3×D12—21no—
128C3×D12—21no—
129C23×C9—21no—
130C23×C9—61no—
131C23×C9#6711yescharacteristic
132C3×C3⋊C8—21no—
133C3×C3⋊C8C3×D412yes—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137Q8×C9—41no—
138D4×C9—61no—
139D4×C9—61no—
140D4×C9—122no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—22no—
144C2×C4×C9—22no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—21no—
149C2×C4×C9—41no—
150C2×C4×C9—41no—
151C2×C4×C9—61no—
152C2×C4×C9#6711yescharacteristic
153C2×C4×C9C22×D311yescharacteristic
154C8×C9—32no—
155C8×C9—61no—
156#465—31noSylow
157C2×C3×C9—22no—
158C2×C3×C9#12111yescharacteristic
159C2×C3×C9C2×SD1611yescharacteristic
160C2×C3×C9#13511yescharacteristic
161D3×C9—81no—
162C2×C3×Q8—21no—
163C2×C3×Q8C22×C911yescharacteristic
164C2×C3×Q8C2×C3×D311yescharacteristic
165C2×C3×D4—31no—
166C2×C3×D4—61no—
167C22×C3×C4—21no—
168C22×C3×C4C22×C911yescharacteristic
169C22×C3×C4C2×C3×D311yescharacteristic
170C2×#67—21no—
171C2×D12C22×C911yescharacteristic
172C3×SD16—62no—
173C2×C3×C8—31no—
174C2×C3×C8—31no—
175C2×C3×C8—62no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4C22×C911yescharacteristic
182C3×C4⋊C4C2×C3×D311yescharacteristic
183C3×C22⋊C4—22no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—61no—
187#215—22no—
188(C2×D3)⋊C4—21no—
189C4⋊Dic3C22×C911yescharacteristic
190C2×C3⋊C8—22no—
191C2×C3⋊C8C22×C911yescharacteristic
192C2×C3⋊C8C22×C911yescharacteristic
193C22×C32—21no—
194C22×C32—21no—
195C22×C32C3×SD1612yes—
196C22×C32C2×C3×D411yescharacteristic
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C32×C4—21no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4C2×C3×D411yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3×Dic3—41no—
206C22×C9—21no—
207C22×C9—21no—
208C22×C9—22no—
209C22×C9—21no—
210C22×C9—41no—
211C22×C9—41no—
212C22×C9—121no—
213C22×C9—121no—
214C22×C9—121no—
215C22×C9#21512yescomplemented
216C22×C9C2×#6711yescharacteristic
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—41no—
221C4×C9—42no—
222C4×C9—42no—
223C4×C9—41no—
224C4×C9—41no—
225C4×C9—121no—
226C4×C9D3×D411yescharacteristic
227C4×C9#23711yescharacteristic
228C2×SD16—31no—
229C22×C8—31no—
230C22⋊Q8D3×C911yescharacteristic, complemented
231#121—31no—
232#106—31no—
233#103—31no—
234#102—31no—
235C3×C9#46511yescharacteristic, complemented, Sylow
236C23×C3—21no—
237C23×C3—61no—
238C23×C3D4×C911yescharacteristic
239C23×C3C3×#6711yescharacteristic
240C22×D3—21no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×D4—61no—
248C3×D4—61no—
249C3×D4—122no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—61no—
265C2×C3×C4D4×C911yescharacteristic
266C2×C3×C4C23×C911yescharacteristic
267C2×C3×C4C3×#6711yescharacteristic
268C2×C3×C4C22×C3×D311yescharacteristic
269#67—42no—
270C2×Dic3—21no—
271D12—21no—
272D12—21no—
273C3×C8—32no—
274C3×C8—61no—
275C3⋊C8—21no—
276C3⋊C8D4×C912yescomplemented
277C2×C32—22no—
278C2×C32C3×#12111yescharacteristic
279C2×C32C2×C3×SD1611yescharacteristic
280C2×C32C3×#13511yescharacteristic
281C3×D3—81no—
282C2×C9—22no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—42no—
287C2×C9—241no—
288C2×C9#93911yescharacteristic
289C2×C9C2×#21511yescharacteristic
290C2×C9#95011yescharacteristic
291C2×Q8C2×D3×C911yescharacteristic
292C2×D4—31no—
293C2×D4—61no—
294C22×C4C2×D3×C911yescharacteristic
295SD16—62no—
296C2×C8—31no—
297C2×C8—31no—
298C2×C8—62no—
299C4⋊C4—21no—
300C4⋊C4—31no—
301C4⋊C4C2×D3×C911yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—61no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—121no—
315C22×C3C9×SD1612yescomplemented
316C22×C3C2×D4×C911yescharacteristic
317C22×C3C3×#21512yes—
318C22×C3C2×C3×#6711yescharacteristic
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—42no—
328C3×C4—42no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—121no—
334C3×C4C2×D4×C911yescharacteristic
335C3×C4C9×#4111yescharacteristic
336C3×C4C3×D3×D411yescharacteristic
337C3×C4C3×#23711yescharacteristic
338Dic3—41no—
339C32C3×#46511yescharacteristic
340C9—21no—
341C9#504811yescharacteristic, direct product factor
342C23—61no—
343C23C9×#6711yescharacteristic
344Q8—21no—
345Q8—21no—
346D4—61no—
347D4—61no—
348D4—122no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—61no—
354C2×C4C9×#6711yescharacteristic
355C2×C4C22×D3×C911yescharacteristic
356C8—32no—
357C8—61no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—42no—
364C2×C3—241no—
365C2×C3C9×#12111yescharacteristic
366C2×C3C2×C9×SD1611yescharacteristic
367C2×C3C9×#13511yescharacteristic
368C2×C3C3×#93911yescharacteristic
369C2×C3C2×C3×#21511yescharacteristic
370C2×C3C3×#95011yescharacteristic
371D3—81no—
372C22—21no—
373C22—21no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22C9×#21512yescomplemented
378C22C2×C9×#6711yescharacteristic
379C4—21no—
380C4—41no—
381C4—41no—
382C4—121no—
383C4D3×D4×C911yescharacteristic
384C4C9×#23711yescharacteristic
385C3—21no—
386C3C9×#46511yescharacteristic, complemented
387C3C3×#504811yescharacteristic
388C2—22no—
389C2—241no—
390C2C9×#93911yescharacteristic
391C2C2×C9×#21511yescharacteristic
392C2C9×#95011yescharacteristic
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×C3×D4
prime radicalC3×#930C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×C22⋊Q8D3
hypocenterC3C9×#465
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
1. C3×C9×C22⋊Q8C2semidirect
2. C2×C9×#215C2semidirect
4. C22×C9×C3⋊C8C2semidirect
5. C9×#835C2semidirect
6. C9×#809C2semidirect
7. C9×#808C2semidirect
33. C9×C22⋊Q8D3semidirect
53. C9×C3⋊C8D4semidirect
55. #5048C9direct (N × H)
89. C22×C3×C9SD16semidirect
103. C3×C22⋊Q8C2×C9semidirect
106. C2×#215C2×C9semidirect
108. C22×C3⋊C8C2×C9semidirect
112. #835C2×C9semidirect
113. #809C2×C9semidirect
114. #808C2×C9semidirect
215. C22×C9#215semidirect
230. C22⋊Q8D3×C9semidirect
235. C3×C9#465semidirect
276. C3⋊C8D4×C9semidirect
315. C22×C3C9×SD16semidirect
341. C9#5048direct (N × H)
377. C22C9×#215semidirect
386. C3C9×#465semidirect

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×#2152
C9×#9302
C22×C9×C3⋊C82
C9×#8352
C9×#8092
C9×#8082
C3×#50483
C9×#4653