Finite Group Databasebeta

← C9×#5057 #13142914

Subgroups — C9×#5057 #13142914

Order 1728.

448 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×#124C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3C2×C9×#213C211yesmaximal, characteristic
4C4×C9×D12C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6C9×#832C211yesmaximal, characteristic, complemented
7C9×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5057C311yesmaximal, characteristic
9C9×#463—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C4×D3×C9—21no—
14C3×C9×C22⋊C4—21no—
15C3×C9×C22⋊C4—21no—
16C3×C42×C9C2211yescharacteristic
17C9×#215—22no—
18C9×#213—22no—
19C9×(C2×D3)⋊C4—21no—
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23C32×#124C2×C311yescharacteristic
24C2×C3×#215C2×C311yescharacteristic
25C2×C3×#213C2×C311yescharacteristic
26C3×C4×D12C2×C311yescharacteristic
27C3×#835C2×C311yescharacteristic
28C3×#832C2×C311yescharacteristic
29C3×C4⋊(C3⋊C8)C2×C311yescharacteristic
30C2×C9×SD16—31no—
31C2×C9×D8—31no—
32C9×#124—21no—
33C9×#124D311yescharacteristic, complemented
34C4×D4×C9—31no—
35C9×C4⋊C8—31no—
36C9×#103—31no—
37C9×#102—31no—
38C23×C3×C9—21no—
39C22×D3×C9—21no—
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C3×D4×C9—21no—
43C3×D4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9D411yescharacteristic
46C2×C3×C4×C9D411yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48C2×C9×Dic3—21no—
49C9×D12—21no—
50C9×D12D412yes—
51C4×D3×C9—42no—
52C9×C3⋊C8—21no—
53C9×C3⋊C8—21no—
54C3×#463—31no—
55#5057C911yescharacteristic, direct product factor
56C2×C32×Q8C22×C311yescharacteristic
57C2×C32×D4C22×C311yescharacteristic
58C2×C3×D12C22×C311yescharacteristic
59C2×C3×C4×D3—21no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C42C22×C311yescharacteristic
63C3×#215—22no—
64C3×#213—22no—
65C3×(C2×D3)⋊C4—21no—
66C3×C4⋊Dic3C22×C311yescharacteristic
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C2×C3×C3⋊C8C22×C311yescharacteristic
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—21no—
72C2×D4×C9—31no—
73C2×D4×C9C2×D311yescharacteristic
74C22×C4×C9—61no—
75C9×SD16—62no—
76C9×D8—62no—
77C2×C8×C9—31no—
78C2×C8×C9—31no—
79C9×C4⋊C4—31no—
80C9×C22⋊C4—22no—
81C9×C22⋊C4—22no—
82C9×C22⋊C4—21no—
83C9×C22⋊C4—21no—
84C9×C22⋊C4—61no—
85C42×C9—21no—
86C42×C9C2×D311yescharacteristic
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9—41no—
90C22×C3×C9C2×D411yescharacteristic
91C2×D3×C9—22no—
92C2×D3×C9—41no—
93C2×D3×C9—41no—
94C3×C4×C9—21no—
95C3×C4×C9—21no—
96C3×C4×C9—41no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9#4111yescharacteristic
99C9×Dic3—41no—
100C2×C3×SD16—31no—
101C2×C3×D8—31no—
102C3×#124—21no—
103C3×#124C2×C911yescharacteristic, complemented
104C3×#124C3×D311yescharacteristic
105C3×C4×D4—31no—
106C2×#215C2×C911yescharacteristic, complemented
107C2×#213C2×C911yescharacteristic
108C4×D12C2×C911yescharacteristic, complemented
109C3×C4⋊C8—31no—
110C3×#103—31no—
111C3×#102—31no—
112#835C2×C911yescharacteristic, complemented
113#832C2×C911yescharacteristic, complemented
114C4⋊(C3⋊C8)C2×C911yescharacteristic, complemented
115C23×C32—21no—
116C22×C3×D3—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×D4—21no—
120C32×D4—21no—
121C2×C32×C4—21no—
122C2×C32×C4C3×D411yescharacteristic
123C2×C32×C4C3×D411yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C2×C3×Dic3—21no—
126C3×D12—21no—
127C3×D12C3×D412yes—
128C3×C4×D3—42no—
129C23×C9—22no—
130C23×C9—21no—
131C23×C9—61no—
132C3×C3⋊C8—21no—
133C3×C3⋊C8—21no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137Q8×C9—41no—
138D4×C9—21no—
139D4×C9—21no—
140D4×C9—32no—
141D4×C9—41no—
142D4×C9—41no—
143D4×C9—61no—
144C2×C4×C9—22no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—21no—
149C2×C4×C9—61no—
150C2×C4×C9—122no—
151C2×C4×C9#6711yescharacteristic
152C2×C4×C9#6711yescharacteristic
153C2×C4×C9C22×D311yescharacteristic
154C8×C9—61no—
155C8×C9—61no—
156#463—31noSylow
157C2×C3×C9—81no—
158C2×C3×C9#12111yescharacteristic
159C2×C3×C9#13511yescharacteristic
160C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
161D3×C9—42no—
162C2×C3×Q8—21no—
163C2×C3×Q8C22×C911yescharacteristic
164C2×C3×Q8C2×C3×D311yescharacteristic
165C2×C3×D4—21no—
166C2×C3×D4—31no—
167C2×C3×D4C22×C911yescharacteristic
168C2×C3×D4C2×C3×D311yescharacteristic
169C22×C3×C4—61no—
170C2×D12C22×C911yescharacteristic
171C2×C4×D3—21no—
172C3×SD16—62no—
173C3×D8—62no—
174C2×C3×C8—31no—
175C2×C3×C8—31no—
176C3×C4⋊C4—31no—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—21no—
180C3×C22⋊C4—21no—
181C3×C22⋊C4—21no—
182C3×C22⋊C4—21no—
183C3×C22⋊C4—61no—
184C3×C42—21no—
185C3×C42C22×C911yescharacteristic
186C3×C42C2×C3×D311yescharacteristic
187#215—22no—
188#213—22no—
189(C2×D3)⋊C4—21no—
190C4⋊Dic3C22×C911yescharacteristic
191C2×C3⋊C8C22×C911yescharacteristic
192C2×C3⋊C8C22×C911yescharacteristic
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32C2×C3×D411yescharacteristic
197C2×C3×D3—22no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—41no—
203C32×C4C2×C3×D411yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3×Dic3—41no—
206C22×C9—21no—
207C22×C9—42no—
208C22×C9—42no—
209C22×C9—42no—
210C22×C9—41no—
211C22×C9—41no—
212C22×C9—41no—
213C22×C9—62no—
214C22×C9—121no—
215C22×C9—121no—
216C22×C9C2×#6711yescharacteristic
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—41no—
222C4×C9—41no—
223C4×C9—42no—
224C4×C9—41no—
225C4×C9—121no—
226C4×C9D3×D411yescharacteristic
227C4×C9#23711yescharacteristic
228C2×SD16—31no—
229C2×D8—31no—
230#124D3×C911yescharacteristic, complemented
231C4×D4—31no—
232C4⋊C8—31no—
233#103—31no—
234#102—31no—
235C3×C9#46311yescharacteristic, complemented, Sylow
236C23×C3—22no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—61no—
240C22×D3—21no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—32no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—61no—
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—61no—
262C2×C3×C4—122no—
263C2×C3×C4D4×C911yescharacteristic
264C2×C3×C4D4×C911yescharacteristic
265C2×C3×C4C23×C911yescharacteristic
266C2×C3×C4C3×#6711yescharacteristic
267C2×C3×C4C3×#6711yescharacteristic
268C2×C3×C4C22×C3×D311yescharacteristic
269C2×Dic3—21no—
270D12—21no—
271D12D4×C912yes—
272C4×D3—42no—
273C3×C8—61no—
274C3×C8—61no—
275C3⋊C8—21no—
276C3⋊C8—21no—
277C2×C32—81no—
278C2×C32C3×#12111yescharacteristic
279C2×C32C3×#13511yescharacteristic
280C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
281C3×D3—42no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—82no—
286C2×C9—81no—
287C2×C9—122no—
288C2×C9#93811yescharacteristic
289C2×C9#94911yescharacteristic
290C2×C9#95011yescharacteristic
291C2×Q8C2×D3×C911yescharacteristic
292C2×D4—31no—
293C2×D4C2×D3×C911yescharacteristic
294C22×C4—61no—
295SD16—62no—
296D8—62no—
297C2×C8—31no—
298C2×C8—31no—
299C4⋊C4—31no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—61no—
303C42C2×D3×C911yescharacteristic
304C22×C3—21no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—62no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3C2×D4×C911yescharacteristic
318C22×C3C2×C3×#6711yescharacteristic
319C2×D3—22no—
320C2×D3—41no—
321C2×D3—41no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—42no—
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×#46311yescharacteristic
340C9—21no—
341C9#505711yescharacteristic, direct product factor
342C23—21no—
343C23—61no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—32no—
349D4—61no—
350C2×C4—21no—
351C2×C4—61no—
352C2×C4—122no—
353C2×C4C9×#6711yescharacteristic
354C2×C4C9×#6711yescharacteristic
355C2×C4C22×D3×C911yescharacteristic
356C8—61no—
357C8—61no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—82no—
362C2×C3—81no—
363C2×C3—81no—
364C2×C3—122no—
365C2×C3C9×#12111yescharacteristic
366C2×C3C9×#13511yescharacteristic
367C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
368C2×C3C3×#93811yescharacteristic
369C2×C3C3×#94911yescharacteristic
370C2×C3C3×#95011yescharacteristic
371D3—42no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—62no—
376C22—121no—
377C22—121no—
378C22C2×C9×#6711yescharacteristic
379C4—21no—
380C4—21no—
381C4—41no—
382C4—121no—
383C4D3×D4×C911yescharacteristic
384C4C9×#23711yescharacteristic
385C3—21no—
386C3C9×#46311yescharacteristic, complemented
387C3C3×#505711yescharacteristic
388C2—81no—
389C2—122no—
390C2C9×#93811yescharacteristic
391C2C9×#94911yescharacteristic
392C2C9×#95011yescharacteristic
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#124C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×#213C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#124D3
hypocenterC3C9×#463
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C42×C9 ◁ C9×#124

Complements

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

NH ≅ G/Ntype
1. C3×C9×#124C2semidirect
2. C2×C9×#215C2semidirect
4. C4×C9×D12C2semidirect
5. C9×#835C2semidirect
6. C9×#832C2semidirect
7. C9×C4⋊(C3⋊C8)C2semidirect
33. C9×#124D3semidirect
55. #5057C9direct (N × H)
103. C3×#124C2×C9semidirect
106. C2×#215C2×C9semidirect
108. C4×D12C2×C9semidirect
112. #835C2×C9semidirect
113. #832C2×C9semidirect
114. C4⋊(C3⋊C8)C2×C9semidirect
230. #124D3×C9semidirect
235. C3×C9#463semidirect
341. C9#5057direct (N × H)
386. C3C9×#463semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1242
C2×C9×#2152
C2×C9×#2132
C4×C9×D122
C9×#8352
C9×#8322
C9×C4⋊(C3⋊C8)2
C3×#50573
C9×#4633