Finite Group Databasebeta

← C9×#4787 #13141726

Subgroups — C9×#4787 #13141726

Order 1728.

422 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#936C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic
3C2×C9×#204C211yesmaximal, characteristic, complemented
4C9×#893C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C9×#810C211yesmaximal, characteristic, complemented
8C3×#4787C311yesmaximal, characteristic
9C9×#463—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C4×D3×C9—21no—
13C2×C9×Dic6C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C9×#217—21no—
17C9×#217—21no—
18C9×#213—22no—
19C9×(C2×D3)⋊C4—21no—
20C4×C9×Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C9×#204—22no—
23C3×#936C2×C311yescharacteristic
24C2×C3×#213C2×C311yescharacteristic
25C2×C3×#204C2×C311yescharacteristic
26C3×#893C2×C311yescharacteristic
27C32×#102C2×C311yescharacteristic
28C3×Dic3⋊C8C2×C311yescharacteristic
29C3×#810C2×C311yescharacteristic
30C2×C9×SD16—31no—
31C2×C9×D8—31no—
32C9×#124—31no—
33C4×D4×C9—31no—
34C9×C4⋊C8—31no—
35C9×#103—31no—
36C9×#102—21no—
37C9×#102D311yescharacteristic, complemented
38C23×C3×C9—21no—
39C22×D3×C9—21no—
40C3×D4×C9—21no—
41C3×D4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C9×Dic3—21no—
45C2×C9×Dic3D411yescharacteristic
46C2×C9×Dic3D411yescharacteristic
47C9×D12—21no—
48C9×D12D412yes—
49C4×D3×C9—42no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52C3×C8×C9—21no—
53C9×C3⋊C8—21no—
54C3×#463—31no—
55#4787C911yescharacteristic, direct product factor
56C2×C32×D4C22×C311yescharacteristic
57C2×C3×D12C22×C311yescharacteristic
58C2×C3×C4×D3—21no—
59C2×C3×Dic6C22×C311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C32×C4⋊C4C22×C311yescharacteristic
62C3×#217—21no—
63C3×#217—21no—
64C3×#213—22no—
65C3×(C2×D3)⋊C4—21no—
66C3×C4×Dic3C22×C311yescharacteristic
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C3×#204—22no—
69C2×Q8×C9—31no—
70C2×D4×C9—21no—
71C2×D4×C9—31no—
72C2×D4×C9C2×D311yescharacteristic
73C22×C4×C9—61no—
74C9×SD16—62no—
75C9×D8—62no—
76C2×C8×C9—21no—
77C2×C8×C9—31no—
78C2×C8×C9C2×D311yescharacteristic
79C9×C4⋊C4—21no—
80C9×C4⋊C4C2×D311yescharacteristic
81C9×C22⋊C4—61no—
82C9×C22⋊C4—61no—
83C9×C22⋊C4—61no—
84C42×C9—31no—
85C22×C3×C9—41no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9C2×D411yescharacteristic
89C2×D3×C9—22no—
90C2×D3×C9—41no—
91C2×D3×C9—41no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C9×Dic3—21no—
96C9×Dic3—21no—
97C9×Dic3—41no—
98C2×C3×SD16—31no—
99C2×C3×D8—31no—
100C3×#124—31no—
101C3×C4×D4—31no—
102#936C2×C911yescharacteristic, complemented
103C2×#213C2×C911yescharacteristic
104C2×#204C2×C911yescharacteristic, complemented
105#893C2×C911yescharacteristic, complemented
106C3×C4⋊C8—31no—
107C3×#103—31no—
108C3×#102—21no—
109C3×#102C2×C911yescharacteristic, complemented
110C3×#102C3×D311yescharacteristic
111Dic3⋊C8C2×C911yescharacteristic, complemented
112#810C2×C911yescharacteristic, complemented
113C23×C32—21no—
114C22×C3×D3—21no—
115C32×D4—21no—
116C32×D4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C23×C311yescharacteristic
119C2×C3×Dic3—21no—
120C2×C3×Dic3C3×D411yescharacteristic
121C2×C3×Dic3C3×D411yescharacteristic
122C3×D12—21no—
123C3×D12C3×D412yes—
124C3×C4×D3—42no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C23×C9—21no—
128C23×C9—41no—
129C23×C9—61no—
130C32×C8—21no—
131C3×C3⋊C8—21no—
132Q8×C9—61no—
133Q8×C9—61no—
134D4×C9—22no—
135D4×C9—21no—
136D4×C9—21no—
137D4×C9—32no—
138D4×C9—41no—
139D4×C9—61no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—31no—
143C2×C4×C9—31no—
144C2×C4×C9—41no—
145C2×C4×C9—61no—
146C2×C4×C9—122no—
147C2×C4×C9C22×D311yescharacteristic
148C8×C9—21no—
149C8×C9—41no—
150C8×C9—61no—
151#463—31noSylow
152C2×C3×C9—81no—
153C2×C3×C9#12111yescharacteristic
154C2×C3×C9#13511yescharacteristic
155C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
156D3×C9—42no—
157C2×C3×Q8—31no—
158C2×C3×D4—21no—
159C2×C3×D4—31no—
160C2×C3×D4C22×C911yescharacteristic
161C2×C3×D4C2×C3×D311yescharacteristic
162C22×C3×C4—61no—
163C2×D12C22×C911yescharacteristic
164C2×C4×D3—21no—
165C2×Dic6C22×C911yescharacteristic
166C3×SD16—62no—
167C3×D8—62no—
168C2×C3×C8—21no—
169C2×C3×C8—31no—
170C2×C3×C8C22×C911yescharacteristic
171C2×C3×C8C2×C3×D311yescharacteristic
172C3×C4⋊C4—21no—
173C3×C4⋊C4C22×C911yescharacteristic
174C3×C4⋊C4C2×C3×D311yescharacteristic
175C3×C22⋊C4—61no—
176C3×C22⋊C4—61no—
177C3×C22⋊C4—61no—
178C3×C42—31no—
179#217—21no—
180#217—21no—
181#213—22no—
182(C2×D3)⋊C4—21no—
183C4×Dic3C22×C911yescharacteristic
184C2×C3⋊C8C22×C911yescharacteristic
185#204—22no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32C2×C3×D411yescharacteristic
190C2×C3×D3—22no—
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C32×C4—41no—
194C32×C4C2×C3×D411yescharacteristic
195C32×C4C3×#4111yescharacteristic
196C3×Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—41no—
199C22×C9—21no—
200C22×C9—42no—
201C22×C9—41no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9—62no—
205C22×C9—81no—
206C22×C9—81no—
207C22×C9—121no—
208C22×C9—121no—
209C22×C9D3×D411yescharacteristic
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—41no—
213C4×C9—61no—
214C4×C9—61no—
215C4×C9—81no—
216C4×C9—121no—
217C4×C9#23511yescharacteristic
218C4×C9D3×D411yescharacteristic
219C2×SD16—31no—
220C2×D8—31no—
221#124—31no—
222C4×D4—31no—
223C4⋊C8—31no—
224#103—31no—
225#102D3×C911yescharacteristic, complemented
226C3×C9#46311yescharacteristic, complemented, Sylow
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—41no—
230C23×C3—61no—
231C22×D3—21no—
232C3×Q8—61no—
233C3×Q8—61no—
234C3×D4—22no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—32no—
240C3×D4—41no—
241C3×D4—61no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—41no—
248C2×C3×C4—61no—
249C2×C3×C4—122no—
250C2×C3×C4C23×C911yescharacteristic
251C2×C3×C4C22×C3×D311yescharacteristic
252C2×Dic3—21no—
253C2×Dic3D4×C911yescharacteristic
254C2×Dic3D4×C911yescharacteristic
255D12—21no—
256D12D4×C912yes—
257C4×D3—42no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—21no—
261C3×C8—21no—
262C3×C8—41no—
263C3×C8—61no—
264C3⋊C8—21no—
265C2×C32—81no—
266C2×C32C3×#12111yescharacteristic
267C2×C32C3×#13511yescharacteristic
268C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
269C3×D3—42no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—82no—
274C2×C9—81no—
275C2×C9—122no—
276C2×C9#88411yescharacteristic
277C2×C9#91111yescharacteristic
278C2×C9#91611yescharacteristic
279C2×Q8—31no—
280C2×D4—31no—
281C2×D4C2×D3×C911yescharacteristic
282C22×C4—61no—
283SD16—62no—
284D8—62no—
285C2×C8—31no—
286C2×C8C2×D3×C911yescharacteristic
287C4⋊C4C2×D3×C911yescharacteristic
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C42—31no—
292C22×C3—21no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—62no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3C2×D4×C911yescharacteristic
306C22×C3C3×D3×D411yescharacteristic
307C2×D3—22no—
308C2×D3—41no—
309C2×D3—41no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—61no—
315C3×C4—61no—
316C3×C4—81no—
317C3×C4—121no—
318C3×C4C2×D4×C911yescharacteristic
319C3×C4C9×#4111yescharacteristic
320C3×C4C3×#23511yescharacteristic
321C3×C4C3×D3×D411yescharacteristic
322Dic3—21no—
323Dic3—21no—
324Dic3—41no—
325C32C3×#46311yescharacteristic
326C9—21no—
327C9#478711yescharacteristic, direct product factor
328C23—21no—
329C23—61no—
330Q8—61no—
331Q8—61no—
332D4—21no—
333D4—21no—
334D4—32no—
335D4—61no—
336C2×C4—21no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—61no—
340C2×C4—122no—
341C2×C4C22×D3×C911yescharacteristic
342C8—21no—
343C8—61no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—82no—
348C2×C3—81no—
349C2×C3—81no—
350C2×C3—122no—
351C2×C3C9×#12111yescharacteristic
352C2×C3C9×#13511yescharacteristic
353C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
354C2×C3C3×#88411yescharacteristic
355C2×C3C3×#91111yescharacteristic
356C2×C3C3×#91611yescharacteristic
357D3—42no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—62no—
362C22—121no—
363C22—121no—
364C22D3×D4×C911yescharacteristic
365C4—41no—
366C4—61no—
367C4—61no—
368C4—121no—
369C4C9×#23511yescharacteristic
370C4D3×D4×C911yescharacteristic
371C3—21no—
372C3C9×#46311yescharacteristic, complemented
373C3C3×#478711yescharacteristic
374C2—81no—
375C2—122no—
376C2C9×#88411yescharacteristic
377C2C9×#91111yescharacteristic
378C2C9×#91611yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×#213C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#102D3
hypocenterC3C9×#463
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#102

Complements

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

NH ≅ G/Ntype
1. C9×#936C2semidirect
3. C2×C9×#204C2semidirect
4. C9×#893C2semidirect
5. C3×C9×#102C2semidirect
6. C9×Dic3⋊C8C2semidirect
7. C9×#810C2semidirect
37. C9×#102D3semidirect
55. #4787C9direct (N × H)
102. #936C2×C9semidirect
104. C2×#204C2×C9semidirect
105. #893C2×C9semidirect
109. C3×#102C2×C9semidirect
111. Dic3⋊C8C2×C9semidirect
112. #810C2×C9semidirect
225. #102D3×C9semidirect
226. C3×C9#463semidirect
327. C9#4787direct (N × H)
372. C3C9×#463semidirect

Maximal subgroups

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

subgroupindex
C9×#9362
C2×C9×#2132
C2×C9×#2042
C9×#8932
C3×C9×#1022
C9×Dic3⋊C82
C9×#8102
C3×#47873
C9×#4633