Finite Group Databasebeta

← C9×#4818 #13141848

Subgroups — C9×#4818 #13141848

Order 1728.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#944C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic
3C2×C9×#204C211yesmaximal, characteristic
4C9×#893C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C9×#810C211yesmaximal, characteristic, complemented
8C3×#4818C311yesmaximal, characteristic
9C9×#461—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C4×D3×C9—21no—
13C2×C9×Dic6C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C9×#215—22no—
17C9×(C2×D3)⋊C4—21no—
18C9×Dic3⋊C4—21no—
19C9×Dic3⋊C4—21no—
20C4×C9×Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C9×#204—22no—
23C3×#944C2×C311yescharacteristic
24C2×C3×#215C2×C311yescharacteristic
25C2×C3×#204C2×C311yescharacteristic
26C3×#893C2×C311yescharacteristic
27C32×#103C2×C311yescharacteristic
28C3×Dic3⋊C8C2×C311yescharacteristic
29C3×#810C2×C311yescharacteristic
30C2×C9×SD16—31no—
31C2×C9×SD16—31no—
32C9×C4⋊Q8—31no—
33C4×D4×C9—31no—
34C9×C4⋊C8—31no—
35C9×#103—21no—
36C9×#103—31no—
37C9×#103D311yescharacteristic, complemented
38C22×D3×C9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×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×#461—31no—
55#4818C911yescharacteristic, direct product factor
56C2×C32×Q8C22×C311yescharacteristic
57C2×C3×D12C22×C311yescharacteristic
58C2×C3×C4×D3—21no—
59C2×C3×Dic6C22×C311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C32×C4⋊C4C22×C311yescharacteristic
62C3×#215—22no—
63C3×(C2×D3)⋊C4—21no—
64C3×Dic3⋊C4—21no—
65C3×Dic3⋊C4—21no—
66C3×C4×Dic3C22×C311yescharacteristic
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C3×#204—22no—
69C2×Q8×C9—21no—
70C2×Q8×C9—31no—
71C2×Q8×C9C2×D311yescharacteristic
72C2×D4×C9—31no—
73C22×C4×C9—61no—
74C9×SD16—62no—
75C9×SD16—62no—
76C2×C8×C9—21no—
77C2×C8×C9—31no—
78C2×C8×C9C2×D311yescharacteristic
79C9×C4⋊C4—21no—
80C9×C4⋊C4—61no—
81C9×C4⋊C4—61no—
82C9×C4⋊C4C2×D311yescharacteristic
83C9×C22⋊C4—61no—
84C42×C9—31no—
85C22×C3×C9C2×D411yescharacteristic
86C2×D3×C9—22no—
87C2×D3×C9—41no—
88C2×D3×C9—41no—
89C3×C4×C9—41no—
90C3×C4×C9—41no—
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C9×Dic3—21no—
94C9×Dic3—41no—
95C9×Dic3SD1612yescomplemented
96C2×C3×SD16—31no—
97C2×C3×SD16—31no—
98C3×C4⋊Q8—31no—
99C3×C4×D4—31no—
100#944C2×C911yescharacteristic, complemented
101C2×#215C2×C911yescharacteristic
102C2×#204C2×C911yescharacteristic
103#893C2×C911yescharacteristic
104C3×C4⋊C8—31no—
105C3×#103—21no—
106C3×#103—31no—
107C3×#103C2×C911yescharacteristic, complemented
108C3×#103C3×D311yescharacteristic
109Dic3⋊C8C2×C911yescharacteristic, complemented
110#810C2×C911yescharacteristic, complemented
111C22×C3×D3—21no—
112C32×Q8—21no—
113C32×Q8—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4C23×C311yescharacteristic
117C2×C3×Dic3—21no—
118C2×C3×Dic3C3×D411yescharacteristic
119C2×C3×Dic3C3×D411yescharacteristic
120C3×D12—21no—
121C3×D12C3×D412yes—
122C3×C4×D3—42no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C23×C9—61no—
126C32×C8—21no—
127C3×C3⋊C8—21no—
128Q8×C9—22no—
129Q8×C9—21no—
130Q8×C9—21no—
131Q8×C9—41no—
132Q8×C9—61no—
133Q8×C9—61no—
134D4×C9—32no—
135D4×C9—61no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—31no—
140C2×C4×C9—31no—
141C2×C4×C9—41no—
142C2×C4×C9—41no—
143C2×C4×C9—61no—
144C2×C4×C9—122no—
145C2×C4×C9C22×D311yescharacteristic
146C8×C9—21no—
147C8×C9—41no—
148C8×C9—61no—
149#461—31noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9C2×SD1611yescharacteristic
152C2×C3×C9#13711yescharacteristic
153D3×C9—42no—
154C2×C3×Q8—21no—
155C2×C3×Q8—31no—
156C2×C3×Q8C22×C911yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—31no—
159C22×C3×C4—61no—
160C2×D12C22×C911yescharacteristic
161C2×C4×D3—21no—
162C2×Dic6C22×C911yescharacteristic
163C3×SD16—62no—
164C3×SD16—62no—
165C2×C3×C8—21no—
166C2×C3×C8—31no—
167C2×C3×C8C22×C911yescharacteristic
168C2×C3×C8C2×C3×D311yescharacteristic
169C3×C4⋊C4—21no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4C22×C911yescharacteristic
173C3×C4⋊C4C2×C3×D311yescharacteristic
174C3×C22⋊C4—61no—
175C3×C42—31no—
176#215—22no—
177(C2×D3)⋊C4—21no—
178Dic3⋊C4—21no—
179Dic3⋊C4—21no—
180C4×Dic3C22×C911yescharacteristic
181C2×C3⋊C8C22×C911yescharacteristic
182#204—22no—
183C22×C32C2×C3×D411yescharacteristic
184C2×C3×D3—22no—
185C2×C3×D3—41no—
186C2×C3×D3—41no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4C2×C3×D411yescharacteristic
190C32×C4C3×#4111yescharacteristic
191C3×Dic3—21no—
192C3×Dic3—41no—
193C3×Dic3C3×SD1612yes—
194C22×C9—21no—
195C22×C9—62no—
196C22×C9—121no—
197C22×C9—121no—
198C22×C9D3×D411yescharacteristic
199C4×C9—21no—
200C4×C9—21no—
201C4×C9—32no—
202C4×C9—42no—
203C4×C9—41no—
204C4×C9—41no—
205C4×C9—61no—
206C4×C9—81no—
207C4×C9—121no—
208C4×C9#23511yescharacteristic
209C4×C9D3×D411yescharacteristic
210C2×SD16—31no—
211C2×SD16—31no—
212C4⋊Q8—31no—
213C4×D4—31no—
214C4⋊C8—31no—
215#103—31no—
216#103D3×C911yescharacteristic, complemented
217C3×C9#46111yescharacteristic, complemented, Sylow
218C23×C3—61no—
219C22×D3—21no—
220C3×Q8—22no—
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—21no—
225C3×Q8—41no—
226C3×Q8—61no—
227C3×Q8—61no—
228C3×D4—32no—
229C3×D4—61no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—61no—
240C2×C3×C4—122no—
241C2×C3×C4C23×C911yescharacteristic
242C2×C3×C4C22×C3×D311yescharacteristic
243C2×Dic3—21no—
244C2×Dic3D4×C911yescharacteristic
245C2×Dic3D4×C911yescharacteristic
246D12—21no—
247D12D4×C912yes—
248C4×D3—42no—
249Dic6—21no—
250Dic6—21no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8—41no—
254C3×C8—61no—
255C3⋊C8—21no—
256C2×C32C3×#12111yescharacteristic
257C2×C32C2×C3×SD1611yescharacteristic
258C2×C32C3×#13711yescharacteristic
259C3×D3—42no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9—122no—
264C2×C9#88411yescharacteristic
265C2×C9D3×SD1611yescharacteristic
266C2×C9#91811yescharacteristic
267C2×Q8—31no—
268C2×Q8C2×D3×C911yescharacteristic
269C2×D4—31no—
270C22×C4—61no—
271SD16—62no—
272SD16—62no—
273C2×C8—31no—
274C2×C8C2×D3×C911yescharacteristic
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4C2×D3×C911yescharacteristic
278C22⋊C4—61no—
279C42—31no—
280C22×C3—21no—
281C22×C3—62no—
282C22×C3—121no—
283C22×C3—121no—
284C22×C3C2×D4×C911yescharacteristic
285C22×C3C3×D3×D411yescharacteristic
286C2×D3—22no—
287C2×D3—41no—
288C2×D3—41no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—32no—
292C3×C4—42no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—61no—
298C3×C4—81no—
299C3×C4—121no—
300C3×C4C2×D4×C911yescharacteristic
301C3×C4C9×#4111yescharacteristic
302C3×C4C3×#23511yescharacteristic
303C3×C4C3×D3×D411yescharacteristic
304Dic3—21no—
305Dic3—41no—
306Dic3C9×SD1612yescomplemented
307C32C3×#46111yescharacteristic
308C9—21no—
309C9#481811yescharacteristic, direct product factor
310C23—61no—
311Q8—21no—
312Q8—21no—
313Q8—61no—
314Q8—61no—
315D4—32no—
316D4—61no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4C22×D3×C911yescharacteristic
324C8—21no—
325C8—61no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—122no—
330C2×C3C9×#12111yescharacteristic
331C2×C3C2×C9×SD1611yescharacteristic
332C2×C3C9×#13711yescharacteristic
333C2×C3C3×#88411yescharacteristic
334C2×C3C3×D3×SD1611yescharacteristic
335C2×C3C3×#91811yescharacteristic
336D3—42no—
337C22—62no—
338C22—121no—
339C22—121no—
340C22D3×D4×C911yescharacteristic
341C4—32no—
342C4—41no—
343C4—41no—
344C4—61no—
345C4—121no—
346C4C9×#23511yescharacteristic
347C4D3×D4×C911yescharacteristic
348C3—21no—
349C3C9×#46111yescharacteristic, complemented
350C3C3×#481811yescharacteristic
351C2—122no—
352C2C9×#88411yescharacteristic
353C2D3×C9×SD1611yescharacteristic
354C2C9×#91811yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#103C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×D12C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#103D3
hypocenterC3C9×#461
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#103

Complements

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

NH ≅ G/Ntype
1. C9×#944C2semidirect
5. C3×C9×#103C2semidirect
6. C9×Dic3⋊C8C2semidirect
7. C9×#810C2semidirect
37. C9×#103D3semidirect
55. #4818C9direct (N × H)
95. C9×Dic3SD16semidirect
100. #944C2×C9semidirect
107. C3×#103C2×C9semidirect
109. Dic3⋊C8C2×C9semidirect
110. #810C2×C9semidirect
216. #103D3×C9semidirect
217. C3×C9#461semidirect
306. Dic3C9×SD16semidirect
309. C9#4818direct (N × H)
349. C3C9×#461semidirect

Maximal subgroups

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

subgroupindex
C9×#9442
C2×C9×#2152
C2×C9×#2042
C9×#8932
C3×C9×#1032
C9×Dic3⋊C82
C9×#8102
C3×#48183
C9×#4613