Finite Group Databasebeta

← C9×#5711 #13153229

Subgroups — C9×#5711 #13153229

Order 1728.

816 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#126C211yesmaximal, characteristic, complemented
2C9×#895C213yesmaximal, complemented
3C9×#894C213yesmaximal, complemented
4C9×#884C213yesmaximal, complemented
5C9×#882C213yesmaximal, complemented
6C9×#876C211yesmaximal, characteristic, complemented
7C9×#874C211yesmaximal, characteristic, complemented
8C3×#5711C311yesmaximal, characteristic
9C9×#560—31nomaximal
10C23×D3×C9C2211yescharacteristic
11C2×C9×#67C2213yes—
12C2×C9×D12C2213yes—
13C2×C9×D12C2213yes—
14C2×C9×D12C2213yescomplemented
15C2×C4×D3×C9C2213yescomplemented
16C3×C9×C4⋊C4C2213yescomplemented
17C3×C9×C22⋊C4C2213yes—
18C3×C42×C9C2211yescharacteristic, complemented
19C9×(C2×D3)⋊C4C2213yes—
20C9×(C2×D3)⋊C4C2213yes—
21C9×(C2×D3)⋊C4C2213yescomplemented
22C9×Dic3⋊C4C2213yescomplemented
23C32×#126C2×C311yescharacteristic
24C3×#895C2×C313yes—
25C3×#894C2×C313yes—
26C3×#884C2×C313yes—
27C3×#882C2×C313yes—
28C3×#876C2×C311yescharacteristic
29C3×#874C2×C311yescharacteristic
30C9×#127—31no—
31C9×#126—21no—
32C9×#126—31no—
33C9×#126D311yescharacteristic, complemented
34C9×#123—33no—
35C9×#121—33no—
36C9×#121—33no—
37C9×#120—33no—
38C23×C3×C9C2311yescharacteristic
39C22×D3×C9—43no—
40C22×D3×C9C2311yescharacteristic
41C22×D3×C9C2311yescharacteristic
42C22×D3×C9C2313yes—
43C2×C3×C4×C9C2313yes—
44C2×C3×C4×C9C2313yes—
45C9×#67—43no—
46C2×C9×Dic3C2313yes—
47C9×D12—43no—
48C9×D12—43no—
49C9×D12—43no—
50C4×D3×C9—43no—
51C3×#560—31no—
52#5711C911yescharacteristic, direct product factor
53C23×C3×D3C22×C311yescharacteristic
54C2×C3×#67C22×C313yes—
55C2×C3×D12C22×C313yes—
56C2×C3×D12C22×C313yes—
57C2×C3×D12C22×C313yes—
58C2×C3×C4×D3C22×C313yes—
59C24×C9—31no—
60C32×C4⋊C4C22×C313yes—
61C32×C22⋊C4C22×C313yes—
62C32×C42C22×C311yescharacteristic
63C3×(C2×D3)⋊C4C22×C313yes—
64C3×(C2×D3)⋊C4C22×C313yes—
65C3×(C2×D3)⋊C4C22×C313yes—
66C3×Dic3⋊C4C22×C313yes—
67C2×D4×C9—33no—
68C2×D4×C9—33no—
69C2×D4×C9—33no—
70C2×D4×C9—33no—
71C22×C4×C9—33no—
72C9×C4⋊C4—23no—
73C9×C4⋊C4—33no—
74C9×C4⋊C4C2×D313yescomplemented
75C9×C22⋊C4—23no—
76C9×C22⋊C4—33no—
77C9×C22⋊C4—33no—
78C9×C22⋊C4—33no—
79C9×C22⋊C4C2×D313yes—
80C42×C9—21no—
81C42×C9C2×D311yescharacteristic, complemented
82C22×C3×C9—23no—
83C22×C3×C9C2411yescharacteristic
84C2×D3×C9—23no—
85C2×D3×C9—23no—
86C2×D3×C9—23no—
87C2×D3×C9—23no—
88C2×D3×C9—23no—
89C2×D3×C9—44no—
90C3×C4×C9—23no—
91C3×C4×C9—23no—
92C9×Dic3—23no—
93C3×#127—31no—
94C3×#126—21no—
95C3×#126—31no—
96C3×#126C2×C911yescharacteristic, complemented
97C3×#126C3×D311yescharacteristic
98C3×#123—33no—
99C3×#121—33no—
100C3×#121—33no—
101C3×#120—33no—
102#895C2×C913yescomplemented
103#894C2×C913yescomplemented
104#884C2×C913yescomplemented
105#882C2×C913yescomplemented
106#876C2×C911yescharacteristic, complemented
107#874C2×C911yescharacteristic, complemented
108C23×C32C23×C311yescharacteristic
109C22×C3×D3—43no—
110C22×C3×D3C23×C311yescharacteristic
111C22×C3×D3C23×C311yescharacteristic
112C22×C3×D3C23×C313yes—
113C2×C32×C4C23×C313yes—
114C2×C32×C4C23×C313yes—
115C3×#67—43no—
116C2×C3×Dic3C23×C313yes—
117C3×D12—43no—
118C3×D12—43no—
119C3×D12—43no—
120C3×C4×D3—43no—
121C23×C9—21no—
122C23×C9—31no—
123C23×C9—31no—
124C23×C9—33no—
125C23×C9—123no—
126C23×C9C22×D311yescharacteristic
127D4×C9—123no—
128D4×C9—123no—
129D4×C9—123no—
130D4×C9—123no—
131C2×C4×C9—23no—
132C2×C4×C9—23no—
133C2×C4×C9—33no—
134C2×C4×C9—123no—
135C2×C4×C9C22×D313yes—
136C2×C4×C9C22×D313yes—
137#560—31noSylow
138C2×C3×C9—41no—
139C2×C3×C9H(5,𝔽2)13yes—
140D3×C9—43no—
141D3×C9—41no—
142D3×C9—41no—
143C24×C3—31no—
144C23×D3C22×C911yescharacteristic
145C2×C3×D4—33no—
146C2×C3×D4—33no—
147C2×C3×D4—33no—
148C2×C3×D4—33no—
149C22×C3×C4—33no—
150C2×#67C22×C913yes—
151C2×D12C22×C913yes—
152C2×D12C22×C913yes—
153C2×D12C22×C913yescomplemented
154C2×C4×D3C22×C913yescomplemented
155C3×C4⋊C4—23no—
156C3×C4⋊C4—33no—
157C3×C4⋊C4C22×C913yescomplemented
158C3×C4⋊C4C2×C3×D313yes—
159C3×C22⋊C4—23no—
160C3×C22⋊C4—33no—
161C3×C22⋊C4—33no—
162C3×C22⋊C4—33no—
163C3×C22⋊C4C22×C913yes—
164C3×C22⋊C4C2×C3×D313yes—
165C3×C42—21no—
166C3×C42C22×C911yescharacteristic, complemented
167C3×C42C2×C3×D311yescharacteristic
168(C2×D3)⋊C4C22×C913yes—
169(C2×D3)⋊C4C22×C913yes—
170(C2×D3)⋊C4C22×C913yescomplemented
171Dic3⋊C4C22×C913yescomplemented
172C22×C32—23no—
173C22×C32C24×C311yescharacteristic
174C2×C3×D3—23no—
175C2×C3×D3—23no—
176C2×C3×D3—23no—
177C2×C3×D3—23no—
178C2×C3×D3—23no—
179C2×C3×D3—44no—
180C32×C4—23no—
181C32×C4—23no—
182C3×Dic3—23no—
183C22×C9—23no—
184C22×C9—21no—
185C22×C9—43no—
186C22×C9—63no—
187C22×C9—63no—
188C22×C9—63no—
189C22×C9—63no—
190C22×C9—63no—
191C22×C9—124no—
192C22×C9C23×D311yescharacteristic
193C4×C9—23no—
194C4×C9—23no—
195C4×C9—43no—
196C4×C9—43no—
197C4×C9—63no—
198#127—31no—
199#126—31no—
200#126D3×C911yescharacteristic, complemented
201#123—33no—
202#121—33no—
203#121—33no—
204#120—33no—
205C3×C9#56011yescharacteristic, complemented, Sylow
206C23×C3—21no—
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—33no—
210C23×C3—123no—
211C23×C3C23×C911yescharacteristic
212C23×C3C22×C3×D311yescharacteristic
213C22×D3—43no—
214C22×D3C23×C911yescharacteristic
215C22×D3C23×C911yescharacteristic
216C22×D3C23×C913yes—
217C3×D4—123no—
218C3×D4—123no—
219C3×D4—123no—
220C3×D4—123no—
221C2×C3×C4—23no—
222C2×C3×C4—23no—
223C2×C3×C4—33no—
224C2×C3×C4—123no—
225C2×C3×C4C23×C913yes—
226C2×C3×C4C23×C913yes—
227C2×C3×C4C22×C3×D313yes—
228C2×C3×C4C22×C3×D313yes—
229#67—43no—
230C2×Dic3C23×C913yes—
231D12—43no—
232D12—43no—
233D12—43no—
234C4×D3—43no—
235C2×C32—41no—
236C2×C32C3×H(5,𝔽2)13yes—
237C3×D3—43no—
238C3×D3—41no—
239C3×D3—41no—
240C2×C9—23no—
241C2×C9—41no—
242C2×C9—81no—
243C2×C9—123no—
244C2×C9—121no—
245C2×C9—121no—
246C2×C9#100913yes—
247C24—31no—
248C2×D4—33no—
249C2×D4—33no—
250C2×D4—33no—
251C2×D4—33no—
252C22×C4—33no—
253C4⋊C4—33no—
254C4⋊C4C2×D3×C913yescomplemented
255C22⋊C4—33no—
256C22⋊C4—33no—
257C22⋊C4—33no—
258C22⋊C4C2×D3×C913yes—
259C42C2×D3×C911yescharacteristic, complemented
260C22×C3—23no—
261C22×C3—23no—
262C22×C3—21no—
263C22×C3—43no—
264C22×C3—63no—
265C22×C3—63no—
266C22×C3—63no—
267C22×C3—63no—
268C22×C3—63no—
269C22×C3—124no—
270C22×C3C24×C911yescharacteristic
271C22×C3C23×C3×D311yescharacteristic
272C2×D3—23no—
273C2×D3—23no—
274C2×D3—23no—
275C2×D3—23no—
276C2×D3—23no—
277C2×D3—44no—
278C3×C4—23no—
279C3×C4—23no—
280C3×C4—23no—
281C3×C4—23no—
282C3×C4—43no—
283C3×C4—43no—
284C3×C4—63no—
285Dic3—23no—
286C32C3×#56011yescharacteristic
287C9—21no—
288C9#571111yescharacteristic, direct product factor
289C23—31no—
290C23—31no—
291C23—33no—
292C23—123no—
293C23C22×D3×C911yescharacteristic
294D4—123no—
295D4—123no—
296D4—123no—
297D4—123no—
298C2×C4—33no—
299C2×C4—123no—
300C2×C4C22×D3×C913yes—
301C2×C4C22×D3×C913yes—
302C2×C3—23no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—81no—
306C2×C3—121no—
307C2×C3—121no—
308C2×C3—123no—
309C2×C3C9×H(5,𝔽2)13yes—
310C2×C3C3×#100913yes—
311D3—41no—
312D3—41no—
313D3—43no—
314C22—23no—
315C22—63no—
316C22—63no—
317C22—63no—
318C22—63no—
319C22—63no—
320C22—124no—
321C22C23×D3×C911yescharacteristic
322C4—23no—
323C4—23no—
324C4—63no—
325C3—21no—
326C3C9×#56011yescharacteristic, complemented
327C3C3×#571111yescharacteristic
328C2—41no—
329C2—121no—
330C2—121no—
331C2—123no—
332C2C9×#100913yes—
subgroupG/H
centerC22×C9C23×D3
commutatorC22×C3C24×C9
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#126C2
socle, cosocleC22×C32C24×C3
prime radicalC3×#5711C3
hypercenterC9×#126D3
hypocenterC3C9×#560
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C9×#126

Complements

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

NH ≅ G/Ntype
1. C3×C9×#126C2semidirect
2. C9×#895C2semidirect
3. C9×#894C2semidirect
4. C9×#884C2semidirect
5. C9×#882C2semidirect
6. C9×#876C2semidirect
7. C9×#874C2semidirect
14. C2×C9×D12C22semidirect
15. C2×C4×D3×C9C22semidirect
16. C3×C9×C4⋊C4C22semidirect
18. C3×C42×C9C22semidirect
21. C9×(C2×D3)⋊C4C22semidirect
22. C9×Dic3⋊C4C22semidirect
33. C9×#126D3semidirect
52. #5711C9direct (N × H)
74. C9×C4⋊C4C2×D3semidirect
81. C42×C9C2×D3semidirect
96. C3×#126C2×C9semidirect
102. #895C2×C9semidirect
103. #894C2×C9semidirect
104. #884C2×C9semidirect
105. #882C2×C9semidirect
106. #876C2×C9semidirect
107. #874C2×C9semidirect
153. C2×D12C22×C9semidirect
154. C2×C4×D3C22×C9semidirect
157. C3×C4⋊C4C22×C9semidirect
166. C3×C42C22×C9semidirect
170. (C2×D3)⋊C4C22×C9semidirect
171. Dic3⋊C4C22×C9semidirect
200. #126D3×C9semidirect
205. C3×C9#560semidirect
254. C4⋊C4C2×D3×C9semidirect
259. C42C2×D3×C9semidirect
288. C9#5711direct (N × H)
326. C3C9×#560semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1262
C9×#8952
C9×#8942
C9×#8842
C9×#8822
C9×#8762
C9×#8742
C3×#57113
C9×#5603