Finite Group Databasebeta

← C9×#5750 #13153405

Subgroups — C9×#5750 #13153405

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×C9×#41C211yesmaximal, characteristic
2C3×C9×#131C211yesmaximal, characteristic, complemented
3C9×#948C211yesmaximal, characteristic, complemented
4C9×#907C213yesmaximal, complemented
5D3×C9×M16C213yesmaximal, complemented
6C9×#901C213yesmaximal, complemented
7C2×C9×ΓL(1,𝔽25)C213yesmaximal, complemented
8C3×#5750C311yesmaximal, characteristic
9C9×#568—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C9×#239C411yescharacteristic
12D3×Q8×C9C411yescharacteristic
13C9×#237C413yes—
14D3×D4×C9C413yes—
15C9×#235C2213yes—
16C2×C4×D3×C9C2213yes—
17C3×C9×M16C2213yescomplemented
18C2×C3×C8×C9C2213yescomplemented
19C9×#208C2213yescomplemented
20C2×C9×C3⋊C8C2213yescomplemented
21C9×ΓL(1,𝔽25)C2211yescharacteristic
22C9×ΓL(1,𝔽25)C2213yes—
23C9×ΓL(1,𝔽25)C2213yescomplemented
24C9×ΓL(1,𝔽25)C2213yescomplemented
25D3×C8×C9C2216yescomplemented
26C3×D3×#41C2×C311yescharacteristic
27C32×#131C2×C311yescharacteristic
28C3×#948C2×C311yescharacteristic
29C3×#907C2×C313yes—
30C3×D3×M16C2×C313yes—
31C3×#901C2×C313yes—
32C2×C3×ΓL(1,𝔽25)C2×C313yes—
33C2×C9×#41—31no—
34C9×#131—21no—
35C9×#131—31no—
36C9×#131—33no—
37C9×#131—33no—
38C9×#131D311yescharacteristic, complemented
39C2×C9×M16—33no—
40C2×C9×M16—33no—
41C22×D3×C9C2×C413yes—
42C3×Q8×C9C2×C411yescharacteristic
43C3×D4×C9C2×C413yes—
44C2×C3×C4×C9C2313yes—
45C9×#67C2×C416yes—
46C2×C9×Dic3C2×C413yes—
47C9×D12C2×C413yes—
48C4×D3×C9C2×C413yes—
49C4×D3×C9C2×C413yes—
50C4×D3×C9C2311yescharacteristic
51C4×D3×C9C2313yes—
52C9×Dic6C2×C413yes—
53C3×C8×C9C2313yes—
54C3×C8×C9C2311yescharacteristic
55C9×C3⋊C8C2313yes—
56C9×C3⋊C8C2311yescharacteristic
57C3×#568—31no—
58#5750C911yescharacteristic, direct product factor
59C32×#41C22×C311yescharacteristic
60C3×#239C3×C411yescharacteristic
61C3×D3×Q8C3×C411yescharacteristic
62C3×#237C3×C413yes—
63C3×D3×D4C3×C413yes—
64C3×#235C22×C313yes—
65C2×C3×C4×D3C22×C313yes—
66C32×M16C22×C313yes—
67C2×C32×C8C22×C313yes—
68C3×#208C22×C313yes—
69C2×C3×C3⋊C8C22×C313yes—
70C3×ΓL(1,𝔽25)C22×C311yescharacteristic
71C3×ΓL(1,𝔽25)C22×C313yes—
72C3×ΓL(1,𝔽25)C22×C313yes—
73C3×ΓL(1,𝔽25)C22×C313yes—
74C3×D3×C8C22×C316yes—
75C9×#41—21no—
76C9×#41—31no—
77C9×#41—33no—
78C9×#41—33no—
79C9×#41C2×D311yescharacteristic
80C2×Q8×C9—31no—
81C2×D4×C9—33no—
82C22×C4×C9—33no—
83C9×M16—23no—
84C9×M16—33no—
85C9×M16—33no—
86C9×M16—31no—
87C9×M16—33no—
88C9×M16—33no—
89C9×M16C2×D313yescomplemented
90C2×C8×C9—23no—
91C2×C8×C9—33no—
92C2×C8×C9—36no—
93C2×C8×C9C2×D313yescomplemented
94C22×C3×C9C22×C413yes—
95C2×D3×C9—43no—
96C2×D3×C9C22×C411yescharacteristic
97C2×D3×C9C22×C413yes—
98C3×C4×C9C22×C413yes—
99C3×C4×C9C2411yescharacteristic
100C9×Dic3C22×C413yes—
101C9×Dic3C22×C411yescharacteristic
102C2×C3×#41—31no—
103D3×#41C2×C911yescharacteristic
104C3×#131—21no—
105C3×#131—31no—
106C3×#131—33no—
107C3×#131—33no—
108C3×#131C2×C911yescharacteristic, complemented
109C3×#131C3×D311yescharacteristic
110C2×C3×M16—33no—
111C2×C3×M16—33no—
112#948C2×C911yescharacteristic, complemented
113#907C2×C913yescomplemented
114D3×M16C2×C913yescomplemented
115#901C2×C913yescomplemented
116C2×ΓL(1,𝔽25)C2×C913yescomplemented
117C22×C3×D3C2×C3×C413yes—
118C32×Q8C2×C3×C411yescharacteristic
119C32×D4C2×C3×C413yes—
120C2×C32×C4C23×C313yes—
121C3×#67C2×C3×C416yes—
122C2×C3×Dic3C2×C3×C413yes—
123C3×D12C2×C3×C413yes—
124C3×C4×D3C2×C3×C413yes—
125C3×C4×D3C2×C3×C413yes—
126C3×C4×D3C23×C313yes—
127C3×C4×D3C23×C311yescharacteristic
128C3×Dic6C2×C3×C413yes—
129C23×C9—33no—
130C32×C8C23×C313yes—
131C32×C8C23×C311yescharacteristic
132C3×C3⋊C8C23×C313yes—
133C3×C3⋊C8C23×C311yescharacteristic
134Q8×C9—21no—
135Q8×C9—33no—
136Q8×C9C4×D311yescharacteristic
137D4×C9—23no—
138D4×C9—36no—
139D4×C9—33no—
140D4×C9C4×D313yes—
141C2×C4×C9—23no—
142C2×C4×C9—33no—
143C2×C4×C9—33no—
144C2×C4×C9—33no—
145C2×C4×C9—33no—
146C2×C4×C9—31no—
147C2×C4×C9C22×D313yes—
148C8×C9—21no—
149C8×C9—23no—
150C8×C9—33no—
151C8×C9—31no—
152C8×C9C22×D313yes—
153C8×C9C22×D311yescharacteristic
154#568—31noSylow
155C2×C3×C9—23no—
156C2×C3×C9C23×C411yescharacteristic
157D3×C9—21no—
158D3×C9—23no—
159C3×#41—21no—
160C3×#41—31no—
161C3×#41—33no—
162C3×#41—33no—
163C3×#41C22×C911yescharacteristic
164C3×#41C2×C3×D311yescharacteristic
165C2×C3×Q8—31no—
166C2×C3×D4—33no—
167C22×C3×C4—33no—
168#239C4×C911yescharacteristic
169D3×Q8C4×C911yescharacteristic
170#237C4×C913yes—
171D3×D4C4×C913yes—
172#235C22×C913yes—
173C2×C4×D3C22×C913yes—
174C3×M16—23no—
175C3×M16—33no—
176C3×M16—33no—
177C3×M16—31no—
178C3×M16—33no—
179C3×M16—33no—
180C3×M16C22×C913yescomplemented
181C3×M16C2×C3×D313yes—
182C2×C3×C8—23no—
183C2×C3×C8—33no—
184C2×C3×C8—36no—
185C2×C3×C8C22×C913yescomplemented
186C2×C3×C8C2×C3×D313yes—
187#208C22×C913yescomplemented
188C2×C3⋊C8C22×C913yescomplemented
189ΓL(1,𝔽25)C22×C911yescharacteristic
190ΓL(1,𝔽25)C22×C913yes—
191ΓL(1,𝔽25)C22×C913yescomplemented
192ΓL(1,𝔽25)C22×C913yescomplemented
193D3×C8C22×C916yescomplemented
194C22×C32C22×C3×C413yes—
195C2×C3×D3—43no—
196C2×C3×D3C22×C3×C411yescharacteristic
197C2×C3×D3C22×C3×C413yes—
198C32×C4C22×C3×C413yes—
199C32×C4C24×C311yescharacteristic
200C3×Dic3C22×C3×C413yes—
201C3×Dic3C22×C3×C411yescharacteristic
202C22×C9—23no—
203C22×C9—31no—
204C22×C9—33no—
205C22×C9—123no—
206C22×C9C2×C4×D313yes—
207C4×C9—21no—
208C4×C9—23no—
209C4×C9—31no—
210C4×C9—33no—
211C4×C9C2×C4×D313yes—
212C4×C9C23×D311yescharacteristic
213C2×#41—31no—
214#131—31no—
215#131—33no—
216#131—33no—
217#131D3×C911yescharacteristic, complemented
218C2×M16—33no—
219C2×M16—33no—
220C3×C9#56811yescharacteristic, complemented, Sylow
221C23×C3—33no—
222C22×D3C2×C4×C913yes—
223C3×Q8—21no—
224C3×Q8—33no—
225C3×Q8C2×C4×C911yescharacteristic
226C3×Q8C3×C4×D311yescharacteristic
227C3×D4—23no—
228C3×D4—36no—
229C3×D4—33no—
230C3×D4C2×C4×C913yes—
231C3×D4C3×C4×D313yes—
232C2×C3×C4—23no—
233C2×C3×C4—33no—
234C2×C3×C4—33no—
235C2×C3×C4—31no—
236C2×C3×C4—33no—
237C2×C3×C4—33no—
238C2×C3×C4C23×C913yes—
239C2×C3×C4C22×C3×D313yes—
240#67C2×C4×C916yes—
241C2×Dic3C2×C4×C913yes—
242D12C2×C4×C913yes—
243C4×D3C2×C4×C913yes—
244C4×D3C2×C4×C913yes—
245C4×D3C23×C913yes—
246C4×D3C23×C911yescharacteristic
247Dic6C2×C4×C913yes—
248C3×C8—21no—
249C3×C8—23no—
250C3×C8—33no—
251C3×C8—31no—
252C3×C8C23×C913yes—
253C3×C8C23×C911yescharacteristic
254C3×C8C22×C3×D313yes—
255C3×C8C22×C3×D311yescharacteristic
256C3⋊C8C23×C913yes—
257C3⋊C8C23×C911yescharacteristic
258C2×C32—23no—
259C2×C32C23×C3×C411yescharacteristic
260C3×D3—21no—
261C3×D3—23no—
262C2×C9—23no—
263C2×C9—21no—
264C2×C9—43no—
265C2×C9—63no—
266C2×C9—61no—
267C2×C9C22×C4×D311yescharacteristic
268#41—31no—
269#41—33no—
270#41—33no—
271#41C2×D3×C911yescharacteristic
272C2×Q8—31no—
273C2×D4—33no—
274C22×C4—33no—
275M16—33no—
276M16—33no—
277M16—31no—
278M16—33no—
279M16—33no—
280M16C2×D3×C913yescomplemented
281C2×C8—33no—
282C2×C8—36no—
283C2×C8C2×D3×C913yescomplemented
284C22×C3—23no—
285C22×C3—31no—
286C22×C3—33no—
287C22×C3—123no—
288C22×C3C22×C4×C913yes—
289C22×C3C2×C3×C4×D313yes—
290C2×D3—43no—
291C2×D3C22×C4×C911yescharacteristic
292C2×D3C22×C4×C913yes—
293C3×C4—21no—
294C3×C4—23no—
295C3×C4—31no—
296C3×C4—33no—
297C3×C4C22×C4×C913yes—
298C3×C4C24×C911yescharacteristic
299C3×C4C2×C3×C4×D313yes—
300C3×C4C23×C3×D311yescharacteristic
301Dic3C22×C4×C913yes—
302Dic3C22×C4×C911yescharacteristic
303C32C3×#56811yescharacteristic
304C9—21no—
305C9#575011yescharacteristic, direct product factor
306C23—33no—
307Q8—33no—
308Q8C4×D3×C911yescharacteristic
309D4—36no—
310D4—33no—
311D4C4×D3×C913yes—
312C2×C4—33no—
313C2×C4—33no—
314C2×C4—31no—
315C2×C4—33no—
316C2×C4—33no—
317C2×C4C22×D3×C913yes—
318C8—31no—
319C8—33no—
320C8C22×D3×C913yes—
321C8C22×D3×C911yescharacteristic
322C2×C3—23no—
323C2×C3—23no—
324C2×C3—21no—
325C2×C3—43no—
326C2×C3—61no—
327C2×C3—63no—
328C2×C3C23×C4×C911yescharacteristic
329C2×C3C22×C3×C4×D311yescharacteristic
330D3—21no—
331D3—23no—
332C22—31no—
333C22—33no—
334C22—123no—
335C22C2×C4×D3×C913yes—
336C4—31no—
337C4—33no—
338C4C2×C4×D3×C913yes—
339C4C23×D3×C911yescharacteristic
340C3—21no—
341C3C9×#56811yescharacteristic, complemented
342C3C3×#575011yescharacteristic
343C2—23no—
344C2—61no—
345C2—63no—
346C2C22×C4×D3×C911yescharacteristic
subgroupG/H
centerC4×C9C23×D3
commutatorC2×C3C23×C4×C9
FrattiniC3×C4C23×C3×D3
FittingC3×C9×#131C2
socleC2×C32C23×C3×C4
prime radicalC3×D3×#41C2×C3
cosocleC32×C4C24×C3
hypercenterC9×#131D3
hypocenterC3C9×#568
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C9 ◁ C9×#131

Complements

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

NH ≅ G/Ntype
2. C3×C9×#131C2semidirect
3. C9×#948C2semidirect
4. C9×#907C2semidirect
5. D3×C9×M16C2semidirect
6. C9×#901C2semidirect
7. C2×C9×ΓL(1,𝔽25)C2semidirect
17. C3×C9×M16C22semidirect
18. C2×C3×C8×C9C22semidirect
19. C9×#208C22semidirect
20. C2×C9×C3⋊C8C22semidirect
23. C9×ΓL(1,𝔽25)C22semidirect
24. C9×ΓL(1,𝔽25)C22semidirect
25. D3×C8×C9C22semidirect
38. C9×#131D3semidirect
58. #5750C9direct (N × H)
89. C9×M16C2×D3semidirect
93. C2×C8×C9C2×D3semidirect
108. C3×#131C2×C9semidirect
112. #948C2×C9semidirect
113. #907C2×C9semidirect
114. D3×M16C2×C9semidirect
115. #901C2×C9semidirect
116. C2×ΓL(1,𝔽25)C2×C9semidirect
180. C3×M16C22×C9semidirect
185. C2×C3×C8C22×C9semidirect
187. #208C22×C9semidirect
188. C2×C3⋊C8C22×C9semidirect
191. ΓL(1,𝔽25)C22×C9semidirect
192. ΓL(1,𝔽25)C22×C9semidirect
193. D3×C8C22×C9semidirect
217. #131D3×C9semidirect
220. C3×C9#568semidirect
280. M16C2×D3×C9semidirect
283. C2×C8C2×D3×C9semidirect
305. C9#5750direct (N × H)
341. C3C9×#568semidirect

Maximal subgroups

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

subgroupindex
D3×C9×#412
C3×C9×#1312
C9×#9482
C9×#9072
D3×C9×M162
C9×#9012
C2×C9×ΓL(1,𝔽25)2
C3×#57503
C9×#5683