Finite Group Databasebeta

← C9×#5834 #13153708

Subgroups — C9×#5834 #13153708

Order 1728.

1100 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×#41C211yesmaximal, characteristic, complemented
2C2×C9×#239C211yesmaximal, characteristic, complemented
3Q8×C9×Dic3C211yesmaximal, characteristic, complemented
4C9×#940C213yesmaximal, complemented
5C9×#938C213yesmaximal, complemented
6C9×#930C213yesmaximal, complemented
7C4×C9×#67C213yesmaximal, complemented
8C3×#5834C311yesmaximal, characteristic
9C9×#550—31nomaximal
10C3×C9×#41—24no—
11C2×C3×Q8×C9C2211yescharacteristic
12C2×C3×D4×C9C2213yes—
13C22×C3×C4×C9C2213yes—
14C2×C9×#67C2213yes—
15C2×C9×#67C2216yes—
16C9×#239—24no—
17C2×C9×D12C2213yes—
18C2×C4×D3×C9C2213yes—
19C9×#217C2213yes—
20C9×(C2×D3)⋊C4C2213yes—
21C9×C4⋊Dic3C2213yes—
22C9×Dic3⋊C4C2211yescharacteristic
23C4×C9×Dic3C2213yes—
24C2×C32×#41C2×C311yescharacteristic
25C2×C3×#239C2×C311yescharacteristic
26C3×Q8×Dic3C2×C311yescharacteristic
27C3×#940C2×C313yes—
28C3×#938C2×C313yes—
29C3×#930C2×C313yes—
30C3×C4×#67C2×C313yes—
31C2×C9×#41—21no—
32C2×C9×#41—31no—
33C2×C9×#41D311yescharacteristic, complemented
34C9×#127—33no—
35C9×#121—33no—
36C9×#121—33no—
37C4×Q8×C9—31no—
38C4×D4×C9—33no—
39C23×C3×C9C2313yes—
40C22×D3×C9C2313yes—
41C3×Q8×C9D414yescomplemented
42C3×D4×C9—26no—
43C2×C3×C4×C9—26no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3×C4×C9C2313yes—
46C9×#67—26no—
47C9×#67—46no—
48C2×C9×Dic3C2311yescharacteristic
49C2×C9×Dic3C2313yes—
50C2×C9×Dic3C2311yescharacteristic
51C9×D12—26no—
52C4×D3×C9—26no—
53C3×#550—31no—
54#5834C911yescharacteristic, direct product factor
55C32×#41—24no—
56C2×C32×Q8C22×C311yescharacteristic
57C2×C32×D4C22×C313yes—
58C22×C32×C4C22×C313yes—
59C2×C3×#67C22×C313yes—
60C2×C3×#67C22×C316yes—
61C3×#239—24no—
62C2×C3×D12C22×C313yes—
63C2×C3×C4×D3C22×C313yes—
64C3×#217C22×C313yes—
65C3×(C2×D3)⋊C4C22×C313yes—
66C3×C4⋊Dic3C22×C313yes—
67C3×Dic3⋊C4C22×C311yescharacteristic
68C3×C4×Dic3C22×C313yes—
69C9×#41—28no—
70C9×#41—24no—
71C9×#41—64no—
72C2×Q8×C9—21no—
73C2×Q8×C9C2×D311yescharacteristic
74C2×D4×C9—23no—
75C2×D4×C9—33no—
76C2×D4×C9—36no—
77C2×D4×C9—33no—
78C2×D4×C9C2×D313yes—
79C22×C4×C9—23no—
80C22×C4×C9—33no—
81C22×C4×C9C2×D313yes—
82C9×C4⋊C4—31no—
83C9×C4⋊C4—33no—
84C9×C22⋊C4—33no—
85C9×C22⋊C4—33no—
86C42×C9—33no—
87C22×C3×C9—23no—
88C22×C3×C9—23no—
89C22×C3×C9—23no—
90C22×C3×C9C2411yescharacteristic
91C2×D3×C9—23no—
92C2×D3×C9—23no—
93C2×D3×C9—23no—
94C3×C4×C9—21no—
95C3×C4×C9C2×D416yes—
96C9×Dic3—21no—
97C9×Dic3—23no—
98C9×Dic3#4112yescomplemented
99C2×C3×#41—21no—
100C2×C3×#41—31no—
101C2×C3×#41C2×C911yescharacteristic, complemented
102C2×C3×#41C3×D311yescharacteristic
103C2×#239C2×C911yescharacteristic, complemented
104C3×#127—33no—
105C3×#121—33no—
106C3×#121—33no—
107C3×C4×Q8—31no—
108C3×C4×D4—33no—
109Q8×Dic3C2×C911yescharacteristic, complemented
110#940C2×C913yescomplemented
111#938C2×C913yescomplemented
112#930C2×C913yescomplemented
113C4×#67C2×C913yescomplemented
114C23×C32C23×C313yes—
115C22×C3×D3C23×C313yes—
116C32×Q8C3×D414yes—
117C32×D4—26no—
118C2×C32×C4—26no—
119C2×C32×C4C23×C311yescharacteristic
120C2×C32×C4C23×C313yes—
121C3×#67—26no—
122C3×#67—46no—
123C2×C3×Dic3C23×C313yes—
124C2×C3×Dic3C23×C311yescharacteristic
125C2×C3×Dic3C23×C311yescharacteristic
126C3×D12—26no—
127C3×C4×D3—26no—
128C23×C9—23no—
129C23×C9—33no—
130C23×C9C22×D313yes—
131Q8×C9—24no—
132Q8×C9#6714yescomplemented
133D4×C9—212no—
134D4×C9—26no—
135D4×C9—66no—
136D4×C9—66no—
137D4×C9—126no—
138C2×C4×C9—212no—
139C2×C4×C9—26no—
140C2×C4×C9—21no—
141C2×C4×C9—23no—
142C2×C4×C9—31no—
143C2×C4×C9—33no—
144C2×C4×C9—31no—
145C2×C4×C9—66no—
146C2×C4×C9C22×D311yescharacteristic
147C2×C4×C9C22×D313yes—
148#550—31noSylow
149C2×C3×C9—43no—
150C2×C3×C9C22×D411yescharacteristic
151C2×C3×C9C2×#4111yescharacteristic
152C2×C3×C9H(5,𝔽2)11yescharacteristic
153D3×C9—43no—
154C3×#41—28no—
155C3×#41—24no—
156C3×#41—24no—
157C3×#41—64no—
158C2×C3×Q8—21no—
159C2×C3×Q8C22×C911yescharacteristic
160C2×C3×Q8C2×C3×D311yescharacteristic
161C2×C3×D4—23no—
162C2×C3×D4—33no—
163C2×C3×D4—36no—
164C2×C3×D4—33no—
165C2×C3×D4C22×C913yes—
166C2×C3×D4C2×C3×D313yes—
167C22×C3×C4—23no—
168C22×C3×C4—33no—
169C22×C3×C4C22×C913yes—
170C22×C3×C4C2×C3×D313yes—
171C2×#67C22×C913yes—
172C2×#67C22×C916yes—
173#239—24no—
174C2×D12C22×C913yes—
175C2×C4×D3C22×C913yes—
176C3×C4⋊C4—31no—
177C3×C4⋊C4—33no—
178C3×C22⋊C4—33no—
179C3×C22⋊C4—33no—
180C3×C42—33no—
181#217C22×C913yes—
182(C2×D3)⋊C4C22×C913yes—
183C4⋊Dic3C22×C913yes—
184Dic3⋊C4C22×C911yescharacteristic
185C4×Dic3C22×C913yes—
186C22×C32—23no—
187C22×C32—23no—
188C22×C32—23no—
189C22×C32C24×C311yescharacteristic
190C2×C3×D3—23no—
191C2×C3×D3—23no—
192C2×C3×D3—23no—
193C32×C4—21no—
194C32×C4C2×C3×D416yes—
195C3×Dic3—21no—
196C3×Dic3—23no—
197C3×Dic3C3×#4112yes—
198C22×C9—26no—
199C22×C9—23no—
200C22×C9—23no—
201C22×C9—23no—
202C22×C9—21no—
203C22×C9—43no—
204C22×C9—43no—
205C22×C9—63no—
206C22×C9—63no—
207C22×C9—63no—
208C22×C9C23×D311yescharacteristic
209C4×C9—22no—
210C4×C9—21no—
211C4×C9—26no—
212C4×C9—32no—
213C4×C9—63no—
214C4×C9—61no—
215C4×C9C2×#6716yes—
216C2×#41—31no—
217C2×#41D3×C911yescharacteristic, complemented
218#127—33no—
219#121—33no—
220#121—33no—
221C4×Q8—31no—
222C4×D4—33no—
223C3×C9#55011yescharacteristic, complemented, Sylow
224C23×C3—23no—
225C23×C3—33no—
226C23×C3C23×C913yes—
227C23×C3C22×C3×D313yes—
228C22×D3C23×C913yes—
229C3×Q8—24no—
230C3×Q8D4×C914yescomplemented
231C3×Q8C3×#6714yes—
232C3×D4—212no—
233C3×D4—26no—
234C3×D4—26no—
235C3×D4—66no—
236C3×D4—66no—
237C3×D4—126no—
238C2×C3×C4—212no—
239C2×C3×C4—26no—
240C2×C3×C4—26no—
241C2×C3×C4—21no—
242C2×C3×C4—23no—
243C2×C3×C4—31no—
244C2×C3×C4—33no—
245C2×C3×C4—31no—
246C2×C3×C4—66no—
247C2×C3×C4C23×C911yescharacteristic
248C2×C3×C4C23×C913yes—
249C2×C3×C4C22×C3×D311yescharacteristic
250C2×C3×C4C22×C3×D313yes—
251#67—26no—
252#67—46no—
253C2×Dic3C23×C913yes—
254C2×Dic3C23×C911yescharacteristic
255C2×Dic3C23×C911yescharacteristic
256D12—26no—
257C4×D3—26no—
258C2×C32—43no—
259C2×C32C22×C3×D411yescharacteristic
260C2×C32C2×C3×#4111yescharacteristic
261C2×C32C3×H(5,𝔽2)11yescharacteristic
262C3×D3—43no—
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—46no—
267C2×C9—43no—
268C2×C9—123no—
269C2×C9D3×#4111yescharacteristic
270C2×C9#100911yescharacteristic
271C2×C9C22×#6711yescharacteristic
272#41—24no—
273#41—64no—
274C2×Q8C2×D3×C911yescharacteristic
275C2×D4—33no—
276C2×D4—36no—
277C2×D4—33no—
278C2×D4C2×D3×C913yes—
279C22×C4—33no—
280C22×C4C2×D3×C913yes—
281C4⋊C4—31no—
282C4⋊C4—33no—
283C22⋊C4—33no—
284C22⋊C4—33no—
285C42—33no—
286C22×C3—26no—
287C22×C3—23no—
288C22×C3—23no—
289C22×C3—23no—
290C22×C3—23no—
291C22×C3—23no—
292C22×C3—23no—
293C22×C3—21no—
294C22×C3—43no—
295C22×C3—43no—
296C22×C3—63no—
297C22×C3—63no—
298C22×C3—63no—
299C22×C3C24×C911yescharacteristic
300C22×C3C23×C3×D311yescharacteristic
301C2×D3—23no—
302C2×D3—23no—
303C2×D3—23no—
304C3×C4—22no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—26no—
308C3×C4—32no—
309C3×C4—63no—
310C3×C4—61no—
311C3×C4C2×D4×C916yes—
312C3×C4C2×C3×#6716yes—
313Dic3—21no—
314Dic3—23no—
315Dic3C9×#4112yescomplemented
316C32C3×#55011yescharacteristic
317C9—21no—
318C9#583411yescharacteristic, direct product factor
319C23—33no—
320C23C22×D3×C913yes—
321Q8C9×#6714yescomplemented
322D4—26no—
323D4—66no—
324D4—66no—
325D4—126no—
326C2×C4—26no—
327C2×C4—31no—
328C2×C4—33no—
329C2×C4—31no—
330C2×C4—66no—
331C2×C4C22×D3×C911yescharacteristic
332C2×C4C22×D3×C913yes—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—46no—
337C2×C3—43no—
338C2×C3—43no—
339C2×C3—123no—
340C2×C3C22×D4×C911yescharacteristic
341C2×C3C2×C9×#4111yescharacteristic
342C2×C3C9×H(5,𝔽2)11yescharacteristic
343C2×C3C3×D3×#4111yescharacteristic
344C2×C3C3×#100911yescharacteristic
345C2×C3C22×C3×#6711yescharacteristic
346D3—43no—
347C22—23no—
348C22—23no—
349C22—23no—
350C22—63no—
351C22—63no—
352C22—63no—
353C22C23×D3×C911yescharacteristic
354C4—21no—
355C4—32no—
356C4—63no—
357C4—61no—
358C4C2×C9×#6716yes—
359C3—21no—
360C3C9×#55011yescharacteristic, complemented
361C3C3×#583411yescharacteristic
362C2—43no—
363C2—123no—
364C2D3×C9×#4111yescharacteristic
365C2C9×#100911yescharacteristic
366C2C22×C9×#6711yescharacteristic
subgroupG/H
centerC22×C9C23×D3
commutatorC22×C3C24×C9
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×#41C2
socle, cosocleC22×C32C24×C3
prime radicalC3×#5834C3
hypercenterC2×C9×#41D3
hypocenterC3C9×#550
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C9×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×#41C2semidirect
2. C2×C9×#239C2semidirect
3. Q8×C9×Dic3C2semidirect
4. C9×#940C2semidirect
5. C9×#938C2semidirect
6. C9×#930C2semidirect
7. C4×C9×#67C2semidirect
33. C2×C9×#41D3semidirect
41. C3×Q8×C9D4semidirect
54. #5834C9direct (N × H)
98. C9×Dic3#41semidirect
101. C2×C3×#41C2×C9semidirect
103. C2×#239C2×C9semidirect
109. Q8×Dic3C2×C9semidirect
110. #940C2×C9semidirect
111. #938C2×C9semidirect
112. #930C2×C9semidirect
113. C4×#67C2×C9semidirect
132. Q8×C9#67semidirect
217. C2×#41D3×C9semidirect
223. C3×C9#550semidirect
230. C3×Q8D4×C9semidirect
315. Dic3C9×#41semidirect
318. C9#5834direct (N × H)
321. Q8C9×#67semidirect
360. C3C9×#550semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×#412
C2×C9×#2392
Q8×C9×Dic32
C9×#9402
C9×#9382
C9×#9302
C4×C9×#672
C3×#58343
C9×#5503