Finite Group Databasebeta

← C9×#5836 #13153746

Subgroups — C9×#5836 #13153746

Order 1728.

940 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C9×#1010C211yesmaximal, characteristic, complemented
3C9×#951C213yesmaximal, complemented
4C9×#950C213yesmaximal, complemented
5C9×#948C211yesmaximal, characteristic, complemented
6C2×C9×#214C213yesmaximal, complemented
7C9×#932C213yesmaximal, complemented
8C3×#5836C311yesmaximal, characteristic
9C9×#577—31nomaximal
10C3×C9×#41—21no—
11C3×C9×#41C2213yes—
12C3×C9×#41C2211yescharacteristic
13C2×C3×D4×C9—23no—
14C2×C3×D4×C9C2213yes—
15D3×Q8×C9—21no—
16C9×#237—23no—
17C9×#235C2213yescomplemented
18C2×C9×Dic6C2213yescomplemented
19C9×#216C2213yescomplemented
20C9×#215C2211yescharacteristic, complemented
21C9×#214C2216yescomplemented
22C9×#214C2213yescomplemented
23C9×#213C2213yescomplemented
24C9×#208C2213yescomplemented
25C2×C9×C3⋊C8C2213yescomplemented
26C32×H(5,𝔽2)C2×C311yescharacteristic
27C3×#1010C2×C311yescharacteristic
28C3×#951C2×C313yes—
29C3×#950C2×C313yes—
30C3×#948C2×C311yescharacteristic
31C2×C3×#214C2×C313yes—
32C3×#932C2×C313yes—
33C9×2−1+4—31no—
34C9×H(5,𝔽2)—21no—
35C9×H(5,𝔽2)D311yescharacteristic, complemented
36C9×#137—33no—
37C9×AGL(1,ℤ/8)—33no—
38C9×#135—33no—
39C2×C9×SD16—33no—
40C9×#131—31no—
41C23×C3×C9—23no—
42C3×Q8×C9D411yescharacteristic
43C3×Q8×C9C2311yescharacteristic
44C3×D4×C9—23no—
45C3×D4×C9—23no—
46C3×D4×C9D413yes—
47C3×D4×C9C2313yes—
48C2×C3×C4×C9—23no—
49C2×C3×C4×C9C2313yes—
50C9×#67—23no—
51C2×C9×Dic3—23no—
52C9×D12C2311yescharacteristic
53C4×D3×C9—23no—
54C9×Dic6—23no—
55C9×Dic6C2313yes—
56C9×C3⋊C8C2311yescharacteristic
57C9×C3⋊C8C2313yes—
58C3×#577—31no—
59#5836C911yescharacteristic, direct product factor
60C32×#41—21no—
61C32×#41C22×C313yes—
62C32×#41C22×C311yescharacteristic
63C2×C32×D4—23no—
64C2×C32×D4C22×C313yes—
65C3×D3×Q8—21no—
66C3×#237—23no—
67C3×#235C22×C313yes—
68C2×C3×Dic6C22×C313yes—
69C3×#216C22×C313yes—
70C3×#215C22×C311yescharacteristic
71C3×#214C22×C316yes—
72C3×#214C22×C313yes—
73C3×#213C22×C313yes—
74C3×#208C22×C313yes—
75C2×C3×C3⋊C8C22×C313yes—
76C9×#41—22no—
77C9×#41—23no—
78C9×#41—21no—
79C9×#41—21no—
80C9×#41—33no—
81C9×#41—63no—
82C9×#41C2×D313yes—
83C9×#41C2×D311yescharacteristic
84C2×Q8×C9—33no—
85C2×Q8×C9—61no—
86C2×D4×C9—26no—
87C2×D4×C9—23no—
88C2×D4×C9—23no—
89C2×D4×C9C2×D313yes—
90C9×Q16—33no—
91C9×SD16—31no—
92C9×SD16—33no—
93C9×SD16—36no—
94C9×D8—33no—
95C9×M16—33no—
96C2×C8×C9—33no—
97C22×C3×C9—23no—
98C22×C3×C9—83no—
99C22×C3×C9C2×D413yes—
100C2×D3×C9—21no—
101C3×C4×C9—21no—
102C3×C4×C9C2×D413yes—
103C3×C4×C9C2411yescharacteristic
104C9×Dic3—23no—
105C3×2−1+4—31no—
106C3×H(5,𝔽2)—21no—
107C3×H(5,𝔽2)C2×C911yescharacteristic, complemented
108C3×H(5,𝔽2)C3×D311yescharacteristic
109#1010C2×C911yescharacteristic, complemented
110C3×#137—33no—
111C3×AGL(1,ℤ/8)—33no—
112C3×#135—33no—
113C2×C3×SD16—33no—
114C3×#131—31no—
115#951C2×C913yescomplemented
116#950C2×C913yescomplemented
117#948C2×C911yescharacteristic, complemented
118C2×#214C2×C913yescomplemented
119#932C2×C913yescomplemented
120C23×C32—23no—
121C32×Q8C3×D411yescharacteristic
122C32×Q8C23×C311yescharacteristic
123C32×D4—23no—
124C32×D4—23no—
125C32×D4C3×D413yes—
126C32×D4C23×C313yes—
127C2×C32×C4—23no—
128C2×C32×C4C23×C313yes—
129C3×#67—23no—
130C2×C3×Dic3—23no—
131C3×D12C23×C311yescharacteristic
132C3×C4×D3—23no—
133C3×Dic6—23no—
134C3×Dic6C23×C313yes—
135C23×C9—26no—
136C23×C9—23no—
137C3×C3⋊C8C23×C311yescharacteristic
138C3×C3⋊C8C23×C313yes—
139Q8×C9—21no—
140Q8×C9—21no—
141Q8×C9—33no—
142Q8×C9—63no—
143Q8×C9#6711yescharacteristic
144Q8×C9C22×D311yescharacteristic
145D4×C9—26no—
146D4×C9—26no—
147D4×C9—23no—
148D4×C9—23no—
149D4×C9—23no—
150D4×C9—23no—
151D4×C9—31no—
152D4×C9—63no—
153D4×C9#6713yes—
154D4×C9C22×D313yes—
155C2×C4×C9—26no—
156C2×C4×C9—23no—
157C2×C4×C9—23no—
158C2×C4×C9—63no—
159C2×C4×C9—63no—
160C2×C4×C9C22×D313yes—
161C8×C9—31no—
162C8×C9—33no—
163#577—31noSylow
164C2×C3×C9—23no—
165C2×C3×C9—43no—
166C2×C3×C9C22×D411yescharacteristic
167D3×C9—41no—
168C3×#41—22no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—23no—
172C3×#41—21no—
173C3×#41—33no—
174C3×#41—63no—
175C3×#41C22×C913yes—
176C3×#41C22×C911yescharacteristic
177C3×#41C2×C3×D313yes—
178C3×#41C2×C3×D311yescharacteristic
179C2×C3×Q8—33no—
180C2×C3×Q8—61no—
181C2×C3×D4—26no—
182C2×C3×D4—23no—
183C2×C3×D4—23no—
184C2×C3×D4—23no—
185C2×C3×D4C22×C913yes—
186C2×C3×D4C2×C3×D313yes—
187D3×Q8—21no—
188#237—23no—
189#235C22×C913yescomplemented
190C2×Dic6C22×C913yescomplemented
191C3×Q16—33no—
192C3×SD16—31no—
193C3×SD16—33no—
194C3×SD16—36no—
195C3×D8—33no—
196C3×M16—33no—
197C2×C3×C8—33no—
198#216C22×C913yescomplemented
199#215C22×C911yescharacteristic, complemented
200#214C22×C916yescomplemented
201#214C22×C913yescomplemented
202#213C22×C913yescomplemented
203#208C22×C913yescomplemented
204C2×C3⋊C8C22×C913yescomplemented
205C22×C32—23no—
206C22×C32—83no—
207C22×C32C2×C3×D413yes—
208C2×C3×D3—21no—
209C32×C4—21no—
210C32×C4C2×C3×D413yes—
211C32×C4C24×C311yescharacteristic
212C3×Dic3—23no—
213C22×C9—26no—
214C22×C9—23no—
215C22×C9—23no—
216C22×C9—61no—
217C22×C9—83no—
218C22×C9—86no—
219C22×C9C2×#6713yes—
220C4×C9—22no—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—23no—
224C4×C9—63no—
225C4×C9C2×#6713yes—
226C4×C9C23×D311yescharacteristic
2272−1+4—31no—
228H(5,𝔽2)D3×C911yescharacteristic, complemented
229#137—33no—
230AGL(1,ℤ/8)—33no—
231#135—33no—
232C2×SD16—33no—
233#131—31no—
234C3×C9#57711yescharacteristic, complemented, Sylow
235C23×C3—26no—
236C23×C3—23no—
237C23×C3—23no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—33no—
241C3×Q8—63no—
242C3×Q8D4×C911yescharacteristic
243C3×Q8C23×C911yescharacteristic
244C3×Q8C3×#6711yescharacteristic
245C3×Q8C22×C3×D311yescharacteristic
246C3×D4—26no—
247C3×D4—26no—
248C3×D4—23no—
249C3×D4—23no—
250C3×D4—23no—
251C3×D4—23no—
252C3×D4—23no—
253C3×D4—23no—
254C3×D4—31no—
255C3×D4—63no—
256C3×D4D4×C913yes—
257C3×D4C23×C913yes—
258C3×D4C3×#6713yes—
259C3×D4C22×C3×D313yes—
260C2×C3×C4—26no—
261C2×C3×C4—23no—
262C2×C3×C4—23no—
263C2×C3×C4—23no—
264C2×C3×C4—63no—
265C2×C3×C4—63no—
266C2×C3×C4C23×C913yes—
267C2×C3×C4C22×C3×D313yes—
268#67—23no—
269C2×Dic3—23no—
270D12C23×C911yescharacteristic
271C4×D3—23no—
272Dic6—23no—
273Dic6C23×C913yes—
274C3×C8—31no—
275C3×C8—33no—
276C3⋊C8C23×C911yescharacteristic
277C3⋊C8C23×C913yes—
278C2×C32—23no—
279C2×C32—43no—
280C2×C32C22×C3×D411yescharacteristic
281C3×D3—41no—
282C2×C9—23no—
283C2×C9—21no—
284C2×C9—43no—
285C2×C9—46no—
286C2×C9—43no—
287C2×C9—121no—
288C2×C9C22×#6711yescharacteristic
289#41—21no—
290#41—33no—
291#41—63no—
292#41C2×D3×C913yes—
293#41C2×D3×C911yescharacteristic
294C2×Q8—33no—
295C2×Q8—61no—
296C2×D4—23no—
297C2×D4C2×D3×C913yes—
298Q16—33no—
299SD16—31no—
300SD16—33no—
301SD16—36no—
302D8—33no—
303M16—33no—
304C2×C8—33no—
305C22×C3—26no—
306C22×C3—23no—
307C22×C3—23no—
308C22×C3—23no—
309C22×C3—61no—
310C22×C3—83no—
311C22×C3—83no—
312C22×C3—86no—
313C22×C3C2×D4×C913yes—
314C22×C3C2×C3×#6713yes—
315C2×D3—21no—
316C3×C4—22no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—23no—
321C3×C4—63no—
322C3×C4C2×D4×C913yes—
323C3×C4C24×C911yescharacteristic
324C3×C4C2×C3×#6713yes—
325C3×C4C23×C3×D311yescharacteristic
326Dic3—23no—
327C32C3×#57711yescharacteristic
328C9—21no—
329C9#583611yescharacteristic, direct product factor
330C23—23no—
331Q8—33no—
332Q8—63no—
333Q8C9×#6711yescharacteristic
334Q8C22×D3×C911yescharacteristic
335D4—23no—
336D4—23no—
337D4—31no—
338D4—63no—
339D4C9×#6713yes—
340D4C22×D3×C913yes—
341C2×C4—23no—
342C2×C4—63no—
343C2×C4—63no—
344C2×C4C22×D3×C913yes—
345C8—31no—
346C8—33no—
347C2×C3—23no—
348C2×C3—23no—
349C2×C3—21no—
350C2×C3—43no—
351C2×C3—46no—
352C2×C3—43no—
353C2×C3—43no—
354C2×C3—121no—
355C2×C3C22×D4×C911yescharacteristic
356C2×C3C22×C3×#6711yescharacteristic
357D3—41no—
358C22—23no—
359C22—61no—
360C22—83no—
361C22C2×C9×#6713yes—
362C4—21no—
363C4—63no—
364C4C2×C9×#6713yes—
365C4C23×D3×C911yescharacteristic
366C3—21no—
367C3C9×#57711yescharacteristic, complemented
368C3C3×#583611yescharacteristic
369C2—23no—
370C2—43no—
371C2—121no—
372C2C22×C9×#6711yescharacteristic
subgroupG/H
centerC2×C9C22×#67
commutatorC3×C4C24×C9
FrattiniC3×C4C23×C3×D3
FittingC3×C9×H(5,𝔽2)C2
socleC2×C32C22×C3×D4
prime radicalC3×#5836C3
cosocleC32×C4C24×C3
hypercenterC9×H(5,𝔽2)D3
hypocenterC3C9×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C9×#41 ◁ C9×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×C9×H(5,𝔽2)C2semidirect
2. C9×#1010C2semidirect
3. C9×#951C2semidirect
4. C9×#950C2semidirect
5. C9×#948C2semidirect
6. C2×C9×#214C2semidirect
7. C9×#932C2semidirect
17. C9×#235C22semidirect
18. C2×C9×Dic6C22semidirect
19. C9×#216C22semidirect
20. C9×#215C22semidirect
21. C9×#214C22semidirect
22. C9×#214C22semidirect
23. C9×#213C22semidirect
24. C9×#208C22semidirect
25. C2×C9×C3⋊C8C22semidirect
35. C9×H(5,𝔽2)D3semidirect
59. #5836C9direct (N × H)
107. C3×H(5,𝔽2)C2×C9semidirect
109. #1010C2×C9semidirect
115. #951C2×C9semidirect
116. #950C2×C9semidirect
117. #948C2×C9semidirect
118. C2×#214C2×C9semidirect
119. #932C2×C9semidirect
189. #235C22×C9semidirect
190. C2×Dic6C22×C9semidirect
198. #216C22×C9semidirect
199. #215C22×C9semidirect
200. #214C22×C9semidirect
201. #214C22×C9semidirect
202. #213C22×C9semidirect
203. #208C22×C9semidirect
204. C2×C3⋊C8C22×C9semidirect
228. H(5,𝔽2)D3×C9semidirect
234. C3×C9#577semidirect
329. C9#5836direct (N × H)
367. C3C9×#577semidirect

Maximal subgroups

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

subgroupindex
C3×C9×H(5,𝔽2)2
C9×#10102
C9×#9512
C9×#9502
C9×#9482
C2×C9×#2142
C9×#9322
C3×#58363
C9×#5773