Finite Group Databasebeta

← C9×#5202 #13143498

Subgroups — C9×#5202 #13143498

Order 1728.

516 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#1009C211yesmaximal, characteristic
2C3×C9×#137C211yesmaximal, characteristic, complemented
3C9×#947C211yesmaximal, characteristic, complemented
4C9×#942C211yesmaximal, characteristic, complemented
5C9×#837C211yesmaximal, characteristic, complemented
6C9×#825C211yesmaximal, characteristic, complemented
7C9×#823C211yesmaximal, characteristic, complemented
8C3×#5202C311yesmaximal, characteristic
9C9×#455—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C3×Q8×C9C2211yescharacteristic, complemented
12C9×#239—21no—
13D3×D4×C9—22no—
14D3×D4×C9—21no—
15C9×#235—21no—
16C9×#235C2211yescharacteristic
17C2×C9×D12—21no—
18C2×C9×D12C2211yescharacteristic
19C3×C9×Q16—21no—
20C3×C9×SD16—21no—
21C3×C9×M16C2211yescharacteristic, complemented
22C9×#216—21no—
23C9×#215—21no—
24C9×(C2×D3)⋊C4—22no—
25C4×C9×Dic3C2211yescharacteristic, complemented
26C9×#208C2211yescharacteristic, complemented
27C3×#1009C2×C311yescharacteristic
28C32×#137C2×C311yescharacteristic
29C3×#947C2×C311yescharacteristic
30C3×#942C2×C311yescharacteristic
31C9×H(5,𝔽2)—31no—
32C3×#837C2×C311yescharacteristic
33C3×#825C2×C311yescharacteristic
34C3×#823C2×C311yescharacteristic
35C9×#137—21no—
36C9×#137—31no—
37C9×#137D311yescharacteristic, complemented
38C9×#124—31no—
39C9×#104—31no—
40C9×#104—31no—
41C9×#100—31no—
42C22×D3×C9—22no—
43C22×D3×C9D412yes—
44C3×Q8×C9—21no—
45C3×Q8×C9—21no—
46C3×Q8×C9D411yescharacteristic
47C3×D4×C9—21no—
48C3×D4×C9D411yescharacteristic
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9C2311yescharacteristic
52C9×#67—22no—
53C9×#67—21no—
54C2×C9×Dic3—21no—
55C9×D12—22no—
56C9×D12—21no—
57C9×D12—21no—
58C9×D12D411yescharacteristic
59C4×D3×C9—22no—
60C4×D3×C9—21no—
61C9×Dic6D411yescharacteristic
62C3×C8×C9—21no—
63C9×C3⋊C8—21no—
64C3×#455—31no—
65#5202C911yescharacteristic, direct product factor
66C32×#41C22×C311yescharacteristic
67C2×C32×Q8C22×C311yescharacteristic
68C3×#239—21no—
69C3×D3×D4—22no—
70C3×D3×D4—21no—
71C3×#235—21no—
72C3×#235C22×C311yescharacteristic
73C2×C3×D12—21no—
74C2×C3×D12C22×C311yescharacteristic
75C32×Q16—21no—
76C32×SD16—21no—
77C32×M16C22×C311yescharacteristic
78C3×#216—21no—
79C3×#215—21no—
80C3×(C2×D3)⋊C4—22no—
81C3×C4×Dic3C22×C311yescharacteristic
82C3×#208C22×C311yescharacteristic
83C9×#41—21no—
84C9×#41—31no—
85C9×#41—61no—
86C9×#41—61no—
87C9×#41C2×D311yescharacteristic
88C2×Q8×C9—21no—
89C2×Q8×C9C2×D311yescharacteristic, complemented
90C2×D4×C9—31no—
91C2×D4×C9—61no—
92C2×D4×C9—62no—
93C2×D4×C9—61no—
94C9×Q16—22no—
95C9×Q16—21no—
96C9×Q16—61no—
97C9×SD16—22no—
98C9×SD16—21no—
99C9×SD16—61no—
100C9×M16—21no—
101C9×M16—31no—
102C9×M16C2×D311yescharacteristic, complemented
103C9×C22⋊C4—62no—
104C42×C9—31no—
105C22×C3×C9—21no—
106C22×C3×C9C2×D411yescharacteristic
107C2×D3×C9—22no—
108C2×D3×C9—21no—
109C2×D3×C9—42no—
110C2×D3×C9—82no—
111C3×C4×C9—21no—
112C3×C4×C9—41no—
113C3×C4×C9C2×D411yescharacteristic
114C3×C4×C9C2×D411yescharacteristic
115C9×Dic3—21no—
116C9×Dic3—41no—
117C3×H(5,𝔽2)—31no—
118#1009C2×C911yescharacteristic
119C3×#137—21no—
120C3×#137—31no—
121C3×#137C2×C911yescharacteristic, complemented
122C3×#137C3×D311yescharacteristic
123C3×#124—31no—
124#947C2×C911yescharacteristic, complemented
125#942C2×C911yescharacteristic, complemented
126C3×#104—31no—
127C3×#104—31no—
128C3×#100—31no—
129#837C2×C911yescharacteristic, complemented
130#825C2×C911yescharacteristic, complemented
131#823C2×C911yescharacteristic, complemented
132C22×C3×D3—22no—
133C22×C3×D3C3×D412yes—
134C32×Q8—21no—
135C32×Q8—21no—
136C32×Q8C3×D411yescharacteristic
137C32×D4—21no—
138C32×D4C3×D411yescharacteristic
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4C23×C311yescharacteristic
142C3×#67—22no—
143C3×#67—21no—
144C2×C3×Dic3—21no—
145C3×D12—22no—
146C3×D12—21no—
147C3×D12—21no—
148C3×D12C3×D411yescharacteristic
149C3×C4×D3—22no—
150C3×C4×D3—21no—
151C3×Dic6C3×D411yescharacteristic
152C23×C9—32no—
153C23×C9—62no—
154C32×C8—21no—
155C3×C3⋊C8—21no—
156Q8×C9—22no—
157Q8×C9—21no—
158Q8×C9—21no—
159Q8×C9—21no—
160Q8×C9—31no—
161Q8×C9—41no—
162Q8×C9#6711yescharacteristic
163D4×C9—21no—
164D4×C9—21no—
165D4×C9—31no—
166D4×C9—41no—
167D4×C9—62no—
168D4×C9—62no—
169D4×C9—61no—
170D4×C9—61no—
171D4×C9—61no—
172D4×C9#6711yescharacteristic
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—41no—
177C2×C4×C9—41no—
178C2×C4×C9—62no—
179C2×C4×C9—61no—
180C2×C4×C9—61no—
181C2×C4×C9C22×D311yescharacteristic
182C8×C9—22no—
183C8×C9—21no—
184C8×C9—61no—
185#455—31noSylow
186C2×C3×C9—21no—
187C2×C3×C9—41no—
188C2×C3×C9#12011yescharacteristic
189D3×C9—42no—
190D3×C9—41no—
191C3×#41—21no—
192C3×#41—31no—
193C3×#41—61no—
194C3×#41—61no—
195C3×#41C22×C911yescharacteristic
196C3×#41C2×C3×D311yescharacteristic
197C2×C3×Q8—21no—
198C2×C3×Q8C22×C911yescharacteristic, complemented
199C2×C3×Q8C2×C3×D311yescharacteristic
200C2×C3×D4—31no—
201C2×C3×D4—61no—
202C2×C3×D4—62no—
203C2×C3×D4—61no—
204#239—21no—
205D3×D4—22no—
206D3×D4—21no—
207#235—21no—
208#235C22×C911yescharacteristic
209C2×D12—21no—
210C2×D12C22×C911yescharacteristic
211C3×Q16—22no—
212C3×Q16—21no—
213C3×Q16—21no—
214C3×Q16—61no—
215C3×SD16—22no—
216C3×SD16—21no—
217C3×SD16—21no—
218C3×SD16—61no—
219C3×M16—21no—
220C3×M16—31no—
221C3×M16C22×C911yescharacteristic, complemented
222C3×M16C2×C3×D311yescharacteristic
223C3×C22⋊C4—62no—
224C3×C42—31no—
225#216—21no—
226#215—21no—
227(C2×D3)⋊C4—22no—
228C4×Dic3C22×C911yescharacteristic, complemented
229#208C22×C911yescharacteristic, complemented
230C22×C32—21no—
231C22×C32C2×C3×D411yescharacteristic
232C2×C3×D3—22no—
233C2×C3×D3—21no—
234C2×C3×D3—42no—
235C2×C3×D3—82no—
236C32×C4—21no—
237C32×C4—41no—
238C32×C4C2×C3×D411yescharacteristic
239C32×C4C2×C3×D411yescharacteristic
240C3×Dic3—21no—
241C3×Dic3—41no—
242C22×C9—21no—
243C22×C9—21no—
244C22×C9—41no—
245C22×C9—61no—
246C22×C9—62no—
247C22×C9—122no—
248C22×C9—242no—
249C22×C9D3×D411yescharacteristic
250C4×C9—21no—
251C4×C9—21no—
252C4×C9—21no—
253C4×C9—41no—
254C4×C9—42no—
255C4×C9—41no—
256C4×C9—61no—
257C4×C9—121no—
258C4×C9D3×D411yescharacteristic
259C4×C9C2×#6711yescharacteristic
260H(5,𝔽2)—31no—
261#137—31no—
262#137D3×C911yescharacteristic, complemented
263#124—31no—
264#104—31no—
265#104—31no—
266#100—31no—
267C3×C9#45511yescharacteristic, complemented, Sylow
268C23×C3—32no—
269C23×C3—62no—
270C22×D3—22no—
271C22×D3D4×C912yes—
272C3×Q8—22no—
273C3×Q8—21no—
274C3×Q8—21no—
275C3×Q8—21no—
276C3×Q8—21no—
277C3×Q8—21no—
278C3×Q8—31no—
279C3×Q8—41no—
280C3×Q8D4×C911yescharacteristic
281C3×Q8C3×#6711yescharacteristic
282C3×D4—21no—
283C3×D4—21no—
284C3×D4—21no—
285C3×D4—31no—
286C3×D4—41no—
287C3×D4—62no—
288C3×D4—62no—
289C3×D4—61no—
290C3×D4—61no—
291C3×D4—61no—
292C3×D4D4×C911yescharacteristic
293C3×D4C3×#6711yescharacteristic
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—62no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4C23×C911yescharacteristic
305C2×C3×C4C22×C3×D311yescharacteristic
306#67—22no—
307#67—21no—
308C2×Dic3—21no—
309D12—22no—
310D12—21no—
311D12—21no—
312D12D4×C911yescharacteristic
313C4×D3—22no—
314C4×D3—21no—
315Dic6D4×C911yescharacteristic
316C3×C8—22no—
317C3×C8—21no—
318C3×C8—21no—
319C3×C8—61no—
320C3⋊C8—21no—
321C2×C32—21no—
322C2×C32—41no—
323C2×C32C3×#12011yescharacteristic
324C3×D3—42no—
325C3×D3—41no—
326C2×C9—21no—
327C2×C9—21no—
328C2×C9—41no—
329C2×C9—41no—
330C2×C9—81no—
331C2×C9—122no—
332C2×C9—121no—
333C2×C9#93711yescharacteristic
334#41—31no—
335#41—61no—
336#41—61no—
337#41C2×D3×C911yescharacteristic
338C2×Q8C2×D3×C911yescharacteristic, complemented
339C2×D4—31no—
340C2×D4—61no—
341C2×D4—62no—
342C2×D4—61no—
343Q16—21no—
344Q16—61no—
345SD16—21no—
346SD16—61no—
347M16—31no—
348M16C2×D3×C911yescharacteristic, complemented
349C22⋊C4—62no—
350C42—31no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—41no—
355C22×C3—61no—
356C22×C3—62no—
357C22×C3—122no—
358C22×C3—242no—
359C22×C3C2×D4×C911yescharacteristic
360C22×C3C3×D3×D411yescharacteristic
361C2×D3—22no—
362C2×D3—21no—
363C2×D3—42no—
364C2×D3—82no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—41no—
370C3×C4—42no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—61no—
374C3×C4—121no—
375C3×C4C2×D4×C911yescharacteristic
376C3×C4C2×D4×C911yescharacteristic
377C3×C4C3×D3×D411yescharacteristic
378C3×C4C2×C3×#6711yescharacteristic
379Dic3—21no—
380Dic3—41no—
381C32C3×#45511yescharacteristic
382C9—21no—
383C9#520211yescharacteristic, direct product factor
384C23—32no—
385C23—62no—
386Q8—21no—
387Q8—21no—
388Q8—31no—
389Q8C9×#6711yescharacteristic
390D4—21no—
391D4—31no—
392D4—62no—
393D4—62no—
394D4—61no—
395D4—61no—
396D4—61no—
397D4C9×#6711yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—62no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4C22×D3×C911yescharacteristic
404C8—21no—
405C8—61no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—81no—
413C2×C3—122no—
414C2×C3—121no—
415C2×C3C9×#12011yescharacteristic
416C2×C3C3×#93711yescharacteristic
417D3—42no—
418D3—41no—
419C22—21no—
420C22—61no—
421C22—62no—
422C22—122no—
423C22—242no—
424C22D3×D4×C911yescharacteristic
425C4—21no—
426C4—41no—
427C4—61no—
428C4—121no—
429C4D3×D4×C911yescharacteristic
430C4C2×C9×#6711yescharacteristic
431C3—21no—
432C3C9×#45511yescharacteristic, complemented
433C3C3×#520211yescharacteristic
434C2—21no—
435C2—41no—
436C2—122no—
437C2—121no—
438C2C9×#93711yescharacteristic
subgroupG/H
centerC2×C9#937
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#137C2
socleC2×C32C3×#120
prime radicalC3×#1009C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#137D3
hypocenterC3C9×#455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#137

Complements

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

NH ≅ G/Ntype
2. C3×C9×#137C2semidirect
3. C9×#947C2semidirect
4. C9×#942C2semidirect
5. C9×#837C2semidirect
6. C9×#825C2semidirect
7. C9×#823C2semidirect
11. C2×C3×Q8×C9C22semidirect
21. C3×C9×M16C22semidirect
25. C4×C9×Dic3C22semidirect
26. C9×#208C22semidirect
37. C9×#137D3semidirect
65. #5202C9direct (N × H)
89. C2×Q8×C9C2×D3semidirect
102. C9×M16C2×D3semidirect
121. C3×#137C2×C9semidirect
124. #947C2×C9semidirect
125. #942C2×C9semidirect
129. #837C2×C9semidirect
130. #825C2×C9semidirect
131. #823C2×C9semidirect
198. C2×C3×Q8C22×C9semidirect
221. C3×M16C22×C9semidirect
228. C4×Dic3C22×C9semidirect
229. #208C22×C9semidirect
262. #137D3×C9semidirect
267. C3×C9#455semidirect
338. C2×Q8C2×D3×C9semidirect
348. M16C2×D3×C9semidirect
383. C9#5202direct (N × H)
432. C3C9×#455semidirect

Maximal subgroups

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

subgroupindex
C9×#10092
C3×C9×#1372
C9×#9472
C9×#9422
C9×#8372
C9×#8252
C9×#8232
C3×#52023
C9×#4553