Finite Group Databasebeta

← C9×#5205 #13143499

Subgroups — C9×#5205 #13143499

Order 1728.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#1010C211yesmaximal, characteristic
2C3×C9×#137C211yesmaximal, characteristic, complemented
3C9×#944C211yesmaximal, characteristic, complemented
4C9×#942C211yesmaximal, characteristic, complemented
5C9×#837C211yesmaximal, characteristic, complemented
6C9×#825C211yesmaximal, characteristic, complemented
7C9×#824C211yesmaximal, characteristic, complemented
8C3×#5205C311yesmaximal, characteristic
9C9×#456—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C3×Q8×C9C2211yescharacteristic
12D3×Q8×C9—21no—
13C9×#237—21no—
14C9×#237—22no—
15C9×#235—21no—
16C9×#235C2211yescharacteristic
17C2×C9×Dic6—21no—
18C2×C9×Dic6C2211yescharacteristic
19C3×C9×Q16—21no—
20C3×C9×SD16—21no—
21C3×C9×M16C2211yescharacteristic
22C9×#216—21no—
23C9×#215—21no—
24C9×Dic3⋊C4—22no—
25C4×C9×Dic3C2211yescharacteristic
26C9×#208C2211yescharacteristic
27C3×#1010C2×C311yescharacteristic
28C32×#137C2×C311yescharacteristic
29C3×#944C2×C311yescharacteristic
30C3×#942C2×C311yescharacteristic
31C9×2−1+4—31no—
32C3×#837C2×C311yescharacteristic
33C3×#825C2×C311yescharacteristic
34C3×#824C2×C311yescharacteristic
35C9×#137—21no—
36C9×#137—31no—
37C9×#137D311yescharacteristic, complemented
38C9×C4⋊Q8—31no—
39C9×#104—31no—
40C9×#104—31no—
41C9×#101—31no—
42C3×Q8×C9—21no—
43C3×Q8×C9—21no—
44C3×Q8×C9D411yescharacteristic
45C3×D4×C9—21no—
46C3×D4×C9D411yescharacteristic
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50C9×#67—22no—
51C9×#67—21no—
52C2×C9×Dic3—22no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3D412yes—
55C9×D12D411yescharacteristic
56C4×D3×C9—22no—
57C4×D3×C9—21no—
58C9×Dic6—22no—
59C9×Dic6—21no—
60C9×Dic6—21no—
61C9×Dic6D411yescharacteristic
62C3×C8×C9—21no—
63C9×C3⋊C8—21no—
64C3×#456—31no—
65#5205C911yescharacteristic, direct product factor
66C32×#41C22×C311yescharacteristic
67C2×C32×Q8C22×C311yescharacteristic
68C3×D3×Q8—21no—
69C3×#237—21no—
70C3×#237—22no—
71C3×#235—21no—
72C3×#235C22×C311yescharacteristic
73C2×C3×Dic6—21no—
74C2×C3×Dic6C22×C311yescharacteristic
75C32×Q16—21no—
76C32×SD16—21no—
77C32×M16C22×C311yescharacteristic
78C3×#216—21no—
79C3×#215—21no—
80C3×Dic3⋊C4—22no—
81C3×C4×Dic3C22×C311yescharacteristic
82C3×#208C22×C311yescharacteristic
83C9×#41—21no—
84C9×#41—31no—
85C9×#41—61no—
86C9×#41—62no—
87C9×#41—61no—
88C9×#41C2×D311yescharacteristic
89C2×Q8×C9—21no—
90C2×Q8×C9—31no—
91C2×Q8×C9—61no—
92C2×Q8×C9—61no—
93C2×Q8×C9C2×D311yescharacteristic
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
103C9×C4⋊C4—62no—
104C42×C9—31no—
105C22×C3×C9—21no—
106C22×C3×C9C2×D411yescharacteristic
107C2×D3×C9—21no—
108C3×C4×C9—21no—
109C3×C4×C9—41no—
110C3×C4×C9C2×D411yescharacteristic
111C3×C4×C9C2×D411yescharacteristic
112C9×Dic3—22no—
113C9×Dic3—22no—
114C9×Dic3—21no—
115C3×2−1+4—31no—
116#1010C2×C911yescharacteristic
117C3×#137—21no—
118C3×#137—31no—
119C3×#137C2×C911yescharacteristic, complemented
120C3×#137C3×D311yescharacteristic
121C3×C4⋊Q8—31no—
122#944C2×C911yescharacteristic, complemented
123#942C2×C911yescharacteristic, complemented
124C3×#104—31no—
125C3×#104—31no—
126C3×#101—31no—
127#837C2×C911yescharacteristic, complemented
128#825C2×C911yescharacteristic, complemented
129#824C2×C911yescharacteristic, complemented
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8C3×D411yescharacteristic
133C32×D4—21no—
134C32×D4C3×D411yescharacteristic
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C23×C311yescharacteristic
138C3×#67—22no—
139C3×#67—21no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3C3×D412yes—
143C3×D12C3×D411yescharacteristic
144C3×C4×D3—22no—
145C3×C4×D3—21no—
146C3×Dic6—22no—
147C3×Dic6—21no—
148C3×Dic6—21no—
149C3×Dic6C3×D411yescharacteristic
150C32×C8—21no—
151C3×C3⋊C8—21no—
152Q8×C9—22no—
153Q8×C9—21no—
154Q8×C9—21no—
155Q8×C9—21no—
156Q8×C9—31no—
157Q8×C9—41no—
158Q8×C9—62no—
159Q8×C9—61no—
160Q8×C9—61no—
161Q8×C9#6711yescharacteristic
162D4×C9—21no—
163D4×C9—21no—
164D4×C9—31no—
165D4×C9—41no—
166D4×C9—62no—
167D4×C9—61no—
168D4×C9#6711yescharacteristic
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—32no—
173C2×C4×C9—41no—
174C2×C4×C9—41no—
175C2×C4×C9—62no—
176C2×C4×C9—62no—
177C2×C4×C9—61no—
178C2×C4×C9—61no—
179C2×C4×C9C22×D311yescharacteristic
180C8×C9—22no—
181C8×C9—21no—
182C8×C9—61no—
183#456—31noSylow
184C2×C3×C9—21no—
185C2×C3×C9—41no—
186C2×C3×C9#12011yescharacteristic
187D3×C9—41no—
188C3×#41—21no—
189C3×#41—31no—
190C3×#41—61no—
191C3×#41—62no—
192C3×#41—61no—
193C3×#41C22×C911yescharacteristic
194C3×#41C2×C3×D311yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—31no—
197C2×C3×Q8—61no—
198C2×C3×Q8—61no—
199C2×C3×Q8C22×C911yescharacteristic
200C2×C3×Q8C2×C3×D311yescharacteristic
201D3×Q8—21no—
202#237—21no—
203#237—22no—
204#235—21no—
205#235C22×C911yescharacteristic
206C2×Dic6—21no—
207C2×Dic6C22×C911yescharacteristic
208C3×Q16—22no—
209C3×Q16—21no—
210C3×Q16—21no—
211C3×Q16—61no—
212C3×SD16—22no—
213C3×SD16—21no—
214C3×SD16—21no—
215C3×SD16—61no—
216C3×M16—21no—
217C3×M16—31no—
218C3×M16C22×C911yescharacteristic
219C3×M16C2×C3×D311yescharacteristic
220C3×C4⋊C4—62no—
221C3×C42—31no—
222#216—21no—
223#215—21no—
224Dic3⋊C4—22no—
225C4×Dic3C22×C911yescharacteristic
226#208C22×C911yescharacteristic
227C22×C32—21no—
228C22×C32C2×C3×D411yescharacteristic
229C2×C3×D3—21no—
230C32×C4—21no—
231C32×C4—41no—
232C32×C4C2×C3×D411yescharacteristic
233C32×C4C2×C3×D411yescharacteristic
234C3×Dic3—22no—
235C3×Dic3—22no—
236C3×Dic3—21no—
237C22×C9—21no—
238C22×C9—21no—
239C22×C9—41no—
240C22×C9—61no—
241C22×C9D3×D411yescharacteristic
242C4×C9—21no—
243C4×C9—21no—
244C4×C9—21no—
245C4×C9—41no—
246C4×C9—42no—
247C4×C9—41no—
248C4×C9—62no—
249C4×C9—62no—
250C4×C9—61no—
251C4×C9D3×D411yescharacteristic
252C4×C9C2×#6711yescharacteristic
2532−1+4—31no—
254#137—31no—
255#137D3×C911yescharacteristic, complemented
256C4⋊Q8—31no—
257#104—31no—
258#104—31no—
259#101—31no—
260C3×C9#45611yescharacteristic, complemented, Sylow
261C3×Q8—22no—
262C3×Q8—21no—
263C3×Q8—21no—
264C3×Q8—21no—
265C3×Q8—21no—
266C3×Q8—21no—
267C3×Q8—31no—
268C3×Q8—41no—
269C3×Q8—62no—
270C3×Q8—61no—
271C3×Q8—61no—
272C3×Q8D4×C911yescharacteristic
273C3×Q8C3×#6711yescharacteristic
274C3×D4—21no—
275C3×D4—21no—
276C3×D4—21no—
277C3×D4—31no—
278C3×D4—41no—
279C3×D4—62no—
280C3×D4—61no—
281C3×D4D4×C911yescharacteristic
282C3×D4C3×#6711yescharacteristic
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—32no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4C23×C911yescharacteristic
296C2×C3×C4C22×C3×D311yescharacteristic
297#67—22no—
298#67—21no—
299C2×Dic3—22no—
300C2×Dic3—21no—
301C2×Dic3D4×C912yes—
302D12D4×C911yescharacteristic
303C4×D3—22no—
304C4×D3—21no—
305Dic6—22no—
306Dic6—21no—
307Dic6—21no—
308Dic6D4×C911yescharacteristic
309C3×C8—22no—
310C3×C8—21no—
311C3×C8—21no—
312C3×C8—61no—
313C3⋊C8—21no—
314C2×C32—21no—
315C2×C32—41no—
316C2×C32C3×#12011yescharacteristic
317C3×D3—41no—
318C2×C9—21no—
319C2×C9—21no—
320C2×C9—41no—
321C2×C9—41no—
322C2×C9—81no—
323C2×C9—121no—
324C2×C9#93711yescharacteristic
325#41—31no—
326#41—61no—
327#41—62no—
328#41—61no—
329#41C2×D3×C911yescharacteristic
330C2×Q8—31no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8C2×D3×C911yescharacteristic
334Q16—21no—
335Q16—61no—
336SD16—21no—
337SD16—61no—
338M16—31no—
339M16C2×D3×C911yescharacteristic
340C4⋊C4—62no—
341C42—31no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—61no—
347C22×C3C2×D4×C911yescharacteristic
348C22×C3C3×D3×D411yescharacteristic
349C2×D3—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—41no—
355C3×C4—42no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—61no—
361C3×C4C2×D4×C911yescharacteristic
362C3×C4C2×D4×C911yescharacteristic
363C3×C4C3×D3×D411yescharacteristic
364C3×C4C2×C3×#6711yescharacteristic
365Dic3—22no—
366Dic3—22no—
367Dic3—21no—
368C32C3×#45611yescharacteristic
369C9—21no—
370C9#520511yescharacteristic, direct product factor
371Q8—21no—
372Q8—21no—
373Q8—31no—
374Q8—62no—
375Q8—61no—
376Q8—61no—
377Q8C9×#6711yescharacteristic
378D4—21no—
379D4—31no—
380D4—62no—
381D4—61no—
382D4C9×#6711yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—32no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4C22×D3×C911yescharacteristic
391C8—21no—
392C8—61no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—81no—
400C2×C3—121no—
401C2×C3C9×#12011yescharacteristic
402C2×C3C3×#93711yescharacteristic
403D3—41no—
404C22—21no—
405C22—61no—
406C22D3×D4×C911yescharacteristic
407C4—21no—
408C4—41no—
409C4—62no—
410C4—62no—
411C4—61no—
412C4D3×D4×C911yescharacteristic
413C4C2×C9×#6711yescharacteristic
414C3—21no—
415C3C9×#45611yescharacteristic, complemented
416C3C3×#520511yescharacteristic
417C2—21no—
418C2—41no—
419C2—121no—
420C2C9×#93711yescharacteristic
subgroupG/H
centerC2×C9#937
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#137C2
socleC2×C32C3×#120
prime radicalC3×#1010C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#137D3
hypocenterC3C9×#456
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×#944C2semidirect
4. C9×#942C2semidirect
5. C9×#837C2semidirect
6. C9×#825C2semidirect
7. C9×#824C2semidirect
37. C9×#137D3semidirect
65. #5205C9direct (N × H)
119. C3×#137C2×C9semidirect
122. #944C2×C9semidirect
123. #942C2×C9semidirect
127. #837C2×C9semidirect
128. #825C2×C9semidirect
129. #824C2×C9semidirect
255. #137D3×C9semidirect
260. C3×C9#456semidirect
370. C9#5205direct (N × H)
415. C3C9×#456semidirect

Maximal subgroups

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

subgroupindex
C9×#10102
C3×C9×#1372
C9×#9442
C9×#9422
C9×#8372
C9×#8252
C9×#8242
C3×#52053
C9×#4563