Finite Group Databasebeta

← C9×#4826 #13141978

Subgroups — C9×#4826 #13141978

Order 1728.

470 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#1010C211yesmaximal, characteristic
2C9×#951C211yesmaximal, characteristic, complemented
3C9×#909C211yesmaximal, characteristic, complemented
4C9×#869C211yesmaximal, characteristic, complemented
5C3×C9×#104C211yesmaximal, characteristic, complemented
6C9×#824C211yesmaximal, characteristic, complemented
7C9×#805C211yesmaximal, characteristic, complemented
8C3×#4826C311yesmaximal, characteristic
9C9×#456—31nomaximal
10C3×C9×#41C2211yescharacteristic
11D3×Q8×C9—21no—
12C9×#237—21no—
13C9×#237—22no—
14C9×#235—21no—
15C9×#235C2211yescharacteristic
16C2×C9×Dic6—21no—
17C2×C9×Dic6C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C3×C9×M16C2211yescharacteristic
20C3×C42×C9C2211yescharacteristic
21C9×#216—21no—
22C9×#214—21no—
23C9×C4⋊Dic3—22no—
24C9×#208C2211yescharacteristic
25C9×Dic12—21no—
26C9×#204—21no—
27C3×#1010C2×C311yescharacteristic
28C3×#951C2×C311yescharacteristic
29C3×#909C2×C311yescharacteristic
30C3×#869C2×C311yescharacteristic
31C9×2−1+4—31no—
32C32×#104C2×C311yescharacteristic
33C3×#824C2×C311yescharacteristic
34C3×#805C2×C311yescharacteristic
35C9×#137—31no—
36C9×#137—31no—
37C9×C4⋊Q8—31no—
38C9×#104—21no—
39C9×#104—31no—
40C9×#104D311yescharacteristic, complemented
41C9×#101—31no—
42C3×Q8×C9D411yescharacteristic
43C3×D4×C9—21no—
44C3×D4×C9D411yescharacteristic
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9C2311yescharacteristic
48C9×#67—22no—
49C9×#67—21no—
50C2×C9×Dic3—22no—
51C2×C9×Dic3—21no—
52C2×C9×Dic3D412yes—
53C9×D12D411yescharacteristic
54C4×D3×C9—22no—
55C4×D3×C9—21no—
56C9×Dic6—22no—
57C9×Dic6—21no—
58C9×Dic6—21no—
59C9×Dic6—21no—
60C9×Dic6—21no—
61C9×Dic6D411yescharacteristic
62C3×C8×C9—21no—
63C9×C3⋊C8—21no—
64C3×#456—31no—
65#4826C911yescharacteristic, direct product factor
66C32×#41C22×C311yescharacteristic
67C3×D3×Q8—21no—
68C3×#237—21no—
69C3×#237—22no—
70C3×#235—21no—
71C3×#235C22×C311yescharacteristic
72C2×C3×Dic6—21no—
73C2×C3×Dic6C22×C311yescharacteristic
74C2×C3×Dic6C22×C311yescharacteristic
75C32×M16C22×C311yescharacteristic
76C32×C42C22×C311yescharacteristic
77C3×#216—21no—
78C3×#214—21no—
79C3×C4⋊Dic3—22no—
80C3×#208C22×C311yescharacteristic
81C3×Dic12—21no—
82C3×#204—21no—
83C9×#41—21no—
84C9×#41—31no—
85C9×#41—61no—
86C9×#41—62no—
87C9×#41—61no—
88C9×#41C2×D311yescharacteristic
89C2×Q8×C9—31no—
90C2×Q8×C9—31no—
91C2×Q8×C9—61no—
92C2×Q8×C9—61no—
93C9×Q16—61no—
94C9×Q16—61no—
95C9×SD16—61no—
96C9×SD16—61no—
97C9×M16—21no—
98C9×M16—31no—
99C9×M16C2×D311yescharacteristic
100C9×C4⋊C4—62no—
101C42×C9—21no—
102C42×C9C2×D311yescharacteristic
103C22×C3×C9—21no—
104C22×C3×C9C2×D411yescharacteristic
105C2×D3×C9—21no—
106C3×C4×C9—22no—
107C3×C4×C9—21no—
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C9×Dic3—22no—
111C9×Dic3—21no—
112C9×Dic3—41no—
113C3×2−1+4—31no—
114#1010C2×C911yescharacteristic
115C3×#137—31no—
116C3×#137—31no—
117C3×C4⋊Q8—31no—
118#951C2×C911yescharacteristic, complemented
119#909C2×C911yescharacteristic, complemented
120#869C2×C911yescharacteristic, complemented
121C3×#104—21no—
122C3×#104—31no—
123C3×#104C2×C911yescharacteristic, complemented
124C3×#104C3×D311yescharacteristic
125C3×#101—31no—
126#824C2×C911yescharacteristic, complemented
127#805C2×C911yescharacteristic, complemented
128C32×Q8C3×D411yescharacteristic
129C32×D4—21no—
130C32×D4C3×D411yescharacteristic
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4C23×C311yescharacteristic
134C3×#67—22no—
135C3×#67—21no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3C3×D412yes—
139C3×D12C3×D411yescharacteristic
140C3×C4×D3—22no—
141C3×C4×D3—21no—
142C3×Dic6—22no—
143C3×Dic6—21no—
144C3×Dic6—21no—
145C3×Dic6—21no—
146C3×Dic6—21no—
147C3×Dic6C3×D411yescharacteristic
148C32×C8—21no—
149C3×C3⋊C8—21no—
150Q8×C9—21no—
151Q8×C9—31no—
152Q8×C9—62no—
153Q8×C9—61no—
154Q8×C9—61no—
155Q8×C9—61no—
156Q8×C9—61no—
157Q8×C9D1211yescharacteristic
158D4×C9—21no—
159D4×C9—21no—
160D4×C9—31no—
161D4×C9—41no—
162D4×C9—62no—
163D4×C9—61no—
164D4×C9D1211yescharacteristic
165C2×C4×C9—21no—
166C2×C4×C9—21no—
167C2×C4×C9—21no—
168C2×C4×C9—32no—
169C2×C4×C9—41no—
170C2×C4×C9—41no—
171C2×C4×C9—62no—
172C2×C4×C9—62no—
173C2×C4×C9—61no—
174C2×C4×C9—61no—
175C2×C4×C9C22×D311yescharacteristic
176C8×C9—21no—
177C8×C9—41no—
178C8×C9—61no—
179#456—31noSylow
180C2×C3×C9—21no—
181C2×C3×C9—41no—
182C2×C3×C9#12011yescharacteristic
183D3×C9—41no—
184C3×#41—21no—
185C3×#41—31no—
186C3×#41—61no—
187C3×#41—62no—
188C3×#41—61no—
189C3×#41C22×C911yescharacteristic
190C3×#41C2×C3×D311yescharacteristic
191C2×C3×Q8—31no—
192C2×C3×Q8—31no—
193C2×C3×Q8—61no—
194C2×C3×Q8—61no—
195D3×Q8—21no—
196#237—21no—
197#237—22no—
198#235—21no—
199#235C22×C911yescharacteristic
200C2×Dic6—21no—
201C2×Dic6C22×C911yescharacteristic
202C2×Dic6C22×C911yescharacteristic
203C3×Q16—61no—
204C3×Q16—61no—
205C3×SD16—61no—
206C3×SD16—61no—
207C3×M16—21no—
208C3×M16—31no—
209C3×M16C22×C911yescharacteristic
210C3×M16C2×C3×D311yescharacteristic
211C3×C4⋊C4—62no—
212C3×C42—21no—
213C3×C42C22×C911yescharacteristic
214C3×C42C2×C3×D311yescharacteristic
215#216—21no—
216#214—21no—
217C4⋊Dic3—22no—
218#208C22×C911yescharacteristic
219Dic12—21no—
220#204—21no—
221C22×C32—21no—
222C22×C32C2×C3×D411yescharacteristic
223C2×C3×D3—21no—
224C32×C4—22no—
225C32×C4—21no—
226C32×C4C2×C3×D411yescharacteristic
227C32×C4C2×C3×D411yescharacteristic
228C3×Dic3—22no—
229C3×Dic3—21no—
230C3×Dic3—41no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—41no—
234C22×C9—61no—
235C22×C9D3×D411yescharacteristic
236C4×C9—22no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—21no—
240C4×C9—41no—
241C4×C9—42no—
242C4×C9—62no—
243C4×C9—61no—
244C4×C9—121no—
245C4×C9C2×D1211yescharacteristic
246C4×C9D3×D411yescharacteristic
2472−1+4—31no—
248#137—31no—
249#137—31no—
250C4⋊Q8—31no—
251#104—31no—
252#104D3×C911yescharacteristic, complemented
253#101—31no—
254C3×C9#45611yescharacteristic, complemented, Sylow
255C3×Q8—21no—
256C3×Q8—31no—
257C3×Q8—62no—
258C3×Q8—61no—
259C3×Q8—61no—
260C3×Q8—61no—
261C3×Q8—61no—
262C3×Q8D4×C911yescharacteristic
263C3×Q8C3×D1211yescharacteristic
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—31no—
268C3×D4—41no—
269C3×D4—62no—
270C3×D4—61no—
271C3×D4D4×C911yescharacteristic
272C3×D4C3×D1211yescharacteristic
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—32no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4C23×C911yescharacteristic
286C2×C3×C4C22×C3×D311yescharacteristic
287#67—22no—
288#67—21no—
289C2×Dic3—22no—
290C2×Dic3—21no—
291C2×Dic3D4×C912yes—
292D12D4×C911yescharacteristic
293C4×D3—22no—
294C4×D3—21no—
295Dic6—22no—
296Dic6—21no—
297Dic6—21no—
298Dic6—21no—
299Dic6—21no—
300Dic6D4×C911yescharacteristic
301C3×C8—21no—
302C3×C8—21no—
303C3×C8—41no—
304C3×C8—61no—
305C3⋊C8—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32C3×#12011yescharacteristic
309C3×D3—41no—
310C2×C9—21no—
311C2×C9—21no—
312C2×C9—41no—
313C2×C9—41no—
314C2×C9—81no—
315C2×C9—121no—
316C2×C9#88211yescharacteristic
317#41—31no—
318#41—61no—
319#41—62no—
320#41—61no—
321#41C2×D3×C911yescharacteristic
322C2×Q8—31no—
323C2×Q8—31no—
324C2×Q8—61no—
325C2×Q8—61no—
326Q16—61no—
327Q16—61no—
328SD16—61no—
329SD16—61no—
330M16—31no—
331M16C2×D3×C911yescharacteristic
332C4⋊C4—62no—
333C42C2×D3×C911yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—61no—
339C22×C3C2×D4×C911yescharacteristic
340C22×C3C3×D3×D411yescharacteristic
341C2×D3—21no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—42no—
350C3×C4—62no—
351C3×C4—61no—
352C3×C4—121no—
353C3×C4C2×D4×C911yescharacteristic
354C3×C4C2×D4×C911yescharacteristic
355C3×C4C2×C3×D1211yescharacteristic
356C3×C4C3×D3×D411yescharacteristic
357Dic3—22no—
358Dic3—21no—
359Dic3—41no—
360C32C3×#45611yescharacteristic
361C9—21no—
362C9#482611yescharacteristic, direct product factor
363Q8—31no—
364Q8—62no—
365Q8—61no—
366Q8—61no—
367Q8—61no—
368Q8—61no—
369Q8C9×D1211yescharacteristic
370D4—21no—
371D4—31no—
372D4—62no—
373D4—61no—
374D4C9×D1211yescharacteristic
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—32no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4C22×D3×C911yescharacteristic
383C8—21no—
384C8—61no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—81no—
392C2×C3—121no—
393C2×C3C9×#12011yescharacteristic
394C2×C3C3×#88211yescharacteristic
395D3—41no—
396C22—21no—
397C22—61no—
398C22D3×D4×C911yescharacteristic
399C4—22no—
400C4—21no—
401C4—62no—
402C4—61no—
403C4—121no—
404C4C2×C9×D1211yescharacteristic
405C4D3×D4×C911yescharacteristic
406C3—21no—
407C3C9×#45611yescharacteristic, complemented
408C3C3×#482611yescharacteristic
409C2—21no—
410C2—41no—
411C2—121no—
412C2C9×#88211yescharacteristic
subgroupG/H
centerC2×C9#882
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#104C2
socleC2×C32C3×#120
prime radicalC3×#1010C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#104D3
hypocenterC3C9×#456
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#104

Complements

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

NH ≅ G/Ntype
2. C9×#951C2semidirect
3. C9×#909C2semidirect
4. C9×#869C2semidirect
5. C3×C9×#104C2semidirect
6. C9×#824C2semidirect
7. C9×#805C2semidirect
40. C9×#104D3semidirect
65. #4826C9direct (N × H)
118. #951C2×C9semidirect
119. #909C2×C9semidirect
120. #869C2×C9semidirect
123. C3×#104C2×C9semidirect
126. #824C2×C9semidirect
127. #805C2×C9semidirect
252. #104D3×C9semidirect
254. C3×C9#456semidirect
362. C9#4826direct (N × H)
407. C3C9×#456semidirect

Maximal subgroups

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

subgroupindex
C9×#10102
C9×#9512
C9×#9092
C9×#8692
C3×C9×#1042
C9×#8242
C9×#8052
C3×#48263
C9×#4563