Finite Group Databasebeta

← C9×#4783 #13141766

Subgroups — C9×#4783 #13141766

Order 1728.

506 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3C2×C9×Dic12C211yesmaximal, characteristic, complemented
4C9×#897C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7C9×#810C211yesmaximal, characteristic, complemented
8C3×#4783C311yesmaximal, characteristic
9C9×#452—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C22×C9×Dic3—21no—
13C9×#237—22no—
14C9×#237—22no—
15C2×C4×D3×C9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×C3×C8×C9C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C9×#214—22no—
21C9×(C2×D3)⋊C4—21no—
22C9×C4⋊Dic3—21no—
23C2×C9×C3⋊C8C2211yescharacteristic
24C9×Dic12—22no—
25C2×C3×#237C2×C311yescharacteristic
26C2×C3×#214C2×C311yescharacteristic
27C2×C3×Dic12C2×C311yescharacteristic
28C3×#897C2×C311yescharacteristic
29C2×C9×#41—31no—
30C32×#102C2×C311yescharacteristic
31C3×(C2×D3)⋊C8C2×C311yescharacteristic
32C3×#810C2×C311yescharacteristic
33C2×C9×Q16—31no—
34C2×C9×SD16—31no—
35C9×C22⋊Q8—31no—
36C9×#103—31no—
37C9×#102—21no—
38C9×#102D311yescharacteristic, complemented
39C9×C22⋊C8—31no—
40C23×C3×C9—21no—
41C22×D3×C9D411yescharacteristic
42C3×D4×C9—21no—
43C3×D4×C9D412yes—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C9×#67—24no—
47C2×C9×Dic3—24no—
48C2×C9×Dic3—21no—
49C2×C9×Dic3—21no—
50C2×C9×Dic3D411yescharacteristic
51C4×D3×C9—21no—
52C4×D3×C9—21no—
53C9×Dic6—21no—
54C9×Dic6—21no—
55C9×Dic6—21no—
56C9×Dic6D412yes—
57C3×C8×C9—21no—
58C9×C3⋊C8—21no—
59C3×#452—31no—
60#4783C911yescharacteristic, direct product factor
61C2×C32×D4C22×C311yescharacteristic
62C2×C3×#67—21no—
63C22×C3×Dic3—21no—
64C3×#237—22no—
65C3×#237—22no—
66C2×C3×C4×D3C22×C311yescharacteristic
67C2×C3×Dic6C22×C311yescharacteristic
68C2×C3×Dic6C22×C311yescharacteristic
69C2×C32×C8C22×C311yescharacteristic
70C32×C4⋊C4C22×C311yescharacteristic
71C3×#214—22no—
72C3×(C2×D3)⋊C4—21no—
73C3×C4⋊Dic3—21no—
74C2×C3×C3⋊C8C22×C311yescharacteristic
75C3×Dic12—22no—
76C9×#41—62no—
77C9×#41—62no—
78C2×Q8×C9—31no—
79C2×Q8×C9—31no—
80C2×D4×C9—21no—
81C2×D4×C9—61no—
82C2×D4×C9C2×D311yescharacteristic
83C22×C4×C9—31no—
84C22×C4×C9—61no—
85C9×Q16—62no—
86C9×SD16—62no—
87C2×C8×C9—21no—
88C2×C8×C9—31no—
89C2×C8×C9C2×D311yescharacteristic
90C9×C4⋊C4—21no—
91C9×C4⋊C4—61no—
92C9×C4⋊C4C2×D311yescharacteristic
93C9×C22⋊C4—61no—
94C22×C3×C9—22no—
95C22×C3×C9—41no—
96C22×C3×C9—41no—
97C22×C3×C9C2×D411yescharacteristic
98C2×D3×C9—21no—
99C2×D3×C9—21no—
100C2×D3×C9—21no—
101C3×C4×C9—41no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C9×Dic3—22no—
105C9×Dic3—21no—
106C9×Dic3—41no—
107C2×C3×#41—31no—
108C2×#237C2×C911yescharacteristic
109C2×C3×Q16—31no—
110C2×C3×SD16—31no—
111C3×C22⋊Q8—31no—
112C2×#214C2×C911yescharacteristic, complemented
113C2×Dic12C2×C911yescharacteristic, complemented
114#897C2×C911yescharacteristic, complemented
115C3×#103—31no—
116C3×#102—21no—
117C3×#102C2×C911yescharacteristic, complemented
118C3×#102C3×D311yescharacteristic
119C3×C22⋊C8—31no—
120(C2×D3)⋊C8C2×C911yescharacteristic, complemented
121#810C2×C911yescharacteristic, complemented
122C23×C32—21no—
123C22×C3×D3C3×D411yescharacteristic
124C32×D4—21no—
125C32×D4C3×D412yes—
126C2×C32×C4—21no—
127C2×C32×C4C23×C311yescharacteristic
128C3×#67—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3C3×D411yescharacteristic
133C3×C4×D3—21no—
134C3×C4×D3—21no—
135C3×Dic6—21no—
136C3×Dic6—21no—
137C3×Dic6—21no—
138C3×Dic6C3×D412yes—
139C23×C9—21no—
140C23×C9—31no—
141C23×C9—41no—
142C32×C8—21no—
143C3×C3⋊C8—21no—
144Q8×C9—32no—
145Q8×C9—61no—
146Q8×C9—61no—
147Q8×C9—61no—
148D4×C9—21no—
149D4×C9—22no—
150D4×C9—41no—
151D4×C9—64no—
152D4×C9D1212yes—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9—31no—
156C2×C4×C9—41no—
157C2×C4×C9—64no—
158C2×C4×C9—61no—
159C2×C4×C9—61no—
160C2×C4×C9—61no—
161C2×C4×C9—61no—
162C2×C4×C9C22×D311yescharacteristic
163C8×C9—21no—
164C8×C9—41no—
165C8×C9—61no—
166#452—31noSylow
167C2×C3×C9—42no—
168C2×C3×C9#12011yescharacteristic
169C2×C3×C9#13511yescharacteristic
170C2×C3×C9#13711yescharacteristic
171D3×C9—41no—
172C3×#41—62no—
173C3×#41—62no—
174C2×C3×Q8—31no—
175C2×C3×Q8—31no—
176C2×C3×D4—21no—
177C2×C3×D4—61no—
178C2×C3×D4C22×C911yescharacteristic
179C2×C3×D4C2×C3×D311yescharacteristic
180C22×C3×C4—31no—
181C22×C3×C4—61no—
182C2×#67—21no—
183C22×Dic3—21no—
184#237—22no—
185#237—22no—
186C2×C4×D3C22×C911yescharacteristic
187C2×Dic6C22×C911yescharacteristic
188C2×Dic6C22×C911yescharacteristic
189C3×Q16—62no—
190C3×SD16—62no—
191C2×C3×C8—21no—
192C2×C3×C8—31no—
193C2×C3×C8C22×C911yescharacteristic
194C2×C3×C8C2×C3×D311yescharacteristic
195C3×C4⋊C4—21no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4C22×C911yescharacteristic
198C3×C4⋊C4C2×C3×D311yescharacteristic
199C3×C22⋊C4—61no—
200#214—22no—
201(C2×D3)⋊C4—21no—
202C4⋊Dic3—21no—
203C2×C3⋊C8C22×C911yescharacteristic
204Dic12—22no—
205C22×C32—22no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32C2×C3×D411yescharacteristic
209C2×C3×D3—21no—
210C2×C3×D3—21no—
211C2×C3×D3—21no—
212C32×C4—41no—
213C32×C4C2×C3×D411yescharacteristic
214C32×C4C2×C3×D411yescharacteristic
215C3×Dic3—22no—
216C3×Dic3—21no—
217C3×Dic3—41no—
218C22×C9—22no—
219C22×C9—21no—
220C22×C9—42no—
221C22×C9—41no—
222C22×C9—41no—
223C22×C9—61no—
224C22×C9—61no—
225C22×C9—61no—
226C22×C9—81no—
227C22×C9—81no—
228C22×C9D3×D411yescharacteristic
229C4×C9—21no—
230C4×C9—21no—
231C4×C9—41no—
232C4×C9—61no—
233C4×C9—62no—
234C4×C9—81no—
235C4×C9—121no—
236C4×C9C2×D1211yescharacteristic
237C4×C9D3×D411yescharacteristic
238C2×#41—31no—
239C2×Q16—31no—
240C2×SD16—31no—
241C22⋊Q8—31no—
242#103—31no—
243#102D3×C911yescharacteristic, complemented
244C22⋊C8—31no—
245C3×C9#45211yescharacteristic, complemented, Sylow
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—31no—
249C23×C3—41no—
250C22×D3D4×C911yescharacteristic
251C3×Q8—32no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×D4—21no—
256C3×D4—21no—
257C3×D4—22no—
258C3×D4—41no—
259C3×D4—64no—
260C3×D4D4×C912yes—
261C3×D4C3×D1212yes—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—41no—
267C2×C3×C4—64no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4C23×C911yescharacteristic
273C2×C3×C4C22×C3×D311yescharacteristic
274#67—24no—
275C2×Dic3—24no—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3D4×C911yescharacteristic
279C4×D3—21no—
280C4×D3—21no—
281Dic6—21no—
282Dic6—21no—
283Dic6—21no—
284Dic6D4×C912yes—
285C3×C8—21no—
286C3×C8—21no—
287C3×C8—41no—
288C3×C8—61no—
289C3⋊C8—21no—
290C2×C32—42no—
291C2×C32C3×#12011yescharacteristic
292C2×C32C3×#13511yescharacteristic
293C2×C32C3×#13711yescharacteristic
294C3×D3—41no—
295C2×C9—21no—
296C2×C9—21no—
297C2×C9—21no—
298C2×C9—42no—
299C2×C9—82no—
300C2×C9—121no—
301C2×C9#88211yescharacteristic
302C2×C9#91211yescharacteristic
303C2×C9#91511yescharacteristic
304#41—62no—
305#41—62no—
306C2×Q8—31no—
307C2×Q8—31no—
308C2×D4—61no—
309C2×D4C2×D3×C911yescharacteristic
310C22×C4—31no—
311C22×C4—61no—
312Q16—62no—
313SD16—62no—
314C2×C8—31no—
315C2×C8C2×D3×C911yescharacteristic
316C4⋊C4—61no—
317C4⋊C4C2×D3×C911yescharacteristic
318C22⋊C4—61no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—21no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—61no—
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3C2×D4×C911yescharacteristic
333C22×C3C3×D3×D411yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—61no—
342C3×C4—62no—
343C3×C4—81no—
344C3×C4—121no—
345C3×C4C2×D4×C911yescharacteristic
346C3×C4C2×D4×C911yescharacteristic
347C3×C4C2×C3×D1211yescharacteristic
348C3×C4C3×D3×D411yescharacteristic
349Dic3—22no—
350Dic3—21no—
351Dic3—41no—
352C32C3×#45211yescharacteristic
353C9—21no—
354C9#478311yescharacteristic, direct product factor
355C23—21no—
356C23—31no—
357Q8—32no—
358Q8—61no—
359Q8—61no—
360Q8—61no—
361D4—21no—
362D4—64no—
363D4C9×D1212yes—
364C2×C4—21no—
365C2×C4—31no—
366C2×C4—64no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4C22×D3×C911yescharacteristic
372C8—21no—
373C8—61no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—42no—
378C2×C3—42no—
379C2×C3—82no—
380C2×C3—121no—
381C2×C3C9×#12011yescharacteristic
382C2×C3C9×#13511yescharacteristic
383C2×C3C9×#13711yescharacteristic
384C2×C3C3×#88211yescharacteristic
385C2×C3C3×#91211yescharacteristic
386C2×C3C3×#91511yescharacteristic
387D3—41no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—61no—
392C22—61no—
393C22—61no—
394C22D3×D4×C911yescharacteristic
395C4—41no—
396C4—61no—
397C4—62no—
398C4—121no—
399C4C2×C9×D1211yescharacteristic
400C4D3×D4×C911yescharacteristic
401C3—21no—
402C3C9×#45211yescharacteristic, complemented
403C3C3×#478311yescharacteristic
404C2—42no—
405C2—121no—
406C2C9×#88211yescharacteristic
407C2C9×#91211yescharacteristic
408C2C9×#91511yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×#237C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#102D3
hypocenterC3C9×#452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#102

Complements

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

NH ≅ G/Ntype
2. C2×C9×#214C2semidirect
3. C2×C9×Dic12C2semidirect
4. C9×#897C2semidirect
5. C3×C9×#102C2semidirect
6. C9×(C2×D3)⋊C8C2semidirect
7. C9×#810C2semidirect
38. C9×#102D3semidirect
60. #4783C9direct (N × H)
112. C2×#214C2×C9semidirect
113. C2×Dic12C2×C9semidirect
114. #897C2×C9semidirect
117. C3×#102C2×C9semidirect
120. (C2×D3)⋊C8C2×C9semidirect
121. #810C2×C9semidirect
243. #102D3×C9semidirect
245. C3×C9#452semidirect
354. C9#4783direct (N × H)
402. C3C9×#452semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C2×C9×#2142
C2×C9×Dic122
C9×#8972
C3×C9×#1022
C9×(C2×D3)⋊C82
C9×#8102
C3×#47833
C9×#4523