Finite Group Databasebeta

← C9×#5047 #13142874

Subgroups — C9×#5047 #13142874

Order 1728.

530 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic
2C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C9×#216C211yesmaximal, characteristic, complemented
4C2×C9×#215C211yesmaximal, characteristic, complemented
5C9×#830C211yesmaximal, characteristic, complemented
6C9×#810C211yesmaximal, characteristic, complemented
7C9×#809C211yesmaximal, characteristic, complemented
8C3×#5047C311yesmaximal, characteristic
9C9×#452—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×C9×#67—21no—
13C9×#235—22no—
14C9×#235—22no—
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9—21no—
17C2×C9×Dic6C2211yescharacteristic
18C3×C9×C4⋊C4—21no—
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C9×C22⋊C4—21no—
21C9×#216—22no—
22C9×#215—22no—
23C2×C9×C3⋊C8C2211yescharacteristic
24C2×C9×C3⋊C8C2211yescharacteristic
25C2×C3×#235C2×C311yescharacteristic
26C32×C22⋊Q8C2×C311yescharacteristic
27C2×C3×#216C2×C311yescharacteristic
28C2×C3×#215C2×C311yescharacteristic
29C2×C9×#41—31no—
30C3×#830C2×C311yescharacteristic
31C3×#810C2×C311yescharacteristic
32C3×#809C2×C311yescharacteristic
33C2×C9×Q16—31no—
34C2×C9×SD16—31no—
35C9×C22⋊Q8—21no—
36C9×C22⋊Q8D311yescharacteristic, complemented
37C9×#103—31no—
38C9×#102—31no—
39C9×C22⋊C8—31no—
40C23×C3×C9D411yescharacteristic
41C22×D3×C9—21no—
42C3×Q8×C9—21no—
43C3×Q8×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9D411yescharacteristic
49C2×C3×C4×C9C2311yescharacteristic
50C9×#67—24no—
51C2×C9×Dic3—21no—
52C9×D12—21no—
53C9×D12D412yes—
54C4×D3×C9—24no—
55C9×Dic6—21no—
56C9×Dic6D412yes—
57C9×C3⋊C8—21no—
58C9×C3⋊C8—21no—
59C3×#452—31no—
60#5047C911yescharacteristic, direct product factor
61C2×C32×Q8C22×C311yescharacteristic
62C22×C32×C4C22×C311yescharacteristic
63C2×C3×#67—21no—
64C3×#235—22no—
65C3×#235—22no—
66C2×C3×D12C22×C311yescharacteristic
67C2×C3×C4×D3—21no—
68C2×C3×Dic6C22×C311yescharacteristic
69C32×C4⋊C4—21no—
70C32×C4⋊C4C22×C311yescharacteristic
71C32×C22⋊C4—21no—
72C3×#216—22no—
73C3×#215—22no—
74C2×C3×C3⋊C8C22×C311yescharacteristic
75C2×C3×C3⋊C8C22×C311yescharacteristic
76C9×#41—62no—
77C9×#41—62no—
78C2×Q8×C9—21no—
79C2×Q8×C9—31no—
80C2×Q8×C9C2×D311yescharacteristic
81C2×D4×C9—31no—
82C2×D4×C9—61no—
83C22×C4×C9—21no—
84C22×C4×C9—61no—
85C22×C4×C9C2×D311yescharacteristic
86C9×Q16—62no—
87C9×SD16—62no—
88C2×C8×C9—31no—
89C2×C8×C9—31no—
90C9×C4⋊C4—22no—
91C9×C4⋊C4—21no—
92C9×C4⋊C4—21no—
93C9×C4⋊C4C2×D311yescharacteristic
94C9×C22⋊C4—22no—
95C9×C22⋊C4—21no—
96C22×C3×C9—21no—
97C22×C3×C9—21no—
98C22×C3×C9—21no—
99C22×C3×C9C2×D411yescharacteristic
100C2×D3×C9—22no—
101C2×D3×C9—41no—
102C2×D3×C9—41no—
103C3×C4×C9—21no—
104C3×C4×C9—41no—
105C3×C4×C9—41no—
106C3×C4×C9C2×D411yescharacteristic
107C3×C4×C9C2×D411yescharacteristic
108C9×Dic3—22no—
109C2×C3×#41—31no—
110C2×#235C2×C911yescharacteristic
111C2×C3×Q16—31no—
112C2×C3×SD16—31no—
113C3×C22⋊Q8—21no—
114C3×C22⋊Q8C2×C911yescharacteristic, complemented
115C3×C22⋊Q8C3×D311yescharacteristic
116C2×#216C2×C911yescharacteristic, complemented
117C2×#215C2×C911yescharacteristic, complemented
118C3×#103—31no—
119C3×#102—31no—
120C3×C22⋊C8—31no—
121#830C2×C911yescharacteristic, complemented
122#810C2×C911yescharacteristic, complemented
123#809C2×C911yescharacteristic, complemented
124C23×C32C3×D411yescharacteristic
125C22×C3×D3—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C23×C311yescharacteristic
134C3×#67—24no—
135C2×C3×Dic3—21no—
136C3×D12—21no—
137C3×D12C3×D412yes—
138C3×C4×D3—24no—
139C3×Dic6—21no—
140C3×Dic6C3×D412yes—
141C23×C9—21no—
142C23×C9—61no—
143C23×C9#6711yescharacteristic
144C3×C3⋊C8—21no—
145C3×C3⋊C8—21no—
146Q8×C9—21no—
147Q8×C9—21no—
148Q8×C9—32no—
149Q8×C9—41no—
150Q8×C9—41no—
151Q8×C9—61no—
152D4×C9—32no—
153D4×C9—64no—
154D4×C9—61no—
155C2×C4×C9—21no—
156C2×C4×C9—21no—
157C2×C4×C9—22no—
158C2×C4×C9—22no—
159C2×C4×C9—21no—
160C2×C4×C9—21no—
161C2×C4×C9—21no—
162C2×C4×C9—21no—
163C2×C4×C9—41no—
164C2×C4×C9—41no—
165C2×C4×C9—64no—
166C2×C4×C9—61no—
167C2×C4×C9#6711yescharacteristic
168C2×C4×C9C22×D311yescharacteristic
169C8×C9—61no—
170C8×C9—61no—
171#452—31noSylow
172C2×C3×C9—41no—
173C2×C3×C9#12011yescharacteristic
174C2×C3×C9#13511yescharacteristic
175C2×C3×C9#13711yescharacteristic
176D3×C9—42no—
177C3×#41—62no—
178C3×#41—62no—
179C2×C3×Q8—21no—
180C2×C3×Q8—31no—
181C2×C3×Q8C22×C911yescharacteristic
182C2×C3×Q8C2×C3×D311yescharacteristic
183C2×C3×D4—31no—
184C2×C3×D4—61no—
185C22×C3×C4—21no—
186C22×C3×C4—61no—
187C22×C3×C4C22×C911yescharacteristic
188C22×C3×C4C2×C3×D311yescharacteristic
189C2×#67—21no—
190#235—22no—
191#235—22no—
192C2×D12C22×C911yescharacteristic
193C2×C4×D3—21no—
194C2×Dic6C22×C911yescharacteristic
195C3×Q16—62no—
196C3×SD16—62no—
197C2×C3×C8—31no—
198C2×C3×C8—31no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4C22×C911yescharacteristic
204C3×C4⋊C4C2×C3×D311yescharacteristic
205C3×C22⋊C4—22no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208#216—22no—
209#215—22no—
210C2×C3⋊C8C22×C911yescharacteristic
211C2×C3⋊C8C22×C911yescharacteristic
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32C2×C3×D411yescharacteristic
216C2×C3×D3—22no—
217C2×C3×D3—41no—
218C2×C3×D3—41no—
219C32×C4—21no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4C2×C3×D411yescharacteristic
223C32×C4C2×C3×D411yescharacteristic
224C3×Dic3—22no—
225C22×C9—22no—
226C22×C9—21no—
227C22×C9—21no—
228C22×C9—21no—
229C22×C9—21no—
230C22×C9—41no—
231C22×C9—41no—
232C22×C9—62no—
233C22×C9—121no—
234C22×C9—121no—
235C22×C9C2×#6711yescharacteristic
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—41no—
240C4×C9—42no—
241C4×C9—42no—
242C4×C9—41no—
243C4×C9—41no—
244C4×C9—62no—
245C4×C9D3×D411yescharacteristic
246C4×C9D3×D411yescharacteristic
247C2×#41—31no—
248C2×Q16—31no—
249C2×SD16—31no—
250C22⋊Q8D3×C911yescharacteristic, complemented
251#103—31no—
252#102—31no—
253C22⋊C8—31no—
254C3×C9#45211yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—61no—
257C23×C3D4×C911yescharacteristic
258C23×C3C3×#6711yescharacteristic
259C22×D3—21no—
260C3×Q8—21no—
261C3×Q8—21no—
262C3×Q8—21no—
263C3×Q8—21no—
264C3×Q8—32no—
265C3×Q8—41no—
266C3×Q8—41no—
267C3×Q8—61no—
268C3×D4—32no—
269C3×D4—64no—
270C3×D4—61no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—64no—
286C2×C3×C4—61no—
287C2×C3×C4D4×C911yescharacteristic
288C2×C3×C4C23×C911yescharacteristic
289C2×C3×C4C3×#6711yescharacteristic
290C2×C3×C4C22×C3×D311yescharacteristic
291#67—24no—
292C2×Dic3—21no—
293D12—21no—
294D12D4×C912yes—
295C4×D3—24no—
296Dic6—21no—
297Dic6D4×C912yes—
298C3×C8—61no—
299C3×C8—61no—
300C3⋊C8—21no—
301C3⋊C8—21no—
302C2×C32—41no—
303C2×C32C3×#12011yescharacteristic
304C2×C32C3×#13511yescharacteristic
305C2×C32C3×#13711yescharacteristic
306C3×D3—42no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—21no—
310C2×C9—42no—
311C2×C9—41no—
312C2×C9—122no—
313C2×C9#93711yescharacteristic
314C2×C9#94211yescharacteristic
315C2×C9#95011yescharacteristic
316#41—62no—
317#41—62no—
318C2×Q8—31no—
319C2×Q8C2×D3×C911yescharacteristic
320C2×D4—31no—
321C2×D4—61no—
322C22×C4—61no—
323C22×C4C2×D3×C911yescharacteristic
324Q16—62no—
325SD16—62no—
326C2×C8—31no—
327C2×C8—31no—
328C4⋊C4—21no—
329C4⋊C4C2×D3×C911yescharacteristic
330C22⋊C4—21no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—62no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3C2×D4×C911yescharacteristic
345C22×C3C2×C3×#6711yescharacteristic
346C2×D3—22no—
347C2×D3—41no—
348C2×D3—41no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—62no—
361C3×C4C2×D4×C911yescharacteristic
362C3×C4C2×D4×C911yescharacteristic
363C3×C4C3×D3×D411yescharacteristic
364C3×C4C3×D3×D411yescharacteristic
365Dic3—22no—
366C32C3×#45211yescharacteristic
367C9—21no—
368C9#504711yescharacteristic, direct product factor
369C23—61no—
370C23C9×#6711yescharacteristic
371Q8—21no—
372Q8—21no—
373Q8—32no—
374Q8—61no—
375D4—32no—
376D4—64no—
377D4—61no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—64no—
383C2×C4—61no—
384C2×C4C9×#6711yescharacteristic
385C2×C4C22×D3×C911yescharacteristic
386C8—61no—
387C8—61no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—122no—
395C2×C3C9×#12011yescharacteristic
396C2×C3C9×#13511yescharacteristic
397C2×C3C9×#13711yescharacteristic
398C2×C3C3×#93711yescharacteristic
399C2×C3C3×#94211yescharacteristic
400C2×C3C3×#95011yescharacteristic
401D3—42no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—62no—
406C22—121no—
407C22—121no—
408C22C2×C9×#6711yescharacteristic
409C4—21no—
410C4—41no—
411C4—41no—
412C4—62no—
413C4D3×D4×C911yescharacteristic
414C4D3×D4×C911yescharacteristic
415C3—21no—
416C3C9×#45211yescharacteristic, complemented
417C3C3×#504711yescharacteristic
418C2—41no—
419C2—122no—
420C2C9×#93711yescharacteristic
421C2C9×#94211yescharacteristic
422C2C9×#95011yescharacteristic
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×#235C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×C22⋊Q8D3
hypocenterC3C9×#452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×C4×C9 ◁ C9×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×C9×C22⋊Q8C2semidirect
3. C2×C9×#216C2semidirect
4. C2×C9×#215C2semidirect
5. C9×#830C2semidirect
6. C9×#810C2semidirect
7. C9×#809C2semidirect
36. C9×C22⋊Q8D3semidirect
60. #5047C9direct (N × H)
114. C3×C22⋊Q8C2×C9semidirect
116. C2×#216C2×C9semidirect
117. C2×#215C2×C9semidirect
121. #830C2×C9semidirect
122. #810C2×C9semidirect
123. #809C2×C9semidirect
250. C22⋊Q8D3×C9semidirect
254. C3×C9#452semidirect
368. C9#5047direct (N × H)
416. C3C9×#452semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C3×C9×C22⋊Q82
C2×C9×#2162
C2×C9×#2152
C9×#8302
C9×#8102
C9×#8092
C3×#50473
C9×#4523