Finite Group Databasebeta

← C9×#5173 #13143358

Subgroups — C9×#5173 #13143358

Order 1728.

524 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic
2C2×C3×C9×SD16C211yesmaximal, characteristic, complemented
3C9×#946C211yesmaximal, characteristic, complemented
4C2×C9×#216C211yesmaximal, characteristic, complemented
5C9×#832C211yesmaximal, characteristic, complemented
6C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7C9×#816C211yesmaximal, characteristic, complemented
8C3×#5173C311yesmaximal, characteristic
9C9×#452—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C9×#67—21no—
13C22×C9×Dic3—21no—
14C9×#237—22no—
15C9×#237—22no—
16C2×C4×D3×C9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C3×C9×SD16—22no—
19C2×C3×C8×C9C2211yescharacteristic
20C9×#216—22no—
21C9×(C2×D3)⋊C4—21no—
22C9×C4⋊Dic3C2211yescharacteristic
23C9×Dic3⋊C4—21no—
24C2×C9×C3⋊C8C2211yescharacteristic
25C2×C3×#237C2×C311yescharacteristic
26C2×C32×SD16C2×C311yescharacteristic
27C3×#946C2×C311yescharacteristic
28C2×C3×#216C2×C311yescharacteristic
29C2×C9×#41—31no—
30C3×#832C2×C311yescharacteristic
31C3×(C2×D3)⋊C8C2×C311yescharacteristic
32C3×#816C2×C311yescharacteristic
33C2×C9×Q16—31no—
34C2×C9×SD16—21no—
35C2×C9×SD16D311yescharacteristic, complemented
36C9×C22⋊Q8—31no—
37C9×#103—31no—
38C9×#102—31no—
39C9×C22⋊C8—31no—
40C23×C3×C9—21no—
41C22×D3×C9D411yescharacteristic
42C3×Q8×C9—21no—
43C3×Q8×C9—21no—
44C3×D4×C9—21no—
45C3×D4×C9D412yes—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9C2311yescharacteristic
48C9×#67—24no—
49C2×C9×Dic3—24no—
50C2×C9×Dic3—21no—
51C2×C9×Dic3—21no—
52C2×C9×Dic3D411yescharacteristic
53C4×D3×C9—21no—
54C4×D3×C9—21no—
55C9×Dic6—21no—
56C9×Dic6D412yes—
57C3×C8×C9—21no—
58C9×C3⋊C8—21no—
59C3×#452—31no—
60#5173C911yescharacteristic, direct product factor
61C2×C32×Q8C22×C311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×C3×#67—21no—
64C22×C3×Dic3—21no—
65C3×#237—22no—
66C3×#237—22no—
67C2×C3×C4×D3C22×C311yescharacteristic
68C2×C3×Dic6C22×C311yescharacteristic
69C32×SD16—22no—
70C2×C32×C8C22×C311yescharacteristic
71C3×#216—22no—
72C3×(C2×D3)⋊C4—21no—
73C3×C4⋊Dic3C22×C311yescharacteristic
74C3×Dic3⋊C4—21no—
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—21no—
82C2×D4×C9—61no—
83C2×D4×C9C2×D311yescharacteristic
84C22×C4×C9—31no—
85C22×C4×C9—61no—
86C9×Q16—62no—
87C9×SD16—24no—
88C9×SD16—22no—
89C2×C8×C9—21no—
90C2×C8×C9—31no—
91C2×C8×C9C2×D311yescharacteristic
92C9×C4⋊C4—31no—
93C9×C4⋊C4—61no—
94C9×C22⋊C4—61no—
95C22×C3×C9—22no—
96C22×C3×C9—41no—
97C22×C3×C9—41no—
98C22×C3×C9C2×D411yescharacteristic
99C2×D3×C9—21no—
100C2×D3×C9—21no—
101C2×D3×C9—21no—
102C3×C4×C9—41no—
103C3×C4×C9C2×D411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C9×Dic3—22no—
106C9×Dic3—21no—
107C9×Dic3—41no—
108C2×C3×#41—31no—
109C2×#237C2×C911yescharacteristic
110C2×C3×Q16—31no—
111C2×C3×SD16—21no—
112C2×C3×SD16C2×C911yescharacteristic, complemented
113C2×C3×SD16C3×D311yescharacteristic
114C3×C22⋊Q8—31no—
115#946C2×C911yescharacteristic, complemented
116C2×#216C2×C911yescharacteristic, complemented
117C3×#103—31no—
118C3×#102—31no—
119C3×C22⋊C8—31no—
120#832C2×C911yescharacteristic, complemented
121(C2×D3)⋊C8C2×C911yescharacteristic, complemented
122#816C2×C911yescharacteristic, complemented
123C23×C32—21no—
124C22×C3×D3C3×D411yescharacteristic
125C32×Q8—21no—
126C32×Q8—21no—
127C32×D4—21no—
128C32×D4C3×D412yes—
129C2×C32×C4—21no—
130C2×C32×C4C23×C311yescharacteristic
131C3×#67—24no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3C3×D411yescharacteristic
136C3×C4×D3—21no—
137C3×C4×D3—21no—
138C3×Dic6—21no—
139C3×Dic6C3×D412yes—
140C23×C9—21no—
141C23×C9—31no—
142C23×C9—41no—
143C32×C8—21no—
144C3×C3⋊C8—21no—
145Q8×C9—22no—
146Q8×C9—21no—
147Q8×C9—21no—
148Q8×C9—32no—
149Q8×C9—41no—
150Q8×C9—61no—
151D4×C9—21no—
152D4×C9—22no—
153D4×C9—41no—
154D4×C9—64no—
155D4×C9#6712yes—
156C2×C4×C9—21no—
157C2×C4×C9—21no—
158C2×C4×C9—31no—
159C2×C4×C9—41no—
160C2×C4×C9—64no—
161C2×C4×C9—61no—
162C2×C4×C9—61no—
163C2×C4×C9—61no—
164C2×C4×C9—61no—
165C2×C4×C9C22×D311yescharacteristic
166C8×C9—22no—
167C8×C9—21no—
168C8×C9—61no—
169#452—31noSylow
170C2×C3×C9—42no—
171C2×C3×C9#12011yescharacteristic
172C2×C3×C9#13511yescharacteristic
173C2×C3×C9#13711yescharacteristic
174D3×C9—41no—
175C3×#41—62no—
176C3×#41—62no—
177C2×C3×Q8—21no—
178C2×C3×Q8—31no—
179C2×C3×Q8C22×C911yescharacteristic
180C2×C3×Q8C2×C3×D311yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4—61no—
183C2×C3×D4C22×C911yescharacteristic
184C2×C3×D4C2×C3×D311yescharacteristic
185C22×C3×C4—31no—
186C22×C3×C4—61no—
187C2×#67—21no—
188C22×Dic3—21no—
189#237—22no—
190#237—22no—
191C2×C4×D3C22×C911yescharacteristic
192C2×Dic6C22×C911yescharacteristic
193C3×Q16—62no—
194C3×SD16—24no—
195C3×SD16—22no—
196C3×SD16—22no—
197C2×C3×C8—21no—
198C2×C3×C8—31no—
199C2×C3×C8C22×C911yescharacteristic
200C2×C3×C8C2×C3×D311yescharacteristic
201C3×C4⋊C4—31no—
202C3×C4⋊C4—61no—
203C3×C22⋊C4—61no—
204#216—22no—
205(C2×D3)⋊C4—21no—
206C4⋊Dic3C22×C911yescharacteristic
207Dic3⋊C4—21no—
208C2×C3⋊C8C22×C911yescharacteristic
209C22×C32—22no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32C2×C3×D411yescharacteristic
213C2×C3×D3—21no—
214C2×C3×D3—21no—
215C2×C3×D3—21no—
216C32×C4—41no—
217C32×C4C2×C3×D411yescharacteristic
218C32×C4C2×C3×D411yescharacteristic
219C3×Dic3—22no—
220C3×Dic3—21no—
221C3×Dic3—41no—
222C22×C9—22no—
223C22×C9—21no—
224C22×C9—42no—
225C22×C9—41no—
226C22×C9—41no—
227C22×C9—61no—
228C22×C9—61no—
229C22×C9—61no—
230C22×C9—81no—
231C22×C9—81no—
232C22×C9D3×D411yescharacteristic
233C4×C9—21no—
234C4×C9—21no—
235C4×C9—42no—
236C4×C9—41no—
237C4×C9—61no—
238C4×C9—62no—
239C4×C9—121no—
240C4×C9D3×D411yescharacteristic
241C4×C9C2×#6711yescharacteristic
242C2×#41—31no—
243C2×Q16—31no—
244C2×SD16D3×C911yescharacteristic, complemented
245C22⋊Q8—31no—
246#103—31no—
247#102—31no—
248C22⋊C8—31no—
249C3×C9#45211yescharacteristic, complemented, Sylow
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—31no—
253C23×C3—41no—
254C22×D3D4×C911yescharacteristic
255C3×Q8—22no—
256C3×Q8—21no—
257C3×Q8—21no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—32no—
261C3×Q8—41no—
262C3×Q8—61no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—22no—
266C3×D4—41no—
267C3×D4—64no—
268C3×D4D4×C912yes—
269C3×D4C3×#6712yes—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—31no—
274C2×C3×C4—41no—
275C2×C3×C4—64no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4C23×C911yescharacteristic
281C2×C3×C4C22×C3×D311yescharacteristic
282#67—24no—
283C2×Dic3—24no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3D4×C911yescharacteristic
287C4×D3—21no—
288C4×D3—21no—
289Dic6—21no—
290Dic6D4×C912yes—
291C3×C8—22no—
292C3×C8—21no—
293C3×C8—21no—
294C3×C8—61no—
295C3⋊C8—21no—
296C2×C32—42no—
297C2×C32C3×#12011yescharacteristic
298C2×C32C3×#13511yescharacteristic
299C2×C32C3×#13711yescharacteristic
300C3×D3—41no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—21no—
304C2×C9—42no—
305C2×C9—82no—
306C2×C9—121no—
307C2×C9#91511yescharacteristic
308C2×C9#91611yescharacteristic
309C2×C9#93711yescharacteristic
310#41—62no—
311#41—62no—
312C2×Q8—31no—
313C2×Q8C2×D3×C911yescharacteristic
314C2×D4—61no—
315C2×D4C2×D3×C911yescharacteristic
316C22×C4—31no—
317C22×C4—61no—
318Q16—62no—
319SD16—22no—
320C2×C8—31no—
321C2×C8C2×D3×C911yescharacteristic
322C4⋊C4—31no—
323C4⋊C4—61no—
324C22⋊C4—61no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—61no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3C2×D4×C911yescharacteristic
339C22×C3C3×D3×D411yescharacteristic
340C2×D3—21no—
341C2×D3—21no—
342C2×D3—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—42no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—61no—
349C3×C4—62no—
350C3×C4—121no—
351C3×C4C2×D4×C911yescharacteristic
352C3×C4C2×D4×C911yescharacteristic
353C3×C4C3×D3×D411yescharacteristic
354C3×C4C2×C3×#6711yescharacteristic
355Dic3—22no—
356Dic3—21no—
357Dic3—41no—
358C32C3×#45211yescharacteristic
359C9—21no—
360C9#517311yescharacteristic, direct product factor
361C23—21no—
362C23—31no—
363Q8—21no—
364Q8—21no—
365Q8—32no—
366Q8—61no—
367D4—21no—
368D4—64no—
369D4C9×#6712yes—
370C2×C4—21no—
371C2×C4—31no—
372C2×C4—64no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4C22×D3×C911yescharacteristic
378C8—21no—
379C8—61no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—42no—
384C2×C3—42no—
385C2×C3—82no—
386C2×C3—121no—
387C2×C3C9×#12011yescharacteristic
388C2×C3C9×#13511yescharacteristic
389C2×C3C9×#13711yescharacteristic
390C2×C3C3×#91511yescharacteristic
391C2×C3C3×#91611yescharacteristic
392C2×C3C3×#93711yescharacteristic
393D3—41no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—61no—
398C22—61no—
399C22—61no—
400C22D3×D4×C911yescharacteristic
401C4—41no—
402C4—61no—
403C4—62no—
404C4—121no—
405C4D3×D4×C911yescharacteristic
406C4C2×C9×#6711yescharacteristic
407C3—21no—
408C3C9×#45211yescharacteristic, complemented
409C3C3×#517311yescharacteristic
410C2—42no—
411C2—121no—
412C2C9×#91511yescharacteristic
413C2C9×#91611yescharacteristic
414C2C9×#93711yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×SD16C2
socleC22×C32C2×C3×D4
prime radicalC2×C3×#237C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC2×C9×SD16D3
hypocenterC3C9×#452
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C2×C9×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×SD16C2semidirect
3. C9×#946C2semidirect
4. C2×C9×#216C2semidirect
5. C9×#832C2semidirect
6. C9×(C2×D3)⋊C8C2semidirect
7. C9×#816C2semidirect
35. C2×C9×SD16D3semidirect
60. #5173C9direct (N × H)
112. C2×C3×SD16C2×C9semidirect
115. #946C2×C9semidirect
116. C2×#216C2×C9semidirect
120. #832C2×C9semidirect
121. (C2×D3)⋊C8C2×C9semidirect
122. #816C2×C9semidirect
244. C2×SD16D3×C9semidirect
249. C3×C9#452semidirect
360. C9#5173direct (N × H)
408. 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×C3×C9×SD162
C9×#9462
C2×C9×#2162
C9×#8322
C9×(C2×D3)⋊C82
C9×#8162
C3×#51733
C9×#4523