Finite Group Databasebeta

← #11887197

Subgroups — #11887197

Order 1152.

612 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#546C211yesmaximal, characteristic
2#10589643C211yesmaximal, characteristic, complemented
3#10588918C211yesmaximal, characteristic, complemented
4#10588917C211yesmaximal, characteristic, complemented
5#10588897C211yesmaximal, characteristic, complemented
6C2×#63392C211yesmaximal, characteristic, complemented
7#10588736C211yesmaximal, characteristic, complemented
8#10588735C211yesmaximal, characteristic, complemented
9#10588727C211yesmaximal, characteristic, complemented
10#10588723C211yesmaximal, characteristic, complemented
11#10588717C211yesmaximal, characteristic, complemented
12#10588715C211yesmaximal, characteristic, complemented
13C4×#3672C211yesmaximal, characteristic, complemented
14#10588692C211yesmaximal, characteristic, complemented
15D4×C9⋊C8C211yesmaximal, characteristic, complemented
16#83494—31nomaximal
17C2×C9×#41C2211yescharacteristic
18C22×D4×C9C2211yescharacteristic
19C9×#124C2211yescharacteristic
20C9×#123—21no—
21C9×C22⋊Q8C2211yescharacteristic
22C9×#121—21no—
23C9×#121C2211yescharacteristic
24C9×#120—21no—
25C4×D4×C9C2211yescharacteristic
26C4×D4×C9C2211yescharacteristic
27C2×C9×C22⋊C4—21no—
28C2×#3674C2211yescharacteristic, complemented
29#63497—21no—
30C2×#3672C2211yescharacteristic
31C2×#3672C2211yescharacteristic
32#63488C2211yescharacteristic, complemented
33C2×C4⋊Dic9C2211yescharacteristic
34#63486C2211yescharacteristic
35#63486C2211yescharacteristic, complemented
36C22×C9⋊C8C2211yescharacteristic
37C22×C9⋊C8C2211yescharacteristic, complemented
38#63451—21no—
39#63441—21no—
40#63439—21no—
41#63432C2211yescharacteristic, complemented
42C4×Dic18C2211yescharacteristic, complemented
43#63395C2211yescharacteristic, complemented
44#63395C2211yescharacteristic, complemented
45#63395C2211yescharacteristic, complemented
46#63392C2211yescharacteristic
47#63392C2211yescharacteristic
48#63392C2211yescharacteristic
49#63392C2211yescharacteristic
50#63392C2211yescharacteristic
51#63392C2211yescharacteristic
52#63389C2211yescharacteristic
53#63389C2211yescharacteristic, complemented
54#63368C2211yescharacteristic, complemented
55#63367C2211yescharacteristic, complemented
56#63367C2211yescharacteristic, complemented
57#63366C2211yescharacteristic, complemented
58C4⋊(C9⋊C8)C2211yescharacteristic, complemented
59C4×C9⋊C8C2211yescharacteristic, complemented
60C3×#546D311yescharacteristic
61#5541—31no—
62#5239—31no—
63#5233—31no—
64#5218—31no—
65C2×#832—31no—
66#5056—31no—
67#5054—31no—
68#5051—31no—
69#5043—31no—
70#5041—31no—
71#5035—31no—
72C4×#214—31no—
73D4×C3⋊C8—31no—
74#5009—31no—
75C24×C9—21no—
76C9×#41—22no—
77C9×#41—41no—
78C2×Q8×C9C2311yescharacteristic
79C2×D4×C9—22no—
80C2×D4×C9—21no—
81C2×D4×C9—21no—
82C2×D4×C9—21no—
83C2×D4×C9—41no—
84C2×D4×C9D411yescharacteristic
85C2×D4×C9C2311yescharacteristic
86C2×D4×C9C2311yescharacteristic
87C22×C4×C9—21no—
88C22×C4×C9—21no—
89C22×C4×C9C2311yescharacteristic
90C22×C4×C9C2311yescharacteristic
91C22×Dic9—21no—
92C2×Dic18C2311yescharacteristic
93C9×C4⋊C4—21no—
94C9×C4⋊C4D411yescharacteristic
95C9×C4⋊C4C2311yescharacteristic
96C9×C22⋊C4—21no—
97C9×C22⋊C4—21no—
98C9×C22⋊C4—21no—
99C9×C22⋊C4—21no—
100C9×C22⋊C4—21no—
101C9×C22⋊C4D411yescharacteristic
102C9×C22⋊C4D411yescharacteristic
103C42×C9C2311yescharacteristic
104#3676—21no—
105#3676—21no—
106#3676—21no—
107#3674—41no—
108#3672—22no—
109#3672—41no—
110C4⋊Dic9—21no—
111C4⋊Dic9C2311yescharacteristic
112C4⋊Dic9C2311yescharacteristic
113C4⋊Dic9C2311yescharacteristic
114Dic9⋊C4—21no—
115Dic9⋊C4—21no—
116Dic9⋊C4—21no—
117Dic9⋊C4—21no—
118Dic9⋊C4—21no—
119C4×Dic9—21no—
120C4×Dic9—21no—
121C2×C9⋊C8—41no—
122C2×C9⋊C8—41no—
123C2×C9⋊C8C2311yescharacteristic
124C2×C9⋊C8C2311yescharacteristic
125C2×C9⋊C8C2311yescharacteristic
126C2×C9⋊C8C2311yescharacteristic
127#3296—91noSylow
128C2×C3×#41C2×D311yescharacteristic
129C22×C3×D4C2×D311yescharacteristic
130C3×#124C2×D311yescharacteristic
131C3×#123—21no—
132C3×C22⋊Q8C2×D311yescharacteristic
133C3×#121—21no—
134C3×#121C2×D311yescharacteristic
135C3×#120—21no—
136C3×C4×D4C2×D311yescharacteristic
137C3×C4×D4C2×D311yescharacteristic
138C2×C3×C22⋊C4—21no—
139C2×#216—31no—
140#935—61no—
141C2×#214—31no—
142C2×#214—31no—
143#926—31no—
144C2×C4⋊Dic3—31no—
145#924—31no—
146#924—31no—
147C22×C3⋊C8—31no—
148C22×C3⋊C8—31no—
149#889—61no—
150#879—61no—
151#877—61no—
152#870—31no—
153C4×Dic6—31no—
154#835—31no—
155#835—31no—
156#835—31no—
157#832—31no—
158#832—31no—
159#832—31no—
160#832—31no—
161#832—31no—
162#832—31no—
163#830—31no—
164#830—31no—
165#810—31no—
166#808—31no—
167#808—31no—
168#807—31no—
169C4⋊(C3⋊C8)—31no—
170C4×C3⋊C8—31no—
171C23×C9—21no—
172C23×C9—21no—
173C23×C9—42no—
174C23×C9—42no—
175C23×C9—81no—
176C23×C9C2×D411yescharacteristic
177C23×C9C2×D411yescharacteristic
178Q8×C9—21no—
179Q8×C9—21no—
180D4×C9—22no—
181D4×C9—21no—
182D4×C9—21no—
183D4×C9—21no—
184D4×C9—42no—
185D4×C9—42no—
186D4×C9—81no—
187D4×C9#4112yes—
188C2×C4×C9—21no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—42no—
195C2×C4×C9—81no—
196C2×C4×C9C2×D411yescharacteristic
197C2×C4×C9C2×D411yescharacteristic
198C2×C4×C9C2×D411yescharacteristic
199C2×C4×C9C2×D411yescharacteristic
200C2×C4×C9C2411yescharacteristic
201C2×Dic9—21no—
202C2×Dic9—21no—
203C2×Dic9—21no—
204C2×Dic9—21no—
205C2×Dic9—42no—
206Dic18—21no—
207Dic18—21no—
208C9⋊C8—21no—
209C9⋊C8—21no—
210C9⋊C8—21no—
211C9⋊C8—21no—
212#552—91no—
213#546D911yescharacteristic
214#488—91no—
215#483—91no—
216#482—91no—
217#478—91no—
218#467—91no—
219#465—91no—
220#464—91no—
221#452—91no—
222#450—91no—
223C4×SD16—91no—
224C8×D4—91no—
225#416—91no—
226C2×#102—91no—
227C24×C3—21no—
228C3×#41—22no—
229C3×#41—41no—
230C2×C3×Q8C22×D311yescharacteristic
231C2×C3×D4—22no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—41no—
236C2×C3×D4#6711yescharacteristic
237C2×C3×D4C22×D311yescharacteristic
238C2×C3×D4C22×D311yescharacteristic
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4C22×D311yescharacteristic
242C22×C3×C4C22×D311yescharacteristic
243C22×Dic3—61no—
244C2×Dic6—31no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4#6711yescharacteristic
247C3×C4⋊C4C22×D311yescharacteristic
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4#6711yescharacteristic
254C3×C22⋊C4#6711yescharacteristic
255C3×C42C22×D311yescharacteristic
256#217—61no—
257#217—61no—
258#217—61no—
259#216—121no—
260#214—62no—
261#214—121no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266Dic3⋊C4—61no—
267Dic3⋊C4—61no—
268Dic3⋊C4—61no—
269Dic3⋊C4—61no—
270Dic3⋊C4—61no—
271C4×Dic3—61no—
272C4×Dic3—61no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—121no—
278C2×C3⋊C8—121no—
279C22×C9—21no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—22no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—41no—
286C22×C9—41no—
287C22×C9—41no—
288C22×C9—41no—
289C22×C9—41no—
290C22×C9—84no—
291C22×C9#13512yes—
292C22×C9C22×D411yescharacteristic
293C4×C9—21no—
294C4×C9—21no—
295C4×C9—21no—
296C4×C9—21no—
297C4×C9—41no—
298C4×C9—41no—
299C4×C9C2×#4111yescharacteristic
300C4×C9H(5,𝔽2)11yescharacteristic
301Dic9—41no—
302Dic9—41no—
303Dic9—41no—
304Dic9—41no—
305C2×#41C2×D911yescharacteristic
306C22×D4C2×D911yescharacteristic
307C2×Q16—91no—
308C2×SD16—91no—
309C2×SD16—91no—
310C22×C8—91no—
311C22×C8—91no—
312#126—181no—
313#125—91no—
314#125—181no—
315#124C2×D911yescharacteristic
316#123—21no—
317#123—181no—
318C22⋊Q8—91no—
319C22⋊Q8—91no—
320C22⋊Q8C2×D911yescharacteristic
321#121—21no—
322#121C2×D911yescharacteristic
323#120—21no—
324C4×Q8—91no—
325C4×D4C2×D911yescharacteristic
326C4×D4C2×D911yescharacteristic
327#117—91no—
328#117—181no—
329C2×C4⋊C4—91no—
330C2×C22⋊C4—21no—
331#107—91no—
332#106—91no—
333#106—91no—
334C4⋊C8—91no—
335#103—91no—
336#103—91no—
337#103—91no—
338#103—91no—
339#102—91no—
340#102—91no—
341#102—91no—
342#102—91no—
343#102—91no—
344#102—91no—
345C22⋊C8—91no—
346C22⋊C8—91no—
347C4×C8—91no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—81no—
353C23×C3C2×#6711yescharacteristic
354C23×C3C2×#6711yescharacteristic
355C3×Q8—21no—
356C3×Q8—21no—
357C3×D4—22no—
358C3×D4—21no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—42no—
362C3×D4—42no—
363C3×D4—81no—
364C3×D4#23712yes—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—42no—
372C2×C3×C4—81no—
373C2×C3×C4C2×#6711yescharacteristic
374C2×C3×C4C2×#6711yescharacteristic
375C2×C3×C4C2×#6711yescharacteristic
376C2×C3×C4C2×#6711yescharacteristic
377C2×C3×C4C23×D311yescharacteristic
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—122no—
383Dic6—61no—
384Dic6—61no—
385C3⋊C8—61no—
386C3⋊C8—61no—
387C3⋊C8—61no—
388C3⋊C8—61no—
389C2×C9—22no—
390C2×C9—41no—
391C2×C9—42no—
392C2×C9—81no—
393C2×C9#54611yescharacteristic
394C2×C9C2×#13511yescharacteristic
395C2×C9#57711yescharacteristic
396C24—21no—
397#41—22no—
398#41—41no—
399C2×Q8—91no—
400C2×Q8C22×D911yescharacteristic
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—41no—
406C2×D4#61111yescharacteristic
407C2×D4C22×D911yescharacteristic
408C2×D4C22×D911yescharacteristic
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—181no—
412C22×C4C22×D911yescharacteristic
413C22×C4C22×D911yescharacteristic
414Q16—361no—
415SD16—182no—
416SD16—361no—
417C2×C8—91no—
418C2×C8—91no—
419C2×C8—91no—
420C2×C8—91no—
421C2×C8—361no—
422C2×C8—361no—
423C4⋊C4—21no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—181no—
428C4⋊C4—181no—
429C4⋊C4—181no—
430C4⋊C4—181no—
431C4⋊C4—181no—
432C4⋊C4—181no—
433C4⋊C4#61111yescharacteristic
434C4⋊C4C22×D911yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—181no—
441C22⋊C4—181no—
442C22⋊C4—181no—
443C22⋊C4#61111yescharacteristic
444C22⋊C4#61111yescharacteristic
445C42—181no—
446C42—181no—
447C42C22×D911yescharacteristic
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—84no—
460C22×C3#95012yes—
461C22×C3C22×#6711yescharacteristic
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4C2×#23711yescharacteristic
469C3×C4#100411yescharacteristic
470Dic3—121no—
471Dic3—121no—
472Dic3—121no—
473Dic3—121no—
474C9#329611yescharacteristic, complemented, Sylow
475C23—21no—
476C23—21no—
477C23—42no—
478C23—42no—
479C23—81no—
480C23C2×#61111yescharacteristic
481C23C2×#61111yescharacteristic
482Q8—21no—
483Q8—21no—
484Q8—181no—
485Q8—181no—
486D4—22no—
487D4—21no—
488D4—21no—
489D4—21no—
490D4—42no—
491D4—42no—
492D4—81no—
493D4#369912yes—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—42no—
501C2×C4—81no—
502C2×C4—181no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—181no—
506C2×C4—362no—
507C2×C4C2×#61111yescharacteristic
508C2×C4C2×#61111yescharacteristic
509C2×C4C2×#61111yescharacteristic
510C2×C4C2×#61111yescharacteristic
511C2×C4C23×D911yescharacteristic
512C8—181no—
513C8—181no—
514C8—181no—
515C8—181no—
516C2×C3—22no—
517C2×C3—41no—
518C2×C3—42no—
519C2×C3—81no—
520C2×C3#580611yescharacteristic
521C2×C3C2×#95011yescharacteristic
522C2×C3#583611yescharacteristic
523C22—21no—
524C22—21no—
525C22—21no—
526C22—22no—
527C22—21no—
528C22—21no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—84no—
535C22#6351312yes—
536C22C22×#61111yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—41no—
542C4—41no—
543C4—361no—
544C4—361no—
545C4—361no—
546C4—361no—
547C4C2×#369911yescharacteristic
548C4#6371011yescharacteristic
549C3#8349411yescharacteristic
550C2—22no—
551C2—41no—
552C2—42no—
553C2—81no—
554C2#1058990811yescharacteristic
555C2C2×#6351311yescharacteristic
556C2#1058993811yescharacteristic
subgroupG/H
centerC22C22×#611
commutator, cosocleC2×C4×C9C24
FrattiniC2×C3×C4C23×D3
FittingC9×#546C2
socleC22×C3C22×#67
prime radicalC3×#546D3
hypercenter#546D9
hypocenterC9#3296
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
2. #10589643C2semidirect
3. #10588918C2semidirect
4. #10588917C2semidirect
5. #10588897C2semidirect
6. C2×#63392C2semidirect
7. #10588736C2semidirect
8. #10588735C2semidirect
9. #10588727C2semidirect
10. #10588723C2semidirect
11. #10588717C2semidirect
12. #10588715C2semidirect
13. C4×#3672C2semidirect
14. #10588692C2semidirect
15. D4×C9⋊C8C2semidirect
28. C2×#3674C22semidirect
32. #63488C22semidirect
35. #63486C22semidirect
37. C22×C9⋊C8C22semidirect
41. #63432C22semidirect
42. C4×Dic18C22semidirect
43. #63395C22semidirect
44. #63395C22semidirect
45. #63395C22semidirect
53. #63389C22semidirect
54. #63368C22semidirect
55. #63367C22semidirect
56. #63367C22semidirect
57. #63366C22semidirect
58. C4⋊(C9⋊C8)C22semidirect
59. C4×C9⋊C8C22semidirect
474. C9#3296semidirect

Maximal subgroups

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

subgroupindex
C9×#5462
#105896432
#105889182
#105889172
#105888972
C2×#633922
#105887362
#105887352
#105887272
#105887232
#105887172
#105887152
C4×#36722
#105886922
D4×C9⋊C82
#834943