Finite Group Databasebeta

← #11887074

Subgroups — #11887074

Order 1152.

693 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#611C211yesmaximal, characteristic
2#10589737C211yesmaximal, characteristic, complemented
3C9×#450C211yesmaximal, characteristic, complemented
4C2×#63392C211yesmaximal, characteristic, complemented
5#10588851C211yesmaximal, characteristic, complemented
6#10588849C211yesmaximal, characteristic, complemented
7#10588843C211yesmaximal, characteristic, complemented
8SD16×Dic9C211yesmaximal, characteristic, complemented
9#10588731C211yesmaximal, characteristic, complemented
10#10588457C211yesmaximal, characteristic, complemented
11D9×#102C211yesmaximal, characteristic, complemented
12#10588441C211yesmaximal, characteristic, complemented
13#10588440C211yesmaximal, characteristic, complemented
14#10588417C211yesmaximal, characteristic, complemented
15#611⋊C8C211yesmaximal, characteristic, complemented
16#80158—31nomaximal
17C22×D4×C9C2211yescharacteristic
18C22×#611—21no—
19C2×D4×D9C2211yescharacteristic
20C2×C9×SD16C2211yescharacteristic
21C2×C9×SD16C2211yescharacteristic, complemented
22C9×C22⋊Q8C2211yescharacteristic, complemented
23#63515—21no—
24#63508C2211yescharacteristic, complemented
25Q8×Dic9C2211yescharacteristic, complemented
26C2×#3675C2211yescharacteristic, complemented
27#63502C2211yescharacteristic
28#63501—21no—
29#63500C2211yescharacteristic
30#63499—21no—
31D4×Dic9C2211yescharacteristic
32#63492C2211yescharacteristic
33C4×#611C2211yescharacteristic
34C2×C4⋊Dic9C2211yescharacteristic, complemented
35C22×C9⋊C8C2211yescharacteristic, complemented
36C2×C8×D9C2211yescharacteristic, complemented
37#63459—21no—
38C4⋊C4×D9C2211yescharacteristic, complemented
39#63451—21no—
40#63450C2211yescharacteristic, complemented
41#611⋊C4—21no—
42#63440—21no—
43C9×#102C2211yescharacteristic
44C9×#102C2211yescharacteristic, complemented
45C9×C22⋊C8C2211yescharacteristic, complemented
46#63395C2211yescharacteristic, complemented
47#63392C2211yescharacteristic
48#63392C2211yescharacteristic
49#63392C2211yescharacteristic, complemented
50#63392C2211yescharacteristic, complemented
51#63380C2211yescharacteristic, complemented
52#63380C2211yescharacteristic, complemented
53(C2×D9)⋊C8C2211yescharacteristic, complemented
54#63377C2211yescharacteristic, complemented
55#63377C2211yescharacteristic, complemented
56Dic9⋊C8C2211yescharacteristic, complemented
57C8×Dic9C2211yescharacteristic, complemented
58#63369C2211yescharacteristic, complemented
59#63367C2211yescharacteristic, complemented
60D4×#67—31no—
61#5635—31no—
62C3×#450D311yescharacteristic
63C2×#832—31no—
64#5171—31no—
65#5169—31no—
66#5163—31no—
67Dic3×SD16—31no—
68#5048—31no—
69#4777—31no—
70D3×#102—31no—
71#4761—31no—
72#4760—31no—
73#4735—31no—
74#67⋊C8—31no—
75C24×C9—21no—
76C23×D9—21no—
77C2×Q8×C9C2311yescharacteristic
78C2×D4×C9—22no—
79C2×D4×C9—21no—
80C2×D4×C9—41no—
81C2×D4×C9C2311yescharacteristic
82C2×D4×C9C2311yescharacteristic
83C22×C4×C9C2311yescharacteristic
84C2×#611—21no—
85C2×#611—21no—
86C2×#611—21no—
87C2×#611—21no—
88C2×#611—21no—
89C2×#611—44no—
90C2×#611D411yescharacteristic
91C22×Dic9—21no—
92C22×Dic9—21no—
93D4×D9—22no—
94D4×D9—41no—
95C2×D36C2311yescharacteristic
96C2×C4×D9—21no—
97C2×C4×D9C2311yescharacteristic
98C2×Dic18—21no—
99C9×SD16—22no—
100C9×SD16—41no—
101C2×C8×C9C2311yescharacteristic
102C2×C8×C9C2311yescharacteristic
103C9×C4⋊C4—21no—
104C9×C4⋊C4C2311yescharacteristic
105C9×C22⋊C4—21no—
106#3676—21no—
107#3676—21no—
108#3676D411yescharacteristic
109#3675—41no—
110(C2×D9)⋊C4—21no—
111(C2×D9)⋊C4—21no—
112(C2×D9)⋊C4D411yescharacteristic
113C4⋊Dic9—21no—
114C4⋊Dic9—21no—
115C4⋊Dic9C2311yescharacteristic
116C4⋊Dic9C2311yescharacteristic
117C4⋊Dic9C2311yescharacteristic
118Dic9⋊C4—21no—
119Dic9⋊C4—21no—
120Dic9⋊C4—21no—
121Dic9⋊C4D411yescharacteristic
122C4×Dic9—21no—
123C4×Dic9C2311yescharacteristic
124C2×C9⋊C8—41no—
125C2×C9⋊C8C2311yescharacteristic
126C2×C9⋊C8C2311yescharacteristic
127C8×D9—41no—
128#3295—91noSylow
129C22×C3×D4C2×D311yescharacteristic
130C22×#67—61no—
131C2×D3×D4—31no—
132C2×C3×SD16C2×D311yescharacteristic
133C2×C3×SD16C2×D311yescharacteristic
134C3×C22⋊Q8C2×D311yescharacteristic
135#953—61no—
136#946—31no—
137Q8×Dic3—31no—
138C2×#215—31no—
139#940—31no—
140#939—61no—
141#938—31no—
142#937—61no—
143D4×Dic3—31no—
144#930—31no—
145C4×#67—31no—
146C2×C4⋊Dic3—31no—
147C22×C3⋊C8—31no—
148C2×D3×C8—31no—
149#897—61no—
150D3×C4⋊C4—31no—
151#889—61no—
152#888—31no—
153#67⋊C4—61no—
154#878—61no—
155C3×#102C2×D311yescharacteristic
156C3×#102C2×D311yescharacteristic
157C3×C22⋊C8C2×D311yescharacteristic
158#835—31no—
159#832—31no—
160#832—31no—
161#832—31no—
162#832—31no—
163#821—31no—
164#821—31no—
165(C2×D3)⋊C8—31no—
166#817—31no—
167#817—31no—
168Dic3⋊C8—31no—
169C8×Dic3—31no—
170#809—31no—
171#808—31no—
172C23×C9—21no—
173C23×C9—21no—
174C23×C9—42no—
175C23×C9—42no—
176C23×C9—81no—
177C23×C9C2×D411yescharacteristic
178C22×D9—21no—
179C22×D9—42no—
180C22×D9—42no—
181C22×D9—81no—
182C22×D9C2×D411yescharacteristic
183Q8×C9—21no—
184Q8×C9—21no—
185D4×C9—21no—
186D4×C9—21no—
187D4×C9—21no—
188D4×C9—42no—
189D4×C9#4112yes—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9C2×D411yescharacteristic
195C2×C4×C9C2411yescharacteristic
196#611—42no—
197#611—42no—
198#611—42no—
199#611—42no—
200#611—81no—
201#611SD1614yescomplemented
202C2×Dic9—21no—
203C2×Dic9—21no—
204C2×Dic9—21no—
205C2×Dic9—42no—
206C2×Dic9—81no—
207C2×Dic9C2×D411yescharacteristic
208C2×Dic9C2×D411yescharacteristic
209C2×Dic9C2×D411yescharacteristic
210D36—21no—
211D36—21no—
212C4×D9—21no—
213C4×D9—21no—
214C4×D9—81no—
215Dic18—81no—
216C8×C9—21no—
217C8×C9—21no—
218C9⋊C8—21no—
219C9⋊C8—21no—
220#554—91no—
221D42—91no—
222#486—91no—
223#481—91no—
224#481—91no—
225#476—91no—
226#465—91no—
227#465—91no—
228#460—91no—
229#450—91no—
230#450D911yescharacteristic, complemented
231C4×SD16—91no—
232C8×D4—91no—
233C2×#102—91no—
234C2×#102—91no—
235C24×C3—21no—
236C23×D3—61no—
237C2×C3×Q8C22×D311yescharacteristic
238C2×C3×D4—22no—
239C2×C3×D4—21no—
240C2×C3×D4—41no—
241C2×C3×D4C22×D311yescharacteristic
242C2×C3×D4C22×D311yescharacteristic
243C22×C3×C4C22×D311yescharacteristic
244C2×#67—31no—
245C2×#67—61no—
246C2×#67—61no—
247C2×#67—61no—
248C2×#67—61no—
249C2×#67—61no—
250C2×#67—124no—
251C22×Dic3—61no—
252C22×Dic3—61no—
253D3×D4—62no—
254D3×D4—121no—
255C2×D12—31no—
256C2×C4×D3—31no—
257C2×C4×D3—61no—
258C2×Dic6—61no—
259C3×SD16—22no—
260C3×SD16—41no—
261C2×C3×C8C22×D311yescharacteristic
262C2×C3×C8C22×D311yescharacteristic
263C3×C4⋊C4—21no—
264C3×C4⋊C4C22×D311yescharacteristic
265C3×C22⋊C4—21no—
266#217—31no—
267#217—61no—
268#217—61no—
269#215—121no—
270(C2×D3)⋊C4—31no—
271(C2×D3)⋊C4—61no—
272(C2×D3)⋊C4—61no—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—31no—
276C4⋊Dic3—61no—
277C4⋊Dic3—61no—
278Dic3⋊C4—31no—
279Dic3⋊C4—61no—
280Dic3⋊C4—61no—
281Dic3⋊C4—61no—
282C4×Dic3—31no—
283C4×Dic3—61no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—121no—
287D3×C8—121no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—22no—
291C22×C9—41no—
292C22×C9—41no—
293C22×C9—41no—
294C22×C9—41no—
295C22×C9—41no—
296C22×C9—84no—
297C22×C9C2×SD1612yes—
298C22×C9C22×D411yescharacteristic
299C2×D9—21no—
300C2×D9—21no—
301C2×D9—41no—
302C2×D9—41no—
303C2×D9—41no—
304C2×D9—84no—
305C2×D9C2×SD1612yes—
306C4×C9—21no—
307C4×C9—41no—
308C4×C9—41no—
309C4×C9C2×#4111yescharacteristic
310C4×C9H(5,𝔽2)11yescharacteristic
311Dic9—21no—
312Dic9—21no—
313Dic9—41no—
314Dic9—41no—
315Dic9—41no—
316Dic9C2×SD1612yes—
317C22×D4—91no—
318C22×D4—181no—
319C22×D4C2×D911yescharacteristic
320C2×SD16—91no—
321C2×SD16C2×D911yescharacteristic
322C2×SD16C2×D911yescharacteristic, complemented
323C22×C8—91no—
324C22×C8—91no—
325C4⋊Q8—91no—
326#127—91no—
327#125—181no—
328C22⋊Q8—91no—
329C22⋊Q8—181no—
330C22⋊Q8—181no—
331C22⋊Q8C2×D911yescharacteristic, complemented
332#121—91no—
333#121—91no—
334#121—181no—
335#120—181no—
336#120—181no—
337C4×Q8—91no—
338C4×D4—91no—
339C4×D4—91no—
340C4×D4—181no—
341C2×C4⋊C4—91no—
342C2×C4⋊C4—91no—
343#106—91no—
344#106—91no—
345#106—91no—
346C4⋊C8—91no—
347#103—91no—
348#102—91no—
349#102—91no—
350#102—91no—
351#102—91no—
352#102—91no—
353#102—91no—
354#102—91no—
355#102C2×D911yescharacteristic
356#102C2×D911yescharacteristic, complemented
357C22⋊C8—91no—
358C22⋊C8C2×D911yescharacteristic, complemented
359C4×C8—91no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—42no—
363C23×C3—42no—
364C23×C3—81no—
365C23×C3D3×D411yescharacteristic
366C22×D3—31no—
367C22×D3—61no—
368C22×D3—122no—
369C22×D3—122no—
370C22×D3—241no—
371C3×Q8—21no—
372C3×Q8—21no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—21no—
376C3×D4—42no—
377C3×D4#23712yes—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4D3×D411yescharacteristic
383C2×C3×C4C23×D311yescharacteristic
384#67—34no—
385#67—122no—
386#67—122no—
387#67—122no—
388#67—122no—
389#67—241no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—122no—
397C2×Dic3—241no—
398D12—61no—
399D12—61no—
400C4×D3—61no—
401C4×D3—61no—
402C4×D3—241no—
403Dic6—241no—
404C3×C8—21no—
405C3×C8—21no—
406C3⋊C8—61no—
407C3⋊C8—61no—
408C2×C9—22no—
409C2×C9—42no—
410C2×C9—81no—
411C2×C9#54611yescharacteristic
412C2×C9C22×SD1611yescharacteristic
413C2×C9#57611yescharacteristic
414D9—22no—
415D9—81no—
416C24—21no—
417C24—181no—
418C2×Q8—181no—
419C2×Q8C22×D911yescharacteristic
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—41no—
423C2×D4—91no—
424C2×D4—91no—
425C2×D4—181no—
426C2×D4—181no—
427C2×D4—181no—
428C2×D4—182no—
429C2×D4—181no—
430C2×D4—181no—
431C2×D4—361no—
432C2×D4—364no—
433C2×D4C22×D911yescharacteristic
434C2×D4C22×D911yescharacteristic
435C22×C4—91no—
436C22×C4—181no—
437C22×C4—181no—
438C22×C4—181no—
439C22×C4C22×D911yescharacteristic
440SD16—22no—
441SD16—41no—
442SD16—361no—
443C2×C8—91no—
444C2×C8—91no—
445C2×C8—361no—
446C2×C8—361no—
447C2×C8C22×D911yescharacteristic
448C2×C8C22×D911yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—91no—
451C4⋊C4—91no—
452C4⋊C4—91no—
453C4⋊C4—91no—
454C4⋊C4—181no—
455C4⋊C4—181no—
456C4⋊C4—181no—
457C4⋊C4—181no—
458C4⋊C4—181no—
459C4⋊C4C22×D911yescharacteristic
460C22⋊C4—21no—
461C22⋊C4—91no—
462C22⋊C4—91no—
463C22⋊C4—181no—
464C22⋊C4—181no—
465C22⋊C4—181no—
466C22⋊C4—181no—
467C42—91no—
468C42—181no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—22no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—84no—
478C22×C3D3×SD1612yes—
479C22×C3C2×D3×D411yescharacteristic
480C2×D3—32no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—244no—
487C3×C4—21no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4C2×#23711yescharacteristic
491C3×C4#100411yescharacteristic
492Dic3—32no—
493Dic3—61no—
494Dic3—61no—
495Dic3—121no—
496Dic3—121no—
497Dic3—121no—
498C9#329511yescharacteristic, complemented, Sylow
499C23—21no—
500C23—21no—
501C23—42no—
502C23—42no—
503C23—81no—
504C23—91no—
505C23—181no—
506C23—362no—
507C23—362no—
508C23—721no—
509C23D4×D911yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—721no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—42no—
517D4—94no—
518D4—181no—
519D4—181no—
520D4—362no—
521D4—362no—
522D4—362no—
523D4—362no—
524D4—721no—
525D4#369912yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—91no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—181no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—362no—
539C2×C4—721no—
540C2×C4—721no—
541C2×C4D4×D911yescharacteristic
542C2×C4C23×D911yescharacteristic
543C8—21no—
544C8—21no—
545C8—181no—
546C8—181no—
547C2×C3—22no—
548C2×C3—42no—
549C2×C3—81no—
550C2×C3#559111yescharacteristic
551C2×C3C2×D3×SD1611yescharacteristic
552C2×C3#577411yescharacteristic
553D3—62no—
554D3—241no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—84no—
564C22—92no—
565C22—181no—
566C22—181no—
567C22—361no—
568C22—361no—
569C22—361no—
570C22—724no—
571C22SD16×D912yes—
572C22C2×D4×D911yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—92no—
577C4—181no—
578C4—181no—
579C4—361no—
580C4—361no—
581C4—361no—
582C4C2×#369911yescharacteristic
583C4#6371011yescharacteristic
584C3#8015811yescharacteristic
585C2—22no—
586C2—42no—
587C2—81no—
588C2—182no—
589C2—721no—
590C2#1058969311yescharacteristic
591C2C2×SD16×D911yescharacteristic
592C2#1058987611yescharacteristic
subgroupG/H
centerC22C2×D4×D9
commutator, cosocleC2×C4×C9C24
FrattiniC2×C3×C4C23×D3
FittingC9×#450C2
socleC22×C3C2×D3×D4
prime radicalD4×#611C2
hypercenter#450D9
hypocenterC9#3295
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #10589737C2semidirect
3. C9×#450C2semidirect
4. C2×#63392C2semidirect
5. #10588851C2semidirect
6. #10588849C2semidirect
7. #10588843C2semidirect
8. SD16×Dic9C2semidirect
9. #10588731C2semidirect
10. #10588457C2semidirect
11. D9×#102C2semidirect
12. #10588441C2semidirect
13. #10588440C2semidirect
14. #10588417C2semidirect
15. #611⋊C8C2semidirect
21. C2×C9×SD16C22semidirect
22. C9×C22⋊Q8C22semidirect
24. #63508C22semidirect
25. Q8×Dic9C22semidirect
26. C2×#3675C22semidirect
34. C2×C4⋊Dic9C22semidirect
35. C22×C9⋊C8C22semidirect
36. C2×C8×D9C22semidirect
38. C4⋊C4×D9C22semidirect
40. #63450C22semidirect
44. C9×#102C22semidirect
45. C9×C22⋊C8C22semidirect
46. #63395C22semidirect
49. #63392C22semidirect
50. #63392C22semidirect
51. #63380C22semidirect
52. #63380C22semidirect
53. (C2×D9)⋊C8C22semidirect
54. #63377C22semidirect
55. #63377C22semidirect
56. Dic9⋊C8C22semidirect
57. C8×Dic9C22semidirect
58. #63369C22semidirect
59. #63367C22semidirect
201. #611SD16semidirect
230. #450D9semidirect
322. C2×SD16C2×D9semidirect
331. C22⋊Q8C2×D9semidirect
356. #102C2×D9semidirect
358. C22⋊C8C2×D9semidirect
498. C9#3295semidirect

Maximal subgroups

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

subgroupindex
D4×#6112
#105897372
C9×#4502
C2×#633922
#105888512
#105888492
#105888432
SD16×Dic92
#105887312
#105884572
D9×#1022
#105884412
#105884402
#105884172
#611⋊C82
#801583