Finite Group Databasebeta

← #29836494

Subgroups — #29836494

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C211yesmaximal, characteristic, complemented
2C22×#14519720C211yesmaximal, characteristic, complemented
3Q16×C22⋊D61C211yesmaximal, characteristic, complemented
4#21457501C211yesmaximal, characteristic
5#21448087C211yesmaximal, characteristic, complemented
6#21447935C211yesmaximal, characteristic, complemented
7#21447908C211yesmaximal, characteristic, complemented
8#21447883C211yesmaximal, characteristic, complemented
9#21447863C211yesmaximal, characteristic, complemented
10#21447758C211yesmaximal, characteristic, complemented
11#21447750C211yesmaximal, characteristic, complemented
12#21447737C211yesmaximal, characteristic, complemented
13#21447686C211yesmaximal, characteristic, complemented
14#21447641C211yesmaximal, characteristic, complemented
15C61×#2188C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic, complemented
18C22×D488C2211yescharacteristic
19Q8×C22⋊D61—21no—
20#16840403—21no—
21#16840204—24no—
22C2×#14519723C2211yescharacteristic, complemented
23C2×#14519722C2211yescharacteristic
24C2×#14519722C2211yescharacteristic
25C2×#14519720—24no—
26C2×#14519720C2211yescharacteristic
27C2×#14519720C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×#14519720C2211yescharacteristic
30C2×#14519716C2211yescharacteristic, complemented
31C22×#11758185C2211yescharacteristic
32#16840147C2211yescharacteristic
33#16840140C2211yescharacteristic, complemented
34#16840003C2211yescharacteristic
35#16840000C2211yescharacteristic, complemented
36#16839999C2211yescharacteristic
37#16839980—21no—
38#16839970—21no—
39#16839918—21no—
40#16839916—21no—
41#16839912—21no—
42#16839910—21no—
43Q16×Dic61C2211yescharacteristic, complemented
44#16839844C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839565C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic
51#16839564C2211yescharacteristic
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C22×Q16×C61C2211yescharacteristic
55C2×Q32×C61C2211yescharacteristic, complemented
56C2×Q32×C61C2211yescharacteristic, complemented
57C61×#467C2211yescharacteristic
58C61×#358C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×C22⋊C16C2211yescharacteristic
61C2×#11758226—21no—
62C2×Q8×D61—21no—
63C22×D244—21no—
64#14519800—22no—
65#14519800—41no—
66C2×Dic244C2311yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—21no—
69Q16×D61—22no—
70Q16×D61—41no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×D488—22no—
74C2×D488—21no—
75C2×D488—41no—
76C2×D488C2311yescharacteristic
77C2×D488C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519775—21no—
80#14519775—41no—
81#14519766—21no—
82#14519762—41no—
83#14519756—21no—
84#14519756—41no—
85#14519753—21no—
86#14519753—41no—
87#14519752—21no—
88#14519752—41no—
89#14519747—41no—
90Q8×Dic61—21no—
91#14519739—21no—
92C4×C22⋊D61—41no—
93C4×C22⋊D61D411yescharacteristic
94C22⋊C4×D61—41no—
95#14519723—22no—
96#14519722—22no—
97#14519722—41no—
98#14519720—22no—
99#14519720—22no—
100#14519720—22no—
101#14519720D414yescomplemented
102#14519716—22no—
103C2×#11758185—22no—
104C2×#11758185C2311yescharacteristic
105C2×#11758185C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519702—21no—
109#14519701—21no—
110#14519701—21no—
111#14519697—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×Q16×C61—22no—
118C2×Q16×C61—21no—
119C2×Q16×C61—41no—
120C2×Q16×C61C2311yescharacteristic
121C2×Q16×C61C2311yescharacteristic
122C2×Q16×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125Q32×C61—22no—
126Q32×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#103×C61—21no—
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—42no—
136C2×Q8×C61—41no—
137C2×Q8×C61—81no—
138C22×C4×C61—41no—
139C22×C4×C61C2×D411yescharacteristic
140C2×C22⋊D61D811yescharacteristic
141#11758226—42no—
142#11758226—81no—
143Q8×D61—42no—
144Q8×D61—81no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—42no—
148C2×D244—41no—
149C2×D244—41no—
150C2×D244—41no—
151C2×D244—81no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—41no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—21no—
163Q16×C61—42no—
164Q16×C61C2×D412yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C22⋊Dic61—41no—
173C22⋊Dic61D811yescharacteristic
174#11758202—42no—
175#11758202—81no—
176#11758201—42no—
177#11758201—81no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198D811yescharacteristic
184C4⋊Dic61—21no—
185C4⋊Dic61—41no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189#11758196—41no—
190#11758196D811yescharacteristic
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—21no—
196Dic244—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—42no—
201D488C2×D412yes—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61C2×D811yescharacteristic
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—161no—
214C22×D61C2×D811yescharacteristic
215Q8×C61—22no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61C2×D811yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C22⋊D61—22no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61C2×D811yescharacteristic
235C2×Dic61C2×D811yescharacteristic
236C2×Dic61C2×D811yescharacteristic
237D244—22no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—82no—
242D244—82no—
243D244—161no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—82no—
247C4×D61—82no—
248C4×D61—161no—
249Dic122—41no—
250Dic122—41no—
251Dic122—82no—
252C8×C61—21no—
253C8×C61C22×D411yescharacteristic
254C8×C61H(5,𝔽2)11yescharacteristic
255#92418—21no—
256#92418—21no—
257#33832—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61#51012yescomplemented
261C22×C61C22×D811yescharacteristic
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—164no—
272C4×C61—21no—
273C4×C61—42no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61D4211yescharacteristic
278C4×C61#57611yescharacteristic
279Dic61—21no—
280Dic61—21no—
281Dic61—21no—
282Dic61—81no—
283Dic61—81no—
284C2×#510—611no—
285C22×SD32—611no—
286D4×Q16—611no—
287#3280—611no—
288#2251—611no—
289#2220—611no—
290#2216—611no—
291#2211—611no—
292#2208—611no—
293#2189—611no—
294#2188D6111yescharacteristic, complemented
295#2187—611no—
296#2185—611no—
297#2175—611no—
298#2168—611no—
299C2×C61—22no—
300C2×C61D4×D811yescharacteristic
301C2×C61C2×#51011yescharacteristic
302C2×C61#341611yescharacteristic
303D61—41no—
304D61—82no—
305D61—161no—
306C2×#135—611no—
307C22×Q16—611no—
308C22×Q16C2×D6111yescharacteristic
309C22×D8—611no—
310D4×Q8—1221no—
311#546—1221no—
312#510—1224no—
313C2×Q32—611no—
314C2×Q32C2×D6111yescharacteristic, complemented
315C2×Q32C2×D6111yescharacteristic, complemented
316C2×SD32—611no—
317C2×SD32—611no—
318C2×SD32—611no—
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—1224no—
323C2×M32—611no—
324C22×C16—611no—
325#495—611no—
326#494—611no—
327#467—611no—
328#467—611no—
329#467C2×D6111yescharacteristic
330#466—611no—
331#463—1221no—
332#462—1221no—
333#451—1221no—
334#451—1221no—
335#450—1221no—
336#449—1221no—
337C4×Q16—611no—
338C4×D8—611no—
339C8×D4—611no—
340#366—611no—
341C4⋊C16—611no—
342#358—611no—
343#358—611no—
344#358C2×D6111yescharacteristic, complemented
345#358C2×D6111yescharacteristic, complemented
346#357—611no—
347#357—611no—
348C22⋊C16—611no—
349C22⋊C16C2×D6111yescharacteristic
350#346—611no—
351C61#3383211yescharacteristic, complemented, Sylow
352C2×#41—1221no—
353C22×Q8—21no—
354C22×Q8—1221no—
355C22×D4—1221no—
356#135—1222no—
357#135—2441no—
358C2×Q16—22no—
359C2×Q16—21no—
360C2×Q16—41no—
361C2×Q16—611no—
362C2×Q16—1221no—
363C2×Q16—1221no—
364C2×Q16—1222no—
365C2×Q16—2441no—
366C2×Q16C22×D6111yescharacteristic
367C2×Q16C22×D6111yescharacteristic
368C2×Q16C22×D6111yescharacteristic
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—611no—
372C2×D8—611no—
373C2×D8—1222no—
374C2×D8—1221no—
375C2×D8—2441no—
376C22×C8—611no—
377C22×C8C22×D6111yescharacteristic
378C4⋊Q8—1221no—
379C4⋊Q8—2441no—
380#124—1221no—
381#123—2441no—
382C22⋊Q8—21no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—2441no—
386C22⋊Q8—2441no—
387#121—1221no—
388#121—2441no—
389#120—2441no—
390C4×Q8—1221no—
391C4×D4—611no—
392C4×D4—1221no—
393C4×D4—2441no—
394C2×C22⋊C4—2441no—
395Q32—22no—
396Q32—41no—
397Q32—1222no—
398SD32—614no—
399SD32—1222no—
400SD32—1222no—
401SD32—1222no—
402SD32—1222no—
403SD32—2441no—
404M32—1222no—
405C2×C16—611no—
406C2×C16—611no—
407C2×C16—1222no—
408C2×C16C22×D6111yescharacteristic
409C2×C16C22×D6111yescharacteristic
410#107—611no—
411#107C22×D6111yescharacteristic
412C4⋊C8—611no—
413#103—21no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8—611no—
420C22⋊C8—611no—
421C4×C8—611no—
422C24—2441no—
423#41—4881no—
424#41—2442no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—41no—
430C2×Q8—81no—
431C2×Q8—1221no—
432C2×Q8—4881no—
433C2×Q8—2441no—
434C2×Q8—2442no—
435C2×D4—611no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—4881no—
439C2×D4—2442no—
440C2×D4—2441no—
441C2×D4—2441no—
442C2×D4—2441no—
443C22×C4—41no—
444C22×C4—611no—
445C22×C4—2441no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4D4×D6111yescharacteristic
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—21no—
454Q16—42no—
455Q16—1221no—
456Q16—1221no—
457Q16—4881no—
458Q16—2442no—
459Q16D4×D6112yes—
460SD16—4881no—
461SD16—2442no—
462D8—612no—
463D8—1221no—
464D8—1221no—
465D8—1221no—
466D8—2442no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8D4×D6111yescharacteristic
474C2×C8C23×D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—611no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C22⋊C4—41no—
485C22⋊C4—611no—
486C22⋊C4—611no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C42—611no—
494C42—2441no—
495C16—21no—
496C16—21no—
497C16—612no—
498C16—1221no—
499C23—611no—
500C23—9761no—
501C23—4882no—
502C23—4882no—
503C23—2441no—
504C23—2441no—
505C23D8×D6111yescharacteristic
506Q8—22no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—4882no—
514Q8—2441no—
515Q8—2441no—
516D4—1222no—
517D4—1222no—
518D4—9761no—
519D4—4882no—
520D4—4882no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—82no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—9761no—
537C2×C4—4882no—
538C2×C4—4882no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4D8×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C8—21no—
544C8—1221no—
545C8—1221no—
546C8C2×D4×D6111yescharacteristic
547C8#1451982211yescharacteristic
548C22—21no—
549C22—21no—
550C22—1221no—
551C22—1221no—
552C22—1221no—
553C22—9764no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—2442no—
560C22#1684020412yescomplemented
561C22C2×D8×D6111yescharacteristic
562C4—21no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—1221no—
568C4—1221no—
569C4—1221no—
570C4—4881no—
571C4—4881no—
572C4#1684039411yescharacteristic
573C4#1684056611yescharacteristic
574C2—22no—
575C2—9761no—
576C2—4882no—
577C2—2441no—
578C2#2145748911yescharacteristic
579C2C2×#1684020411yescharacteristic
580C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2188C2
socleC22×C61C22×D8
prime radical#21457501C2
hypercenter#2188D61
hypocenterC61#33832
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2188

Complements

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

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
2. C22×#14519720C2semidirect
3. Q16×C22⋊D61C2semidirect
5. #21448087C2semidirect
6. #21447935C2semidirect
7. #21447908C2semidirect
8. #21447883C2semidirect
9. #21447863C2semidirect
10. #21447758C2semidirect
11. #21447750C2semidirect
12. #21447737C2semidirect
13. #21447686C2semidirect
14. #21447641C2semidirect
15. C61×#2188C2semidirect
17. C2×Q16×D61C22semidirect
22. C2×#14519723C22semidirect
30. C2×#14519716C22semidirect
33. #16840140C22semidirect
35. #16840000C22semidirect
43. Q16×Dic61C22semidirect
46. #16839587C22semidirect
47. #16839580C22semidirect
48. #16839566C22semidirect
49. #16839565C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
55. C2×Q32×C61C22semidirect
56. C2×Q32×C61C22semidirect
58. C61×#358C22semidirect
59. C61×#358C22semidirect
101. #14519720D4semidirect
260. C22×C61#510semidirect
294. #2188D61semidirect
314. C2×Q32C2×D61semidirect
315. C2×Q32C2×D61semidirect
344. #358C2×D61semidirect
345. #358C2×D61semidirect
351. C61#33832semidirect
560. C22#16840204semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
C22×#145197202
Q16×C22⋊D612
#214575012
#214480872
#214479352
#214479082
#214478832
#214478632
#214477582
#214477502
#214477372
#214476862
#214476412
C61×#21882
#3383261