Finite Group Databasebeta

← #29836522

Subgroups — #29836522

Order 15616.

692 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic, complemented
2C2×#16840193C211yesmaximal, characteristic, complemented
3#21457539C211yesmaximal, characteristic, complemented
4#21457505C211yesmaximal, characteristic, complemented
5#21448088C211yesmaximal, characteristic, complemented
6#21447926C211yesmaximal, characteristic, complemented
7#21447918C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447762C211yesmaximal, characteristic, complemented
10#21447756C211yesmaximal, characteristic, complemented
11#21447746C211yesmaximal, characteristic, complemented
12#21447745C211yesmaximal, characteristic, complemented
13#21447693C211yesmaximal, characteristic, complemented
14#21447643C211yesmaximal, characteristic, complemented
15C61×#2208C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519800C2211yescharacteristic, complemented
18C2×#14519796C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840415—21no—
21#16840407—21no—
22#16840199—24no—
23#16840193—24no—
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519723C2211yescharacteristic, complemented
26C2×#14519722C2211yescharacteristic, complemented
27C2×#14519720C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×D976C2211yescharacteristic
30C2×#14519718C2211yescharacteristic
31C2×#14519716C2211yescharacteristic
32C2×C16×D61C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic
34#16840004C2211yescharacteristic
35#16840000C2211yescharacteristic
36#16840000C2211yescharacteristic, complemented
37#16839999C2211yescharacteristic
38#16839977—21no—
39#16839964—21no—
40#16839923—21no—
41#16839923—21no—
42#16839914—21no—
43#16839908—21no—
44C4×D488C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839564C2211yescharacteristic
49#16839563C2211yescharacteristic
50#16839562C2211yescharacteristic, complemented
51#16839541C2211yescharacteristic
52#16839541C2211yescharacteristic, complemented
53#16839538C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic
56C61×#494C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#358C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×#11758226—21no—
63C2×#11758224—21no—
64C2×D4×D61—21no—
65#14519800—22no—
66#14519800—22no—
67#14519800—41no—
68#14519800—41no—
69#14519796—22no—
70#14519796—41no—
71C2×Dic244C2311yescharacteristic
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×D488C2311yescharacteristic
77C2×D488C2311yescharacteristic
78C2×#11758199—21no—
79D8×D61—22no—
80D8×D61—41no—
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83#14519765—21no—
84#14519762—41no—
85#14519762—41no—
86#14519761—41no—
87#14519757—21no—
88#14519753—21no—
89#14519753—21no—
90#14519752—21no—
91#14519752—41no—
92#14519751—41no—
93#14519747—41no—
94C4×D244—21no—
95C4×D244D411yescharacteristic
96C4⋊C4×D61—41no—
97C2×#11758198—41no—
98#14519723—22no—
99#14519723—41no—
100#14519722—22no—
101#14519720—22no—
102#14519720—41no—
103D976—22no—
104#14519718D414yescomplemented
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519701—21no—
112#14519701—21no—
113#14519700—21no—
114#14519699—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61—21no—
119C2×Q16×C61C2311yescharacteristic
120C2×Q16×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122C4⋊Q8×C61—21no—
123C4×D4×C61—21no—
124SD32×C61—22no—
125SD32×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C22×C4×C61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C22×Dic61—41no—
139#11758226—42no—
140#11758226—42no—
141#11758226—81no—
142#11758226—81no—
143#11758224—42no—
144#11758224—81no—
145D4×D61—42no—
146D4×D61—81no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—41no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244D811yescharacteristic
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×Dic122—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—21no—
163Q16×C61—81no—
164D8×C61—21no—
165D8×C61C2×D412yes—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C42×C61C2×D411yescharacteristic
173#11758201—42no—
174#11758201—42no—
175#11758200—42no—
176#11758199—42no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198D811yescharacteristic
185#11758198D811yescharacteristic
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188C4⋊Dic61D811yescharacteristic
189#11758196—41no—
190#11758196—41no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488C2×D412yes—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—161no—
214C22×D61C2×D811yescharacteristic
215C22×D61C2×D811yescharacteristic
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220D4×C61—22no—
221D4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—161no—
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C2×D811yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C22⋊D61—82no—
230C22⋊D61—82no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—82no—
234C2×Dic61C2×D811yescharacteristic
235C2×Dic61C2×D811yescharacteristic
236D244—22no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—82no—
242D244—82no—
243D244—161no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—82no—
249C4×D61—82no—
250C4×D61—82no—
251Dic122—41no—
252Dic122—41no—
253C8×C61—21no—
254C8×C61C22×D411yescharacteristic
255C8×C61H(5,𝔽2)11yescharacteristic
256#92418—21no—
257#92418—21no—
258#33833—611nomaximal, Sylow
259C22×C61—42no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61C22×D811yescharacteristic
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—42no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—164no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61D4211yescharacteristic
282C4×C61#57611yescharacteristic
283Dic61—21no—
284Dic61—21no—
285Dic61—81no—
286Dic61—81no—
287C2×#509—611no—
288C2×#508—611no—
289#3283—611no—
290#3280—611no—
291#2251—611no—
292#2219—611no—
293#2217—611no—
294#2210—611no—
295#2208D6111yescharacteristic, complemented
296#2189—611no—
297#2189—611no—
298#2187—611no—
299#2186—611no—
300#2177—611no—
301#2168—611no—
302C2×C61—82no—
303C2×C61D4×D811yescharacteristic
304C2×C61#341411yescharacteristic
305C2×C61#341611yescharacteristic
306D61—41no—
307D61—41no—
308D61—82no—
309D61—161no—
310C2×#135—611no—
311C2×#135—611no—
312C2×#135—611no—
313C22×D8—611no—
314#547—1221no—
315#546—1221no—
316#509—1224no—
317#508—1224no—
318C2×Q32—611no—
319C2×Q32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32C2×D6111yescharacteristic
324C2×SD32C2×D6111yescharacteristic
325C2×D16—611no—
326C2×D16—611no—
327C2×M32—611no—
328C22×C16—611no—
329#495—611no—
330#494C2×D6111yescharacteristic, complemented
331#467—611no—
332#467—611no—
333#467—611no—
334#466—611no—
335#463—1221no—
336#461—1221no—
337#452—1221no—
338#452—1221no—
339#450—1221no—
340#449—1221no—
341C4×D8—611no—
342C4×D8C2×D6111yescharacteristic
343C8×D4—611no—
344#367—611no—
345C4⋊C16C2×D6111yescharacteristic, complemented
346#358—611no—
347#358C2×D6111yescharacteristic, complemented
348#358C2×D6111yescharacteristic, complemented
349#357—611no—
350#357—611no—
351#357—611no—
352C22⋊C16—611no—
353C22⋊C16—611no—
354#346—611no—
355C61#3383311yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1221no—
359C22×D4—1221no—
360#135—1222no—
361#135—1222no—
362#135—1222no—
363#135—2441no—
364#135—2441no—
365#135—2441no—
366C2×Q16—21no—
367C2×Q16—611no—
368C2×Q16C22×D6111yescharacteristic
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—611no—
375C2×D8—611no—
376C2×D8—1221no—
377C2×D8—1222no—
378C2×D8—2441no—
379C2×D8C22×D6111yescharacteristic
380C22×C8—611no—
381C22×C8—611no—
382C4⋊Q8—21no—
383#124—1221no—
384#123—2441no—
385#123—2441no—
386#123—2441no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390#121—1221no—
391#121—2441no—
392#121—2441no—
393#120—2441no—
394C4×D4—21no—
395C4×D4—611no—
396C4×D4—1221no—
397C2×C4⋊C4—2441no—
398C2×C22⋊C4—2441no—
399Q32—1222no—
400Q32—2441no—
401SD32—22no—
402SD32—22no—
403SD32—1222no—
404SD32—1222no—
405SD32—2441no—
406D16—614no—
407D16—1222no—
408M32—1222no—
409C2×C16—611no—
410C2×C16—611no—
411C2×C16—1222no—
412C2×C16C22×D6111yescharacteristic
413C2×C16C22×D6111yescharacteristic
414#107—611no—
415#107C22×D6111yescharacteristic
416C4⋊C8—611no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—1221no—
422#102—1221no—
423C22⋊C8—611no—
424C22⋊C8—611no—
425C4×C8C22×D6111yescharacteristic
426C24—2441no—
427#41—4881no—
428#41—4881no—
429#41—4881no—
430#41—2442no—
431#41—2442no—
432#41—2442no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1221no—
436C2×D4—21no—
437C2×D4—611no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—4881no—
441C2×D4—2441no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4—2441no—
445C2×D4—2442no—
446C2×D4—2441no—
447C22×C4—41no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4—2441no—
454Q16—21no—
455Q16—21no—
456Q16—21no—
457Q16—21no—
458Q16—81no—
459Q16—1221no—
460Q16—1221no—
461SD16—4881no—
462SD16—2442no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—612no—
467D8—1221no—
468D8—1221no—
469D8—1221no—
470D8—2442no—
471D8D4×D6112yes—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8D4×D6111yescharacteristic
479C2×C8C23×D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—611no—
484C4⋊C4—1221no—
485C4⋊C4—2441no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
488C22⋊C4—41no—
489C22⋊C4—611no—
490C22⋊C4—611no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C42D4×D6111yescharacteristic
499C16—21no—
500C16—21no—
501C16—612no—
502C16—1221no—
503C23—41no—
504C23—611no—
505C23—611no—
506C23—9761no—
507C23—4882no—
508C23—4882no—
509C23—2441no—
510C23—2441no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—2441no—
516Q8—2441no—
517D4—22no—
518D4—41no—
519D4—1222no—
520D4—9761no—
521D4—4882no—
522D4—4882no—
523D4—4882no—
524D4—4882no—
525D4—2441no—
526D4—2441no—
527D4—2441no—
528D4—2441no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—161no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—4882no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4D8×D6111yescharacteristic
546C2×C4D8×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C8—21no—
549C8—1221no—
550C8—1221no—
551C8C2×D4×D6111yescharacteristic
552C8#1451982311yescharacteristic
553C22—42no—
554C22—81no—
555C22—81no—
556C22—1221no—
557C22—1221no—
558C22—1221no—
559C22—1221no—
560C22—1221no—
561C22—1221no—
562C22—9764no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22—2442no—
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—21no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—1221no—
576C4—1221no—
577C4—4881no—
578C4—4881no—
579C4#1684039611yescharacteristic
580C4#1684056611yescharacteristic
581C2—82no—
582C2—9761no—
583C2—4882no—
584C2—2441no—
585C2—2441no—
586C2#2145749111yescharacteristic
587C2#2145895411yescharacteristic
588C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2208C2
socleC22×C61C22×D8
hypercenter#2208D61
hypocenterC61#33833
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×#16840193C2semidirect
3. #21457539C2semidirect
4. #21457505C2semidirect
5. #21448088C2semidirect
6. #21447926C2semidirect
7. #21447918C2semidirect
8. #21447875C2semidirect
9. #21447762C2semidirect
10. #21447756C2semidirect
11. #21447746C2semidirect
12. #21447745C2semidirect
13. #21447693C2semidirect
14. #21447643C2semidirect
15. C61×#2208C2semidirect
17. C2×#14519800C22semidirect
24. C2×#14519723C22semidirect
25. C2×#14519723C22semidirect
26. C2×#14519722C22semidirect
32. C2×C16×D61C22semidirect
36. #16840000C22semidirect
44. C4×D488C22semidirect
46. #16839589C22semidirect
47. #16839567C22semidirect
50. #16839562C22semidirect
52. #16839541C22semidirect
53. #16839538C22semidirect
56. C61×#494C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#358C22semidirect
104. #14519718D4semidirect
295. #2208D61semidirect
330. #494C2×D61semidirect
345. C4⋊C16C2×D61semidirect
347. #358C2×D61semidirect
348. #358C2×D61semidirect
355. C61#33833semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×#168401932
#214575392
#214575052
#214480882
#214479262
#214479182
#214478752
#214477622
#214477562
#214477462
#214477452
#214476932
#214476432
C61×#22082
#3383361