Finite Group Databasebeta

← #29836697

Subgroups — #29836697

Order 15616.

696 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×#16840193C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457559C211yesmaximal, characteristic, complemented
4#21457503C211yesmaximal, characteristic, complemented
5#21447912C211yesmaximal, characteristic, complemented
6#21447912C211yesmaximal, characteristic, complemented
7#21447878C211yesmaximal, characteristic, complemented
8#21447871C211yesmaximal, characteristic, complemented
9#21447758C211yesmaximal, characteristic, complemented
10#21447751C211yesmaximal, characteristic, complemented
11#21447739C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13C4×#14519722C211yesmaximal, characteristic, complemented
14C16×D244C211yesmaximal, characteristic, complemented
15C61×#2249C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519796C2211yescharacteristic, complemented
18C2×Q16×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840420—21no—
21#16840405—21no—
22#16840193—24no—
23C2×#14519723C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519720C2211yescharacteristic
27SD32×D61—24no—
28C2×#14519718C2211yescharacteristic
29C2×C16×D61C2211yescharacteristic
30C2×C16×D61C2211yescharacteristic, complemented
31#16840000C2211yescharacteristic
32#16840000C2211yescharacteristic, complemented
33#16839995C2211yescharacteristic
34#16839995C2211yescharacteristic
35#16839981—21no—
36#16839974—21no—
37#16839921—21no—
38#16839911—21no—
39#16839906—21no—
40#16839899—21no—
41C4×Dic244C2211yescharacteristic, complemented
42C4×D488C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#16839590C2211yescharacteristic, complemented
45C4⋊#11758185C2211yescharacteristic, complemented
46#16839567C2211yescharacteristic, complemented
47#16839566C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic
50#16839563C2211yescharacteristic
51#16839563C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic
53#16839541C2211yescharacteristic, complemented
54C2×Q32×C61C2211yescharacteristic, complemented
55C2×SD32×C61C2211yescharacteristic
56C2×SD32×C61C2211yescharacteristic, complemented
57C2×D16×C61C2211yescharacteristic
58C61×#495C2211yescharacteristic
59C61×#495C2211yescharacteristic, complemented
60C4×C16×C61C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×#11758224—21no—
63C2×Q8×D61—21no—
64C2×D4×D61—21no—
65#14519800—22no—
66#14519800—41no—
67#14519796—22no—
68#14519796—41no—
69C2×Dic244C2311yescharacteristic
70C2×#11758202—21no—
71Q16×D61—22no—
72Q16×D61—41no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×D488C2311yescharacteristic
76C2×#11758199—21no—
77D8×D61—22no—
78D8×D61—41no—
79C2×C8×D61C2311yescharacteristic
80C2×C8×D61C2311yescharacteristic
81#14519763—41no—
82#14519759—41no—
83#14519754—41no—
84#14519753—21no—
85#14519753—21no—
86#14519750—41no—
87#14519750—41no—
88#14519748—21no—
89#14519748—21no—
90#14519747—41no—
91C4×Dic122—21no—
92C4×D244—21no—
93C4×D244D411yescharacteristic
94#14519737—41no—
95C22⋊C4×D61—41no—
96#14519723—22no—
97#14519722D414yescomplemented
98#14519721—22no—
99#14519720—22no—
100#14519718—22no—
101C2×#11758185C2311yescharacteristic
102C2×#11758185C2311yescharacteristic
103C16×D61—22no—
104C16×D61—22no—
105#14519711C2311yescharacteristic
106#14519711C2311yescharacteristic
107C4⋊#92418D411yescharacteristic
108#14519702—21no—
109#14519701—21no—
110#14519701—21no—
111#14519699—21no—
112#14519698—21no—
113#14519698—21no—
114#14519689D411yescharacteristic
115#14519689D411yescharacteristic
116C2×Q16×C61C2311yescharacteristic
117C2×Q16×C61C2311yescharacteristic
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#124×C61—21no—
123#124×C61—21no—
124Q32×C61—41no—
125SD32×C61—22no—
126SD32×C61—22no—
127D16×C61—41no—
128C2×C16×C61C2311yescharacteristic
129C2×C16×C61C2311yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C22×Dic61—41no—
141#11758226—42no—
142#11758226—81no—
143Q8×D61—42no—
144Q8×D61—81no—
145#11758224—42no—
146#11758224—81no—
147D4×D61—42no—
148D4×D61—81no—
149C2×D244—21no—
150C2×D244—41no—
151C2×D244D811yescharacteristic
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—41no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—21no—
163SD16×C61—81no—
164SD16×C61—81no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2411yescharacteristic
171C22⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C42×C61C2×D411yescharacteristic
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178#11758202—42no—
179#11758201—42no—
180#11758200—42no—
181#11758199—42no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198D811yescharacteristic
186#11758198D811yescharacteristic
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61D811yescharacteristic
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193C4×Dic61—41no—
194C2×#92418C2×D411yescharacteristic
195C2×#92418C2×D411yescharacteristic
196Dic244—21no—
197Dic244C2×D412yes—
198D488—21no—
199D488C2×D412yes—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61—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—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C2×D811yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C22⋊D61—82no—
230C22⋊D61—82no—
231C22⋊D61—82no—
232C22⋊D61—161no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—82no—
237C2×Dic61C2×D811yescharacteristic
238C2×Dic61C2×D811yescharacteristic
239D244—22no—
240D244—22no—
241D244—41no—
242D244—82no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—82no—
248C4×D61—82no—
249C4×D61—161no—
250Dic122—22no—
251Dic122—41no—
252Dic122—82no—
253C8×C61—21no—
254C8×C61C22×D411yescharacteristic
255C8×C61H(5,𝔽2)11yescharacteristic
256#92418—21no—
257#92418—21no—
258#33848—611nomaximal, Sylow
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61C22×D811yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—42no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—164no—
275C2×D61#50812yescomplemented
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61D4211yescharacteristic
281C4×C61#57611yescharacteristic
282Dic61—21no—
283Dic61—21no—
284Dic61—42no—
285Dic61—81no—
286Dic61—81no—
287C2×#508—611no—
288C22×SD32—611no—
289#3285—611no—
290#3280—611no—
291#2249D6111yescharacteristic, complemented
292#2217—611no—
293#2217—611no—
294#2211—611no—
295#2210—611no—
296#2189—611no—
297#2188—611no—
298#2186—611no—
299#2184—611no—
300C4×SD32—611no—
301D4×C16—611no—
302C2×C61—161no—
303C2×C61—161no—
304C2×C61D4×D811yescharacteristic
305C2×C61C2×#50811yescharacteristic
306C2×C61#341611yescharacteristic
307D61—22no—
308D61—41no—
309D61—82no—
310C2×#135—611no—
311C2×#135—611no—
312C22×Q16—611no—
313C22×D8—611no—
314#548—1221no—
315#546—1221no—
316#508—1224no—
317C2×Q32—611no—
318C2×Q32C2×D6111yescharacteristic, complemented
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—1224no—
323C2×SD32C2×D6111yescharacteristic
324C2×SD32C2×D6111yescharacteristic, complemented
325C2×D16—611no—
326C2×D16C2×D6111yescharacteristic
327C22×C16—611no—
328C22×C16—611no—
329#495C2×D6111yescharacteristic
330#495C2×D6111yescharacteristic, complemented
331#467—611no—
332#467—611no—
333#466—611no—
334#466—611no—
335#464—1221no—
336#463—1221no—
337#452—1221no—
338#450—1221no—
339#449—1221no—
340#448—1221no—
341C4×Q16—611no—
342C4×D8—611no—
343C8×D4—611no—
344#367—611no—
345C4⋊C16—611no—
346#358—611no—
347#358—611no—
348#358—611no—
349#357—611no—
350#357—611no—
351#357—611no—
352C22⋊C16—611no—
353C22⋊C16—611no—
354C4×C16C2×D6111yescharacteristic, complemented
355C61#3384811yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×#41—1221no—
358C22×Q8—1221no—
359C22×D4—1221no—
360#135—1222no—
361#135—1222no—
362#135—2441no—
363#135—2441no—
364C2×Q16—611no—
365C2×Q16—1221no—
366C2×Q16—1222no—
367C2×Q16—2441no—
368C2×Q16C22×D6111yescharacteristic
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—611no—
375C2×D8—1221no—
376C2×D8—1222no—
377C2×D8—2441no—
378C2×D8C22×D6111yescharacteristic
379C2×D8C22×D6111yescharacteristic
380C22×C8—611no—
381C22×C8—611no—
382#124—21no—
383#124—21no—
384#124—2441no—
385#123—2441no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—1221no—
390#121—1221no—
391#121—2441no—
392#121—2441no—
393#120—2441no—
394C4×Q8—1221no—
395C4×D4—611no—
396C4×D4—1221no—
397C4×D4—2441no—
398C2×C22⋊C4—2441no—
399Q32—41no—
400Q32—1222no—
401SD32—22no—
402SD32—22no—
403SD32—614no—
404SD32—1222no—
405SD32—1222no—
406D16—41no—
407D16—1222no—
408C2×C16—611no—
409C2×C16—611no—
410C2×C16—1222no—
411C2×C16—1222no—
412C2×C16C22×D6111yescharacteristic
413C2×C16C22×D6111yescharacteristic
414#107—611no—
415#107—611no—
416C4⋊C8—611no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—1221no—
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—2442no—
430#41—2442no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—2441no—
436C2×Q8—2442no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—611no—
440C2×D4—1221no—
441C2×D4—4881no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4—2442no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4—2441no—
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—612no—
459Q16—1221no—
460Q16—2442no—
461SD16—81no—
462SD16—81no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—21no—
468D8—21no—
469D8—612no—
470D8—1221no—
471D8—2442no—
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—611no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—611no—
491C22⋊C4—611no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C42—2441no—
498C42D4×D6111yescharacteristic
499C16—21no—
500C16—1221no—
501C16—1221no—
502C16D4×D6112yes—
503C23—41no—
504C23—41no—
505C23—611no—
506C23—611no—
507C23—9761no—
508C23—4882no—
509C23—4882no—
510C23—2441no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—1222no—
516Q8—4882no—
517Q8—2441no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—1222no—
523D4—1222no—
524D4—9761no—
525D4—4882no—
526D4—4882no—
527D4—4882no—
528D4—4882no—
529D4—2441no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—9761no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—2441no—
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#1451982211yescharacteristic
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—612no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—9764no—
566C22—4881no—
567C22—4881no—
568C22—2442no—
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—21no—
572C4—81no—
573C4—81no—
574C4—1221no—
575C4—1221no—
576C4—4881no—
577C4—4881no—
578C4—2442no—
579C4#1684039711yescharacteristic
580C4#1684056311yescharacteristic
581C2—161no—
582C2—161no—
583C2—1222no—
584C2—4882no—
585C2—2441no—
586C2#2145749411yescharacteristic
587C2#508×D6111yescharacteristic
588C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2249C2
socleC22×C61C22×D8
hypercenter#2249D61
hypocenterC61#33848
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. C2×#16840193C2semidirect
2. C2×SD32×D61C2semidirect
3. #21457559C2semidirect
4. #21457503C2semidirect
5. #21447912C2semidirect
6. #21447912C2semidirect
7. #21447878C2semidirect
8. #21447871C2semidirect
9. #21447758C2semidirect
10. #21447751C2semidirect
11. #21447739C2semidirect
12. #21447729C2semidirect
13. C4×#14519722C2semidirect
14. C16×D244C2semidirect
15. C61×#2249C2semidirect
17. C2×#14519796C22semidirect
23. C2×#14519723C22semidirect
30. C2×C16×D61C22semidirect
32. #16840000C22semidirect
41. C4×Dic244C22semidirect
44. #16839590C22semidirect
45. C4⋊#11758185C22semidirect
46. #16839567C22semidirect
47. #16839566C22semidirect
48. #16839566C22semidirect
51. #16839563C22semidirect
53. #16839541C22semidirect
54. C2×Q32×C61C22semidirect
56. C2×SD32×C61C22semidirect
59. C61×#495C22semidirect
60. C4×C16×C61C22semidirect
97. #14519722D4semidirect
275. C2×D61#508semidirect
291. #2249D61semidirect
318. C2×Q32C2×D61semidirect
324. C2×SD32C2×D61semidirect
330. #495C2×D61semidirect
354. C4×C16C2×D61semidirect
355. C61#33848semidirect

Maximal subgroups

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

subgroupindex
C2×#168401932
C2×SD32×D612
#214575592
#214575032
#214479122
#214478782
#214478712
#214477582
#214477512
#214477392
#214477292
C4×#145197222
C16×D2442
C61×#22492
#3384861