Finite Group Databasebeta

← #29836698

Subgroups — #29836698

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
2#21457560C211yesmaximal, characteristic, complemented
3#21457504C211yesmaximal, characteristic, complemented
4#21448079C211yesmaximal, characteristic, complemented
5#21447915C211yesmaximal, characteristic, complemented
6#21447912C211yesmaximal, characteristic, complemented
7#21447879C211yesmaximal, characteristic, complemented
8#21447876C211yesmaximal, characteristic, complemented
9#21447756C211yesmaximal, characteristic, complemented
10#21447753C211yesmaximal, characteristic, complemented
11#21447741C211yesmaximal, characteristic, complemented
12#21447731C211yesmaximal, characteristic, complemented
13SD32×Dic61C211yesmaximal, characteristic, complemented
14C16×C22⋊D61C211yesmaximal, characteristic, complemented
15C22×SD32×C61C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic, complemented
17C2×#14519796C2211yescharacteristic
18#16840423—21no—
19#16840406—21no—
20#16840193—24no—
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519723C2211yescharacteristic, complemented
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519718C2211yescharacteristic
25C2×#14519718C2211yescharacteristic
26C2×C16×D61C2211yescharacteristic, complemented
27#16840147C2211yescharacteristic, complemented
28#16840145C2211yescharacteristic
29#16840003C2211yescharacteristic
30#16840000C2211yescharacteristic, complemented
31#16839998C2211yescharacteristic
32#16839995C2211yescharacteristic
33#16839986—21no—
34#16839975—21no—
35#16839923—21no—
36#16839913—21no—
37#16839904—21no—
38#16839901—21no—
39Q16×Dic61C2211yescharacteristic, complemented
40D8×Dic61C2211yescharacteristic
41C8×C22⋊D61C2211yescharacteristic
42#16839590C2211yescharacteristic, complemented
43#16839580C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839566C2211yescharacteristic, complemented
46#16839566C2211yescharacteristic, complemented
47#16839564C2211yescharacteristic, complemented
48#16839563C2211yescharacteristic
49#16839563C2211yescharacteristic
50C22⋊#11758185C2211yescharacteristic
51#16839541C2211yescharacteristic, complemented
52C16×Dic61C2211yescharacteristic, complemented
53C22×Q16×C61C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic
55C2×SD32×C61—24no—
56C2×SD32×C61C2211yescharacteristic
57C2×SD32×C61C2211yescharacteristic
58C2×SD32×C61C2211yescharacteristic
59C2×SD32×C61C2211yescharacteristic
60C22×C16×C61C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758224—21no—
63#14519800—22no—
64#14519800—41no—
65#14519796—22no—
66#14519796—41no—
67C2×Dic244C2311yescharacteristic
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×D488C2311yescharacteristic
73C2×C8×D61C2311yescharacteristic
74#14519766—21no—
75#14519766—41no—
76#14519764—21no—
77#14519758—41no—
78#14519756—21no—
79#14519753—21no—
80#14519753—41no—
81#14519751—21no—
82#14519751—41no—
83#14519749—41no—
84#14519748—21no—
85#14519746—41no—
86Q8×Dic61—21no—
87D4×Dic61—21no—
88C4×C22⋊D61—41no—
89C4×C22⋊D61D411yescharacteristic
90C2×C22⋊Dic61—41no—
91#14519723—22no—
92#14519723—41no—
93#14519722—22no—
94#14519718—22no—
95#14519718—41no—
96C2×#11758185C2311yescharacteristic
97C2×#11758185C2311yescharacteristic
98C16×D61—22no—
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519702—21no—
103#14519701—21no—
104#14519701—21no—
105#14519699—21no—
106#14519698—21no—
107#14519698—21no—
108C22⋊#92418D411yescharacteristic
109#14519689D411yescharacteristic
110C8×Dic61C2311yescharacteristic
111C22×Q8×C61—21no—
112C22×D4×C61—21no—
113C2×Q16×C61—22no—
114C2×Q16×C61—21no—
115C2×Q16×C61—41no—
116C2×Q16×C61C2311yescharacteristic
117C2×Q16×C61C2311yescharacteristic
118C2×D8×C61—22no—
119C2×D8×C61—21no—
120C2×D8×C61—41no—
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124SD32×C61—22no—
125SD32×C61—22no—
126SD32×C61—22no—
127SD32×C61D414yescomplemented
128C2×C16×C61—22no—
129C2×C16×C61C2311yescharacteristic
130C2×C16×C61C2311yescharacteristic
131C24×C61—41no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—42no—
135C2×Q8×C61—41no—
136C2×Q8×C61—81no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—42no—
140C2×D4×C61—41no—
141C2×D4×C61—81no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61D811yescharacteristic
148C22×Dic61—41no—
149C22×Dic61—41no—
150#11758226—42no—
151#11758226—81no—
152#11758224—42no—
153#11758224—81no—
154C2×D244—21no—
155C2×D244—41no—
156C2×C4×D61—41no—
157C2×C4×D61C2×D411yescharacteristic
158C2×Dic122—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—42no—
163Q16×C61C2×D412yes—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—42no—
168D8×C61C2×D412yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2411yescharacteristic
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61D811yescharacteristic
179#11758201—42no—
180#11758201—81no—
181#11758200—42no—
182#11758200—81no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198D811yescharacteristic
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—41no—
191#11758196—41no—
192#11758196—41no—
193#11758196D811yescharacteristic
194C4×Dic61—41no—
195C4×Dic61C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197C2×#92418C2×D411yescharacteristic
198Dic244—21no—
199Dic244—21no—
200D488—21no—
201D488—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—
210C23×C61—82no—
211C23×C61—82no—
212C23×C61—161no—
213C23×C61C2×D811yescharacteristic
214C22×D61—41no—
215C22×D61C2×D811yescharacteristic
216Q8×C61—22no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—82no—
221D4×C61—22no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—82no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C22⋊D61—22no—
234C22⋊D61—82no—
235C22⋊D61—82no—
236C22⋊D61—161no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241C2×Dic61—161no—
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244C2×Dic61C2×D811yescharacteristic
245D244—41no—
246D244—41no—
247D244—82no—
248C4×D61—21no—
249C4×D61—21no—
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#33848—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—164no—
268C22×C61#50812yescomplemented
269C22×C61C22×D811yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—21no—
277C4×C61—42no—
278C4×C61—81no—
279C4×C61D4211yescharacteristic
280C4×C61#57611yescharacteristic
281Dic61—21no—
282Dic61—21no—
283Dic61—21no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287C2×#508—611no—
288C22×SD32D6111yescharacteristic, complemented
289#3285—611no—
290#3280—611no—
291#2249—611no—
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—22no—
303C2×C61—82no—
304C2×C61—161no—
305C2×C61D4×D811yescharacteristic
306C2×C61C2×#50811yescharacteristic
307C2×C61#341611yescharacteristic
308D61—41no—
309D61—161no—
310C2×#135—611no—
311C2×#135—611no—
312C22×Q16C2×D6111yescharacteristic
313C22×D8C2×D6111yescharacteristic
314#548—1221no—
315#546—1221no—
316#508—1224no—
317C2×Q32—611no—
318C2×Q32—611no—
319C2×SD32—24no—
320C2×SD32—611no—
321C2×SD32C2×D6111yescharacteristic
322C2×SD32C2×D6111yescharacteristic
323C2×SD32C2×D6111yescharacteristic
324C2×SD32C2×D6111yescharacteristic
325C2×D16—611no—
326C2×D16—611no—
327C22×C16—611no—
328C22×C16C2×D6111yescharacteristic
329#495—611no—
330#495—611no—
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×C16—611no—
355C61#3384811yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×#41—1221no—
358C22×Q8—21no—
359C22×D4—21no—
360#135—1222no—
361#135—1222no—
362#135—2441no—
363#135—2441no—
364C2×Q16—22no—
365C2×Q16—21no—
366C2×Q16—41no—
367C2×Q16—611no—
368C2×Q16C22×D6111yescharacteristic
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—22no—
375C2×D8—21no—
376C2×D8—41no—
377C2×D8—611no—
378C2×D8C22×D6111yescharacteristic
379C2×D8C22×D6111yescharacteristic
380C22×C8—611no—
381C22×C8C22×D6111yescharacteristic
382#124—1221no—
383#124—1221no—
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—1222no—
400Q32—2441no—
401SD32—22no—
402SD32—22no—
403SD32—22no—
404SD32—1222no—
405SD32C22⋊D6114yescomplemented
406D16—1222no—
407D16—2441no—
408C2×C16—22no—
409C2×C16—611no—
410C2×C16—611no—
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×C8—611no—
426C24—41no—
427#41—4881no—
428#41—4881no—
429#41—2442no—
430#41—2442no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—42no—
434C2×Q8—41no—
435C2×Q8—81no—
436C2×Q8—1221no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—41no—
441C2×D4—81no—
442C2×D4—611no—
443C2×D4—1221no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4—2441no—
448C22×C4—41no—
449C22×C4—611no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4D4×D6111yescharacteristic
454Q16—21no—
455Q16—21no—
456Q16—21no—
457Q16—42no—
458Q16—1221no—
459Q16—1221no—
460Q16C2×C22⋊D6112yes—
461SD16—4881no—
462SD16—4881no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—21no—
468D8—42no—
469D8—1221no—
470D8—1221no—
471D8C2×C22⋊D6112yes—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—611no—
475C2×C8—611no—
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—611no—
487C22⋊C4—611no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C42—611no—
498C42—2441no—
499C16—21no—
500C16—1221no—
501C16—1221no—
502C16C2×C22⋊D6112yes—
503C23—41no—
504C23—41no—
505C23—82no—
506C23—82no—
507C23—161no—
508C23—611no—
509C23—2441no—
510C23D8×D6111yescharacteristic
511Q8—22no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—82no—
516Q8—2441no—
517Q8—2441no—
518D4—22no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—82no—
523D4—1222no—
524D4—9761no—
525D4—4882no—
526D4—4882no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—9761no—
541C2×C4—4882no—
542C2×C4—4882no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4D8×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C8—21no—
549C8—1221no—
550C8—1221no—
551C8C22×C22⋊D6111yescharacteristic
552C8#1451982211yescharacteristic
553C22—21no—
554C22—21no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—164no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22#1684019312yescomplemented
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—42no—
572C4—81no—
573C4—1221no—
574C4—1221no—
575C4—1221no—
576C4—4881no—
577C4—4881no—
578C4—4881no—
579C4D4×C22⋊D6111yescharacteristic
580C4#1684056311yescharacteristic
581C2—22no—
582C2—82no—
583C2—161no—
584C2—9761no—
585C2—2441no—
586C2D8×C22⋊D6111yescharacteristic
587C2C2×#1684019311yescharacteristic
588C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC22×SD32×C61C2
socleC22×C61C22×D8
hypercenterC22×SD32D61
hypocenterC61#33848
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×SD32

Complements

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

NH ≅ G/Ntype
1. C2×#16840193C2semidirect
2. #21457560C2semidirect
3. #21457504C2semidirect
4. #21448079C2semidirect
5. #21447915C2semidirect
6. #21447912C2semidirect
7. #21447879C2semidirect
8. #21447876C2semidirect
9. #21447756C2semidirect
10. #21447753C2semidirect
11. #21447741C2semidirect
12. #21447731C2semidirect
13. SD32×Dic61C2semidirect
14. C16×C22⋊D61C2semidirect
15. C22×SD32×C61C2semidirect
16. C2×#14519800C22semidirect
21. C2×#14519723C22semidirect
22. C2×#14519723C22semidirect
23. C2×#14519722C22semidirect
26. C2×C16×D61C22semidirect
27. #16840147C22semidirect
30. #16840000C22semidirect
39. Q16×Dic61C22semidirect
42. #16839590C22semidirect
43. #16839580C22semidirect
44. #16839567C22semidirect
45. #16839566C22semidirect
46. #16839566C22semidirect
47. #16839564C22semidirect
51. #16839541C22semidirect
52. C16×Dic61C22semidirect
127. SD32×C61D4semidirect
268. C22×C61#508semidirect
288. C22×SD32D61semidirect
355. C61#33848semidirect
405. SD32C22⋊D61semidirect
568. C22#16840193semidirect

Maximal subgroups

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

subgroupindex
C2×#168401932
#214575602
#214575042
#214480792
#214479152
#214479122
#214478792
#214478762
#214477562
#214477532
#214477412
#214477312
SD32×Dic612
C16×C22⋊D612
C22×SD32×C612
#3384861