Finite Group Databasebeta

← #29836702

Subgroups — #29836702

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×#16840191C211yesmaximal, characteristic, complemented
2C22×#14519721C211yesmaximal, characteristic, complemented
3#21457557C211yesmaximal, characteristic, complemented
4#21457504C211yesmaximal, characteristic, complemented
5#21448077C211yesmaximal, characteristic, complemented
6#21447914C211yesmaximal, characteristic, complemented
7#21447913C211yesmaximal, characteristic, complemented
8#21447880C211yesmaximal, characteristic, complemented
9#21447872C211yesmaximal, characteristic, complemented
10#21447760C211yesmaximal, characteristic, complemented
11#21447755C211yesmaximal, characteristic, complemented
12#21447739C211yesmaximal, characteristic, complemented
13#21447674C211yesmaximal, characteristic, complemented
14#21447636C211yesmaximal, characteristic, complemented
15C61×#2184C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×#14519796C2211yescharacteristic, complemented
18C22×Dic244C2211yescharacteristic
19#16840418—21no—
20#16840406—21no—
21#16840191—24no—
22C2×Dic488C2211yescharacteristic, complemented
23C2×Dic488C2211yescharacteristic, complemented
24C2×#14519721—24no—
25C2×#14519721C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519721C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519721C2211yescharacteristic, complemented
30C22×#11758185C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic, complemented
32#16840145C2211yescharacteristic
33#16840145C2211yescharacteristic, complemented
34#16840004C2211yescharacteristic, complemented
35#16840003C2211yescharacteristic
36#16839995C2211yescharacteristic
37#16839983—21no—
38#16839975—21no—
39#16839925—21no—
40#16839913—21no—
41#16839904—21no—
42#16839900—21no—
43#16839856C2211yescharacteristic, complemented
44D8×Dic61C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839567C2211yescharacteristic, complemented
50#16839565C2211yescharacteristic, complemented
51#16839563C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53C16×Dic61C2211yescharacteristic, complemented
54C22×D8×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C2×D16×C61C2211yescharacteristic
57C61×#466C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C61×#357C2211yescharacteristic
60C61×C22⋊C16C2211yescharacteristic
61C2×#11758224—21no—
62C2×#11758224—21no—
63C22×Dic122—21no—
64#14519796—22no—
65#14519796—22no—
66#14519796—41no—
67#14519796—41no—
68C2×Dic244—22no—
69C2×Dic244—21no—
70C2×Dic244—41no—
71C2×Dic244C2311yescharacteristic
72C2×Dic244C2311yescharacteristic
73C2×Dic244C2311yescharacteristic
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×#11758200—21no—
78C2×C8×D61C2311yescharacteristic
79#14519764—21no—
80#14519764—21no—
81#14519763—41no—
82#14519758—41no—
83#14519757—21no—
84#14519756—21no—
85#14519754—41no—
86#14519749—41no—
87#14519749—41no—
88#14519748—21no—
89#14519746—41no—
90#14519742—21no—
91D4×Dic61—21no—
92D4×Dic61—41no—
93C4×C22⋊D61D411yescharacteristic
94C2×C22⋊Dic61—41no—
95Dic488—22no—
96Dic488—41no—
97#14519721—22no—
98#14519721—22no—
99#14519721—22no—
100#14519721—22no—
101#14519721D414yescomplemented
102C2×#11758185—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105C16×D61—22no—
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519702—21no—
109#14519700—21no—
110#14519700—21no—
111#14519698—21no—
112#14519698—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×D4×C61—21no—
117C2×D8×C61—22no—
118C2×D8×C61—21no—
119C2×D8×C61—41no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124#121×C61—21no—
125D16×C61—22no—
126D16×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#102×C61—21no—
131C24×C61—41no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—42no—
136C2×D4×C61—41no—
137C2×D4×C61—41no—
138C2×D4×C61—81no—
139C22×C4×C61C2×D411yescharacteristic
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61D811yescharacteristic
144C22×Dic61—41no—
145C22×Dic61—41no—
146C22×Dic61—41no—
147C22×Dic61—41no—
148#11758224—42no—
149#11758224—42no—
150#11758224—81no—
151#11758224—81no—
152C2×C4×D61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—42no—
157C2×Dic122—41no—
158C2×Dic122—81no—
159D8×C61—21no—
160D8×C61—21no—
161D8×C61—21no—
162D8×C61—21no—
163D8×C61—21no—
164D8×C61—42no—
165D8×C61C2×D412yes—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2411yescharacteristic
170C4⋊C4×C61—21no—
171C22⋊C4×C61—41no—
172C22⋊Dic61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C22⋊Dic61D811yescharacteristic
180#11758200—42no—
181#11758200—42no—
182#11758200—81no—
183#11758200—81no—
184#11758198—41no—
185#11758198D811yescharacteristic
186C4⋊Dic61—21no—
187C4⋊Dic61—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—
197Dic244—21no—
198Dic244—21no—
199Dic244—21no—
200Dic244—42no—
201Dic244C2×D412yes—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—82no—
212C23×C61—82no—
213C23×C61—161no—
214C23×C61C2×D811yescharacteristic
215C22×D61C2×D811yescharacteristic
216D4×C61—22no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—82no—
223D4×C61—82no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61C2×D811yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C22⋊D61—22no—
230C22⋊D61—82no—
231C22⋊D61—82no—
232C22⋊D61—161no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—82no—
238C2×Dic61—82no—
239C2×Dic61—82no—
240C2×Dic61—161no—
241C2×Dic61C2×D811yescharacteristic
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244C4×D61—21no—
245C4×D61—21no—
246Dic122—22no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
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—21no—
260C22×C61—21no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—164no—
271C22×C61#50812yescomplemented
272C22×C61C22×D811yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C4×C61—21no—
277C4×C61—81no—
278C4×C61D4211yescharacteristic
279C4×C61#57611yescharacteristic
280Dic61—21no—
281Dic61—21no—
282Dic61—21no—
283Dic61—42no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287C2×#508—611no—
288C22×SD32—611no—
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#2184D6111yescharacteristic, complemented
300C4×SD32—611no—
301D4×C16—611no—
302C2×C61—22no—
303C2×C61—82no—
304C2×C61—161no—
305C2×C61—161no—
306C2×C61D4×D811yescharacteristic
307C2×C61C2×#50811yescharacteristic
308C2×C61#341611yescharacteristic
309D61—41no—
310C2×#135—611no—
311C2×#135—611no—
312C22×Q16—611no—
313C22×D8C2×D6111yescharacteristic
314#548—1221no—
315#546—1221no—
316#508—1224no—
317C2×Q32—611no—
318C2×Q32—611no—
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32—611no—
324C2×SD32—1224no—
325C2×D16C2×D6111yescharacteristic
326C2×D16C2×D6111yescharacteristic
327C22×C16—611no—
328C22×C16—611no—
329#495—611no—
330#495—611no—
331#467—611no—
332#467—611no—
333#466—611no—
334#466C2×D6111yescharacteristic
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#357C2×D6111yescharacteristic
351#357C2×D6111yescharacteristic
352C22⋊C16—611no—
353C22⋊C16C2×D6111yescharacteristic
354C4×C16—611no—
355C61#3384811yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×#41—1221no—
358C22×Q8—1221no—
359C22×D4—21no—
360#135—1222no—
361#135—1222no—
362#135—2441no—
363#135—2441no—
364C2×Q16—611no—
365C2×Q16—611no—
366C2×Q16—611no—
367C2×Q16—1222no—
368C2×Q16—1221no—
369C2×Q16—2441no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—22no—
375C2×D8—21no—
376C2×D8—41no—
377C2×D8C22×D6111yescharacteristic
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—21no—
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—614no—
402SD32—1222no—
403SD32—1222no—
404SD32—1222no—
405SD32—1222no—
406D16—22no—
407D16—41no—
408C2×C16—611no—
409C2×C16—611no—
410C2×C16—1222no—
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×C8—611no—
426C24—41no—
427#41—4881no—
428#41—4881no—
429#41—2442no—
430#41—2442no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2441no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—42no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—81no—
444C2×D4—611no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4—2441no—
448C22×C4—611no—
449C22×C4—2441no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4D4×D6111yescharacteristic
454Q16—612no—
455Q16—1221no—
456Q16—1221no—
457Q16—1221no—
458Q16—1221no—
459Q16—1221no—
460Q16—2442no—
461SD16—4881no—
462SD16—4881no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—21no—
468D8—21no—
469D8—21no—
470D8—42no—
471D8D4×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—21no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C22⋊C4—41no—
487C22⋊C4—611no—
488C22⋊C4—611no—
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—21no—
501C16—612no—
502C16—1221no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—82no—
507C23—82no—
508C23—161no—
509C23—611no—
510C23D8×D6111yescharacteristic
511Q8—1222no—
512Q8—4882no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518D4—22no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—82no—
525D4—82no—
526D4—1222no—
527D4—9761no—
528D4—4882no—
529D4—4882no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
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×C4—2441no—
546C2×C4D8×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C8—21no—
549C8—1221no—
550C8—1221no—
551C8C2×D4×D6111yescharacteristic
552C8#1451982211yescharacteristic
553C22—21no—
554C22—21no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—164no—
565C22—1221no—
566C22—1221no—
567C22—1221no—
568C22#1684019112yescomplemented
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—81no—
572C4—1221no—
573C4—1221no—
574C4—1221no—
575C4—4881no—
576C4—4881no—
577C4—4881no—
578C4—2442no—
579C4#1684039411yescharacteristic
580C4#1684056611yescharacteristic
581C2—22no—
582C2—82no—
583C2—161no—
584C2—161no—
585C2—2441no—
586C2#2145748911yescharacteristic
587C2C2×#1684019111yescharacteristic
588C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2184C2
socleC22×C61C22×D8
hypercenter#2184D61
hypocenterC61#33848
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. C2×#16840191C2semidirect
2. C22×#14519721C2semidirect
3. #21457557C2semidirect
4. #21457504C2semidirect
5. #21448077C2semidirect
6. #21447914C2semidirect
7. #21447913C2semidirect
8. #21447880C2semidirect
9. #21447872C2semidirect
10. #21447760C2semidirect
11. #21447755C2semidirect
12. #21447739C2semidirect
13. #21447674C2semidirect
14. #21447636C2semidirect
15. C61×#2184C2semidirect
17. C2×#14519796C22semidirect
22. C2×Dic488C22semidirect
23. C2×Dic488C22semidirect
29. C2×#14519721C22semidirect
31. C2×C16×D61C22semidirect
33. #16840145C22semidirect
34. #16840004C22semidirect
43. #16839856C22semidirect
46. #16839589C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839567C22semidirect
50. #16839565C22semidirect
51. #16839563C22semidirect
52. #16839541C22semidirect
53. C16×Dic61C22semidirect
101. #14519721D4semidirect
271. C22×C61#508semidirect
299. #2184D61semidirect
355. C61#33848semidirect
568. C22#16840191semidirect

Maximal subgroups

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

subgroupindex
C2×#168401912
C22×#145197212
#214575572
#214575042
#214480772
#214479142
#214479132
#214478802
#214478722
#214477602
#214477552
#214477392
#214476742
#214476362
C61×#21842
#3384861