Finite Group Databasebeta

← #29836534

Subgroups — #29836534

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840197C211yesmaximal, characteristic, complemented
2C2×#16840195C211yesmaximal, characteristic, complemented
3#21457573C211yesmaximal, characteristic, complemented
4#21457501C211yesmaximal, characteristic, complemented
5#21448084C211yesmaximal, characteristic, complemented
6#21447934C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447883C211yesmaximal, characteristic, complemented
9#21447853C211yesmaximal, characteristic, complemented
10#21447745C211yesmaximal, characteristic, complemented
11#21447741C211yesmaximal, characteristic, complemented
12#21447733C211yesmaximal, characteristic, complemented
13#21447686C211yesmaximal, characteristic, complemented
14#21447647C211yesmaximal, characteristic, complemented
15C61×#2189C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519800C2211yescharacteristic, complemented
18C22×D488C2211yescharacteristic
19#16840431—21no—
20#16840403—21no—
21#16840197—24no—
22#16840195—24no—
23C2×#14519720C2211yescharacteristic
24C2×#14519720C2211yescharacteristic
25C2×#14519720C2211yescharacteristic, complemented
26C2×D976C2211yescharacteristic
27C2×D976C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C2×#14519717C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic, complemented
32#16840147C2211yescharacteristic
33#16840137C2211yescharacteristic, complemented
34#16839999C2211yescharacteristic
35#16839998C2211yescharacteristic
36#16839995C2211yescharacteristic, complemented
37#16839980—21no—
38#16839960—21no—
39#16839912—21no—
40#16839910—21no—
41#16839907—21no—
42#16839906—21no—
43Q16×Dic61C2211yescharacteristic, complemented
44#16839844C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C2×#135×C61C2211yescharacteristic
55C2×Q32×C61C2211yescharacteristic, complemented
56C2×SD32×C61C2211yescharacteristic, complemented
57C61×#467C2211yescharacteristic
58C61×#358C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×C22⋊C16C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758226—21no—
63C22×D244—21no—
64#14519800—22no—
65#14519800—22no—
66#14519800—41no—
67#14519800—41no—
68C2×#11758201—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×D488—22no—
72C2×D488—21no—
73C2×D488—41no—
74C2×D488C2311yescharacteristic
75C2×D488C2311yescharacteristic
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78C2×C8×D61C2311yescharacteristic
79#14519773—21no—
80#14519773—41no—
81#14519766—21no—
82#14519762—41no—
83#14519752—21no—
84#14519752—41no—
85#14519751—21no—
86#14519751—41no—
87#14519748—21no—
88#14519748—41no—
89#14519747—41no—
90Q8×Dic61—21no—
91#14519739—21no—
92C4×C22⋊D61—41no—
93C4×C22⋊D61D411yescharacteristic
94C22⋊C4×D61—41no—
95#14519720—22no—
96#14519720—22no—
97#14519720D414yescomplemented
98D976—22no—
99D976—41no—
100#14519718—22no—
101#14519718—22no—
102#14519717—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105C16×D61—22no—
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519701—21no—
109#14519699—21no—
110#14519698—21no—
111#14519697—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×#41×C61—21no—
117#135×C61—22no—
118#135×C61—41no—
119C2×Q16×C61C2311yescharacteristic
120C2×Q16×C61C2311yescharacteristic
121C2×SD16×C61—21no—
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125Q32×C61—22no—
126SD32×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#103×C61—21no—
131C23×D61—41no—
132#41×C61—42no—
133#41×C61—81no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—41no—
138C22×C4×C61—41no—
139C22×C4×C61C2×D411yescharacteristic
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61D811yescharacteristic
145#11758226—42no—
146#11758226—42no—
147#11758226—81no—
148#11758226—81no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—42no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—41no—
157C2×D244—81no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61C2×D411yescharacteristic
162Q16×C61—21no—
163Q16×C61—21no—
164Q16×C61—21no—
165Q16×C61C2×D412yes—
166SD16×C61—42no—
167D8×C61—21no—
168D8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61D811yescharacteristic
178#11758201—42no—
179#11758201—42no—
180#11758201—81no—
181#11758199—81no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198D811yescharacteristic
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190#11758196D811yescharacteristic
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195D488—21no—
196D488—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×C61—41no—
209C23×C61C2×D811yescharacteristic
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—161no—
216C22×D61C2×D811yescharacteristic
217Q8×C61—22no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—82no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—82no—
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C22⋊D61—41no—
233C22⋊D61—82no—
234C22⋊D61—161no—
235C22⋊D61—161no—
236C22⋊D61—161no—
237C2×Dic61—41no—
238C2×Dic61C2×D811yescharacteristic
239C2×Dic61C2×D811yescharacteristic
240C2×Dic61C2×D811yescharacteristic
241D244—22no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—82no—
248D244—82no—
249D244—82no—
250D244—161no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—82no—
254C4×D61—82no—
255C4×D61—161no—
256C8×C61—21no—
257C8×C61C22×D411yescharacteristic
258C8×C61H(5,𝔽2)11yescharacteristic
259#92418—21no—
260#92418—21no—
261#33834—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61C22×D811yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—42no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—164no—
282C4×C61—21no—
283C4×C61—42no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61D4211yescharacteristic
287C4×C61#57611yescharacteristic
288Dic61—21no—
289Dic61—21no—
290Dic61—21no—
291Dic61—81no—
292C2×#509—611no—
293C2×#508—611no—
294#3287—611no—
295#3280—611no—
296#2250—611no—
297#2220—611no—
298#2215—611no—
299#2211—611no—
300#2206—611no—
301#2189D6111yescharacteristic, complemented
302#2186—611no—
303#2186—611no—
304#2184—611no—
305#2175—611no—
306#2168—611no—
307C2×C61—41no—
308C2×C61—161no—
309C2×C61D4×D811yescharacteristic
310C2×C61#341411yescharacteristic
311C2×C61#341511yescharacteristic
312D61—41no—
313D61—82no—
314D61—161no—
315D61—161no—
316C2×#135—611no—
317C2×#135—611no—
318C2×#135C2×D6111yescharacteristic
319C22×D8—611no—
320#550—1221no—
321#546—1221no—
322#509—1224no—
323#508—1224no—
324C2×Q32C2×D6111yescharacteristic, complemented
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32C2×D6111yescharacteristic, complemented
329C2×D16—611no—
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—611no—
333C2×M32—611no—
334C22×C16—611no—
335#495—611no—
336#493—611no—
337#467C2×D6111yescharacteristic
338#466—611no—
339#466—611no—
340#466—611no—
341#463—1221no—
342#460—1221no—
343#450—1221no—
344#449—1221no—
345#449—1221no—
346#449—1221no—
347C4×Q16—611no—
348C4×D8—611no—
349C8×D4—611no—
350#366—611no—
351C4⋊C16—611no—
352#358—611no—
353#358C2×D6111yescharacteristic, complemented
354#357—611no—
355#357—611no—
356#357—611no—
357#357C2×D6111yescharacteristic, complemented
358C22⋊C16—611no—
359C22⋊C16C2×D6111yescharacteristic
360#346—611no—
361C61#3383411yescharacteristic, complemented, Sylow
362C2×#41—21no—
363C2×#41—1221no—
364C2×#41—1221no—
365C22×D4—1221no—
366#135—22no—
367#135—41no—
368#135—1222no—
369#135—1222no—
370#135—2441no—
371#135—2441no—
372C2×Q16C22×D6111yescharacteristic
373C2×Q16C22×D6111yescharacteristic
374C2×SD16—21no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—611no—
379C2×D8—611no—
380C2×D8—611no—
381C2×D8—1222no—
382C2×D8—1221no—
383C2×D8—1221no—
384C2×D8—2441no—
385C2×D8C22×D6111yescharacteristic
386C22×C8—611no—
387C22×C8C22×D6111yescharacteristic
388#127—1221no—
389#127—2441no—
390#124—1221no—
391#123—2441no—
392C22⋊Q8—21no—
393#121—1221no—
394#121—1221no—
395#121—1221no—
396#121—2441no—
397#121—2441no—
398#121—2441no—
399#120—2441no—
400C4×Q8—1221no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—2441no—
404C2×C22⋊C4—2441no—
405Q32—22no—
406SD32—41no—
407SD32—614no—
408SD32—1222no—
409SD32—1222no—
410D16—1222no—
411D16—1222no—
412D16—1222no—
413D16—2441no—
414M32—1222no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—1222no—
418C2×C16C22×D6111yescharacteristic
419C2×C16C22×D6111yescharacteristic
420#107—611no—
421#107C22×D6111yescharacteristic
422C4⋊C8—611no—
423#103—21no—
424#103—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429C22⋊C8—611no—
430C22⋊C8—611no—
431C4×C8—611no—
432C24—2441no—
433#41—42no—
434#41—81no—
435#41—4881no—
436#41—4881no—
437#41—2442no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—611no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—4881no—
448C2×D4—2442no—
449C2×D4—2441no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2441no—
457C22×C4—41no—
458C22×C4—611no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4D4×D6111yescharacteristic
463Q16—21no—
464Q16—21no—
465Q16—21no—
466Q16D4×D6112yes—
467SD16—42no—
468SD16—4881no—
469SD16—2442no—
470SD16—2442no—
471D8—21no—
472D8—21no—
473D8—612no—
474D8—1221no—
475D8—1221no—
476D8—1221no—
477D8—1221no—
478D8—1221no—
479D8—4881no—
480D8—2442no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8D4×D6111yescharacteristic
488C2×C8C23×D6111yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—611no—
492C4⋊C4—1221no—
493C4⋊C4—2441no—
494C22⋊C4—41no—
495C22⋊C4—611no—
496C22⋊C4—611no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C42—611no—
504C42—2441no—
505C16—21no—
506C16—21no—
507C16—612no—
508C16—1221no—
509C23—41no—
510C23—611no—
511C23—9761no—
512C23—4882no—
513C23—4882no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23D8×D6111yescharacteristic
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522D4—41no—
523D4—41no—
524D4—82no—
525D4—1222no—
526D4—9761no—
527D4—9761no—
528D4—9761no—
529D4—9761no—
530D4—4882no—
531D4—4882no—
532D4—4882no—
533D4—4882no—
534D4—2441no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539D4—2441no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—9761no—
552C2×C4—4882no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4D8×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C8—21no—
558C8—1221no—
559C8—1221no—
560C8C2×D4×D6111yescharacteristic
561C8#1451982211yescharacteristic
562C22—21no—
563C22—21no—
564C22—21no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—1221no—
569C22—1221no—
570C22—1221no—
571C22—9764no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—2442no—
581C22C2×D8×D6111yescharacteristic
582C4—21no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—1221no—
587C4—1221no—
588C4—1221no—
589C4—4881no—
590C4#1684039411yescharacteristic
591C4#1684056611yescharacteristic
592C2—41no—
593C2—161no—
594C2—9761no—
595C2—9761no—
596C2—4882no—
597C2—2441no—
598C2#2145748911yescharacteristic
599C2#2145895411yescharacteristic
600C2#2145895711yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2189C2
socleC22×C61C22×D8
hypercenter#2189D61
hypocenterC61#33834
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. C2×#16840197C2semidirect
2. C2×#16840195C2semidirect
3. #21457573C2semidirect
4. #21457501C2semidirect
5. #21448084C2semidirect
6. #21447934C2semidirect
7. #21447904C2semidirect
8. #21447883C2semidirect
9. #21447853C2semidirect
10. #21447745C2semidirect
11. #21447741C2semidirect
12. #21447733C2semidirect
13. #21447686C2semidirect
14. #21447647C2semidirect
15. C61×#2189C2semidirect
17. C2×#14519800C22semidirect
25. C2×#14519720C22semidirect
31. C2×C16×D61C22semidirect
33. #16840137C22semidirect
36. #16839995C22semidirect
43. Q16×Dic61C22semidirect
46. #16839587C22semidirect
47. #16839580C22semidirect
48. #16839566C22semidirect
51. #16839562C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
55. C2×Q32×C61C22semidirect
56. C2×SD32×C61C22semidirect
58. C61×#358C22semidirect
59. C61×#357C22semidirect
97. #14519720D4semidirect
301. #2189D61semidirect
324. C2×Q32C2×D61semidirect
328. C2×SD32C2×D61semidirect
353. #358C2×D61semidirect
357. #357C2×D61semidirect
361. C61#33834semidirect

Maximal subgroups

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

subgroupindex
C2×#168401972
C2×#168401952
#214575732
#214575012
#214480842
#214479342
#214479042
#214478832
#214478532
#214477452
#214477412
#214477332
#214476862
#214476472
C61×#21892
#3383461