Finite Group Databasebeta

← #29669521

Subgroups — #29669521

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458714C211yesmaximal, characteristic, complemented
2#21458178C211yesmaximal, characteristic, complemented
3#21457537C211yesmaximal, characteristic
4#21445454C211yesmaximal, characteristic, complemented
5#21445311C211yesmaximal, characteristic, complemented
6#21445239C211yesmaximal, characteristic, complemented
7#21445194C211yesmaximal, characteristic, complemented
8#21445192C211yesmaximal, characteristic, complemented
9#21445101C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445025C211yesmaximal, characteristic, complemented
12#21444970C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic, complemented
15C61×#3298C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840472C2211yescharacteristic
18#16840417C2211yescharacteristic
19#16840168—21no—
20#16840160C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840158C2211yescharacteristic
23#16840141C2211yescharacteristic
24#16840114C2211yescharacteristic
25#16840091—21no—
26#16840079—21no—
27#16840062—21no—
28#16840052C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840049—21no—
31#16840047C2211yescharacteristic
32#16840039C2211yescharacteristic
33#16840001—21no—
34#16839988—21no—
35#16839967C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839924—21no—
38#16839922—21no—
39#16839893C2211yescharacteristic
40#16839872—21no—
41C4×#11758199C2211yescharacteristic
42C8×D244C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44#106×D61—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56D4×Q8×C61C2211yescharacteristic
57C61×#547C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#483—21no—
60C61×#475C2211yescharacteristic
61C61×#465—21no—
62C61×#461C2211yescharacteristic
63C61×#448—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C2×#11758222—21no—
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758199D411yescharacteristic
74C22×#92418—21no—
75C2×C8×D61—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519769—21no—
79#14519759—41no—
80#14519756—21no—
81#14519755—41no—
82#14519753—21no—
83#14519748—21no—
84C4×D244—21no—
85C4×D244C2311yescharacteristic
86C4⋊C4×D61—21no—
87C4⋊C4×D61—21no—
88#14519710—21no—
89#14519710D411yescharacteristic
90#14519709—21no—
91#14519709—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708—41no—
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519701—21no—
104#14519700—21no—
105#14519700—41no—
106#14519698—21no—
107#14519698—21no—
108#14519698D411yescharacteristic
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C22×Q8×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C22×C8×C61—21no—
121C4⋊Q8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#123×C61—41no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—21no—
127C22⋊Q8×C61—41no—
128#121×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C2×C22⋊D61—41no—
161#11758222—42no—
162#11758222—81no—
163C2×D244C2×D411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169SD16×C61—41no—
170SD16×C61—81no—
171C2×C8×C61—21no—
172C2×C8×C61—81no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192#11758202—42no—
193#11758202—81no—
194#11758201—42no—
195#11758200—42no—
196#11758199SD1614yescomplemented
197#11758198—21no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—41no—
205#11758196—41no—
206#11758196—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—42no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C8×D61—42no—
215C23×C61—21no—
216C23×C61—21no—
217C22×D61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—82no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—82no—
229D4×C61C2×SD1612yes—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—82no—
241C2×C4×C61—82no—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C22⋊D61—82no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249D244—21no—
250D244C2×SD1612yes—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—82no—
254C4×D61—161no—
255Dic122—41no—
256Dic122—41no—
257Dic122—81no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—41no—
261#92418—21no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#92418C2×SD1612yes—
267#19766—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61D4211yescharacteristic
275C2×D61—22no—
276C2×D61—41no—
277C2×D61—41no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C22×SD1611yescharacteristic
286C4×C61#57511yescharacteristic
287C4×C61#57711yescharacteristic
288C4×C61#57711yescharacteristic
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292#3384—611no—
293#3335—611no—
294#3298D6111yescharacteristic, complemented
295#3283—611no—
296#1716—611no—
297#1690—611no—
298#1679—611no—
299#1671—611no—
300#1671—611no—
301#1654—611no—
302#1651—611no—
303#1643—611no—
304#1635—611no—
305#1582—611no—
306#1580—611no—
307C2×C61—42no—
308C2×C61—81no—
309C2×C61#326711yescharacteristic
310C2×C61D4×SD1611yescharacteristic
311C2×C61#328911yescharacteristic
312D61—42no—
313C2×#135—1221no—
314#554—611no—
315D4×Q8C2×D6111yescharacteristic
316#547—611no—
317#547C2×D6111yescharacteristic
318#501—1221no—
319#498—611no—
320#498—611no—
321#498—611no—
322#494—611no—
323#487—611no—
324#487C2×D6111yescharacteristic
325#483—21no—
326#483—1221no—
327#481—1221no—
328#478—1221no—
329#476—611no—
330#476—611no—
331#476—1221no—
332#475—611no—
333#475C2×D6111yescharacteristic
334#474—611no—
335#467—1221no—
336#465—21no—
337#465—1221no—
338#461—611no—
339#461—611no—
340#461C2×D6111yescharacteristic
341#452—1221no—
342#452—1221no—
343#448—21no—
344#446—611no—
345#442—1221no—
346C4×SD16C2×D6111yescharacteristic
347C4×D8—611no—
348C8×D4—611no—
349C8×D4—611no—
350C8×D4C2×D6111yescharacteristic
351C2×#106—1221no—
352C2×#103—21no—
353#334—611no—
354#334—611no—
355#333—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359Q8⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363#322—611no—
364C61#1976611yescharacteristic, complemented, Sylow
365C2×#41—21no—
366C2×#41—1221no—
367C22×Q8—21no—
368#135—2444no—
369C2×Q16—1221no—
370C2×Q16—1221no—
371C2×SD16—21no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16C22⋊D6111yescharacteristic
375C2×D8—611no—
376C22×C8—21no—
377C22×C8—1221no—
378C22×C8—1221no—
379C4⋊Q8—21no—
380C4⋊Q8—611no—
381C4⋊Q8—611no—
382C4⋊Q8C22×D6111yescharacteristic
383C4⋊Q8C22×D6111yescharacteristic
384#125—1221no—
385#123—41no—
386#123—2441no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—41no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—2441no—
393#121—21no—
394#121—1221no—
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—611no—
398C4×D4—1221no—
399C4×D4C22×D6111yescharacteristic
400C4×D4C22×D6111yescharacteristic
401C2×C4⋊C4—21no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404#107—611no—
405#107—1221no—
406#106—21no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412#106—1221no—
413#106—1221no—
414#106—2441no—
415#106C22⋊D6111yescharacteristic
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421#103—21no—
422#103—21no—
423#103—21no—
424#103—21no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—2441no—
429#103C22⋊D6111yescharacteristic
430#102—611no—
431#102—611no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102C22⋊D6111yescharacteristic
436C22⋊C8—21no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439#97—1221no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8C22×D6111yescharacteristic
443#41—42no—
444#41—81no—
445#41—4881no—
446#41—2442no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—41no—
450C2×Q8—42no—
451C2×Q8—81no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8C2×C22⋊D6111yescharacteristic
455C2×D4—41no—
456C2×D4—611no—
457C2×D4—2441no—
458C2×D4D4×D6111yescharacteristic
459C2×D4C2×C22⋊D6111yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4—2441no—
467Q16—4881no—
468Q16—2442no—
469SD16—41no—
470SD16—81no—
471SD16—2442no—
472SD16—2442no—
473D8—614no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—2442no—
483C2×C8—2442no—
484C2×C8C2×C22⋊D6111yescharacteristic
485C2×C8C2×C22⋊D6111yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—1221no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C42—21no—
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—21no—
515C23—1221no—
516Q8—21no—
517Q8—21no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—82no—
523Q8—4881no—
524Q8—2441no—
525Q8—2441no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—82no—
530D4—612no—
531D4—1221no—
532D4—4882no—
533D4SD16×D6112yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—1221no—
547C2×C4—9761no—
548C2×C4—4882no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4#1451982211yescharacteristic
556C8—21no—
557C8—21no—
558C8—41no—
559C8—612no—
560C8—1221no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C22—22no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—1222no—
572C22—2441no—
573C22—2441no—
574C22D4×C22⋊D6111yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—4881no—
583C4—4881no—
584C4—2441no—
585C4C2×SD16×D6111yescharacteristic
586C4#1684056011yescharacteristic
587C4#1684056811yescharacteristic
588C4#1684057311yescharacteristic
589C2—42no—
590C2—81no—
591C2—2442no—
592C2#2145734011yescharacteristic
593C2SD16×C22⋊D6111yescharacteristic
594C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3298C2
socleC22×C61D42
prime radical#21457537C2
hypercenter#3298D61
hypocenterC61#19766
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3298

Complements

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

NH ≅ G/Ntype
1. #21458714C2semidirect
2. #21458178C2semidirect
4. #21445454C2semidirect
5. #21445311C2semidirect
6. #21445239C2semidirect
7. #21445194C2semidirect
8. #21445192C2semidirect
9. #21445101C2semidirect
10. #21445082C2semidirect
11. #21445025C2semidirect
12. #21444970C2semidirect
13. #21444656C2semidirect
14. #21444641C2semidirect
15. C61×#3298C2semidirect
196. #11758199SD16semidirect
294. #3298D61semidirect
364. C61#19766semidirect

Maximal subgroups

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

subgroupindex
#214587142
#214581782
#214575372
#214454542
#214453112
#214452392
#214451942
#214451922
#214451012
#214450822
#214450252
#214449702
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