Finite Group Databasebeta

← #29669540

Subgroups — #29669540

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458800C211yesmaximal, characteristic, complemented
2#21458738C211yesmaximal, characteristic, complemented
3#21457678C211yesmaximal, characteristic, complemented
4#21445438C211yesmaximal, characteristic, complemented
5#21445288C211yesmaximal, characteristic, complemented
6#21445251C211yesmaximal, characteristic, complemented
7#21445208C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445094C211yesmaximal, characteristic, complemented
11#21445019C211yesmaximal, characteristic, complemented
12#21444921C211yesmaximal, characteristic, complemented
13#21444655C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3284C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840134—21no—
22#16840134C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840092—21no—
26#16840060—21no—
27#16840055—21no—
28#16840047C2211yescharacteristic
29#16840044C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16839989—21no—
32#16839967—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839959—21no—
36#16839959C2211yescharacteristic
37#16839900—21no—
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839863—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519700—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839507C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#550C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#462C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#452—21no—
64C61×#449—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200D411yescharacteristic
74C22×#92418—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519768—21no—
80#14519768—41no—
81#14519756—21no—
82#14519754—41no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86D4×Dic61—21no—
87#14519711—21no—
88#14519711—21no—
89#14519710—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—41no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700D411yescharacteristic
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×#41×C61—21no—
117#135×C61—44no—
118C2×Q16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61D411yescharacteristic
122C2×D8×C61—21no—
123C22×C8×C61—21no—
124C4⋊Q8×C61C2311yescharacteristic
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—81no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61—41no—
163C22×Dic61—41no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122—41no—
168C2×Dic122—81no—
169C2×Dic122C2×D411yescharacteristic
170Q16×C61—42no—
171SD16×C61—42no—
172SD16×C61—81no—
173SD16×C61SD1614yescomplemented
174D8×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—42no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758200—41no—
194#11758200—42no—
195#11758200—81no—
196#11758200—81no—
197#11758200—81no—
198#11758200—81no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—21no—
207#11758196—41no—
208#11758196—41no—
209C4×Dic61—21no—
210C4×Dic61—41no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—81no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—41no—
221Q8×C61—21no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61C2×SD1612yes—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—82no—
233D4×C61—81no—
234D4×C61—82no—
235D4×C61—161no—
236D4×C61C2×SD1612yes—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—82no—
247C2×C4×C61—82no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—82no—
256Dic122—21no—
257Dic122—21no—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261Dic122—41no—
262Dic122—82no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61C2×SD1612yes—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19767—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61D4211yescharacteristic
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61C22×SD1611yescharacteristic
290C4×C61#57511yescharacteristic
291C4×C61#57611yescharacteristic
292C4×C61#57711yescharacteristic
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61—81no—
297#3392—611no—
298#3386—611no—
299#3298—611no—
300#3284D6111yescharacteristic
301#1714—611no—
302#1686—611no—
303#1680—611no—
304#1673—611no—
305#1665—611no—
306#1654—611no—
307#1653—611no—
308#1642—611no—
309#1627—611no—
310#1582—611no—
311SD16⋊C8—611no—
312C2×C61—42no—
313C2×C61—81no—
314C2×C61—161no—
315C2×C61#326811yescharacteristic
316C2×C61D4×SD1611yescharacteristic
317C2×C61#328911yescharacteristic
318C2×#135—21no—
319#557—611no—
320#550C2×D6111yescharacteristic
321D4×Q8—611no—
322#547C2×D6111yescharacteristic
323#501—1221no—
324#500—611no—
325#498—611no—
326#492—611no—
327#492—1221no—
328#492C2×D6111yescharacteristic
329#487—611no—
330#486—611no—
331#483—1221no—
332#478—1221no—
333#477—1221no—
334#475—611no—
335#475—611no—
336#474—611no—
337#465—21no—
338#465—1221no—
339#462C2×D6111yescharacteristic
340#461—611no—
341#461—611no—
342#461—1221no—
343#460—611no—
344#460—1221no—
345#459C2×D6111yescharacteristic
346#452—21no—
347#449—21no—
348#448—1221no—
349#446—611no—
350C8×Q8—611no—
351#440—1221no—
352C4×SD16—611no—
353C4×SD16—1221no—
354C4×SD16C2×D6111yescharacteristic
355C8×D4—611no—
356C8×D4C2×D6111yescharacteristic
357C2×#103—1221no—
358#335—611no—
359#334—611no—
360#332—611no—
361#332—611no—
362#332—611no—
363#331—611no—
364Q8⋊C8—611no—
365Q8⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368#322—611no—
369C61#1976711yescharacteristic, complemented, Sylow
370C2×#41—21no—
371C2×#41—21no—
372C22×Q8—1221no—
373#135—44no—
374C2×Q16—21no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×SD16—611no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×SD16C22⋊D6111yescharacteristic
384C2×D8—21no—
385C22×C8—21no—
386C22×C8—1221no—
387C4⋊Q8—611no—
388C4⋊Q8—611no—
389C4⋊Q8—1221no—
390C4⋊Q8—1221no—
391C4⋊Q8C22×D6111yescharacteristic
392#127—21no—
393#127C22×D6111yescharacteristic
394#125—1221no—
395#125—2441no—
396#123—41no—
397C22⋊Q8—21no—
398C22⋊Q8—1221no—
399C22⋊Q8—2441no—
400#121—21no—
401#121—21no—
402#121—41no—
403C4×Q8—611no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8C22×D6111yescharacteristic
407C4×D4—21no—
408C4×D4—1221no—
409C4×D4C22×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411C2×C4⋊C4—21no—
412#107—1221no—
413#107—1221no—
414#107—1221no—
415#106—611no—
416#106—1221no—
417#106—1221no—
418#106—2441no—
419#106C22⋊D6111yescharacteristic
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8C22×D6111yescharacteristic
426#103—21no—
427#103—611no—
428#103—1221no—
429#103—1221no—
430#103—1221no—
431#103—1221no—
432#103—1221no—
433#103—1221no—
434#103C22⋊D6111yescharacteristic
435#102—21no—
436#102—611no—
437#102—1221no—
438#102—1221no—
439#102—2441no—
440#102C22⋊D6111yescharacteristic
441C22⋊C8—21no—
442C22⋊C8—1221no—
443#97—1221no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—1221no—
447C4×C8C22×D6111yescharacteristic
448#41—42no—
449#41—42no—
450#41—81no—
451#41—81no—
452C2×Q8—21no—
453C2×Q8—611no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—4881no—
457C2×Q8—2441no—
458C2×Q8—2442no—
459C2×Q8C2×C22⋊D6111yescharacteristic
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4D4×D6111yescharacteristic
467C2×D4C2×C22⋊D6111yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—2441no—
473Q16—42no—
474SD16—42no—
475SD16—81no—
476SD16—4881no—
477SD16—4881no—
478SD16—4881no—
479SD16—4881no—
480SD16—2441no—
481SD16—2442no—
482SD16#1175820014yescomplemented
483D8—42no—
484C2×C8—21no—
485C2×C8—42no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—4881no—
493C2×C8C2×C22⋊D6111yescharacteristic
494C2×C8C2×C22⋊D6111yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—611no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—2441no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4D4×D6111yescharacteristic
510C4⋊C4C2×C22⋊D6111yescharacteristic
511C4⋊C4C2×C22⋊D6111yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—2441no—
517C42—21no—
518C42—1221no—
519C42—2441no—
520C42C23×D6111yescharacteristic
521C23—21no—
522C23—21no—
523C23—41no—
524Q8—21no—
525Q8—41no—
526Q8—41no—
527Q8—1221no—
528Q8—1221no—
529Q8—4882no—
530Q8—2441no—
531Q8—2441no—
532Q8—2441no—
533Q8—2441no—
534Q8C2×#1175820012yes—
535D4—21no—
536D4—21no—
537D4—21no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—82no—
543D4—81no—
544D4—82no—
545D4—161no—
546D4C2×#1175820012yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4882no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4#1451982211yescharacteristic
566C8—21no—
567C8—41no—
568C8—1221no—
569C8—1221no—
570C8—2441no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C8C2×#1175820012yes—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22D4×C22⋊D6111yescharacteristic
585C4—22no—
586C4—21no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—4881no—
592C4—4881no—
593C4—2441no—
594C4—2441no—
595C4C22×#1175820011yescharacteristic
596C4#1684055911yescharacteristic
597C4#1684056411yescharacteristic
598C4#1684057311yescharacteristic
599C2—42no—
600C2—81no—
601C2—161no—
602C2#2145735911yescharacteristic
603C2D4×#1175820011yescharacteristic
604C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3284C2
socleC22×C61D42
hypercenter#3284D61
hypocenterC61#19767
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3284

Complements

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

NH ≅ G/Ntype
1. #21458800C2semidirect
2. #21458738C2semidirect
3. #21457678C2semidirect
4. #21445438C2semidirect
5. #21445288C2semidirect
6. #21445251C2semidirect
7. #21445208C2semidirect
8. #21445158C2semidirect
9. #21445103C2semidirect
10. #21445094C2semidirect
11. #21445019C2semidirect
12. #21444921C2semidirect
13. #21444655C2semidirect
14. SD16⋊#92418C2semidirect
173. SD16×C61SD16semidirect
369. C61#19767semidirect
482. SD16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214588002
#214587382
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