Finite Group Databasebeta

← #30610792

Subgroups — #30610792

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462625C211yesmaximal, characteristic, complemented
2#21462519C211yesmaximal, characteristic, complemented
3#21461171C212yesmaximal, complemented
4#21461165C211yesmaximal, characteristic, complemented
5#21461151C212yesmaximal, complemented
6#21461148C211yesmaximal, characteristic, complemented
7#21461131C212yesmaximal, complemented
8#21461125C211yesmaximal, characteristic, complemented
9#21461111C212yesmaximal, complemented
10D61⋊(Q8⋊C8)C211yesmaximal, characteristic, complemented
11#21459415C211yesmaximal, characteristic
12Q8×#92420—21no—
13#16840790—22no—
14#16840790—21no—
15#16840787—22no—
16#16840784C2212yes—
17#16840781C2212yes—
18#16840781C2211yescharacteristic
19#16840781C2212yes—
20#16840766C2211yescharacteristic
21D61⋊(C4⋊C8)C2211yescharacteristic
22#16840737—21no—
23#16840727—21no—
24C4×D61⋊C8C2211yescharacteristic
25C4×D61⋊C8C2212yes—
26#16840612C2211yescharacteristic
27#16840612C2211yescharacteristic
28#16840612C2212yes—
29Q8⋊#92420C2211yescharacteristic
30Q8⋊#92420C2211yescharacteristic
31Q8⋊#92420C2212yes—
32D4⋊#92420C2212yes—
33D4⋊#92420C2212yes—
34D4⋊#92420C2212yes—
35D4⋊#92420C2212yes—
36#16840495—21no—
37#16840490C411yescharacteristic
38#16840485C411yescharacteristic
39#16840481—21no—
40#16840462C412yes—
41#16840457—22no—
42#16840432C411yescharacteristic
43#16840420C412yes—
44#16840311—21no—
45#16840310C2211yescharacteristic
46#16840291C2212yes—
47#16840289—22no—
48C4×#11758226C2211yescharacteristic
49C4×#11758224—21no—
50C4×#11758224C2212yes—
51C4×Q8×D61C2211yescharacteristic
52C61×#559C411yescharacteristic
53C2×#11758213—21no—
54C2×D61⋊C8—21no—
55C2×D61⋊C8—22no—
56C2×D61⋊C8—21no—
57#14519844—21no—
58C4⋊#92420—21no—
59C4⋊#92420—22no—
60C4⋊#92420—21no—
61C4⋊#92420C2311yescharacteristic
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2312yes—
64#14519834—22no—
65#14519834—22no—
66#14519834—21no—
67#14519834—22no—
68#14519832—22no—
69C4×#92420—22no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C2×#11758226D411yescharacteristic
76C2×#11758224—21no—
77C2×#11758224C2×C412yes—
78C2×Q8×D61C2×C411yescharacteristic
79#14519777C2×C411yescharacteristic
80#14519775—21no—
81#14519773—21no—
82#14519772—22no—
83#14519770—22no—
84#14519770—22no—
85#14519769C2×C412yes—
86#14519767—22no—
87#14519765—21no—
88#14519764—21no—
89#14519763—22no—
90#14519760—22no—
91#14519759—22no—
92#14519754—22no—
93#14519753C2×C412yes—
94#14519750—22no—
95#14519750—22no—
96#14519749—22no—
97#14519748—21no—
98#14519748C2×C412yes—
99#14519748C2×C412yes—
100Q8×Dic61D411yescharacteristic
101Q8×Dic61C2311yescharacteristic
102C4×Dic122—21no—
103C4×Dic122—21no—
104C4×Dic122C2×C411yescharacteristic
105C4×Dic122C2×C412yes—
106#14519742—21no—
107#14519742—21no—
108#14519742C2311yescharacteristic
109C4×D244C2×C412yes—
110C4×D244C2×C411yescharacteristic
111D4×Dic61—21no—
112D4×Dic61—21no—
113D4×Dic61C2312yes—
114#14519739D412yes—
115#14519739D411yescharacteristic
116C4×C22⋊D61—22no—
117C4×C22⋊D61—22no—
118#14519737—22no—
119#14519737—22no—
120#14519737—22no—
121#14519735C2×C412yes—
122#14519735D411yescharacteristic
123#14519735D412yes—
124#14519733—22no—
125#14519733—22no—
126C4⋊C4×D61—21no—
127C4⋊C4×D61C2×C411yescharacteristic
128C2×C4×Dic61—22no—
129C42×D61—21no—
130C42×D61C2311yescharacteristic
131C42×D61C2312yes—
132#127×C61C2×C411yescharacteristic
133#124×C61—21no—
134#124×C61C2×C412yes—
135#123×C61—22no—
136C4×Q8×C61C2×C411yescharacteristic
137C4×Q8×C61C2×C411yescharacteristic
138C4×D4×C61—21no—
139C4×D4×C61C2×C412yes—
140C2×Q8×C61C22⋊C411yescharacteristic
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61C22×C412yes—
145C22×C4×C61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151#11758226—22no—
152#11758226—22no—
153Q8×D61—22no—
154Q8×D61—41no—
155#11758224—24no—
156#11758224—42no—
157#11758224—42no—
158#11758224—42no—
159C2×D244C22⋊C412yes—
160C2×D244C22⋊C411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C22×C412yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D411yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122C22×C411yescharacteristic
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—22no—
175C2×#92420—21no—
176C2×#92420—22no—
177C2×#92420—21no—
178#11758213—42no—
179D61⋊C8—44no—
180D61⋊C8—42no—
181D61⋊C8—42no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22⋊C412yes—
185C4⋊C4×C61C22×C411yescharacteristic
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C42×C61—21no—
190C42×C61C22×C412yes—
191C42×C61C22×C411yescharacteristic
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198C22⋊C412yes—
198#11758198C22⋊C412yes—
199#11758198C22⋊C412yes—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C22⋊C411yescharacteristic
203C4⋊Dic61C22⋊C412yes—
204C4⋊Dic61C22×C411yescharacteristic
205C4⋊Dic61C22×C412yes—
206#11758196—21no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—21no—
211#11758196—22no—
212#11758196—21no—
213C4×Dic61—21no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2411yescharacteristic
223C23×C61—22no—
224C22×D61C2×C22⋊C412yes—
225C22×D61C2×C22⋊C411yescharacteristic
226Q8×C61—22no—
227Q8×C61—21no—
228Q8×C61—21no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—42no—
232D4×C61—81no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—82no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C22⋊D61—44no—
242C22⋊D61—44no—
243C22⋊D61—82no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—44no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61C2×C22⋊C412yes—
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C22×D411yescharacteristic
253C2×Dic61C22×D411yescharacteristic
254D244—24no—
255D244—22no—
256C4×D61—22no—
257C4×D61—21no—
258C4×D61—22no—
259C4×D61—21no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—42no—
263C4×D61—41no—
264C4×D61—42no—
265C4×D61—81no—
266C4×D61#13512yes—
267C4×D61#13512yes—
268Dic122—21no—
269Dic122—42no—
270Dic122—42no—
271Dic122#13112yes—
272#92420—22no—
273#92420—42no—
274#92420—41no—
275#92420—41no—
276#92420—41no—
277#92420—42no—
278#17510—611nomaximal, Sylow
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61C22×C22⋊C411yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61#41612yes—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61#41612yes—
295C4×C61C2×#13111yescharacteristic
296C4×C61#56811yescharacteristic
297Dic61—22no—
298Dic61—21no—
299Dic61—22no—
300Dic61—21no—
301Dic61—42no—
302Dic61—41no—
303Dic61C2×#13511yescharacteristic
304Dic61C2×#13511yescharacteristic
305#3480—611no—
306C8×#41—611no—
307#2925—611no—
308#1494—612no—
309#1492—611no—
310#1488—612no—
311#1487—611no—
312#1482—612no—
313#1480—611no—
314#1476—612no—
315C2×Q8⋊C8—611no—
316C2×C61—82no—
317C2×C61#288011yescharacteristic
318C2×C61C2×#41611yescharacteristic
319C2×C61#290111yescharacteristic
320D61—22no—
321D61—42no—
322#559—1221no—
323#559#692311yescharacteristic
324#557—611no—
325#556—611no—
326#556—1221no—
327#553—612no—
328#553—1222no—
329#550—611no—
330#548—612no—
331#532—611no—
332#532—1221no—
333#529—612no—
334#529—1222no—
335C4×#41—611no—
336C4×#41—612no—
337C4×#41—1221no—
338C2×C4×Q8—611no—
339C8×Q8—1221no—
340C8×D4—1222no—
341C8×D4—1221no—
342#433—1222no—
343#432—612no—
344#431—611no—
345#431—612no—
346#431—612no—
347#423—611no—
348C2×C4⋊C8—611no—
349#408—1221no—
350#405—1221no—
351C2×C4×C8—611no—
352C2×C4×C8—612no—
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355Q8⋊C8—612no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—612no—
359D4⋊C8—612no—
360D4⋊C8—612no—
361D4⋊C8—612no—
362D4⋊C8—612no—
363C61#1751011yescharacteristic, complemented, Sylow
364C2×#41—611no—
365C2×#41—612no—
366C2×#41—1221no—
367C22×Q8—611no—
368C2×M16—1221no—
369C22×C8—1221no—
370C22×C8—1222no—
371C22×C8—1221no—
372C4⋊Q8—611no—
373C4⋊Q8—1221no—
374#127—1221no—
375#127C2×#692311yescharacteristic
376#126—1222no—
377#126—1222no—
378#126—1222no—
379#125—612no—
380#125—1222no—
381#124—21no—
382#124—1221no—
383#124—1221no—
384#124—1222no—
385#124C2×#692312yes—
386#123—22no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—612no—
390C22⋊Q8—1222no—
391#121—612no—
392#121—612no—
393#121—1222no—
394#121—1222no—
395#121—1222no—
396#121—1221no—
397C4×Q8—611no—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×Q8—611no—
401C4×Q8—612no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8C2×#692311yescharacteristic
407C4×Q8C2×#692311yescharacteristic
408C4×D4—21no—
409C4×D4—612no—
410C4×D4—611no—
411C4×D4—612no—
412C4×D4—612no—
413C4×D4—611no—
414C4×D4—1221no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—1221no—
420C4×D4—1222no—
421C4×D4C2×#692312yes—
422#117—611no—
423#117—612no—
424#117—612no—
425#117—1222no—
426#117—1222no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C42—611no—
430C2×C42—612no—
431C2×C42—1221no—
432C2×C42—1222no—
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—612no—
436C4⋊C8—1221no—
437C4⋊C8—1222no—
438C4⋊C8—1221no—
439C4⋊C8—1221no—
440C22⋊C8—1222no—
441C22⋊C8—1222no—
442C22⋊C8—1221no—
443C22⋊C8—1222no—
444#97—1222no—
445C4×C8—611no—
446C4×C8—612no—
447C4×C8—611no—
448C4×C8—1222no—
449C4×C8—1221no—
450C4×C8—1222no—
451#41—1222no—
452#41—1222no—
453#41—1224no—
454#41—2442no—
455#41—2442no—
456#41—2442no—
457C2×Q8—611no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1222no—
461C2×Q8—2441no—
462C2×Q8C22⋊#692311yescharacteristic
463C2×Q8C22×#692311yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—612no—
467C2×D4—611no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4C22×#692312yes—
472C22×C4—22no—
473C22×C4—611no—
474C22×C4—612no—
475C22×C4—611no—
476C22×C4—612no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1222no—
482C22×C4—1222no—
483M16—2442no—
484C2×C8—1221no—
485C2×C8—1222no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1222no—
489C2×C8—1221no—
490C2×C8—2444no—
491C2×C8—2442no—
492C2×C8—2442no—
493C4⋊C4—21no—
494C4⋊C4—611no—
495C4⋊C4—612no—
496C4⋊C4—611no—
497C4⋊C4—612no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1221no—
507C4⋊C4C22⋊#692311yescharacteristic
508C4⋊C4C22⋊#692312yes—
509C4⋊C4C22×#692311yescharacteristic
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—612no—
514C22⋊C4—612no—
515C22⋊C4—612no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C42—21no—
522C42—612no—
523C42—612no—
524C42—611no—
525C42—611no—
526C42—611no—
527C42—1221no—
528C42—1222no—
529C42—1221no—
530C42—1222no—
531C42—1221no—
532C42C22×#692312yes—
533C42C22×#692311yescharacteristic
534C23—22no—
535C23—611no—
536C23—612no—
537Q8—22no—
538Q8—21no—
539Q8—21no—
540Q8—612no—
541Q8—1221no—
542Q8—2442no—
543Q8—2442no—
544D4—22no—
545D4—22no—
546D4—42no—
547D4—81no—
548D4—1224no—
549D4—1222no—
550D4—4882no—
551D4—2444no—
552D4—2444no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—82no—
556C2×C4—611no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1222no—
567C2×C4—1221no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—4881no—
572C2×C4—2442no—
573C2×C4—2441no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2444no—
578C2×C4C2×C22⋊#692312yes—
579C2×C4C2×C22⋊#692312yes—
580C2×C4C2×C22⋊#692311yescharacteristic
581C2×C4C2×C22⋊#692311yescharacteristic
582C2×C4C23×#692311yescharacteristic
583C8—1222no—
584C8—2441no—
585C8—2441no—
586C8—2442no—
587C8—2442no—
588C8—2441no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—612no—
593C22—1221no—
594C22—1221no—
595C22—1222no—
596C22—1222no—
597C22—1222no—
598C22C22×C22⋊#692311yescharacteristic
599C4—22no—
600C4—22no—
601C4—21no—
602C4—41no—
603C4—41no—
604C4—611no—
605C4—611no—
606C4—1222no—
607C4—1221no—
608C4—1222no—
609C4—1221no—
610C4—2442no—
611C4—2441no—
612C4#1684075212yes—
613C4D61⋊#13111yescharacteristic
614C4#1684084111yescharacteristic
615C2—82no—
616C2—1222no—
617C2—2442no—
618C2#2146241011yescharacteristic
619C2C2×#1684075211yescharacteristic
620C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#559C4
socleC22×C61C22×C22⋊C4
prime radical#21459415C2
cosocleC4×Dic61C24
hypercenter#559#6923
hypocenterC61#17510
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #21462625C2semidirect
2. #21462519C2semidirect
3. #21461171C2semidirect
4. #21461165C2semidirect
5. #21461151C2semidirect
6. #21461148C2semidirect
7. #21461131C2semidirect
8. #21461125C2semidirect
9. #21461111C2semidirect
10. D61⋊(Q8⋊C8)C2semidirect
363. C61#17510semidirect

Maximal subgroups

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

subgroupindex
#214626252
#214625192
#214611712
#214611652
#214611512
#214611482
#214611312
#214611252
#214611112
D61⋊(Q8⋊C8)2
#214594152
#1751061