Finite Group Databasebeta

← #29650681

Subgroups — #29650681

Order 15616.

600 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456375C211yesmaximal, characteristic
2#21454915C211yesmaximal, characteristic, complemented
3#21454896C211yesmaximal, characteristic
4#21444721C211yesmaximal, characteristic, complemented
5#21444719C211yesmaximal, characteristic, complemented
6#21444711C211yesmaximal, characteristic, complemented
7#21444634C211yesmaximal, characteristic, complemented
8#21444628C211yesmaximal, characteristic, complemented
9SD16×#92418C211yesmaximal, characteristic, complemented
10#21444618C211yesmaximal, characteristic, complemented
11#21444579C211yesmaximal, characteristic, complemented
12#21444290C211yesmaximal, characteristic, complemented
13#21444286C211yesmaximal, characteristic
14#21444101C211yesmaximal, characteristic, complemented
15C61×#1590C211yesmaximal, characteristic, complemented
16#16840316C2211yescharacteristic
17#16840309C2211yescharacteristic
18#16840159C411yescharacteristic
19#16840133C411yescharacteristic
20#16840111C411yescharacteristic
21#16840107C411yescharacteristic
22#16839994C411yescharacteristic
23#16839992C411yescharacteristic
24#16839990C411yescharacteristic
25#16839988C411yescharacteristic
26#16839892C2211yescharacteristic
27#16839890—21no—
28#16839889—21no—
29Q8×#92418C2211yescharacteristic
30#16839886C2211yescharacteristic
31C4×#11758190C2211yescharacteristic
32C4×#11758201C2211yescharacteristic
33C4×#11758200C2211yescharacteristic
34#16839841C2211yescharacteristic
35#16839831—21no—
36#16839830—21no—
37D4×#92418C2211yescharacteristic
38#16839827C2211yescharacteristic
39#16839824—21no—
40#16839822—21no—
41#16839818C2211yescharacteristic
42#16839818C2211yescharacteristic
43#16839813C2211yescharacteristic
44#16839813C2211yescharacteristic
45#16839730—21no—
46#16839727—21no—
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic
49#16839514C2211yescharacteristic
50#16839513C2211yescharacteristic
51#16839494C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54#16839491C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839484C2211yescharacteristic
59C8×#92418C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C4×SD16×C61C2211yescharacteristic
62C61×#436C2211yescharacteristic
63C61×#334C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C82×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758190C2×C411yescharacteristic
69C2×#11758200C2×C411yescharacteristic
70C2×#11758201C2×C411yescharacteristic
71C2×#11758189—21no—
72C22×#92418—21no—
73C2×C8×D61—21no—
74C2×C8×D61C2×C411yescharacteristic
75C2×C8×D61C2×C411yescharacteristic
76#14519776C2×C411yescharacteristic
77#14519776C2×C411yescharacteristic
78#14519775C2×C411yescharacteristic
79#14519773C2×C411yescharacteristic
80#14519767—21no—
81#14519763—21no—
82#14519757C2×C411yescharacteristic
83#14519755—21no—
84#14519753C2×C411yescharacteristic
85#14519752C2×C411yescharacteristic
86#14519750—21no—
87#14519748C2×C411yescharacteristic
88C4×Dic122—21no—
89C4×Dic122C2311yescharacteristic
90C4×D244C2311yescharacteristic
91C4×C22⋊D61—21no—
92#14519734—21no—
93C42×D61C2311yescharacteristic
94#14519709C2×C411yescharacteristic
95#14519708C2×C411yescharacteristic
96#14519708C2×C411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519706—21no—
102#14519706—21no—
103#14519706D411yescharacteristic
104#14519706D411yescharacteristic
105#14519702C2×C411yescharacteristic
106#14519701C2×C411yescharacteristic
107#14519700C2×C411yescharacteristic
108#14519699C2×C411yescharacteristic
109#14519698C2×C411yescharacteristic
110#14519697C2×C411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519689—21no—
117#14519689D411yescharacteristic
118#14519689D411yescharacteristic
119#14519687—21no—
120#14519687—21no—
121#14519687—21no—
122C4×#92418—21no—
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61C2×C411yescharacteristic
128C8×Dic61C2×C411yescharacteristic
129C2×SD16×C61C2×C411yescharacteristic
130C2×M16×C61—21no—
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#106×C61C2×C411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61C2×C411yescharacteristic
138#102×C61C2×C411yescharacteristic
139C22⋊C8×C61—21no—
140#97×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C2×C22⋊D61—21no—
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150#11758222—81no—
151#11758222—81no—
152C2×D244—21no—
153C2×D244C22×C411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61C22×C411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61#4111yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22×C411yescharacteristic
161SD16×C61—22no—
162M16×C61—81no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C22×C411yescharacteristic
167C2×C8×C61C22×C411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C411yescharacteristic
172C4⋊C4×C61C22×C411yescharacteristic
173C22⋊C4×C61—21no—
174C42×C61—21no—
175C42×C61C2411yescharacteristic
176C22⋊Dic61—21no—
177#11758201—22no—
178#11758200—22no—
179#11758198—21no—
180#11758198—21no—
181#11758198—21no—
182#11758198—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61C22×C411yescharacteristic
185C4⋊Dic61C22×C411yescharacteristic
186#11758196—21no—
187#11758196—21no—
188#11758196—21no—
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192C4×Dic61—21no—
193C4×Dic61C22×C411yescharacteristic
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61#4111yescharacteristic
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—81no—
200C2×#92418C22×C411yescharacteristic
201C2×#92418C22×C411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204#11758190—22no—
205#11758189—81no—
206C8×D61—22no—
207C8×D61—22no—
208C8×D61—81no—
209C23×C61—21no—
210C22×D61—21no—
211C22×D61C4×D411yescharacteristic
212Q8×C61—21no—
213Q8×C61—21no—
214D4×C61—21no—
215D4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C23×C411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C22⋊D61—81no—
225C22⋊D61—81no—
226C22⋊D61—81no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61C4×D411yescharacteristic
231C2×Dic61C4×D411yescharacteristic
232C2×Dic61C4×D411yescharacteristic
233D244—21no—
234D244—21no—
235D244—81no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—41no—
243Dic122—21no—
244Dic122—21no—
245Dic122—81no—
246Dic122—81no—
247C8×C61—21no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61#13112yes—
254#92418—21no—
255#92418—21no—
256#92418—21no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#92418#13112yes—
261#17971—611nomaximal, Sylow
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61C2×C4×D411yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—41no—
270C2×D61—41no—
271C2×D61—41no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61C2×#13111yescharacteristic
277C4×C61C2×#13111yescharacteristic
278C4×C61#57711yescharacteristic
279C4×C61#57711yescharacteristic
280Dic61—21no—
281Dic61—21no—
282Dic61—21no—
283Dic61—41no—
284Dic61—41no—
285Dic61—41no—
286#3167—611no—
287#2985—611no—
288#2981—611no—
289#1592—611no—
290#1592—611no—
291#1590—611no—
292#1590D6111yescharacteristic, complemented
293SD16⋊C8—611no—
294SD16⋊C8—611no—
295C8×SD16—611no—
296C8×SD16—611no—
297#1568—611no—
298#1497—611no—
299#1496—611no—
300#1450—611no—
301C2×C61—81no—
302C2×C61#293111yescharacteristic
303C2×C61#294911yescharacteristic
304C2×C61#295711yescharacteristic
305D61—41no—
306D61—81no—
307#533—611no—
308#532—611no—
309#498—611no—
310#492—611no—
311#487—611no—
312#486—611no—
313#465—611no—
314#465—611no—
315#465—611no—
316#465—611no—
317#446—611no—
318#446—1221no—
319#446—1221no—
320#446C2×D6111yescharacteristic
321C8×Q8—611no—
322C8×Q8—611no—
323C4×SD16—611no—
324C4×SD16—611no—
325C4×SD16—611no—
326C4×SD16C2×D6111yescharacteristic
327#436—611no—
328#436C2×D6111yescharacteristic
329#435—1221no—
330#435—1221no—
331C8×D4—611no—
332C8×D4—611no—
333#433—1221no—
334#433—1221no—
335#432—611no—
336#432—611no—
337#431—611no—
338#431—611no—
339#408—1221no—
340#408—1221no—
341C2×C4×C8—611no—
342#335—611no—
343#335—611no—
344#334—611no—
345#334C2×D6111yescharacteristic
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8—611no—
349Q8⋊C8C2×D6111yescharacteristic
350D4⋊C8—611no—
351D4⋊C8—611no—
352D4⋊C8—611no—
353D4⋊C8C2×D6111yescharacteristic
354#322—611no—
355#322—611no—
356C82—611no—
357C82C2×D6111yescharacteristic
358C61#1797111yescharacteristic, complemented, Sylow
359C2×#41—1221no—
360C2×SD16—611no—
361C2×SD16—611no—
362C2×SD16—611no—
363C2×SD16C4×D6111yescharacteristic
364C2×M16—21no—
365C2×M16—1221no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—1221no—
369C22×C8—1221no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8—611no—
373#127—611no—
374#125—1221no—
375#124—1221no—
376C22⋊Q8—611no—
377C22⋊Q8—611no—
378C22⋊Q8—1221no—
379#121—611no—
380#121—611no—
381#121—1221no—
382C4×Q8—611no—
383C4×Q8—1221no—
384C4×Q8C22×D6111yescharacteristic
385C4×D4—611no—
386C4×D4—1221no—
387C4×D4C22×D6111yescharacteristic
388#117—1221no—
389C2×C42—611no—
390#106—611no—
391#106—611no—
392#106—611no—
393#106C4×D6111yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8C22×D6111yescharacteristic
404C4⋊C8C22×D6111yescharacteristic
405#103—611no—
406#103—611no—
407#103—611no—
408#103C4×D6111yescharacteristic
409#102—611no—
410#102—611no—
411#102—611no—
412#102C4×D6111yescharacteristic
413C22⋊C8—21no—
414C22⋊C8—611no—
415C22⋊C8—611no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419C22⋊C8—1221no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422#97—21no—
423#97—1221no—
424#97—1221no—
425#97—1221no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—1221no—
432C4×C8—1221no—
433C4×C8C22×D6111yescharacteristic
434C4×C8C22×D6111yescharacteristic
435C4×C8C22×D6111yescharacteristic
436#41—4881no—
437#41—4881no—
438C2×Q8—611no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×D4—611no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4C2×C4×D6111yescharacteristic
448C22×C4—21no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—1221no—
453SD16—22no—
454SD16—1222no—
455SD16—1222no—
456SD16—1222no—
457M16—81no—
458M16—4881no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—4881no—
472C2×C8—4881no—
473C2×C8C2×C4×D6111yescharacteristic
474C2×C8C2×C4×D6111yescharacteristic
475C2×C8D4×D6111yescharacteristic
476C2×C8D4×D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—611no—
479C4⋊C4—611no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4C2×C4×D6111yescharacteristic
488C4⋊C4C2×C4×D6111yescharacteristic
489C22⋊C4—21no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C22⋊C4—1221no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C42—21no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—1221no—
500C42C23×D6111yescharacteristic
501C23—21no—
502C23—611no—
503C23—1221no—
504Q8—21no—
505Q8—21no—
506Q8—1221no—
507Q8—1221no—
508Q8—4881no—
509Q8—4881no—
510D4—21no—
511D4—21no—
512D4—1221no—
513D4—1221no—
514D4—4881no—
515D4—4881no—
516D4—4881no—
517D4—4881no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C22×C4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4#41×D6111yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—612no—
546C8—1221no—
547C8—1221no—
548C8—1221no—
549C8—2441no—
550C8—2441no—
551C8—2441no—
552C8#1451978512yes—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—1221no—
557C22—1221no—
558C22—1221no—
559C22—2441no—
560C22—2441no—
561C22—2441no—
562C22C4×D4×D6111yescharacteristic
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—1221no—
568C4—1221no—
569C4—1221no—
570C4—2441no—
571C4—2441no—
572C4—2441no—
573C4#131×D6111yescharacteristic
574C4C2×#1451978511yescharacteristic
575C4#1684056811yescharacteristic
576C4#1684056911yescharacteristic
577C2—81no—
578C2—4881no—
579C2—2441no—
580C2#2145452811yescharacteristic
581C2#2145467311yescharacteristic
582C2#2145473111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1590C2
socleC22×C61C2×C4×D4
prime radical#16840309C22
cosocleC42×C61C24
hypercenter#1590D61
hypocenterC61#17971
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #21454915C2semidirect
4. #21444721C2semidirect
5. #21444719C2semidirect
6. #21444711C2semidirect
7. #21444634C2semidirect
8. #21444628C2semidirect
9. SD16×#92418C2semidirect
10. #21444618C2semidirect
11. #21444579C2semidirect
12. #21444290C2semidirect
14. #21444101C2semidirect
15. C61×#1590C2semidirect
292. #1590D61semidirect
358. C61#17971semidirect

Maximal subgroups

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

subgroupindex
#214563752
#214549152
#214548962
#214447212
#214447192
#214447112
#214446342
#214446282
SD16×#924182
#214446182
#214445792
#214442902
#214442862
#214441012
C61×#15902
#1797161