Finite Group Databasebeta

← #29649641

Subgroups — #29649641

Order 15616.

700 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#21454935C211yesmaximal, characteristic, complemented
2#21454840C211yesmaximal, characteristic, complemented
3C4×#14519804C211yesmaximal, characteristic
4#21444685C211yesmaximal, characteristic, complemented
5#21444680C211yesmaximal, characteristic, complemented
6#21444679C211yesmaximal, characteristic, complemented
7#21444656C211yesmaximal, characteristic, complemented
8#21444653C211yesmaximal, characteristic, complemented
9#21444648C211yesmaximal, characteristic, complemented
10#21444647C211yesmaximal, characteristic, complemented
11#21444570C211yesmaximal, characteristic, complemented
12#21444200C211yesmaximal, characteristic, complemented
13D4⋊C8×D61C211yesmaximal, characteristic, complemented
14#21444105C211yesmaximal, characteristic, complemented
15C61×#1585C211yesmaximal, characteristic, complemented
16C2×#14519804C411yescharacteristic
17C4×#11758226C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16839893C2211yescharacteristic, complemented
20#16839873C411yescharacteristic
21#16839871C411yescharacteristic
22#16839865C411yescharacteristic
23#16839863C411yescharacteristic
24C4×#11758201C2211yescharacteristic
25C4×D488C2211yescharacteristic
26C4×#11758199C2211yescharacteristic
27#16839841C2211yescharacteristic, complemented
28#16839839C2211yescharacteristic, complemented
29#16839835—21no—
30#16839832—21no—
31C8×D244C2211yescharacteristic
32D4×#92418C2211yescharacteristic
33#16839827C2211yescharacteristic
34C8×C22⋊D61—21no—
35#16839825—21no—
36#16839817—21no—
37#16839814C2211yescharacteristic
38#16839813C2211yescharacteristic
39#16839801C411yescharacteristic
40#16839784—21no—
41C4⋊C8×D61C2211yescharacteristic
42#16839765C411yescharacteristic
43#16839762C411yescharacteristic
44C2×#14519689—21no—
45C22⋊C8×D61—21no—
46C4×#11758189C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic, complemented
49#16839513C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839491C2211yescharacteristic
53#16839491C2211yescharacteristic, complemented
54D4⋊#92418C2211yescharacteristic
55#16839489C2211yescharacteristic
56#16839489C2211yescharacteristic, complemented
57#16839484C2211yescharacteristic, complemented
58#16839483C2211yescharacteristic, complemented
59#16839483C2211yescharacteristic, complemented
60C61×#446C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic, complemented
67C2×#11758226C2×C411yescharacteristic
68C2×D4×D61C2×C411yescharacteristic
69C22×C4×D61—21no—
70#14519804—24no—
71C2×#11758201C2×C411yescharacteristic
72C2×D488C2×C411yescharacteristic
73C2×#11758199C2×C411yescharacteristic
74C2×#11758189—21no—
75C2×#11758189—21no—
76C2×#11758189C2×C411yescharacteristic
77C2×#11758189C2×C411yescharacteristic
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C2×C8×D61—21no—
83Q8×Dic61C2×C411yescharacteristic
84C4×D244—21no—
85C4×D244C2311yescharacteristic
86C4×D244C2311yescharacteristic
87D4×Dic61C2×C411yescharacteristic
88#14519739—21no—
89#14519739C2×C411yescharacteristic
90#14519739C2×C411yescharacteristic
91C4×C22⋊D61—21no—
92#14519737—21no—
93#14519735—21no—
94#14519735C2×C411yescharacteristic
95C4⋊C4×D61C2×C411yescharacteristic
96C22⋊C4×D61—21no—
97C42×D61—21no—
98C42×D61C2311yescharacteristic
99#14519711C2×C411yescharacteristic
100#14519710C2×C411yescharacteristic
101#14519708C2×C411yescharacteristic
102C4⋊#92418—21no—
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706—21no—
106#14519706D411yescharacteristic
107#14519706D411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519699C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519697C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114C22⋊#92418—21no—
115#14519689—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689D411yescharacteristic
121#14519689D411yescharacteristic
122#14519688—21no—
123#14519687C2×C411yescharacteristic
124#14519687C2×C411yescharacteristic
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×SD16×C61C2×C411yescharacteristic
131C22×C8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#106×C61C2×C411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141#97×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151#11758226—22no—
152#11758226—22no—
153D4×D61—22no—
154D4×D61M1614yescomplemented
155C2×D244—21no—
156C2×D244C22×C411yescharacteristic
157C2×D244C22×C411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—81no—
165C2×C4×D61—81no—
166C2×C4×D61C22×C411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61#4111yescharacteristic
169SD16×C61—22no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—81no—
174C2×C8×C61C22×C411yescharacteristic
175C2×C8×C61C22×C411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C411yescharacteristic
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C42×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—21no—
185#11758201—22no—
186#11758199—22no—
187#11758198—21no—
188#11758198—21no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61C22×C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61C22×C411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—81no—
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209D488—22no—
210#11758189—22no—
211#11758189—22no—
212#11758189—81no—
213#11758189—81no—
214C8×D61—42no—
215C8×D61—81no—
216C8×D61—81no—
217C8×D61—81no—
218C23×C61—21no—
219C22×D61—21no—
220C22×D61—24no—
221C22×D61—21no—
222C22×D61—81no—
223C22×D61—81no—
224C22×D61C4×D411yescharacteristic
225Q8×C61—21no—
226Q8×C61—21no—
227D4×C61—21no—
228D4×C61C2×M1612yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—24no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—81no—
241C2×Dic61C4×D411yescharacteristic
242C2×Dic61C4×D411yescharacteristic
243C2×Dic61C4×D411yescharacteristic
244D244—21no—
245D244—21no—
246D244—21no—
247D244—42no—
248D244C2×M1612yes—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—42no—
257C4×D61—81no—
258C4×D61—81no—
259C4×D61—81no—
260C4×D61C2×M1612yes—
261C4×D61#13112yes—
262C8×C61—22no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—21no—
266C8×C61—21no—
267C8×C61—41no—
268C8×C61—41no—
269#92418—21no—
270#92418—21no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#17873—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C2×C4×D411yescharacteristic
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—22no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—84no—
290C2×D61#43512yes—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61C22×M1611yescharacteristic
296C4×C61C2×#13111yescharacteristic
297C4×C61#57611yescharacteristic
298C4×C61#57711yescharacteristic
299Dic61—21no—
300Dic61—21no—
301Dic61—41no—
302Dic61—41no—
303Dic61#43512yes—
304#2989—611no—
305#2972—611no—
306C4×AGL(1,ℤ/8)—611no—
307#1586—611no—
308#1586—611no—
309#1585—611no—
310#1585D6111yescharacteristic, complemented
311#1582—611no—
312#1582—611no—
313#1581—611no—
314#1581—611no—
315#1565—611no—
316#1476—611no—
317C2×D4⋊C8—611no—
318#1451—611no—
319C2×C61—42no—
320C2×C61C2×#43511yescharacteristic
321C2×C61#294611yescharacteristic
322C2×C61#295611yescharacteristic
323D61—22no—
324D61—42no—
325D61—81no—
326C2×AGL(1,ℤ/8)—611no—
327C4×#41—611no—
328C2×C4×D4—611no—
329#446—611no—
330#446C2×D6111yescharacteristic, complemented
331#442—611no—
332#442—611no—
333#440—611no—
334#440—611no—
335C4×SD16—611no—
336C4×SD16C2×D6111yescharacteristic
337C4×D8—611no—
338C4×D8—611no—
339#436—611no—
340#436—611no—
341#435—1221no—
342#435—1221no—
343C8×D4—611no—
344C8×D4—611no—
345C8×D4—611no—
346C8×D4—1221no—
347C8×D4—1221no—
348C8×D4C2×D6111yescharacteristic
349#432—1221no—
350#431—611no—
351#431—611no—
352#428—611no—
353#423—1221no—
354C2×C4⋊C8—611no—
355#418—611no—
356#417—611no—
357C2×C22⋊C8—1221no—
358C2×C22⋊C8—1221no—
359C4×M16—611no—
360C2×C4×C8—611no—
361#335—611no—
362#335C2×D6111yescharacteristic, complemented
363#334—611no—
364#334—611no—
365Q8⋊C8—611no—
366Q8⋊C8C2×D6111yescharacteristic, complemented
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8C2×D6111yescharacteristic
373#322—611no—
374#322—611no—
375#322—611no—
376#322C2×D6111yescharacteristic, complemented
377C61#1787311yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C22×D4—611no—
380C23×C4—1221no—
381AGL(1,ℤ/8)—1224no—
382C2×SD16—611no—
383C2×SD16C4×D6111yescharacteristic
384C2×D8—611no—
385C2×D8—611no—
386C2×M16—611no—
387C2×M16—611no—
388C2×M16—1221no—
389C2×M16—1221no—
390C22×C8—21no—
391C22×C8—1221no—
392C22×C8—1221no—
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1221no—
396C4×Q8—611no—
397C4×Q8C22×D6111yescharacteristic
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—611no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4C22×D6111yescharacteristic
408#117—611no—
409#117—1221no—
410C2×C4⋊C4—611no—
411C2×C22⋊C4—1221no—
412C2×C42—611no—
413C2×C42—1221no—
414#107—611no—
415#107—611no—
416#106—611no—
417#106C4×D6111yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8C22×D6111yescharacteristic
426C4⋊C8C22×D6111yescharacteristic
427#103—611no—
428#103C4×D6111yescharacteristic
429#102—611no—
430#102—611no—
431#102—611no—
432#102—611no—
433#102—611no—
434#102C4×D6111yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—611no—
437C22⋊C8—611no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444#97—21no—
445#97—611no—
446#97—611no—
447#97—1221no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—611no—
451C4×C8—1221no—
452C4×C8—1221no—
453C4×C8C22×D6111yescharacteristic
454C4×C8C22×D6111yescharacteristic
455C4×C8C22×D6111yescharacteristic
456C24—1221no—
457#41—1222no—
458#41—1222no—
459C2×Q8C2×C4×D6111yescharacteristic
460C2×D4—614no—
461C2×D4—611no—
462C2×D4—611no—
463C2×D4—1221no—
464C2×D4—1222no—
465C2×D4—1221no—
466C2×D4C2×C4×D6111yescharacteristic
467C22×C4—21no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—4881no—
479C22×C4—4881no—
480SD16—22no—
481SD16—1222no—
482D8—1222no—
483D8—1222no—
484M16—1222no—
485M16—1222no—
486M16—4881no—
487M16—4881no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—81no—
492C2×C8—611no—
493C2×C8—611no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—4881no—
500C2×C8—4881no—
501C2×C8—4881no—
502C2×C8—4881no—
503C2×C8—2442no—
504C2×C8C2×C4×D6111yescharacteristic
505C2×C8C2×C4×D6111yescharacteristic
506C2×C8D4×D6111yescharacteristic
507C2×C8D4×D6111yescharacteristic
508C4⋊C4—21no—
509C4⋊C4—611no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4C2×C4×D6111yescharacteristic
514C4⋊C4C2×C4×D6111yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C22⋊C4—1221no—
520C22⋊C4—1221no—
521C22⋊C4—1221no—
522C42—21no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—1221no—
527C42—1221no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—611no—
531C23—1224no—
532C23—1221no—
533C23—1221no—
534C23—4881no—
535C23—4881no—
536Q8—21no—
537Q8—21no—
538D4—21no—
539D4—612no—
540D4—1224no—
541D4—1221no—
542D4—1221no—
543D4—1221no—
544D4—2442no—
545D4M16×D6112yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—611no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—611no—
555C2×C4—611no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4881no—
561C2×C4—4881no—
562C2×C4—4881no—
563C2×C4—4881no—
564C2×C4—2441no—
565C2×C4—2442no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C8—22no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—21no—
578C8—41no—
579C8—41no—
580C8—1221no—
581C8—1221no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—612no—
591C22—1221no—
592C22—1221no—
593C22—1222no—
594C22—4884no—
595C22—2441no—
596C22—2441no—
597C22—2441no—
598C22—2441no—
599C22—2441no—
600C22C4×D4×D6111yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—612no—
606C4—1221no—
607C4—1221no—
608C4—2441no—
609C4—2441no—
610C4C2×M16×D6111yescharacteristic
611C4#131×D6111yescharacteristic
612C4#1684056611yescharacteristic
613C4#1684057111yescharacteristic
614C2—42no—
615C2—1222no—
616C2—4881no—
617C2—2442no—
618C2#435×D6111yescharacteristic
619C2#2145465011yescharacteristic
620C2#2145472411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1585C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519804C4
cosocleC42×C61C24
hypercenter#1585D61
hypocenterC61#17873
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1585

Complements

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

NH ≅ G/Ntype
1. #21454935C2semidirect
2. #21454840C2semidirect
4. #21444685C2semidirect
5. #21444680C2semidirect
6. #21444679C2semidirect
7. #21444656C2semidirect
8. #21444653C2semidirect
9. #21444648C2semidirect
10. #21444647C2semidirect
11. #21444570C2semidirect
12. #21444200C2semidirect
13. D4⋊C8×D61C2semidirect
14. #21444105C2semidirect
15. C61×#1585C2semidirect
19. #16839893C22semidirect
27. #16839841C22semidirect
28. #16839839C22semidirect
48. #16839514C22semidirect
49. #16839513C22semidirect
50. #16839512C22semidirect
51. Q8⋊#92418C22semidirect
53. #16839491C22semidirect
56. #16839489C22semidirect
57. #16839484C22semidirect
58. #16839483C22semidirect
59. #16839483C22semidirect
60. C61×#446C22semidirect
63. C61×#335C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C61×#322C22semidirect
154. D4×D61M16semidirect
310. #1585D61semidirect
330. #446C2×D61semidirect
362. #335C2×D61semidirect
366. Q8⋊C8C2×D61semidirect
376. #322C2×D61semidirect
377. C61#17873semidirect

Maximal subgroups

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

subgroupindex
#214549352
#214548402
C4×#145198042
#214446852
#214446802
#214446792
#214446562
#214446532
#214446482
#214446472
#214445702
#214442002
D4⋊C8×D612
#214441052
C61×#15852
#1787361