Finite Group Databasebeta

← #29649657

Subgroups — #29649657

Order 15616.

698 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#21454926C211yesmaximal, characteristic, complemented
2#21454857C211yesmaximal, characteristic, complemented
3C4×#14519801C211yesmaximal, characteristic
4#21444696C211yesmaximal, characteristic, complemented
5#21444695C211yesmaximal, characteristic, complemented
6#21444684C211yesmaximal, characteristic, complemented
7#21444683C211yesmaximal, characteristic, complemented
8#21444665C211yesmaximal, characteristic, complemented
9#21444664C211yesmaximal, characteristic, complemented
10#21444645C211yesmaximal, characteristic, complemented
11#21444570C211yesmaximal, characteristic, complemented
12#21444198C211yesmaximal, characteristic, complemented
13D4⋊C8×D61C211yesmaximal, characteristic, complemented
14#21444107C211yesmaximal, characteristic, complemented
15C61×#1581C211yesmaximal, characteristic, complemented
16C2×#14519801C411yescharacteristic
17C4×#11758224C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19C8×Dic122C2211yescharacteristic, complemented
20#16839886C2211yescharacteristic, complemented
21#16839874C411yescharacteristic
22#16839871C411yescharacteristic
23#16839861C411yescharacteristic
24#16839859C411yescharacteristic
25C4×#11758190C2211yescharacteristic
26C4×#11758200C2211yescharacteristic
27C4×#11758199C2211yescharacteristic
28#16839841C2211yescharacteristic
29#16839840C2211yescharacteristic
30#16839839C2211yescharacteristic
31#16839838—21no—
32#16839837—21no—
33D4×#92418C2211yescharacteristic, complemented
34C8×C22⋊D61—21no—
35#16839825—21no—
36#16839814C2211yescharacteristic
37#16839813C2211yescharacteristic
38#16839812—21no—
39#16839803C411yescharacteristic
40C2×#14519706—21no—
41C4⋊C8×D61C2211yescharacteristic
42#16839765C411yescharacteristic
43#16839760C411yescharacteristic
44#16839725—21no—
45#16839724—21no—
46C4×#11758189C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic, complemented
49#16839513C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839494C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53#16839491C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic, complemented
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59#16839484C2211yescharacteristic, complemented
60C4×D8×C61C2211yescharacteristic
61C61×#436C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic, complemented
63C61×#335C2211yescharacteristic, complemented
64C61×D4⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic, complemented
66C61×#322C2211yescharacteristic, complemented
67C2×#11758224C2×C411yescharacteristic
68C2×D4×D61C2×C411yescharacteristic
69C22×C4×D61—21no—
70#14519801—24no—
71C2×#11758190C2×C411yescharacteristic
72C2×#11758200C2×C411yescharacteristic
73C2×#11758199C2×C411yescharacteristic
74C2×#11758194—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C2×#11758189C2×C411yescharacteristic
78C2×#11758189C2×C411yescharacteristic
79C22×#92418—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C4×Dic122C2311yescharacteristic
83#14519742C2×C411yescharacteristic
84C4×D244C2311yescharacteristic
85D4×Dic61C2×C411yescharacteristic
86D4×Dic61C2×C411yescharacteristic
87#14519739C2×C411yescharacteristic
88C4×C22⋊D61—21no—
89C4×C22⋊D61—21no—
90#14519737—21no—
91#14519737—21no—
92#14519735C2×C411yescharacteristic
93#14519733—21no—
94C4⋊C4×D61C2×C411yescharacteristic
95C22⋊C4×D61—21no—
96C2×C4×Dic61—21no—
97C42×D61C2311yescharacteristic
98#14519710C2×C411yescharacteristic
99#14519709C2×C411yescharacteristic
100#14519708C2×C411yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706D411yescharacteristic
107#14519706D411yescharacteristic
108#14519702C2×C411yescharacteristic
109#14519700C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114C22⋊#92418—21no—
115C22⋊#92418—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689D411yescharacteristic
120#14519689D411yescharacteristic
121#14519687C2×C411yescharacteristic
122#14519687C2×C411yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×D8×C61C2×C411yescharacteristic
131C2×M16×C61—21no—
132C22×C8×C61—21no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#107×C61C2×C411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#102×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155#11758224—22no—
156#11758224M1614yescomplemented
157D4×D61—22no—
158D4×D61—22no—
159C2×D244C22×C411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C4×D61C22×C411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61#4111yescharacteristic
168C2×Dic122C22×C411yescharacteristic
169D8×C61—22no—
170M16×C61—81no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—81no—
175C2×C8×C61C22×C411yescharacteristic
176C2×C8×C61C22×C411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61C22×C411yescharacteristic
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—21no—
185C22⋊Dic61—21no—
186#11758200—22no—
187#11758199—22no—
188#11758198—21no—
189#11758198—21no—
190#11758198—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61C22×C411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61#4111yescharacteristic
202#11758194—81no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—81no—
207C2×#92418C22×C411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758190—22no—
212#11758189—22no—
213#11758189—22no—
214#11758189—81no—
215#11758189—81no—
216C8×D61—42no—
217C8×D61—81no—
218C23×C61—21no—
219C23×C61—21no—
220C22×D61—21no—
221C22×D61—42no—
222C22×D61—81no—
223C22×D61—81no—
224C22×D61C4×D411yescharacteristic
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61C2×M1612yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C22⋊D61—24no—
239C22⋊D61—42no—
240C2×Dic61—21no—
241C2×Dic61—24no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—81no—
245C2×Dic61—81no—
246C2×Dic61C4×D411yescharacteristic
247C2×Dic61C4×D411yescharacteristic
248C2×Dic61C4×D411yescharacteristic
249D244—21no—
250D244—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—81no—
257C4×D61—81no—
258C4×D61C2×M1612yes—
259C4×D61#13112yes—
260Dic122—21no—
261Dic122C2×M1612yes—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—21no—
266C8×C61—41no—
267C8×C61—41no—
268C8×C61—41no—
269#92418—22no—
270#92418—21no—
271#92418—21no—
272#92418—21no—
273#92418—21no—
274#92418—41no—
275#92418—41no—
276#17875—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61C2×C4×D411yescharacteristic
284C2×D61—21no—
285C2×D61—21no—
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×C61C22×M1611yescharacteristic
295C4×C61C2×#13111yescharacteristic
296C4×C61#57611yescharacteristic
297C4×C61#57711yescharacteristic
298Dic61—22no—
299Dic61—21no—
300Dic61—21no—
301Dic61—41no—
302Dic61—41no—
303Dic61#43512yes—
304#2988—611no—
305#2975—611no—
306C4×AGL(1,ℤ/8)—611no—
307#1588—611no—
308#1588—611no—
309#1586—611no—
310#1586—611no—
311#1583—611no—
312#1583—611no—
313#1581—611no—
314#1581D6111yescharacteristic, complemented
315#1565—611no—
316#1476—611no—
317C2×D4⋊C8—611no—
318#1451—611no—
319C2×C61—42no—
320C2×C61—81no—
321C2×C61C2×#43511yescharacteristic
322C2×C61#294611yescharacteristic
323C2×C61#295611yescharacteristic
324D61—22no—
325D61—81no—
326C2×AGL(1,ℤ/8)—611no—
327C4×#41—611no—
328C2×C4×D4—611no—
329C8×Q8—611no—
330C8×Q8—611no—
331#442—611no—
332#442—611no—
333#440—611no—
334#440—611no—
335C4×SD16—611no—
336C4×SD16—611no—
337C4×D8—611no—
338C4×D8C2×D6111yescharacteristic
339#436—611no—
340#436—611no—
341#436—611no—
342#436—1221no—
343#436—1221no—
344#436C2×D6111yescharacteristic
345C8×D4—611no—
346C8×D4—1221no—
347C8×D4—1221no—
348C8×D4C2×D6111yescharacteristic, complemented
349#431—611no—
350#431—611no—
351#431—1221no—
352#428—611no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8—1221no—
355#418—611no—
356#417—611no—
357#407—1221no—
358#407—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⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8C2×D6111yescharacteristic
372D4⋊C8C2×D6111yescharacteristic, complemented
373#322—611no—
374#322—611no—
375#322—611no—
376#322C2×D6111yescharacteristic, complemented
377C61#1787511yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C22×D4—611no—
380C23×C4—1221no—
381AGL(1,ℤ/8)—1224no—
382C2×SD16—611no—
383C2×SD16—611no—
384C2×D8—611no—
385C2×D8C4×D6111yescharacteristic
386C2×M16—21no—
387C2×M16—611no—
388C2×M16—611no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16—1221no—
392C22×C8—21no—
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1221no—
396C4×Q8—611no—
397C4×Q8—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4C22×D6111yescharacteristic
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#107C4×D6111yescharacteristic
416#106—611no—
417#106—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8C22×D6111yescharacteristic
426C4⋊C8C22×D6111yescharacteristic
427#103—611no—
428#103—611no—
429#102—611no—
430#102—611no—
431#102—611no—
432#102—611no—
433#102C4×D6111yescharacteristic
434#102C4×D6111yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8—611no—
438C22⋊C8—611no—
439C22⋊C8—1221no—
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444#97—611no—
445#97—611no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—611no—
449C4×C8—1221no—
450C4×C8—1221no—
451C4×C8—1221no—
452C4×C8—1221no—
453C4×C8C22×D6111yescharacteristic
454C4×C8C22×D6111yescharacteristic
455C4×C8C22×D6111yescharacteristic
456C24—1221no—
457#41—614no—
458#41—1222no—
459C2×Q8—611no—
460C2×D4—611no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4C2×C4×D6111yescharacteristic
466C2×D4C2×C4×D6111yescharacteristic
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
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—1222no—
481SD16—1222no—
482D8—22no—
483D8—1222no—
484M16—81no—
485M16—1222no—
486M16—1222no—
487M16—4881no—
488M16—4881no—
489M16—4881no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—81no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
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—611no—
509C4⋊C4—611no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4C2×C4×D6111yescharacteristic
514C4⋊C4C2×C4×D6111yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C22⋊C4—1221no—
520C22⋊C4—1221no—
521C22⋊C4—1221no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—1221no—
526C42—1221no—
527C42—1221no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—611no—
532C23—1221no—
533C23—4881no—
534C23—4881no—
535C23—2442no—
536Q8—612no—
537Q8—1221no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—1224no—
542D4—1221no—
543D4—1221no—
544D4—2442no—
545D4M16×D6112yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—611no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—611no—
556C2×C4—611no—
557C2×C4—1224no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—4881no—
564C2×C4—4881no—
565C2×C4—4881no—
566C2×C4—4881no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C8—21no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—1222no—
581C8—1221no—
582C8—1221no—
583C8—1221no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—612no—
594C22—1221no—
595C22—1221no—
596C22—4884no—
597C22—2441no—
598C22—2441no—
599C22—2441no—
600C22C4×D4×D6111yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—612no—
605C4—1221no—
606C4—1221no—
607C4—1222no—
608C4—2441no—
609C4—2441no—
610C4C2×M16×D6111yescharacteristic
611C4#131×D6111yescharacteristic
612C4#1684056311yescharacteristic
613C4#1684056911yescharacteristic
614C2—42no—
615C2—81no—
616C2—1222no—
617C2—4881no—
618C2#435×D6111yescharacteristic
619C2#2145464411yescharacteristic
620C2#2145472711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1581C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519801C4
cosocleC42×C61C24
hypercenter#1581D61
hypocenterC61#17875
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
1. #21454926C2semidirect
2. #21454857C2semidirect
4. #21444696C2semidirect
5. #21444695C2semidirect
6. #21444684C2semidirect
7. #21444683C2semidirect
8. #21444665C2semidirect
9. #21444664C2semidirect
10. #21444645C2semidirect
11. #21444570C2semidirect
12. #21444198C2semidirect
13. D4⋊C8×D61C2semidirect
14. #21444107C2semidirect
15. C61×#1581C2semidirect
19. C8×Dic122C22semidirect
20. #16839886C22semidirect
33. D4×#92418C22semidirect
48. #16839514C22semidirect
49. #16839513C22semidirect
50. #16839512C22semidirect
51. #16839494C22semidirect
52. #16839492C22semidirect
55. D4⋊#92418C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. #16839484C22semidirect
62. C8×D4×C61C22semidirect
63. C61×#335C22semidirect
65. C61×D4⋊C8C22semidirect
66. C61×#322C22semidirect
156. #11758224M16semidirect
314. #1581D61semidirect
348. C8×D4C2×D61semidirect
362. #335C2×D61semidirect
372. D4⋊C8C2×D61semidirect
376. #322C2×D61semidirect
377. C61#17875semidirect

Maximal subgroups

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

subgroupindex
#214549262
#214548572
C4×#145198012
#214446962
#214446952
#214446842
#214446832
#214446652
#214446642
#214446452
#214445702
#214441982
D4⋊C8×D612
#214441072
C61×#15812
#1787561