Finite Group Databasebeta

← #29650202

Subgroups — #29650202

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455817C211yesmaximal, characteristic
2#21454528C211yesmaximal, characteristic, complemented
3#436×D61C211yesmaximal, characteristic, complemented
4#21444705C211yesmaximal, characteristic, complemented
5#21444702C211yesmaximal, characteristic, complemented
6#21444702C211yesmaximal, characteristic, complemented
7#21444651C211yesmaximal, characteristic, complemented
8#21444649C211yesmaximal, characteristic, complemented
9#21444647C211yesmaximal, characteristic, complemented
10#21444645C211yesmaximal, characteristic, complemented
11#21444592C211yesmaximal, characteristic, complemented
12#21444242C211yesmaximal, characteristic, complemented
13#21444216C211yesmaximal, characteristic, complemented
14#21444101C211yesmaximal, characteristic, complemented
15C61×#1589C211yesmaximal, characteristic, complemented
16C2×#14519785—21no—
17C2×M16×D61—21no—
18#16840309C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840077C411yescharacteristic
21#16840074C411yescharacteristic
22#16840041C411yescharacteristic
23#16840040C411yescharacteristic
24#16839956C411yescharacteristic
25#16839955C411yescharacteristic
26#16839897C411yescharacteristic
27#16839894C411yescharacteristic
28C4×D488C2211yescharacteristic
29C4×#11758199C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31#16839841C2211yescharacteristic
32#16839841C2211yescharacteristic, complemented
33#16839840C2211yescharacteristic
34#16839839C2211yescharacteristic
35#16839838—21no—
36#16839838—21no—
37#16839837—21no—
38D4×#92418C2211yescharacteristic, complemented
39#16839827C2211yescharacteristic, complemented
40#16839825—21no—
41#16839813C2211yescharacteristic
42#16839813C2211yescharacteristic
43#16839784C2211yescharacteristic
44C4⋊C8×D61C2211yescharacteristic
45#16839727—21no—
46C22⋊C8×D61—21no—
47C4×C8×D61C2211yescharacteristic
48#16839515C2211yescharacteristic, complemented
49#16839514C2211yescharacteristic, complemented
50#16839514C2211yescharacteristic, complemented
51#16839491C2211yescharacteristic
52#16839491C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55#16839489C2211yescharacteristic
56#16839489C2211yescharacteristic, complemented
57#16839484C2211yescharacteristic, complemented
58#16839484C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C4×D8×C61C2211yescharacteristic
61C61×#436C2211yescharacteristic
62C61×#436C2211yescharacteristic, complemented
63C61×#335C2211yescharacteristic, complemented
64C61×D4⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic, complemented
66C82×C61C2211yescharacteristic, complemented
67C2×#11758222—21no—
68C2×D4×D61C2×C411yescharacteristic
69C22×C4×D61—21no—
70C2×D488C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758199C2×C411yescharacteristic
73#14519785—44no—
74C2×#11758194—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C2×#11758189—21no—
79M16×D61—44no—
80C22×#92418—21no—
81C2×C8×D61—21no—
82C2×C8×D61—21no—
83C2×C8×D61D411yescharacteristic
84C2×C8×D61D411yescharacteristic
85#14519776C2×C411yescharacteristic
86#14519773C2×C411yescharacteristic
87#14519763—21no—
88#14519752C2×C411yescharacteristic
89#14519750—21no—
90#14519748C2×C411yescharacteristic
91C4×D244C2311yescharacteristic
92C4×D244C2311yescharacteristic
93D4×Dic61C2×C411yescharacteristic
94#14519739C2×C411yescharacteristic
95C4×C22⋊D61—21no—
96C4×C22⋊D61—21no—
97#14519737—21no—
98C4⋊C4×D61C2×C411yescharacteristic
99C22⋊C4×D61—21no—
100C42×D61C2311yescharacteristic
101#14519711C2×C411yescharacteristic
102#14519710C2×C411yescharacteristic
103#14519710C2×C411yescharacteristic
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706C2×C411yescharacteristic
110#14519706C2×C411yescharacteristic
111#14519699C2×C411yescharacteristic
112#14519699C2×C411yescharacteristic
113#14519698C2×C411yescharacteristic
114#14519698C2×C411yescharacteristic
115#14519697C2×C411yescharacteristic
116#14519697C2×C411yescharacteristic
117C22⋊#92418—21no—
118C22⋊#92418—21no—
119#14519689—21no—
120#14519689—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689C2×C411yescharacteristic
125#14519689C2×C411yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61D411yescharacteristic
130C8×Dic61D411yescharacteristic
131C2×D8×C61C2×C411yescharacteristic
132C2×M16×C61—21no—
133C2×M16×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#107×C61C2×C411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#102×C61C2×C411yescharacteristic
140#102×C61C2×C411yescharacteristic
141C22⋊C8×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C4×C8×C61C2311yescharacteristic
146C23×D61—21no—
147C2×D4×C61C22×C411yescharacteristic
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C22×Dic61—21no—
156D4×D61—41no—
157D4×D61—41no—
158#11758222—42no—
159#11758222—42no—
160C2×D244—21no—
161C2×D244C22×C411yescharacteristic
162C2×D244C22×C411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61C22×C411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61#4111yescharacteristic
172C2×Dic122—21no—
173D8×C61—41no—
174M16×C61—42no—
175M16×C61—42no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C22×C411yescharacteristic
180C2×C8×C61C22×C411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61C22×C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—21no—
189C22⋊Dic61—21no—
190#11758199—41no—
191#11758199—41no—
192#11758198—21no—
193#11758198—21no—
194#11758198—21no—
195#11758198—21no—
196C4⋊Dic61C22×C411yescharacteristic
197C4⋊Dic61C22×C411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200C4×Dic61C22×C411yescharacteristic
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61#4111yescharacteristic
203#11758194—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—42no—
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212D488—41no—
213#11758189—42no—
214#11758189—42no—
215#11758189—42no—
216#11758189—42no—
217C8×D61—22no—
218C8×D61—42no—
219C8×D61—42no—
220C8×D61D814yescomplemented
221C23×C61—21no—
222C23×C61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61C4×D411yescharacteristic
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—42no—
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—42no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—42no—
249C2×Dic61C4×D411yescharacteristic
250C2×Dic61C4×D411yescharacteristic
251C2×Dic61C4×D411yescharacteristic
252D244—21no—
253D244—21no—
254D244—21no—
255D244—21no—
256D244—42no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—41no—
260C4×D61—42no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—42no—
266C4×D61C2×D812yes—
267C4×D61#13512yes—
268Dic122—42no—
269C8×C61—21no—
270C8×C61—21no—
271C8×C61—21no—
272C8×C61—41no—
273C8×C61—41no—
274C8×C61—41no—
275C8×C61C2×D812yes—
276#92418—21no—
277#92418—21no—
278#92418—21no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#92418C2×D812yes—
283#17928—611nomaximal, Sylow
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61C2×C4×D411yescharacteristic
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—84no—
300C2×D61C4×D812yes—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61#56811yescharacteristic
305C4×C61#56811yescharacteristic
306C4×C61C22×D811yescharacteristic
307C4×C61C2×#13511yescharacteristic
308Dic61—21no—
309Dic61—21no—
310Dic61—41no—
311Dic61—41no—
312Dic61C4×D812yes—
313#3100—611no—
314#2931—611no—
315C2×#436—611no—
316#1589—611no—
317#1589—611no—
318#1589—611no—
319#1589D6111yescharacteristic, complemented
320#1581—611no—
321#1581—611no—
322#1581—611no—
323#1581—611no—
324#1571—611no—
325#1486—611no—
326#1480—611no—
327#1450—611no—
328C2×C61—81no—
329C2×C61—81no—
330C2×C61#293011yescharacteristic
331C2×C61C2×C4×D811yescharacteristic
332C2×C61#295511yescharacteristic
333D61—22no—
334D61—81no—
335D61—81no—
336C2×#131—1221no—
337C22×M16—1221no—
338#532—611no—
339C2×C4×D4—611no—
340#480—611no—
341#480—611no—
342#474—611no—
343#474—611no—
344#459—611no—
345#459—611no—
346#447—611no—
347#447—611no—
348C4×D8—611no—
349C4×D8—611no—
350C4×D8—611no—
351C4×D8C2×D6111yescharacteristic
352#436—611no—
353#436—611no—
354#436—611no—
355#436—611no—
356#436—1221no—
357#436—1221no—
358#436—1221no—
359#436C2×D6111yescharacteristic
360#436C2×D6111yescharacteristic, complemented
361C8×D4—611no—
362C8×D4—611no—
363C8×D4—1221no—
364#431—611no—
365#431—611no—
366#423—611no—
367C2×C4⋊C8—611no—
368#408—1221no—
369C2×C22⋊C8—1221no—
370C2×C4×C8—611no—
371#335—611no—
372#335—611no—
373#335—611no—
374#335C2×D6111yescharacteristic, complemented
375D4⋊C8—611no—
376D4⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380D4⋊C8—611no—
381D4⋊C8C2×D6111yescharacteristic
382D4⋊C8C2×D6111yescharacteristic, complemented
383#322—611no—
384#322—611no—
385C82—611no—
386C82C2×D6111yescharacteristic, complemented
387C61#1792811yescharacteristic, complemented, Sylow
388C2×#41—1221no—
389C22×D4—611no—
390C23×C4—1221no—
391C2×D8—611no—
392C2×D8—611no—
393C2×D8—611no—
394C2×D8C4×D6111yescharacteristic
395#131—2444no—
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—1221no—
399C2×M16—1221no—
400C2×M16—1221no—
401C2×M16—1221no—
402C2×M16—1221no—
403C2×M16—2444no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8—1221no—
407C22×C8—1221no—
408C22×C8—1221no—
409C4⋊Q8—611no—
410#127—611no—
411#124—1221no—
412#121—611no—
413#121—611no—
414#121—1221no—
415C4×D4—611no—
416C4×D4—611no—
417C4×D4—611no—
418C4×D4—611no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4C22×D6111yescharacteristic
423C4×D4C22×D6111yescharacteristic
424C2×C4⋊C4—611no—
425C2×C22⋊C4—1221no—
426C2×C42—611no—
427#107—611no—
428#107—611no—
429#107—611no—
430#107C4×D6111yescharacteristic
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—1221no—
436C4⋊C8—1221no—
437C4⋊C8—1221no—
438C4⋊C8C22×D6111yescharacteristic
439C4⋊C8C22×D6111yescharacteristic
440#102—611no—
441#102—611no—
442#102—611no—
443#102—611no—
444#102—611no—
445#102—611no—
446#102C4×D6111yescharacteristic
447#102C4×D6111yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—611no—
451C22⋊C8—611no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458C22⋊C8—1221no—
459C4×C8—611no—
460C4×C8—611no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—611no—
464C4×C8C22×D6111yescharacteristic
465C4×C8C22×D6111yescharacteristic
466C4×C8C22×D6111yescharacteristic
467C24—1221no—
468#41—2442no—
469#41—2442no—
470C2×Q8—1221no—
471C2×D4—611no—
472C2×D4—611no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—2441no—
479C2×D4—2441no—
480C2×D4C2×C4×D6111yescharacteristic
481C2×D4C2×C4×D6111yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494D8—41no—
495D8—2441no—
496D8—2441no—
497D8—2441no—
498M16—42no—
499M16—42no—
500M16—2442no—
501M16—2442no—
502M16—2442no—
503M16—2442no—
504M16—2442no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—611no—
509C2×C8—614no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—1222no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—2442no—
518C2×C8—2442no—
519C2×C8—2442no—
520C2×C8C2×C4×D6111yescharacteristic
521C2×C8C2×C4×D6111yescharacteristic
522C2×C8D4×D6111yescharacteristic
523C2×C8D4×D6111yescharacteristic
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4C2×C4×D6111yescharacteristic
529C4⋊C4C2×C4×D6111yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—1221no—
533C22⋊C4—1221no—
534C22⋊C4—1221no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42C23×D6111yescharacteristic
542C23—21no—
543C23—21no—
544C23—611no—
545C23—1221no—
546C23—1221no—
547C23—2442no—
548C23—2442no—
549C23—2442no—
550Q8—2442no—
551D4—21no—
552D4—21no—
553D4—21no—
554D4—21no—
555D4—1221no—
556D4—1221no—
557D4—1221no—
558D4—1221no—
559D4—2442no—
560D4—2442no—
561D4—2442no—
562D4—2442no—
563D4—2442no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—611no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—2441no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4C22×C4×D6111yescharacteristic
588C2×C4C2×D4×D6111yescharacteristic
589C2×C4#41×D6111yescharacteristic
590C8—21no—
591C8—21no—
592C8—21no—
593C8—41no—
594C8—41no—
595C8—41no—
596C8—612no—
597C8—1221no—
598C8—1221no—
599C8—1221no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C8D8×D6112yes—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—612no—
611C22—1221no—
612C22—1221no—
613C22—4884no—
614C22—2441no—
615C22—2441no—
616C22—2441no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22C4×D4×D6111yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—612no—
625C4—1221no—
626C4—1221no—
627C4—2441no—
628C4—2441no—
629C4#1684052711yescharacteristic
630C4#1684052811yescharacteristic
631C4C2×D8×D6111yescharacteristic
632C4#135×D6111yescharacteristic
633C2—81no—
634C2—81no—
635C2—1222no—
636C2—4881no—
637C2—4881no—
638C2#2145451911yescharacteristic
639C2C4×D8×D6111yescharacteristic
640C2#2145471711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1589C2
socleC22×C61C2×C4×D4
prime radical#21455817C2
cosocleC42×C61C24
hypercenter#1589D61
hypocenterC61#17928
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1589

Complements

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

NH ≅ G/Ntype
2. #21454528C2semidirect
3. #436×D61C2semidirect
4. #21444705C2semidirect
5. #21444702C2semidirect
6. #21444702C2semidirect
7. #21444651C2semidirect
8. #21444649C2semidirect
9. #21444647C2semidirect
10. #21444645C2semidirect
11. #21444592C2semidirect
12. #21444242C2semidirect
13. #21444216C2semidirect
14. #21444101C2semidirect
15. C61×#1589C2semidirect
32. #16839841C22semidirect
38. D4×#92418C22semidirect
39. #16839827C22semidirect
48. #16839515C22semidirect
49. #16839514C22semidirect
50. #16839514C22semidirect
52. #16839491C22semidirect
54. D4⋊#92418C22semidirect
56. #16839489C22semidirect
57. #16839484C22semidirect
58. #16839484C22semidirect
59. C8×#92418C22semidirect
62. C61×#436C22semidirect
63. C61×#335C22semidirect
65. C61×D4⋊C8C22semidirect
66. C82×C61C22semidirect
220. C8×D61D8semidirect
319. #1589D61semidirect
360. #436C2×D61semidirect
374. #335C2×D61semidirect
382. D4⋊C8C2×D61semidirect
386. C82C2×D61semidirect
387. C61#17928semidirect

Maximal subgroups

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

subgroupindex
#214558172
#214545282
#436×D612
#214447052
#214447022
#214446512
#214446492
#214446472
#214446452
#214445922
#214442422
#214442162
#214441012
C61×#15892
#1792861