Finite Group Databasebeta

← #29647040

Subgroups — #29647040

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459427C211yesmaximal, characteristic
2#21454949C211yesmaximal, characteristic, complemented
3#21454461C211yesmaximal, characteristic, complemented
4#21444281C212yesmaximal, complemented
5#21444256C211yesmaximal, characteristic, complemented
6#21444242C212yesmaximal, complemented
7#21444236C211yesmaximal, characteristic, complemented
8#21444224C212yesmaximal, complemented
9#21444208C212yesmaximal, complemented
10#21444202C211yesmaximal, characteristic, complemented
11C61×#1492C211yesmaximal, characteristic, complemented
12#16840498C411yescharacteristic
13#16840473C411yescharacteristic
14#16840473C411yescharacteristic
15#16840459—21no—
16#16840459C412yes—
17#16840442—22no—
18#16840409—21no—
19#16840409C412yes—
20#16840397C411yescharacteristic
21#16840294—21no—
22#16840294C2212yes—
23#16840269—22no—
24C4×#11758222C2211yescharacteristic
25C4×#11758222C2211yescharacteristic
26C4×D4×D61—21no—
27C4×D4×D61C2212yes—
28#16839886—21no—
29#16839839—21no—
30#16839838—22no—
31#16839821—22no—
32#16839815C2211yescharacteristic, complemented
33#16839814C2212yescomplemented
34#16839813C2212yescomplemented
35#16839813C2212yescomplemented
36#16839788—21no—
37#16839784C2212yescomplemented
38#16839728—21no—
39C4×#11758194C2211yescharacteristic, complemented
40#16839494C2211yescharacteristic, complemented
41#16839494C2211yescharacteristic, complemented
42#16839492C2212yescomplemented
43#16839491C2211yescharacteristic
44#16839491C2211yescharacteristic
45D4⋊#92418C2212yescomplemented
46D4⋊#92418C2212yescomplemented
47#16839489C2212yes—
48C61×#532C2211yescharacteristic
49C61×#423C2211yescharacteristic, complemented
50C2×C4×C8×C61C2211yescharacteristic, complemented
51C61×D4⋊C8C2212yescomplemented
52C61×D4⋊C8C2212yescomplemented
53C2×#11758222C2×C411yescharacteristic
54C2×#11758222D411yescharacteristic
55C2×D4×D61—21no—
56C2×D4×D61C2×C412yes—
57C22×C4×D61—22no—
58C2×#11758194—21no—
59C2×#11758189—22no—
60#14519770—22no—
61#14519770—22no—
62#14519768—21no—
63#14519768C2×C412yes—
64#14519764—21no—
65#14519764C2×C412yes—
66#14519759—22no—
67#14519759—22no—
68#14519758—22no—
69#14519758—22no—
70#14519757—21no—
71#14519757C2×C412yes—
72#14519753—22no—
73#14519753—22no—
74#14519748—22no—
75#14519748—22no—
76#14519748—21no—
77#14519748C2×C412yes—
78#14519745—22no—
79#14519745—22no—
80C4×Dic122C2×C411yescharacteristic
81C4×Dic122C2×C411yescharacteristic
82C4×Dic122D411yescharacteristic
83C4×Dic122C2311yescharacteristic
84C4×D244C2×C411yescharacteristic
85C4×D244C2×C411yescharacteristic
86C4×D244D411yescharacteristic
87C4×D244C2311yescharacteristic
88D4×Dic61—21no—
89D4×Dic61C2×C412yes—
90#14519739—21no—
91#14519739C2×C412yes—
92C4×C22⋊D61—21no—
93C4×C22⋊D61—21no—
94C4×C22⋊D61—22no—
95C4×C22⋊D61—22no—
96C4×C22⋊D61C2×C412yes—
97C4×C22⋊D61D412yes—
98#14519737—22no—
99#14519737—22no—
100#14519736—22no—
101#14519736C2×C411yescharacteristic
102#14519736D411yescharacteristic
103#14519734—21no—
104#14519734D412yes—
105C4⋊C4×D61—21no—
106C4⋊C4×D61C2×C412yes—
107C22⋊C4×D61—22no—
108C22⋊C4×D61—22no—
109C42×D61—21no—
110C42×D61C2312yes—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113#14519706—22no—
114#14519706—22no—
115#14519706—22no—
116C22⋊#92418—21no—
117#14519689—22no—
118#14519689—22no—
119#14519689—22no—
120#14519688—21no—
121C4×#92418—21no—
122C4×#92418C2312yes—
123C8×Dic61—22no—
124C2×#41×C61—21no—
125C2×M16×C61—21no—
126C22×C8×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128#127×C61C2×C411yescharacteristic
129#124×C61—21no—
130#121×C61—21no—
131#121×C61C2×C412yes—
132C4×D4×C61—21no—
133C4×D4×C61C2312yes—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2312yes—
137C4×C8×C61—21no—
138C4×C8×C61C2311yescharacteristic
139C4×C8×C61C2311yescharacteristic
140C23×D61—22no—
141#41×C61—81no—
142#41×C61—81no—
143C2×Q8×C61—21no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—22no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—21no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—22no—
157D4×D61—24no—
158D4×D61—42no—
159D4×D61—44no—
160#11758222—22no—
161#11758222—41no—
162#11758222—42no—
163C2×D244C22⋊C411yescharacteristic
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—82no—
171C2×C4×D61—82no—
172C2×C4×D61C22×C412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×Dic122C22⋊C411yescharacteristic
176C2×Dic122C22×C411yescharacteristic
177M16×C61—81no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—22no—
181C2×C8×C61—42no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2411yescharacteristic
190C22⋊Dic61—21no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61C22⋊C412yes—
196#11758198—22no—
197#11758198—21no—
198#11758198—21no—
199#11758198—22no—
200#11758198C22⋊C412yes—
201#11758198C22⋊C412yes—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22⋊C411yescharacteristic
206C4⋊Dic61C22×C411yescharacteristic
207C4⋊Dic61C22×C411yescharacteristic
208#11758196—21no—
209#11758196—21no—
210#11758196—22no—
211#11758196—22no—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220#11758194—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—22no—
224#11758189—82no—
225C23×C61—22no—
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—21no—
228C22×D61—44no—
229C22×D61—44no—
230C22×D61—82no—
231C22×D61C2×C22⋊C412yes—
232Q8×C61—81no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—42no—
236D4×C61—81no—
237D4×C61—82no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C22⋊D61—42no—
253C22⋊D61—44no—
254C22⋊D61—42no—
255C22⋊D61—44no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—82no—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263D244—21no—
264D244—22no—
265D244#13112yes—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—42no—
275C4×D61—82no—
276C4×D61—82no—
277Dic122—22no—
278Dic122—21no—
279Dic122—21no—
280C8×C61—41no—
281C8×C61—41no—
282C8×C61—42no—
283#92418—42no—
284#92418—41no—
285#92418—41no—
286#17522—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—88no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61#42112yes—
306C4×C61C2×#13111yescharacteristic
307C4×C61#56811yescharacteristic
308C4×C61#57611yescharacteristic
309C4×C61#57711yescharacteristic
310Dic61—22no—
311Dic61—22no—
312Dic61—22no—
313Dic61—41no—
314Dic61—41no—
315Dic61—41no—
316#3481—611no—
317#2991—611no—
318#2922—611no—
319#1495—612no—
320#1492D6111yescharacteristic, complemented
321#1489—611no—
322#1486—612no—
323#1484—611no—
324#1482—612no—
325#1478—612no—
326#1476—611no—
327C2×C61—41no—
328C2×C61—82no—
329C2×C61#288011yescharacteristic
330C2×C61#289711yescharacteristic
331C2×C61C2×#42111yescharacteristic
332D61—42no—
333D61—42no—
334#559—611no—
335#554—611no—
336#554—611no—
337#553—612no—
338#553—1221no—
339#551—1222no—
340#546—612no—
341#546—1221no—
342D42—611no—
343#532C2×D6111yescharacteristic
344#529—612no—
345#529—1221no—
346#525—1222no—
347C4×#41—611no—
348C4×#41—611no—
349C2×C4×D4—612no—
350C2×C4×D4—1221no—
351C8×Q8—1221no—
352#436—1221no—
353#436—1222no—
354#433—1222no—
355#431—612no—
356#431—612no—
357#431—611no—
358#431—612no—
359#424—1221no—
360#423—612no—
361#423C2×D6111yescharacteristic, complemented
362#408—1221no—
363C4×M16—611no—
364C2×C4×C8C2×D6111yescharacteristic, complemented
365Q8⋊C8—612no—
366Q8⋊C8—611no—
367Q8⋊C8—611no—
368D4⋊C8—612no—
369D4⋊C8—612no—
370D4⋊C8—612no—
371D4⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8C2×D6112yescomplemented
374D4⋊C8C2×D6112yescomplemented
375C61#1752211yescharacteristic, complemented, Sylow
376C2×#41—21no—
377C2×#41—611no—
378C2×#41—611no—
379C22×D4—612no—
380C22×D4—1221no—
381C23×C4—1222no—
382C2×M16—21no—
383C2×M16—1222no—
384C2×M16—1221no—
385C22×C8—21no—
386C4⋊Q8C4×D6111yescharacteristic
387#127C4×D6111yescharacteristic
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1221no—
392#124—21no—
393#124—612no—
394#124—1221no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—1222no—
401C22⋊Q8—1221no—
402C22⋊Q8—1222no—
403#121—21no—
404#121—612no—
405#121—1222no—
406#121—1221no—
407#121—1222no—
408#121C4×D6112yes—
409#120—1222no—
410#120—1222no—
411C4×Q8—611no—
412C4×Q8—611no—
413C4×Q8—611no—
414C4×Q8—611no—
415C4×D4—21no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—611no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—611no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—1222no—
430C4×D4—1221no—
431C4×D4—1221no—
432C4×D4C22×D6112yes—
433#117—611no—
434#117—612no—
435#117—611no—
436#117—1222no—
437#117—1221no—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—1221no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C42—612no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445C4⋊C8—21no—
446C4⋊C8—611no—
447C4⋊C8—611no—
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8C22×D6112yes—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455C22⋊C8—1221no—
456#97—1221no—
457C4×C8—21no—
458C4×C8—612no—
459C4×C8—1222no—
460C4×C8—1221no—
461C4×C8C22×D6111yescharacteristic
462C4×C8C22×D6111yescharacteristic
463C24—1222no—
464#41—81no—
465#41—81no—
466#41—1222no—
467#41—2441no—
468#41—2442no—
469C2×Q8—21no—
470C2×Q8—611no—
471C2×Q8—611no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—612no—
476C2×D4—611no—
477C2×D4—611no—
478C2×D4—1222no—
479C2×D4—1224no—
480C2×D4—1222no—
481C2×D4—1221no—
482C2×D4—2442no—
483C2×D4—2444no—
484C2×D4C2×C4×D6112yes—
485C22×C4—22no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—4882no—
496C22×C4—4882no—
497C22×C4C2×C4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500M16—81no—
501M16—4882no—
502M16—2442no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—42no—
507C2×C8—1222no—
508C2×C8—1221no—
509C2×C8—1221no—
510C4⋊C4—21no—
511C4⋊C4—611no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—611no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—612no—
527C22⋊C4—612no—
528C22⋊C4—612no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C42—612no—
539C42—612no—
540C42—612no—
541C42—1221no—
542C42—1221no—
543C42—1221no—
544C42C2×C4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C42C23×D6111yescharacteristic
548C23—22no—
549C23—612no—
550C23—1221no—
551C23—4882no—
552C23—2444no—
553C23—2444no—
554C23C22⋊C4×D6111yescharacteristic
555Q8—81no—
556Q8—1222no—
557Q8—1221no—
558Q8—1221no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—81no—
563D4—82no—
564D4—612no—
565D4—1221no—
566D4—1222no—
567D4—2442no—
568D4—2444no—
569D4—2442no—
570D4—2444no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—4882no—
590C2×C4—4882no—
591C2×C4—4882no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4C22⋊C4×D6111yescharacteristic
598C2×C4C22⋊C4×D6111yescharacteristic
599C2×C4C22⋊C4×D6111yescharacteristic
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C8—41no—
604C8—41no—
605C8—42no—
606C8—2442no—
607C8—2441no—
608C8—2441no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—1222no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—4888no—
620C22—2441no—
621C22—2441no—
622C22C2×C22⋊C4×D6111yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—42no—
627C4—1222no—
628C4—1222no—
629C4—1222no—
630C4—2441no—
631C4—2441no—
632C4—2441no—
633C4#1683977712yes—
634C4#131×D6111yescharacteristic
635C4#1684052711yescharacteristic
636C4#1684056311yescharacteristic
637C4#1684056811yescharacteristic
638C2—41no—
639C2—82no—
640C2—2442no—
641C2—2442no—
642C2#2145426711yescharacteristic
643C2#2145434011yescharacteristic
644C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1492C2
socleC22×C61C22×C22⋊C4
prime radical#21459427C2
cosocleC42×C61C24
hypercenter#1492D61
hypocenterC61#17522
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #21454949C2semidirect
3. #21454461C2semidirect
4. #21444281C2semidirect
5. #21444256C2semidirect
6. #21444242C2semidirect
7. #21444236C2semidirect
8. #21444224C2semidirect
9. #21444208C2semidirect
10. #21444202C2semidirect
11. C61×#1492C2semidirect
32. #16839815C22semidirect
33. #16839814C22semidirect
34. #16839813C22semidirect
35. #16839813C22semidirect
37. #16839784C22semidirect
39. C4×#11758194C22semidirect
40. #16839494C22semidirect
41. #16839494C22semidirect
42. #16839492C22semidirect
45. D4⋊#92418C22semidirect
46. D4⋊#92418C22semidirect
49. C61×#423C22semidirect
50. C2×C4×C8×C61C22semidirect
51. C61×D4⋊C8C22semidirect
52. C61×D4⋊C8C22semidirect
320. #1492D61semidirect
361. #423C2×D61semidirect
364. C2×C4×C8C2×D61semidirect
373. D4⋊C8C2×D61semidirect
374. D4⋊C8C2×D61semidirect
375. C61#17522semidirect

Maximal subgroups

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

subgroupindex
#214594272
#214549492
#214544612
#214442812
#214442562
#214442422
#214442362
#214442242
#214442082
#214442022
C61×#14922
#1752261