Finite Group Databasebeta

← #29647037

Subgroups — #29647037

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#21459429C211yesmaximal, characteristic
2#21454949C211yesmaximal, characteristic, complemented
3#21454464C211yesmaximal, characteristic, complemented
4#21444283C212yesmaximal, complemented
5#21444269C211yesmaximal, characteristic, complemented
6#21444242C212yesmaximal, complemented
7#21444237C211yesmaximal, characteristic, complemented
8#21444226C212yesmaximal, complemented
9#21444208C212yesmaximal, complemented
10#21444201C211yesmaximal, characteristic, complemented
11C61×#1489C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840472C411yescharacteristic
14#16840469C411yescharacteristic
15#16840458—21no—
16#16840456C412yes—
17#16840436—22no—
18#16840408—21no—
19#16840400C412yes—
20D4×D244C411yescharacteristic
21#16840312C2211yescharacteristic
22#16840299C2212yes—
23#16840296—21no—
24#16840271—22no—
25C4×#11758222C2211yescharacteristic
26C2×C4×D244—21no—
27C4×D4×D61C2212yes—
28#16839886—21no—
29#16839839—21no—
30#16839838—22no—
31#16839824—22no—
32#16839815C2211yescharacteristic, complemented
33#16839814C2212yescomplemented
34#16839813C2212yescomplemented
35#16839813C2212yescomplemented
36#16839790—21no—
37#16839784C2212yescomplemented
38#16839730—21no—
39C2×C4×#92418C2211yescharacteristic, complemented
40Q8⋊#92418C2211yescharacteristic, complemented
41Q8⋊#92418C2211yescharacteristic, complemented
42#16839491C2212yescomplemented
43#16839491C2212yescomplemented
44D4⋊#92418C2211yescharacteristic
45D4⋊#92418C2211yescharacteristic
46#16839489C2212yescomplemented
47#16839489C2212yescomplemented
48C4×#41×C61C2211yescharacteristic
49C61×#423C2211yescharacteristic, complemented
50C4×M16×C61C2211yescharacteristic, complemented
51C61×Q8⋊C8C2212yescomplemented
52C61×D4⋊C8C2212yes—
53C2×#11758222—21no—
54C2×#11758222D411yescharacteristic
55C22×D244—21no—
56C2×D4×D61C2×C412yes—
57C22×C4×D61—22no—
58C2×#11758189—22no—
59C22×#92418—21no—
60#14519777C2×C411yescharacteristic
61#14519774C2×C411yescharacteristic
62#14519772—22no—
63#14519771—22no—
64#14519769—21no—
65#14519768C2×C412yes—
66#14519766C2×C412yes—
67#14519765—21no—
68#14519765—21no—
69#14519762—22no—
70#14519762—22no—
71#14519760—22no—
72#14519759—22no—
73#14519756—21no—
74#14519755—22no—
75#14519754—22no—
76#14519753C2×C412yes—
77#14519752C2×C412yes—
78#14519751—21no—
79#14519751—21no—
80#14519751C2×C412yes—
81#14519750—22no—
82#14519749—22no—
83#14519747—22no—
84#14519745—22no—
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122D411yescharacteristic
87C4×D244—21no—
88C4×D244—22no—
89C4×D244—21no—
90C4×D244—22no—
91C4×D244—21no—
92C4×D244C2×C411yescharacteristic
93C4×D244D411yescharacteristic
94C4×D244C2312yes—
95D4×Dic61C2×C412yes—
96#14519739C2×C412yes—
97C4×C22⋊D61—22no—
98C4×C22⋊D61C2×C412yes—
99C4×C22⋊D61D412yes—
100#14519737—22no—
101#14519736D411yescharacteristic
102#14519734—22no—
103#14519734D412yes—
104C2×C4⋊Dic61—21no—
105C4⋊C4×D61C2×C412yes—
106C2×#11758198—22no—
107C22⋊C4×D61—22no—
108C42×D61C2312yes—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111#14519706—22no—
112#14519706—22no—
113#14519706—22no—
114C22⋊#92418—21no—
115#14519689—22no—
116#14519689—22no—
117#14519689—22no—
118C4×#92418—21no—
119C4×#92418—21no—
120C4×#92418C2311yescharacteristic
121C4×#92418C2311yescharacteristic
122C8×Dic61—22no—
123C2×#41×C61C2×C411yescharacteristic
124C2×M16×C61—21no—
125C2×M16×C61—21no—
126C4×Q8×C61C2×C411yescharacteristic
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2×C411yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#117×C61C2×C411yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138#97×C61—21no—
139C4×C8×C61C2312yes—
140C23×D61—22no—
141#41×C61—22no—
142#41×C61—41no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61C22⋊C412yes—
155C22×Dic61—22no—
156D4×D61—24no—
157D4×D61—42no—
158#11758222—42no—
159#11758222—81no—
160#11758222—81no—
161C2×D244—22no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—44no—
165C2×D244C22⋊C411yescharacteristic
166C2×D244C22×C412yes—
167C2×C4×D61—22no—
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×Dic122—21no—
176C2×Dic122C22⋊C411yescharacteristic
177M16×C61—42no—
178M16×C61—81no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2411yescharacteristic
195C22⋊Dic61—22no—
196C22⋊Dic61C22⋊C412yes—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198C22⋊C412yes—
205#11758198C22⋊C412yes—
206C4⋊Dic61—21no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22×C412yes—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196C22⋊C412yes—
215#11758196C22⋊C412yes—
216C4×Dic61C22×C412yes—
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D412yes—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—42no—
224#11758189—82no—
225C23×C61—21no—
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—22no—
228C22×D61—44no—
229C22×D61—44no—
230C22×D61—82no—
231C22×D61C2×C22⋊C412yes—
232Q8×C61—21no—
233Q8×C61—21no—
234D4×C61—21no—
235D4×C61—42no—
236D4×C61#13112yes—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C22⋊D61—42no—
256C22⋊D61—44no—
257C22⋊D61—82no—
258C2×Dic61—22no—
259C2×Dic61—82no—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263D244—22no—
264D244—22no—
265D244—22no—
266D244—44no—
267D244—42no—
268D244—81no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—82no—
277C4×D61—82no—
278Dic122—22no—
279Dic122—81no—
280C8×C61—42no—
281C8×C61—42no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#92418—41no—
286#17522—611nomaximal, Sylow
287C22×C61—22no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—88no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61#42112yes—
308C4×C61C2×#13111yescharacteristic
309C4×C61#56811yescharacteristic
310C4×C61#57611yescharacteristic
311C4×C61#57711yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316#3481—611no—
317#2991—611no—
318#2922—611no—
319#1495—612no—
320#1492—611no—
321#1489D6111yescharacteristic, complemented
322#1486—612no—
323#1484—611no—
324#1482—612no—
325#1478—612no—
326#1476—611no—
327C2×C61—42no—
328C2×C61—41no—
329C2×C61#288011yescharacteristic
330C2×C61#289711yescharacteristic
331C2×C61C2×#42111yescharacteristic
332D61—42no—
333D61—82no—
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#532—611no—
344#529—612no—
345#529—1221no—
346#525—1222no—
347C4×#41—611no—
348C4×#41C2×D6111yescharacteristic
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×M16C2×D6111yescharacteristic, complemented
364C2×C4×C8—611no—
365Q8⋊C8—611no—
366Q8⋊C8—611no—
367Q8⋊C8C2×D6112yescomplemented
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374D4⋊C8C2×D6112yes—
375C61#1752211yescharacteristic, complemented, Sylow
376C2×#41—611no—
377C2×#41—1221no—
378C2×#41C4×D6111yescharacteristic
379C22×D4—612no—
380C22×D4—1221no—
381C23×C4—1222no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—1222no—
385C22×C8—1221no—
386C4⋊Q8—611no—
387#127—611no—
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1221no—
392#124—612no—
393#124—1221no—
394#124—1221no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1221no—
403#121—612no—
404#121—612no—
405#121—1222no—
406#121—1222no—
407#121—1221no—
408#121—1221no—
409#120—1222no—
410#120—1222no—
411C4×Q8—611no—
412C4×Q8—611no—
413C4×Q8C4×D6111yescharacteristic
414C4×Q8C22×D6111yescharacteristic
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—612no—
424C4×D4—1222no—
425C4×D4—1221no—
426C4×D4—1222no—
427C4×D4—1221no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1221no—
431C4×D4C4×D6111yescharacteristic
432C4×D4C22×D6111yescharacteristic
433#117—21no—
434#117—611no—
435#117—612no—
436#117—1222no—
437#117C4×D6111yescharacteristic
438C2×C4⋊C4—612no—
439C2×C4⋊C4—1221no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C42—21no—
443C2×C42—612no—
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—21no—
457C4×C8—611no—
458C4×C8—611no—
459C4×C8—1222no—
460C4×C8—1221no—
461C4×C8—1221no—
462C4×C8C22×D6112yes—
463C24—1222no—
464#41—22no—
465#41—41no—
466#41—4881no—
467#41—4881no—
468#41—2442no—
469C2×Q8—611no—
470C2×Q8—1221no—
471C2×Q8C2×C4×D6111yescharacteristic
472C2×D4—21no—
473C2×D4—612no—
474C2×D4—612no—
475C2×D4—611no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1224no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—2444no—
483C2×D4—2442no—
484C2×D4C2×C4×D6111yescharacteristic
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—4882no—
496C22×C4—4882no—
497C22×C4C2×C4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500M16—42no—
501M16—81no—
502M16—4882no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—2442no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—611no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—611no—
516C4⋊C4—612no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4C2×C4×D6111yescharacteristic
523C4⋊C4C2×C4×D6111yescharacteristic
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—612no—
528C22⋊C4—612no—
529C22⋊C4—612no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—612no—
542C42—612no—
543C42—612no—
544C42C2×C4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C42C23×D6111yescharacteristic
548C23—21no—
549C23—612no—
550C23—1222no—
551C23—4882no—
552C23—2444no—
553C23—2444no—
554C23C22⋊C4×D6111yescharacteristic
555Q8—21no—
556Q8—21no—
557Q8—1222no—
558Q8—4881no—
559D4—21no—
560D4—42no—
561D4—1222no—
562D4—1222no—
563D4—1222no—
564D4—4881no—
565D4—4882no—
566D4—2444no—
567D4—2442no—
568D4—2444no—
569D4—2442no—
570D4#1451978512yes—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—4882no—
592C2×C4—4882no—
593C2×C4—4882no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
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—42no—
604C8—42no—
605C8—2441no—
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—1222no—
616C22—1222no—
617C22—1222no—
618C22—4888no—
619C22—2442no—
620C22—2442no—
621C22—2442no—
622C22C2×C22⋊C4×D6111yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—1222no—
630C4—1222no—
631C4—1222no—
632C4—2442no—
633C4#1683977712yes—
634C4C2×#1451978511yescharacteristic
635C4#1684052811yescharacteristic
636C4#1684056611yescharacteristic
637C4#1684057111yescharacteristic
638C2—41no—
639C2—42no—
640C2—4882no—
641C2—2442no—
642C2#2145426911yescharacteristic
643C2#2145434311yescharacteristic
644C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1489C2
socleC22×C61C22×C22⋊C4
prime radical#21459429C2
cosocleC42×C61C24
hypercenter#1489D61
hypocenterC61#17522
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1489

Complements

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

NH ≅ G/Ntype
2. #21454949C2semidirect
3. #21454464C2semidirect
4. #21444283C2semidirect
5. #21444269C2semidirect
6. #21444242C2semidirect
7. #21444237C2semidirect
8. #21444226C2semidirect
9. #21444208C2semidirect
10. #21444201C2semidirect
11. C61×#1489C2semidirect
32. #16839815C22semidirect
33. #16839814C22semidirect
34. #16839813C22semidirect
35. #16839813C22semidirect
37. #16839784C22semidirect
39. C2×C4×#92418C22semidirect
40. Q8⋊#92418C22semidirect
41. Q8⋊#92418C22semidirect
42. #16839491C22semidirect
43. #16839491C22semidirect
46. #16839489C22semidirect
47. #16839489C22semidirect
49. C61×#423C22semidirect
50. C4×M16×C61C22semidirect
51. C61×Q8⋊C8C22semidirect
321. #1489D61semidirect
361. #423C2×D61semidirect
363. C4×M16C2×D61semidirect
367. Q8⋊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
#214594292
#214549492
#214544642
#214442832
#214442692
#214442422
#214442372
#214442262
#214442082
#214442012
C61×#14892
#1752261