Finite Group Databasebeta

← #29648404

Subgroups — #29648404

Order 15616.

996 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#432×D61C211yesmaximal, characteristic, complemented
3#431×D61C211yesmaximal, characteristic, complemented
4#21444529C211yesmaximal, characteristic, complemented
5#21444525C211yesmaximal, characteristic, complemented
6#21444522C211yesmaximal, characteristic, complemented
7#21444516C212yesmaximal, complemented
8#21444513C212yesmaximal, complemented
9#21444473C212yesmaximal, complemented
10#21444467C212yesmaximal, complemented
11C61×#1554C211yesmaximal, characteristic, complemented
12#16840317C2211yescharacteristic
13#16840317C2211yescharacteristic
14#16840313C2211yescharacteristic
15#16840313C2211yescharacteristic
16#16840312C2211yescharacteristic
17#16840299—21no—
18#16840299C412yes—
19#16840298—21no—
20#16840298C412yes—
21#16840297—21no—
22#16840297—21no—
23#16840275—21no—
24#16840275C412yes—
25#16840274—21no—
26C4×#11758226—21no—
27C4×#11758226C412yes—
28#16840230—21no—
29C2×C42×D61C2211yescharacteristic
30#16839819C2211yescharacteristic
31#16839818C2212yes—
32#16839817—21no—
33#16839816—21no—
34#16839815C2211yescharacteristic
35#16839814C2212yes—
36#16839813—21no—
37#16839812—21no—
38C4⋊C8×D61C2212yes—
39C4⋊C8×D61C2212yes—
40C22⋊C8×D61—21no—
41C22⋊C8×D61—21no—
42#97×D61—21no—
43C4×C8×D61—21no—
44#16839511C2212yescomplemented
45#16839509C2212yescomplemented
46#16839508C2212yescomplemented
47#16839507C2212yescomplemented
48#16839505C2212yescomplemented
49#16839505C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#432C2211yescharacteristic
52C61×#431C2211yescharacteristic
53C61×#333C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×#11758226—21no—
56C2×#11758226C2×C412yes—
57C2×#11758222—21no—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C2×C8×D61—22no—
62C2×C8×D61—22no—
63#14519777C2311yescharacteristic
64#14519774C2311yescharacteristic
65#14519772—22no—
66#14519772—22no—
67#14519768—21no—
68#14519768C2×C412yes—
69#14519766—21no—
70#14519766C2×C412yes—
71#14519765—21no—
72#14519762—22no—
73#14519762—22no—
74#14519761—22no—
75#14519757—21no—
76#14519757—21no—
77#14519753—21no—
78#14519753C2×C412yes—
79#14519752—21no—
80#14519752—21no—
81#14519752—21no—
82#14519752C2×C412yes—
83#14519751—21no—
84#14519751C2×C412yes—
85#14519747—22no—
86Q8×Dic61—21no—
87Q8×Dic61C2×C412yes—
88#14519742—21no—
89#14519742—21no—
90C4×D244—22no—
91C4×D244—22no—
92C4×D244—21no—
93C4×D244C2×C412yes—
94#14519739—22no—
95#14519739—22no—
96#14519739—21no—
97#14519739—21no—
98#14519739—21no—
99#14519739C2×C412yes—
100#14519737—22no—
101#14519737—22no—
102#14519735—21no—
103#14519735—22no—
104#14519735—22no—
105#14519735C2×C412yes—
106#14519734—22no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊#92418C2312yes—
120C4⋊#92418C2312yes—
121#14519706—22no—
122#14519706—22no—
123C22⋊#92418—21no—
124C22⋊#92418—21no—
125#14519689—22no—
126#14519689—22no—
127#14519688—21no—
128#14519687—22no—
129C4×#92418—21no—
130C8×Dic61—22no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133#125×C61—21no—
134C22⋊Q8×C61—21no—
135C22⋊Q8×C61C2×C412yes—
136C4×Q8×C61—21no—
137C4×Q8×C61C2×C412yes—
138#117×C61—21no—
139C2×C42×C61C2311yescharacteristic
140C4⋊C8×C61C2312yes—
141C4⋊C8×C61C2312yes—
142C22⋊C8×C61—21no—
143C22⋊C8×C61—21no—
144#97×C61—21no—
145C4×C8×C61—21no—
146C23×D61C22⋊C411yescharacteristic
147C2×Q8×C61—21no—
148C2×Q8×C61C22×C412yes—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C22×Dic61C22⋊C411yescharacteristic
156#11758226—44no—
157#11758226—44no—
158#11758222—44no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—24no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22×C411yescharacteristic
177C2×C4×D61C22×C411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×Dic122—21no—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C22×C412yes—
191C22⋊C4×C61—22no—
192C42×C61—22no—
193C42×C61C22×C411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2411yescharacteristic
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22×C412yes—
209#11758196—22no—
210#11758196—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C2×#92418—22no—
226C2×#92418—22no—
227C8×D61—44no—
228C8×D61—44no—
229C23×C61C2×C22⋊C411yescharacteristic
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—41no—
234C22×D61C2×C22⋊C411yescharacteristic
235C22×D61C2×C22⋊C411yescharacteristic
236Q8×C61—42no—
237Q8×C61—42no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C22⋊D61—44no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260D244—42no—
261D244—44no—
262D244—44no—
263D244—42no—
264D244—42no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61—44no—
279C4×D61#13514yes—
280Dic122—42no—
281C8×C61—42no—
282C8×C61—42no—
283#92418—42no—
284#92418—42no—
285#17713—611nomaximal, Sylow
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61C22×C22⋊C411yescharacteristic
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—44no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61#41312yes—
299C2×D61#41612yes—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61C2×#13512yes—
307C4×C61#57512yes—
308Dic61—21no—
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—21no—
313Dic61—42no—
314Dic61#41612yes—
315#3444—611no—
316C2×#432—611no—
317C2×#431—611no—
318#1554—611no—
319#1554D6111yescharacteristic, complemented
320#1553—611no—
321#1553—611no—
322#1552—612no—
323#1551—612no—
324#1542—612no—
325#1540—612no—
326C2×C61—41no—
327C2×C61C2×#41311yescharacteristic
328C2×C61C2×#41611yescharacteristic
329C2×C61#289611yescharacteristic
330D61—22no—
331D61—22no—
332D61—82no—
333#533—611no—
334#533—611no—
335#533C2×D6111yescharacteristic
336#532—611no—
337#532—611no—
338#532—611no—
339#529—612no—
340#529—612no—
341#529—1221no—
342#529—1221no—
343#529—1221no—
344#529—1221no—
345#525—612no—
346#525—1221no—
347#525—1221no—
348C4×#41—612no—
349C4×#41—1221no—
350C4×#41—1221no—
351C22×C42—611no—
352#432—611no—
353#432—612no—
354#432—1221no—
355#432—1221no—
356#432C2×D6111yescharacteristic
357#431—611no—
358#431—612no—
359#431—1221no—
360#431—1221no—
361#431C2×D6111yescharacteristic
362C2×C4⋊C8—612no—
363C2×C4⋊C8—612no—
364C2×C22⋊C8—1221no—
365C2×C22⋊C8—1221no—
366C2×#97—1221no—
367C2×C4×C8—1221no—
368#333—612no—
369#333C2×D6112yescomplemented
370#332—612no—
371#332—612no—
372#332—612no—
373#332C2×D6112yescomplemented
374#331—612no—
375#331—612no—
376C61#1771311yescharacteristic, complemented, Sylow
377C2×#41—612no—
378C2×#41—1221no—
379C2×#41—1221no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C22×C8—1222no—
384C22×C8—1222no—
385C4⋊Q8—611no—
386C4⋊Q8C22×D6111yescharacteristic
387C4⋊Q8C22×D6111yescharacteristic
388#127—611no—
389#126—1222no—
390#126—1222no—
391#125—21no—
392#125—612no—
393#125—1221no—
394#124—612no—
395#124—1221no—
396#124—1221no—
397#123—1222no—
398#123—1222no—
399#123—1222no—
400C22⋊Q8—21no—
401C22⋊Q8—612no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405C22⋊Q8C4×D6112yes—
406#121—612no—
407#121—612no—
408#121—1221no—
409#121—1221no—
410#121—1221no—
411#121—1221no—
412#120—1222no—
413C4×Q8—21no—
414C4×Q8—612no—
415C4×Q8—1221no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×Q8C4×D6112yes—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4—1221no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1221no—
431#117—21no—
432#117—612no—
433#117—1222no—
434#117—1222no—
435#117—1222no—
436#117—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—1222no—
449C2×C42C22×D6111yescharacteristic
450C4⋊C8—612no—
451C4⋊C8—612no—
452C4⋊C8—1222no—
453C4⋊C8—1222no—
454C4⋊C8C22×D6112yes—
455C4⋊C8C22×D6112yes—
456C22⋊C8—21no—
457C22⋊C8—21no—
458C22⋊C8—1222no—
459C22⋊C8—1222no—
460C22⋊C8—1221no—
461C22⋊C8—1221no—
462#97—21no—
463#97—1222no—
464#97—1221no—
465C4×C8—21no—
466C4×C8—1222no—
467C4×C8—1221no—
468C24—611no—
469#41—2444no—
470#41—2444no—
471#41—2444no—
472C2×Q8—21no—
473C2×Q8—1221no—
474C2×Q8C2×C4×D6112yes—
475C2×D4—612no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4—1221no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—1224no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4C2×C4×D6111yescharacteristic
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—2444no—
510C2×C8—2444no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—612no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1221no—
522C4⋊C4C2×C4×D6112yes—
523C22⋊C4—22no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C42—22no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—1222no—
546C42—1222no—
547C42C2×C4×D6111yescharacteristic
548C42D4×D6111yescharacteristic
549C42D4×D6111yescharacteristic
550C42C23×D6111yescharacteristic
551C23—611no—
552C23—611no—
553C23—1222no—
554C23—1222no—
555C23—1222no—
556C23—2441no—
557C23C22⋊C4×D6111yescharacteristic
558Q8—42no—
559Q8—42no—
560Q8—2442no—
561D4—2442no—
562D4—2444no—
563D4—2444no—
564D4—2444no—
565D4—2442no—
566D4—2442no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—611no—
573C2×C4—614no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2444no—
593C2×C4—2444no—
594C2×C4—2444no—
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—42no—
604C8—42no—
605C8—2442no—
606C8—2442no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—612no—
611C22—612no—
612C22—1221no—
613C22—1221no—
614C22—1221no—
615C22—1221no—
616C22—2444no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22C2×C22⋊C4×D6111yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—42no—
626C4—42no—
627C4—612no—
628C4—1221no—
629C4—1221no—
630C4—1221no—
631C4—1221no—
632C4—1221no—
633C4—2442no—
634C4#135×D6112yes—
635C4#1684056112yes—
636C2—41no—
637C2—1222no—
638C2—1222no—
639C2—4882no—
640C2#413×D6111yescharacteristic
641C2#416×D6111yescharacteristic
642C2#2145433711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1554C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1554D61
hypocenterC61#17713
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1554

Complements

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

NH ≅ G/Ntype
2. #432×D61C2semidirect
3. #431×D61C2semidirect
4. #21444529C2semidirect
5. #21444525C2semidirect
6. #21444522C2semidirect
7. #21444516C2semidirect
8. #21444513C2semidirect
9. #21444473C2semidirect
10. #21444467C2semidirect
11. C61×#1554C2semidirect
44. #16839511C22semidirect
45. #16839509C22semidirect
46. #16839508C22semidirect
47. #16839507C22semidirect
48. #16839505C22semidirect
49. #16839505C22semidirect
53. C61×#333C22semidirect
54. C61×#332C22semidirect
319. #1554D61semidirect
369. #333C2×D61semidirect
373. #332C2×D61semidirect
376. C61#17713semidirect

Maximal subgroups

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

subgroupindex
#214591162
#432×D612
#431×D612
#214445292
#214445252
#214445222
#214445162
#214445132
#214444732
#214444672
C61×#15542
#1771361