Finite Group Databasebeta

← #29647080

Subgroups — #29647080

Order 15616.

990 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#21459445C211yesmaximal, characteristic
2#21454463C211yesmaximal, characteristic, complemented
3#21444287C212yesmaximal, complemented
4#21444269C211yesmaximal, characteristic, complemented
5#21444256C211yesmaximal, characteristic, complemented
6#21444251C212yesmaximal, complemented
7#21444235C211yesmaximal, characteristic, complemented
8#21444217C212yesmaximal, complemented
9#21444204C212yesmaximal, complemented
10#21444199C211yesmaximal, characteristic, complemented
11C61×#2990C211yesmaximal, characteristic, complemented
12Q8×Dic122C411yescharacteristic
13#16840477C411yescharacteristic
14#16840477C412yes—
15#16840475—21no—
16#16840475—22no—
17#16840434C411yescharacteristic
18#16840433C412yes—
19#16840433C411yescharacteristic
20#16840431—21no—
21#16840312C2212yes—
22#16840312C2211yescharacteristic
23#16840309—21no—
24#16840309—22no—
25C4×#11758226—21no—
26C4×#11758222C2211yescharacteristic
27C4×#11758222C2212yes—
28#16839823—22no—
29#16839815C2212yes—
30#16839815C2212yes—
31#16839789—21no—
32#16839729—21no—
33C4×#11758194C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839492C2211yescharacteristic
36#16839492C2211yescharacteristic
37#16839492C2212yes—
38#16839489C2211yescharacteristic
39#16839489C2211yescharacteristic
40#16839489C2212yes—
41#16839489C2212yes—
42#16839489C2212yes—
43#16839489C2212yes—
44#16839489C2212yes—
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2212yes—
50C61×#431C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C2×C4⋊C8×C61C2212yes—
53C2×#11758226—21no—
54C2×#11758222—22no—
55C2×#11758222—21no—
56C2×#11758222C2×C411yescharacteristic
57C2×#11758222C2×C412yes—
58C2×#11758194—21no—
59C22×#92418—21no—
60#14519777C2×C412yes—
61#14519777C2×C411yescharacteristic
62#14519776—22no—
63#14519776—21no—
64#14519774C2×C412yes—
65#14519774C2×C411yescharacteristic
66#14519773—21no—
67#14519773—22no—
68#14519765—21no—
69#14519765—22no—
70#14519763—22no—
71#14519763—22no—
72#14519763—22no—
73#14519752—22no—
74#14519752—21no—
75#14519751—21no—
76#14519751—22no—
77#14519751C2×C412yes—
78#14519751C2×C412yes—
79#14519751C2×C412yes—
80#14519750—22no—
81#14519750—22no—
82#14519750—22no—
83#14519748—21no—
84#14519748—22no—
85Q8×Dic61—21no—
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C412yes—
88C4×Dic122C2311yescharacteristic
89C4×D244—21no—
90C4×D244—22no—
91C4×D244C2×C411yescharacteristic
92C4×D244C2×C412yes—
93C4×D244C2311yescharacteristic
94C4×D244C2312yes—
95#14519739—22no—
96#14519739—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—21no—
101#14519736C2×C411yescharacteristic
102#14519736C2×C412yes—
103#14519735—22no—
104#14519735—21no—
105#14519734—21no—
106#14519734—22no—
107C42×D61—21no—
108C42×D61—22no—
109C4⋊#92418—22no—
110C4⋊#92418—22no—
111C22⋊#92418—22no—
112C22⋊#92418—22no—
113#14519688—21no—
114C4×#92418—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2312yes—
118C2×#41×C61D411yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—22no—
121C4×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C412yes—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127#117×C61D411yescharacteristic
128#117×C61D412yes—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2312yes—
131C4⋊C8×C61—22no—
132C4⋊C8×C61—21no—
133C4⋊C8×C61C2311yescharacteristic
134C4⋊C8×C61C2312yes—
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—21no—
137C22⋊C8×C61—22no—
138#97×C61—22no—
139C4×C8×C61—21no—
140#41×C61—42no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×D4×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155#11758226—44no—
156#11758222—22no—
157#11758222—24no—
158#11758222—41no—
159#11758222—42no—
160#11758222—82no—
161#11758222—81no—
162#11758222—82no—
163#11758222—81no—
164C2×D244—22no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—22no—
168C2×D244C22×C411yescharacteristic
169C2×D244C22×C412yes—
170C2×C4×D61—21no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—22no—
176C2×Dic122—21no—
177C2×Dic122—22no—
178C2×Dic122C22×C411yescharacteristic
179M16×C61—81no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—22no—
183C2×C8×C61—82no—
184C4⋊C4×C61C22⋊C412yes—
185C4⋊C4×C61C22⋊C411yescharacteristic
186C4⋊C4×C61C22⋊C412yes—
187C4⋊C4×C61C22⋊C411yescharacteristic
188C22⋊C4×C61C22⋊C412yes—
189C22⋊C4×C61C22⋊C412yes—
190C22⋊C4×C61C22⋊C412yes—
191C42×C61C22×C412yes—
192C42×C61C22×C411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D412yes—
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D412yes—
197C42×C61C2411yescharacteristic
198C22⋊Dic61—21no—
199C22⋊Dic61—22no—
200#11758198—21no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—21no—
206#11758198—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61—21no—
209C4⋊Dic61C22×C411yescharacteristic
210C4⋊Dic61C22×C411yescharacteristic
211C4⋊Dic61C22×C412yes—
212#11758196—21no—
213#11758196—22no—
214#11758196—21no—
215C4×Dic61—21no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220#11758194—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—22no—
224C2×#92418—42no—
225C23×C61C2×C22⋊C411yescharacteristic
226C23×C61C2×C22⋊C412yes—
227C22×D61—22no—
228C22×D61—21no—
229Q8×C61—22no—
230D4×C61—22no—
231D4×C61—42no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C412yes—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—42no—
252C22⋊D61—44no—
253C22⋊D61—82no—
254C22⋊D61—82no—
255C22⋊D61—82no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260D244—21no—
261D244—22no—
262D244—21no—
263D244—22no—
264D244—44no—
265D244—42no—
266D244—82no—
267D244—81no—
268C4×D61—42no—
269C4×D61—44no—
270C4×D61—44no—
271C4×D61—41no—
272C4×D61—42no—
273C4×D61—41no—
274C4×D61—42no—
275C4×D61—42no—
276Dic122—21no—
277Dic122—82no—
278Dic122—81no—
279Dic122#13112yes—
280C8×C61—41no—
281C8×C61—42no—
282C8×C61—41no—
283#92418—41no—
284#92418—41no—
285#92418—42no—
286#17526—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—22no—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—42no—
295C2×D61—41no—
296C2×D61—42no—
297C2×D61—41no—
298C2×D61—42no—
299C2×D61—41no—
300C4×C61—21no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—22no—
306C4×C61#42112yes—
307C4×C61C2×#13111yescharacteristic
308C4×C61#56811yescharacteristic
309C4×C61#57611yescharacteristic
310C4×C61#57711yescharacteristic
311Dic61—22no—
312Dic61—41no—
313Dic61—42no—
314Dic61—41no—
315#3483—611no—
316#2990D6111yescharacteristic, complemented
317#2922—611no—
318#1496—612no—
319#1492—611no—
320#1489—611no—
321#1488—612no—
322#1484—611no—
323#1480—612no—
324#1477—612no—
325#1476—611no—
326C2×C61—41no—
327C2×C61—42no—
328C2×C61#288011yescharacteristic
329C2×C61#289711yescharacteristic
330C2×C61C2×#42111yescharacteristic
331D61—82no—
332D61—81no—
333Q82—611no—
334#555—612no—
335#555—611no—
336#555—1221no—
337#555—1222no—
338#550—611no—
339#550—611no—
340#550—612no—
341#550—1221no—
342#532—612no—
343#532—611no—
344#532—1221no—
345#532—1222no—
346C4×#41—611no—
347C4×#41—612no—
348C4×#41—1221no—
349C4×#41C2×D6111yescharacteristic
350#446—21no—
351#435—22no—
352C8×D4—21no—
353#433—1222no—
354#432C2×D6112yes—
355#431—612no—
356#431—612no—
357#431C2×D6111yescharacteristic
358#424—1221no—
359#423C2×D6111yescharacteristic
360C2×C4⋊C8C2×D6112yes—
361#408—1221no—
362C4×M16—611no—
363C2×C4×C8—611no—
364Q8⋊C8—611no—
365Q8⋊C8—611no—
366Q8⋊C8—612no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—612no—
370D4⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374C61#1752611yescharacteristic, complemented, Sylow
375C2×#41—611no—
376C2×#41—612no—
377C2×#41—1221no—
378C2×#41—1222no—
379C2×#41—1221no—
380C2×#41D24411yescharacteristic
381C2×M16—21no—
382C2×M16—1221no—
383C22×C8—22no—
384C22×C8—1221no—
385C4⋊Q8—612no—
386C4⋊Q8—611no—
387C4⋊Q8—1222no—
388C4⋊Q8—1221no—
389#127—612no—
390#127—611no—
391#127—1221no—
392#127—1222no—
393#124—1222no—
394#124—1222no—
395#124—1222no—
396#124—1221no—
397#124—1222no—
398#121—612no—
399#121—612no—
400#121—612no—
401#121—1221no—
402#121—1222no—
403#121—1221no—
404#121—1222no—
405#121—1222no—
406#121—1221no—
407#121—1222no—
408#121—1222no—
409#121—1222no—
410C4×Q8—611no—
411C4×Q8—612no—
412C4×Q8—611no—
413C4×Q8—1221no—
414C4×Q8C4×D6111yescharacteristic
415C4×Q8C22⋊D6111yescharacteristic
416C4×D4—611no—
417C4×D4—612no—
418C4×D4—611no—
419C4×D4—612no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—1222no—
428C4×D4C4×D6111yescharacteristic
429C4×D4C4×D6112yes—
430C4×D4C22⋊D6111yescharacteristic
431C4×D4C22⋊D6112yes—
432#117—611no—
433#117—612no—
434#117—1222no—
435#117—1221no—
436#117—1221no—
437#117—1222no—
438#117D24411yescharacteristic
439#117D24412yes—
440C2×C42—1221no—
441C2×C42—1222no—
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6112yes—
444C4⋊C8—22no—
445C4⋊C8—21no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8C22×D6111yescharacteristic
449C4⋊C8C22×D6112yes—
450C4⋊C8C22×D6111yescharacteristic
451C22⋊C8—21no—
452C22⋊C8—22no—
453C22⋊C8—1222no—
454C22⋊C8—1222no—
455#97—22no—
456#97—1221no—
457C4×C8—21no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8—612no—
461C4×C8—1221no—
462#41—42no—
463#41—1222no—
464#41—1224no—
465#41—4882no—
466#41—4881no—
467#41—4882no—
468#41—4881no—
469#41—2441no—
470#41—2442no—
471#41—2444no—
472C2×Q8—611no—
473C2×Q8—1221no—
474C2×Q8—1222no—
475C2×Q8#1175819811yescharacteristic
476C2×D4—611no—
477C2×D4—612no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—1221no—
482C2×D4—1222no—
483C2×D4—1221no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4#1175819811yescharacteristic
488C2×D4#1175819812yes—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4C2×C4×D6112yes—
497C22×C4C2×D24411yescharacteristic
498C22×C4C2×D24412yes—
499C22×C4C2×C22⋊D6112yes—
500C22×C4C2×C22⋊D6111yescharacteristic
501M16—81no—
502M16—2442no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—82no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1222no—
510C2×C8—2442no—
511C4⋊C4—611no—
512C4⋊C4—611no—
513C4⋊C4—612no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1221no—
519C4⋊C4#1175819812yes—
520C4⋊C4#1175819811yescharacteristic
521C4⋊C4#1175819812yes—
522C4⋊C4#1175819811yescharacteristic
523C22⋊C4—1221no—
524C22⋊C4—1222no—
525C22⋊C4—1221no—
526C22⋊C4—1222no—
527C22⋊C4—1221no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4#1175819812yes—
533C22⋊C4#1175819812yes—
534C22⋊C4#1175819812yes—
535C42—1221no—
536C42—1221no—
537C42—1222no—
538C42—1222no—
539C42—1221no—
540C42C2×C4×D6112yes—
541C42C2×C4×D6111yescharacteristic
542C42C2×D24411yescharacteristic
543C42C2×D24412yes—
544C42C2×C22⋊D6111yescharacteristic
545C42C2×C22⋊D6112yes—
546C42C23×D6111yescharacteristic
547C23—1222no—
548C23—1221no—
549C23C2×#1175819811yescharacteristic
550C23C2×#1175819812yes—
551Q8—22no—
552Q8—612no—
553Q8—1221no—
554Q8—4882no—
555Q8—4881no—
556D4—22no—
557D4—42no—
558D4—1221no—
559D4—1222no—
560D4—1221no—
561D4—1222no—
562D4—4882no—
563D4—4881no—
564D4—4882no—
565D4—4882no—
566D4—4882no—
567D4—2444no—
568D4—2442no—
569D4—2444no—
570D4—2442no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—42no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1221no—
585C2×C4—2442no—
586C2×C4—2444no—
587C2×C4—2444no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2442no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4C2×#1175819812yes—
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819812yes—
596C2×C4C2×#1175819811yescharacteristic
597C2×C4C2×#1175819812yes—
598C2×C4C2×#1175819811yescharacteristic
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C22×C22⋊D6111yescharacteristic
601C2×C4C22×D24411yescharacteristic
602C8—41no—
603C8—42no—
604C8—41no—
605C8—2441no—
606C8—2441no—
607C8—2442no—
608C22—22no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—21no—
613C22—22no—
614C22—2442no—
615C22—2441no—
616C22—2442no—
617C22—2441no—
618C22—2442no—
619C22—2441no—
620C22C22×#1175819811yescharacteristic
621C4—22no—
622C4—21no—
623C4—21no—
624C4—22no—
625C4—21no—
626C4—22no—
627C4—1222no—
628C4—2442no—
629C4—2441no—
630C4—2441no—
631C4#1683977812yes—
632C4#131×D6111yescharacteristic
633C4#1684052711yescharacteristic
634C4#1684056511yescharacteristic
635C4#1684057011yescharacteristic
636C2—41no—
637C2—42no—
638C2—4882no—
639C2—4881no—
640C2#2145426611yescharacteristic
641C2#2145434211yescharacteristic
642C2C2×#1683977811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2990C2
socleC22×C61C22×C22⋊C4
prime radical#21459445C2
cosocleC42×C61C24
hypercenter#2990D61
hypocenterC61#17526
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2990

Complements

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

NH ≅ G/Ntype
2. #21454463C2semidirect
3. #21444287C2semidirect
4. #21444269C2semidirect
5. #21444256C2semidirect
6. #21444251C2semidirect
7. #21444235C2semidirect
8. #21444217C2semidirect
9. #21444204C2semidirect
10. #21444199C2semidirect
11. C61×#2990C2semidirect
316. #2990D61semidirect
374. C61#17526semidirect

Maximal subgroups

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

subgroupindex
#214594452
#214544632
#214442872
#214442692
#214442562
#214442512
#214442352
#214442172
#214442042
#214441992
C61×#29902
#1752661