Finite Group Databasebeta

← #29646904

Subgroups — #29646904

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459416C211yesmaximal, characteristic
2C8×#11758222C211yesmaximal, characteristic, complemented
3#21454487C211yesmaximal, characteristic, complemented
4#21444276C212yesmaximal, complemented
5#21444250C212yesmaximal, complemented
6#21444247C211yesmaximal, characteristic, complemented
7#21444224C212yesmaximal, complemented
8#21444218C211yesmaximal, characteristic, complemented
9#21444198C212yesmaximal, complemented
10C2×#16839494C211yesmaximal, characteristic, complemented
11C61×#1492C211yesmaximal, characteristic, complemented
12#16840498C411yescharacteristic
13#16840494—21no—
14#16840487C411yescharacteristic
15#16840486C411yescharacteristic
16#16840481—21no—
17#16840464C412yes—
18#16840455—22no—
19#16840432C411yescharacteristic
20#16840418C412yes—
21#16840308—21no—
22#16840294C2212yes—
23#16840292—22no—
24#41×Dic61—21no—
25C4×#11758222C2211yescharacteristic
26C4×#11758224C2212yes—
27C2×C4×Dic122C2211yescharacteristic
28C8×Dic122—21no—
29C8×D244—21no—
30C8×C22⋊D61—22no—
31#16839822—22no—
32#16839818C2212yes—
33#16839815C2211yescharacteristic
34#16839814C2212yes—
35#16839813C2212yes—
36C2×C4⋊#92418C2211yescharacteristic
37#16839730—21no—
38#16839720—21no—
39C4×C8×D61C2212yes—
40#16839494C2211yescharacteristic
41#16839494C2211yescharacteristic
42#16839494C2211yescharacteristic
43#16839494C2211yescharacteristic
44#16839492C2212yes—
45#16839492C2212yes—
46D4⋊#92418C2212yes—
47D4⋊#92418C2212yes—
48C61×#532C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C2×C4×C8×C61C2211yescharacteristic
51C61×D4⋊C8C2212yes—
52C61×D4⋊C8C2212yes—
53C2×#11758222D411yescharacteristic
54C2×#11758224C2×C412yes—
55C22×Dic122C2×C411yescharacteristic
56C22×#92418—21no—
57C2×C8×D61—22no—
58#14519775—21no—
59#14519773—21no—
60#14519772—22no—
61#14519770—22no—
62#14519770—22no—
63#14519768C2×C412yes—
64#14519767—22no—
65#14519766—21no—
66#14519764—21no—
67#14519764C2×C412yes—
68#14519763—22no—
69#14519761—22no—
70#14519759—22no—
71#14519758—22no—
72#14519757C2×C412yes—
73#14519754—22no—
74#14519749—22no—
75#14519749—22no—
76#14519749—22no—
77#14519748C2×C412yes—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122C2×C411yescharacteristic
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122D411yescharacteristic
86C4×Dic122C2311yescharacteristic
87C4×Dic122C2311yescharacteristic
88#14519742C2×C412yes—
89C4×D244C2×C411yescharacteristic
90C4×D244D411yescharacteristic
91D4×Dic61—22no—
92D4×Dic61—22no—
93D4×Dic61—21no—
94D4×Dic61—21no—
95D4×Dic61C2×C412yes—
96C4×C22⋊D61—22no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61C2×C412yes—
99C4×C22⋊D61D412yes—
100#14519737—22no—
101#14519736—22no—
102#14519736D411yescharacteristic
103#14519735C2×C412yes—
104#14519734D412yes—
105#14519733—22no—
106C2×C4⋊Dic61C2×C411yescharacteristic
107C2×#11758196—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—22no—
110C42×D61C2312yes—
111C4⋊#92418—21no—
112C4⋊#92418—21no—
113C4⋊#92418C2311yescharacteristic
114C4⋊#92418C2311yescharacteristic
115#14519706—22no—
116C22⋊#92418—21no—
117#14519689—22no—
118#14519689—22no—
119#14519689—22no—
120#14519687—22no—
121C4×#92418C2312yes—
122C8×Dic61—22no—
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
140#41×C61—42no—
141#41×C61—42no—
142C2×Q8×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—22no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61C22⋊C412yes—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158#11758224—24no—
159#11758224—42no—
160#11758222—22no—
161#11758222—22no—
162C2×D244C22⋊C411yescharacteristic
163C2×C4×D61C22×C412yes—
164C2×C4×D61C2×D412yes—
165C2×C4×D61C2×D412yes—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—41no—
169C2×Dic122C22⋊C411yescharacteristic
170C2×Dic122C22×C411yescharacteristic
171C2×Dic122C22×C411yescharacteristic
172M16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—22no—
176C2×C8×C61—42no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61C22×C411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2411yescharacteristic
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61C22⋊C412yes—
190#11758198—22no—
191#11758198—22no—
192#11758198C22⋊C412yes—
193#11758198C22⋊C412yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C411yescharacteristic
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201#11758196—21no—
202#11758196—21no—
203#11758196—21no—
204#11758196—21no—
205#11758196—22no—
206#11758196—22no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218C2×#92418—21no—
219C2×#92418—22no—
220C2×#92418—21no—
221C2×#92418—42no—
222C8×D61—44no—
223C23×C61—22no—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61C2×C22⋊C412yes—
226Q8×C61—42no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—44no—
230D4×C61—42no—
231D4×C61—81no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61#13512yes—
242C2×C4×C61#13512yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C22⋊D61—24no—
247C22⋊D61—44no—
248C22⋊D61—82no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—44no—
254C2×Dic61—42no—
255C2×Dic61—82no—
256C2×Dic61—81no—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C412yes—
260D244—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—42no—
266Dic122—21no—
267Dic122—22no—
268Dic122—21no—
269Dic122—21no—
270Dic122—42no—
271Dic122#13112yes—
272C8×C61—22no—
273C8×C61—41no—
274C8×C61—42no—
275#92418—42no—
276#92418—41no—
277#92418—41no—
278#17510—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61#41612yes—
285C22×C61C22×C22⋊C411yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—42no—
293C4×C61#41612yes—
294C4×C61C2×#13111yescharacteristic
295C4×C61#56811yescharacteristic
296C4×C61C2×#13511yescharacteristic
297C4×C61C2×#13511yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—22no—
301Dic61—22no—
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
305#3480—611no—
306C8×#41—611no—
307#2925—611no—
308#1494—612no—
309#1492D6111yescharacteristic, complemented
310#1488—612no—
311#1487—611no—
312#1482—612no—
313#1480—611no—
314#1476—612no—
315C2×Q8⋊C8—611no—
316C2×C61—22no—
317C2×C61—82no—
318C2×C61#288011yescharacteristic
319C2×C61C2×#41611yescharacteristic
320C2×C61#290111yescharacteristic
321D61—42no—
322#559—611no—
323#559—1221no—
324#557—611no—
325#556—611no—
326#556—1221no—
327#553—612no—
328#553—1222no—
329#550—611no—
330#548—612no—
331#532—1221no—
332#532C2×D6111yescharacteristic
333#529—612no—
334#529—1222no—
335C4×#41—611no—
336C4×#41—612no—
337C4×#41—1221no—
338C2×C4×Q8—611no—
339C8×Q8—1221no—
340C8×D4—1221no—
341C8×D4—1222no—
342#433—1222no—
343#432—612no—
344#431—611no—
345#431—612no—
346#431—612no—
347#423C2×D6111yescharacteristic
348C2×C4⋊C8—611no—
349#408—1221no—
350#405—1221no—
351C2×C4×C8—612no—
352C2×C4×C8C2×D6111yescharacteristic
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359D4⋊C8—612no—
360D4⋊C8—612no—
361D4⋊C8C2×D6112yes—
362D4⋊C8C2×D6112yes—
363C61#1751011yescharacteristic, complemented, Sylow
364C2×#41—21no—
365C2×#41—611no—
366C2×#41—612no—
367C22×Q8—611no—
368C2×M16—21no—
369C22×C8—21no—
370C22×C8—1222no—
371C22×C8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8C4×D6111yescharacteristic
374#127—1221no—
375#127C4×D6111yescharacteristic
376#126—1222no—
377#126—1222no—
378#126—1222no—
379#125—612no—
380#125—1222no—
381#124—21no—
382#124—612no—
383#124—1222no—
384#124—1221no—
385#124—1221no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—612no—
390C22⋊Q8—1222no—
391#121—21no—
392#121—612no—
393#121—1222no—
394#121—1222no—
395#121—1222no—
396#121C4×D6112yes—
397C4×Q8—611no—
398C4×Q8—611no—
399C4×Q8—612no—
400C4×Q8—611no—
401C4×Q8—611no—
402C4×Q8—611no—
403C4×Q8—611no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×D4—21no—
409C4×D4—612no—
410C4×D4—612no—
411C4×D4—612no—
412C4×D4—611no—
413C4×D4—611no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4C22×D6112yes—
422#117—611no—
423#117—612no—
424#117—612no—
425#117—1222no—
426#117—1222no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C42—612no—
430C2×C42—1222no—
431C2×C42—1221no—
432C2×C42C22×D6111yescharacteristic
433C4⋊C8—21no—
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—1222no—
437C4⋊C8—1221no—
438C4⋊C8—1221no—
439C4⋊C8C22×D6112yes—
440C22⋊C8—1222no—
441C22⋊C8—1222no—
442C22⋊C8—1222no—
443C22⋊C8—1221no—
444#97—1222no—
445C4×C8—21no—
446C4×C8—612no—
447C4×C8—1222no—
448C4×C8—1222no—
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451#41—42no—
452#41—42no—
453#41—1222no—
454#41—1222no—
455#41—1224no—
456#41—2442no—
457C2×Q8—21no—
458C2×Q8—611no—
459C2×Q8—611no—
460C2×Q8—611no—
461C2×Q8—1222no—
462C2×Q8—1221no—
463C2×Q8—2441no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—612no—
468C2×D4—611no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4C2×C4×D6112yes—
472C22×C4—22no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483M16—42no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—42no—
488C2×C8—1222no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—2444no—
492C2×C8—2442no—
493C4⋊C4—21no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—611no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4C2×C4×D6112yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—612no—
513C22⋊C4—612no—
514C22⋊C4—612no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C42—612no—
522C42—612no—
523C42—612no—
524C42—1222no—
525C42—1222no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42C2×C4×D6111yescharacteristic
531C42D4×D6111yescharacteristic
532C42D4×D6111yescharacteristic
533C42C23×D6111yescharacteristic
534C23—22no—
535C23—612no—
536C23C22⋊C4×D6111yescharacteristic
537Q8—42no—
538Q8—612no—
539Q8—1222no—
540Q8—1221no—
541Q8—1221no—
542Q8—1221no—
543Q8—2442no—
544D4—22no—
545D4—22no—
546D4—44no—
547D4—42no—
548D4—81no—
549D4—1224no—
550D4—1222no—
551D4—4882no—
552D4—2444no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—1221no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—4882no—
571C2×C4—4881no—
572C2×C4—2442no—
573C2×C4—2444no—
574C2×C4—2442no—
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C22⋊C4×D6111yescharacteristic
578C2×C4#1451979612yes—
579C2×C4#1451979812yes—
580C2×C4C22×C4×D6111yescharacteristic
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C8—22no—
584C8—41no—
585C8—42no—
586C8—2442no—
587C8—2441no—
588C8—2441no—
589C22—21no—
590C22—21no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22#1683975512yes—
598C22C2×C22⋊C4×D6111yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—42no—
603C4—1222no—
604C4—1222no—
605C4—1222no—
606C4—1222no—
607C4—2441no—
608C4—2441no—
609C4—2441no—
610C4#1683975512yes—
611C4#131×D6111yescharacteristic
612C4#1684052711yescharacteristic
613C4C2×#1451979611yescharacteristic
614C4C2×#1451979811yescharacteristic
615C2—22no—
616C2—82no—
617C2—2442no—
618C2#2145426711yescharacteristic
619C2C2×#1683975511yescharacteristic
620C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1492C2
socleC22×C61C22×C22⋊C4
prime radical#21459416C2
cosocleC42×C61C24
hypercenter#1492D61
hypocenterC61#17510
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. C8×#11758222C2semidirect
3. #21454487C2semidirect
4. #21444276C2semidirect
5. #21444250C2semidirect
6. #21444247C2semidirect
7. #21444224C2semidirect
8. #21444218C2semidirect
9. #21444198C2semidirect
10. C2×#16839494C2semidirect
11. C61×#1492C2semidirect
309. #1492D61semidirect
363. C61#17510semidirect

Maximal subgroups

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

subgroupindex
#214594162
C8×#117582222
#214544872
#214442762
#214442502
#214442472
#214442242
#214442182
#214441982
C2×#168394942
C61×#14922
#1751061