Finite Group Databasebeta

← #29656712

Subgroups — #29656712

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457197C211yesmaximal, characteristic, complemented
2#21456215C211yesmaximal, characteristic, complemented
3#21455879C211yesmaximal, characteristic, complemented
4#21444991C212yesmaximal, complemented
5#21444970C212yesmaximal, complemented
6#21444929C212yesmaximal, complemented
7#21444908C212yesmaximal, complemented
8#21444491C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3511C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840124—22no—
14#16840114C2212yes—
15#16840110C2212yes—
16#16840086—21no—
17#16840057C2211yescharacteristic
18#16840052C2211yescharacteristic
19#16840052C2212yes—
20#16840049—21no—
21#16840013—22no—
22#16839962C2211yescharacteristic
23#16839957C2211yescharacteristic
24#16839957C2212yes—
25#16839954—21no—
26#16839907—21no—
27C4×#11758201—21no—
28C4×#11758199—21no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839761—22no—
32C4×#11758194C2211yescharacteristic
33#16839508C2212yes—
34#16839508C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38D4⋊#92418C2212yes—
39C61×#556C2212yes—
40C61×#555—21no—
41C61×#554—21no—
42C61×#553—22no—
43C61×#548—22no—
44C61×#547C2212yes—
45C61×#542—22no—
46C61×#537—21no—
47C61×#533—21no—
48C61×#532—21no—
49C61×#532C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520—21no—
52C4×#41×C61C2211yescharacteristic
53C2×#11758222—21no—
54C2×#11758201—21no—
55C2×#11758199—21no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519710—21no—
67#14519708—22no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2311yescharacteristic
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73#14519701—21no—
74#14519700—24no—
75#14519698—22no—
76#14519698—21no—
77#14519697—22no—
78#14519697—21no—
79#14519697—21no—
80#14519697—24no—
81C22⋊#92418—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61—22no—
86C2×#41×C61—21no—
87C2×#41×C61D411yescharacteristic
88C22×Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61D412yes—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61C2312yes—
94#127×C61D411yescharacteristic
95#126×C61—24no—
96#125×C61—22no—
97#124×C61—24no—
98#124×C61—22no—
99#124×C61—21no—
100#123×C61—24no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105#121×C61—22no—
106#121×C61—22no—
107#121×C61—24no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61D412yes—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61D411yescharacteristic
114C4×Q8×C61C2311yescharacteristic
115C4×D4×C61—22no—
116C4×D4×C61—22no—
117C4×D4×C61—24no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122C4×D4×C61C2312yes—
123C4×D4×C61C2311yescharacteristic
124#117×C61—22no—
125#117×C61—22no—
126#117×C61—21no—
127#117×C61D411yescharacteristic
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—22no—
130C2×C4⋊C4×C61D412yes—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#41×C61—41no—
134#41×C61—41no—
135#41×C61—42no—
136#41×C61—42no—
137#41×C61—82no—
138#41×C61—82no—
139C2×Q8×C61—21no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61—81no—
143C2×Q8×C61—81no—
144C2×Q8×C61C2×D411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61—24no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C2×D4×C61C2×D412yes—
152C22×C4×C61—24no—
153C22×C4×C61—22no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—22no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C2×C22⋊D61—41no—
162#11758222—44no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D414yes—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—22no—
188C42×C61—24no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2411yescharacteristic
196#11758201—42no—
197#11758199—42no—
198#11758199—82no—
199#11758198—41no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203#11758194—42no—
204#11758194—81no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—22no—
208C2×#92418—22no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61#12011yescharacteristic
212C22×D61—41no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—42no—
216Q8×C61—42no—
217Q8×C61—42no—
218Q8×C61—82no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—22no—
222D4×C61—22no—
223D4×C61—42no—
224D4×C61—42no—
225D4×C61—44no—
226D4×C61—42no—
227D4×C61—84no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—24no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—82no—
245C2×C4×C61—84no—
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C22⋊D61—82no—
253C2×Dic61—41no—
254D244—42no—
255D244—41no—
256D244—41no—
257D244—41no—
258D244—41no—
259C4×D61—82no—
260Dic122—42no—
261#92418—42no—
262#92418—41no—
263#92418—41no—
264#92418—42no—
265#18585—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—41no—
270C22×C61—42no—
271C22×C61—41no—
272C22×C61—42no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61C2×#12011yescharacteristic
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61—41no—
285C4×C61—44no—
286C4×C61—41no—
287C4×C61#45412yes—
288C4×C61#57511yescharacteristic
289C4×C61#57511yescharacteristic
290C4×C61#57611yescharacteristic
291C4×C61#57711yescharacteristic
292Dic61—81no—
293#3511D6111yescharacteristic, complemented
294#3253—611no—
295#3141—611no—
296#3106—611no—
297#1638—612no—
298#1635—612no—
299#1628—612no—
300#1625—612no—
301#1546—611no—
302#1489—611no—
303#1484—611no—
304C2×C61—41no—
305C2×C61—81no—
306C2×C61—82no—
307C2×C61#300511yescharacteristic
308C2×C61#301211yescharacteristic
309C2×C61C2×#45411yescharacteristic
310D61—161no—
311#556C2×D6112yes—
312#555—21no—
313#554—21no—
314#553—22no—
315#548—22no—
316#547C2×D6112yes—
317#542—22no—
318#537—21no—
319#533—21no—
320#532—21no—
321#532—611no—
322#532C2×D6111yescharacteristic
323C2×C4⋊Q8C2×D6111yescharacteristic
324#520—21no—
325C4×#41C2×D6111yescharacteristic
326#489—1222no—
327#487—612no—
328#486—612no—
329#482—1221no—
330#477—611no—
331#476—611no—
332#476—612no—
333#476—1221no—
334#469—1222no—
335#460—611no—
336#459—611no—
337#459—612no—
338#459—1221no—
339#449—1221no—
340C4×SD16—1221no—
341C4×D8—1221no—
342#431—611no—
343#423—611no—
344#417—1222no—
345C4×M16—611no—
346#332—612no—
347#332—612no—
348Q8⋊C8—612no—
349D4⋊C8—612no—
350D4⋊C8—612no—
351D4⋊C8—612no—
352C61#1858511yescharacteristic, complemented, Sylow
353C2×#41—22no—
354C2×#41—21no—
355C2×#41—1221no—
356C2×#41C22⋊D6111yescharacteristic
357C22×Q8—21no—
358C2×SD16—1221no—
359C2×D8—1222no—
360C2×D8—1221no—
361C2×M16—1221no—
362C2×M16—1221no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—611no—
366C4⋊Q8C22⋊D6112yes—
367C4⋊Q8C22⋊D6111yescharacteristic
368C4⋊Q8C22×D6112yes—
369#127—611no—
370#127C22⋊D6111yescharacteristic
371#126—24no—
372#125—22no—
373#124—24no—
374#124—22no—
375#124—21no—
376#124—1221no—
377#123—24no—
378C22⋊Q8—22no—
379C22⋊Q8—24no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382#121—22no—
383#121—22no—
384#121—24no—
385#121—22no—
386#121—21no—
387#121—1221no—
388#121—1221no—
389#121C22⋊D6112yes—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8C22⋊D6111yescharacteristic
393C4×Q8C22×D6111yescharacteristic
394C4×D4—22no—
395C4×D4—24no—
396C4×D4—22no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4C22⋊D6111yescharacteristic
403C4×D4C22×D6111yescharacteristic
404C4×D4C22×D6112yes—
405#117—22no—
406#117—22no—
407#117—21no—
408#117C22⋊D6111yescharacteristic
409C2×C4⋊C4—21no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4C22⋊D6112yes—
412C2×C42—21no—
413C2×C42C22×D6111yescharacteristic
414#107—1221no—
415#106—1221no—
416#106—1222no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—612no—
420C4⋊C8—1221no—
421#103—1221no—
422#103—1224no—
423#102—1221no—
424#102—1222no—
425#102—1224no—
426#102—1221no—
427#102—1221no—
428#102—1222no—
429C22⋊C8—1221no—
430#97—1221no—
431C4×C8—612no—
432C4×C8—1221no—
433#41—41no—
434#41—41no—
435#41—42no—
436#41—42no—
437#41—82no—
438#41—82no—
439#41—2444no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—81no—
444C2×Q8—81no—
445C2×Q8—1221no—
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—2441no—
456C2×D4C2×C22⋊D6111yescharacteristic
457C2×D4C2×C22⋊D6112yes—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—2441no—
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468SD16—2442no—
469D8—4882no—
470D8—2442no—
471M16—4881no—
472M16—2442no—
473C2×C8—1222no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1222no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4C2×C22⋊D6114yes—
488C4⋊C4C2×C22⋊D6111yescharacteristic
489C4⋊C4C2×C22⋊D6112yes—
490C4⋊C4C2×C22⋊D6111yescharacteristic
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C42—24no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42C2×C22⋊D6111yescharacteristic
506C42C2×C22⋊D6111yescharacteristic
507C42C2×C22⋊D6111yescharacteristic
508C42C23×D6111yescharacteristic
509C23—21no—
510C23—22no—
511C23—2441no—
512C23#1451974611yescharacteristic
513Q8—21no—
514Q8—21no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—82no—
519Q8—2442no—
520D4—21no—
521D4—21no—
522D4—22no—
523D4—22no—
524D4—42no—
525D4—42no—
526D4—44no—
527D4—42no—
528D4—84no—
529D4—4882no—
530D4—2442no—
531D4—2441no—
532D4—2441no—
533D4—2441no—
534D4—2441no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4#1451974611yescharacteristic
556C2×C4#1451974611yescharacteristic
557C2×C4#1451974611yescharacteristic
558C2×C4C22×C22⋊D6111yescharacteristic
559C2×C4C22×C22⋊D6111yescharacteristic
560C2×C4C22×C22⋊D6111yescharacteristic
561C8—2442no—
562C8—2441no—
563C8—2441no—
564C8—2442no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—42no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22C2×#1451974611yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—44no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—4881no—
587C4#1683993512yes—
588C4#1684056011yescharacteristic
589C4#1684056011yescharacteristic
590C4#1684056511yescharacteristic
591C4#1684057311yescharacteristic
592C2—41no—
593C2—81no—
594C2—82no—
595C2—9761no—
596C2#2145506111yescharacteristic
597C2#2145512311yescharacteristic
598C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#18585
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21457197C2semidirect
2. #21456215C2semidirect
3. #21455879C2semidirect
4. #21444991C2semidirect
5. #21444970C2semidirect
6. #21444929C2semidirect
7. #21444908C2semidirect
8. #21444491C2semidirect
9. #21444258C2semidirect
10. #21444236C2semidirect
11. C61×#3511C2semidirect
293. #3511D61semidirect
352. C61#18585semidirect

Maximal subgroups

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

subgroupindex
#214571972
#214562152
#214558792
#214449912
#214449702
#214449292
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C61×#35112
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