Finite Group Databasebeta

← #30571035

Subgroups — #30571035

Order 15616.

1750 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459681C211yesmaximal, characteristic, complemented
2C2×#16840482C211yesmaximal, characteristic, complemented
3#21458819C214yesmaximal, complemented
4#21458805C214yesmaximal, complemented
5#21458503C214yesmaximal, complemented
6#21458466C214yesmaximal, complemented
7#21456781C214yesmaximal, complemented
8#21456769C211yesmaximal, characteristic, complemented
9#21456741C211yesmaximal, characteristic, complemented
10#21455896C214yesmaximal, complemented
11#21455827C211yesmaximal, characteristic, complemented
12#21454899C211yesmaximal, characteristic, complemented
13C61×#3100C211yesmaximal, characteristic
14#16840498C2212yescomplemented
15#16840487C2212yescomplemented
16#16840482—22no—
17#16840482—22no—
18#16840482C2214yes—
19#16840482C2214yes—
20#16840482C2212yescomplemented
21#16840481—22no—
22#16840469C2212yescomplemented
23#16840452—22no—
24#16840357—22no—
25#16840333—21no—
26#16840319C2211yescharacteristic
27C2×#14519777C2211yescharacteristic
28#16840293C2214yes—
29C2×#14519770—22no—
30C2×#14519764C2212yes—
31C2×#14519756—21no—
32#16840245—21no—
33C2×C4×Dic122C2211yescharacteristic
34C2×#14519742C2212yes—
35C2×#14519736—21no—
36#16840145C2218yes—
37#16840142C2212yescomplemented
38#16840134C2212yescomplemented
39#16840115C2218yes—
40#16840114C2212yescomplemented
41#16840113C2212yescomplemented
42#16840092C2212yes—
43#16840068C2218yescomplemented
44#16840052C2212yes—
45#16839983C2218yes—
46#16839983C2218yescomplemented
47#16839975C2218yes—
48#16839973C2214yescomplemented
49#16839967C2212yes—
50#16839967C2214yescomplemented
51#16839920C2214yes—
52#16839913C2212yes—
53#16839900C2214yes—
54C8×Dic122C2214yescomplemented
55#16839886C2214yescomplemented
56#16839856C2218yescomplemented
57#16839853C2218yescomplemented
58C4×#11758200C2214yes—
59#16839815C2211yescharacteristic
60#16839812C2214yes—
61C61×#532C2211yescharacteristic
62C2×C4×D4×C61C2211yescharacteristic
63C61×#480C2212yes—
64C61×#474C2212yes—
65C61×#459C2212yes—
66C61×#447C2212yes—
67C4×D8×C61C2214yes—
68C61×#431C2211yescharacteristic
69C22×Dic122D411yescharacteristic
70C22×Dic122C2312yes—
71C2×Dic244C2318yes—
72C2×#11758200C2314yes—
73C2×#11758200C2318yes—
74#14519777—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2312yes—
78#14519777C2312yes—
79#14519777C2311yescharacteristic
80#14519775—22no—
81#14519770—24no—
82#14519770—28no—
83#14519770—28no—
84#14519770—24no—
85#14519769—21no—
86#14519768C2314yes—
87#14519767—24no—
88#14519764—22no—
89#14519764—24no—
90#14519764C2318yes—
91#14519764C2314yes—
92#14519758—24no—
93#14519758—24no—
94#14519756—21no—
95#14519756—22no—
96#14519756—22no—
97#14519756—24no—
98#14519756—24no—
99#14519756C2312yes—
100#14519754—24no—
101Q8×Dic61—22no—
102C4×Dic122—21no—
103C4×Dic122D412yes—
104C4×Dic122D412yes—
105C4×Dic122C2312yes—
106C4×Dic122C2312yes—
107C4×Dic122C2312yes—
108#14519742—24no—
109#14519742C2318yes—
110#14519742C2314yes—
111D4×Dic61—24no—
112D4×Dic61—22no—
113D4×Dic61C2314yes—
114#14519736—21no—
115#14519736—22no—
116#14519736—24no—
117#14519736—24no—
118#14519733—24no—
119C2×C4⋊Dic61—21no—
120C2×C4⋊Dic61—21no—
121C2×C4⋊Dic61—22no—
122C2×C4⋊Dic61D411yescharacteristic
123C2×#11758196—22no—
124C2×#11758196D412yes—
125C2×C22⋊Dic61—22no—
126C2×C22⋊Dic61—22no—
127C2×C4×Dic61—22no—
128C2×C4×Dic61C2312yes—
129#14519708C2314yes—
130C4⋊#92418C2312yes—
131#14519706C2318yes—
132#14519702C2318yes—
133#14519700C2314yes—
134#14519700C2318yes—
135#14519698C2314yes—
136C22⋊#92418C2312yes—
137C4×#92418C2312yes—
138C8×Dic61C2318yes—
139C2×#41×C61—21no—
140C22×D4×C61C2311yescharacteristic
141C23×C4×C61—21no—
142C2×D8×C61C2314yes—
143C4⋊Q8×C61C2311yescharacteristic
144#127×C61C2311yescharacteristic
145#124×C61—21no—
146#121×C61—21no—
147#121×C61C2312yes—
148C4×D4×C61—21no—
149C4×D4×C61—21no—
150C4×D4×C61—21no—
151C4×D4×C61C2312yes—
152C4×D4×C61C2312yes—
153C4×D4×C61C2312yes—
154C2×C4⋊C4×C61C2311yescharacteristic
155C2×C22⋊C4×C61—21no—
156C2×C42×C61C2311yescharacteristic
157#107×C61C2314yes—
158C4⋊C8×C61C2312yes—
159#102×C61C2314yes—
160#102×C61C2314yes—
161C22⋊C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C24×C61—21no—
164#41×C61—44no—
165C2×Q8×C61—21no—
166C2×D4×C61—22no—
167C2×D4×C61—21no—
168C2×D4×C61—21no—
169C2×D4×C61—21no—
170C2×D4×C61—42no—
171C2×D4×C61C2412yes—
172C2×D4×C61C2412yes—
173C22×C4×C61—21no—
174C22×C4×C61—22no—
175C22×C4×C61—22no—
176C22×C4×C61—22no—
177C22×C4×C61—44no—
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2411yescharacteristic
181C22×Dic61—22no—
182C22×Dic61—24no—
183C22×Dic61C2×D412yes—
184C22×Dic61C2×D412yes—
185C2×Dic122—24no—
186C2×Dic122—22no—
187C2×Dic122—21no—
188C2×Dic122—44no—
189C2×Dic122C2×D412yes—
190C2×Dic122C2×D412yes—
191C2×Dic122C2414yes—
192C2×Dic122C2412yes—
193D8×C61—44no—
194C2×C8×C61C2414yes—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61C2412yes—
198C4⋊C4×C61C2412yes—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C42×C61C2×D411yescharacteristic
204C42×C61C2×D411yescharacteristic
205C42×C61C2411yescharacteristic
206C42×C61C2411yescharacteristic
207C22⋊Dic61—22no—
208C22⋊Dic61—28no—
209C22⋊Dic61—28no—
210C22⋊Dic61—24no—
211C22⋊Dic61—24no—
212C22⋊Dic61—24no—
213#11758200—44no—
214#11758200—48no—
215C4⋊Dic61—21no—
216C4⋊Dic61—22no—
217C4⋊Dic61—22no—
218C4⋊Dic61—22no—
219C4⋊Dic61—22no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—24no—
223C4⋊Dic61C2×D412yes—
224C4⋊Dic61C2×D412yes—
225C4⋊Dic61C2412yes—
226#11758196—22no—
227#11758196—22no—
228#11758196—28no—
229#11758196—24no—
230#11758196—24no—
231#11758196C2×D414yes—
232#11758196C2×D414yes—
233C4×Dic61—22no—
234C4×Dic61—28no—
235C4×Dic61—24no—
236C4×Dic61C2×D414yes—
237C4×Dic61C2414yes—
238C2×#92418C2414yes—
239Dic244—48no—
240C23×C61—22no—
241C23×C61—22no—
242C23×C61—42no—
243C23×C61—42no—
244C23×C61—42no—
245C23×C61C22×D411yescharacteristic
246Q8×C61—42no—
247D4×C61—24no—
248D4×C61—22no—
249D4×C61—22no—
250D4×C61—42no—
251D4×C61—44no—
252D4×C61—42no—
253D4×C61—81no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—44no—
260C2×C4×C61—44no—
261C2×C4×C61—44no—
262C2×C4×C61#13514yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2511yescharacteristic
269C2×Dic61—24no—
270C2×Dic61—22no—
271C2×Dic61—24no—
272C2×Dic61—22no—
273C2×Dic61—42no—
274C2×Dic61—44no—
275C2×Dic61—84no—
276C2×Dic61C22×D414yes—
277C2×Dic61C22×D414yes—
278Dic122—24no—
279Dic122—24no—
280Dic122—24no—
281Dic122—28no—
282Dic122—44no—
283Dic122—81no—
284C8×C61—24no—
285#92418—24no—
286#61882—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—84no—
296C22×C61C2×#13512yes—
297C22×C61C23×D411yescharacteristic
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61C2×#13512yes—
304C4×C61C2×#13512yes—
305C4×C61#58512yes—
306Dic61—24no—
307Dic61—24no—
308Dic61—42no—
309Dic61—44no—
310Dic61—42no—
311#3507—611no—
312C2×#556—611no—
313#3394—614no—
314#3393—614no—
315#3364—614no—
316#3360—614no—
317#3211—614no—
318#3209—611no—
319#3206—611no—
320#3108—614no—
321#3101—611no—
322#3100D6111yescharacteristic
323#2982—611no—
324C2×C61—22no—
325C2×C61—82no—
326C2×C61—82no—
327C2×C61#348711yescharacteristic
328C2×C61C22×#13511yescharacteristic
329C2×C61#358611yescharacteristic
330#559—612no—
331#557—612no—
332#556—614no—
333#556—614no—
334#556—612no—
335#556—1222no—
336#556—1222no—
337#556—1222no—
338#554—612no—
339#552—1222no—
340#539—1222no—
341#536—1221no—
342#533—611no—
343#532C2×D6111yescharacteristic
344C2×C4⋊Q8—611no—
345#529—614no—
346C2×#126—1222no—
347C2×#124—612no—
348C2×C22⋊Q8—1221no—
349#520—1221no—
350C2×C4×Q8—611no—
351C2×C4×Q8—612no—
352C2×C4×D4C2×D6111yescharacteristic
353C2×#117—1221no—
354#495—618no—
355#494—612no—
356#492—612no—
357#488—618no—
358#487—612no—
359#487—612no—
360#483—612no—
361#480C2×D6112yes—
362#479—618no—
363#476—612no—
364#474C2×D6112yes—
365#464—618no—
366#464—618no—
367#463—618no—
368#462—614no—
369#461—612no—
370#461—614no—
371#459C2×D6112yes—
372#451—614no—
373#450—612no—
374#448—614no—
375#447C2×D6112yes—
376C8×Q8—614no—
377C8×Q8—614no—
378C4×Q16—618no—
379C4×SD16—614no—
380C4×SD16—618no—
381C4×D8C2×D6114yes—
382#431—611no—
383#431—614no—
384#431C2×D6111yescharacteristic
385C61#6188211yescharacteristic, complemented, Sylow
386C2×#41—21no—
387C22×Q8—611no—
388C22×Q8—612no—
389C22×D4C22×D6111yescharacteristic
390C23×C4—21no—
391C2×Q16—618no—
392C2×SD16—618no—
393C2×SD16—614no—
394C2×D8C22×D6114yes—
395C4⋊Q8—611no—
396C4⋊Q8—612no—
397C4⋊Q8—612no—
398C4⋊Q8—611no—
399C4⋊Q8—1222no—
400C4⋊Q8—1221no—
401C4⋊Q8—1221no—
402C4⋊Q8C22×D6111yescharacteristic
403#127C22×D6111yescharacteristic
404#126—1224no—
405#126—1228no—
406#126—1228no—
407#126—1224no—
408#125—614no—
409#125—1224no—
410#125—1221no—
411#124—21no—
412#124—618no—
413#124—614no—
414#124—1224no—
415#124—1222no—
416#123—1224no—
417#123—1224no—
418C22⋊Q8—612no—
419C22⋊Q8—1224no—
420C22⋊Q8—1224no—
421C22⋊Q8—1224no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C22⋊Q8—1221no—
425#121—21no—
426#121C22×D6112yes—
427C4×Q8—612no—
428C4×Q8—612no—
429C4×Q8—612no—
430C4×Q8—612no—
431C4×Q8—618no—
432C4×Q8—614no—
433C4×Q8—612no—
434C4×Q8—1224no—
435C4×Q8—1222no—
436C4×Q8—1221no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—614no—
441C4×D4—1222no—
442C4×D4—1224no—
443C4×D4C22×D6112yes—
444C4×D4C22×D6112yes—
445C4×D4C22×D6112yes—
446#117—1224no—
447#117—1224no—
448#117—1224no—
449#117—1222no—
450#117—1221no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—612no—
453C2×C4⋊C4—1222no—
454C2×C4⋊C4—1222no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C22⋊C4—21no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1222no—
461C2×C42—612no—
462C2×C42—1222no—
463C2×C42C22×D6111yescharacteristic
464#107C22×D6114yes—
465#106—614no—
466C4⋊C8—618no—
467C4⋊C8—612no—
468C4⋊C8C22×D6112yes—
469#103—618no—
470#103—614no—
471#103—618no—
472#102—614no—
473#102C22×D6114yes—
474#102C22×D6114yes—
475C22⋊C8—612no—
476C22⋊C8C22×D6112yes—
477C4×C8—618no—
478C4×C8—612no—
479C4×C8C22×D6112yes—
480C24—21no—
481#41—44no—
482C2×Q8—21no—
483C2×Q8—612no—
484C2×Q8—612no—
485C2×Q8—614no—
486C2×Q8—612no—
487C2×Q8—1221no—
488C2×Q8—1222no—
489C2×Q8—1224no—
490C2×Q8—2444no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4C23×D6112yes—
497C2×D4C23×D6112yes—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—44no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—1224no—
506C22×C4—1222no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—2448no—
511SD16—2448no—
512SD16—2444no—
513D8—44no—
514C2×C8—614no—
515C2×C8C23×D6114yes—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—612no—
519C4⋊C4—614no—
520C4⋊C4—614no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—1228no—
524C4⋊C4—1224no—
525C4⋊C4—1224no—
526C4⋊C4—1224no—
527C4⋊C4—1221no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4C23×D6112yes—
537C4⋊C4C23×D6112yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—1228no—
543C22⋊C4—1228no—
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C22⋊C4—1224no—
547C22⋊C4—1222no—
548C42—614no—
549C42—614no—
550C42—1228no—
551C42—1224no—
552C42—1222no—
553C42D4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C42C23×D6111yescharacteristic
556C42C23×D6111yescharacteristic
557C23—22no—
558C23—22no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C2×D4×D6111yescharacteristic
563Q8—42no—
564Q8—1224no—
565Q8—1224no—
566Q8—1224no—
567Q8—1228no—
568Q8—4881no—
569Q8—2444no—
570D4—22no—
571D4—24no—
572D4—22no—
573D4—42no—
574D4—44no—
575D4—42no—
576D4—81no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—614no—
586C2×C4—614no—
587C2×C4—1224no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1222no—
591C2×C4—4884no—
592C2×C4—2442no—
593C2×C4—2444no—
594C2×C4#1451979614yes—
595C2×C4C2×D4×D6111yescharacteristic
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C24×D6111yescharacteristic
601C8—24no—
602C8—1224no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—84no—
612C22C2×#1451979612yes—
613C22C22×D4×D6111yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—42no—
618C4—42no—
619C4—1224no—
620C4—1224no—
621C4—2444no—
622C4—2442no—
623C4—2442no—
624C4C2×#1451979612yes—
625C4C2×#1451979612yes—
626C4#1684059812yes—
627C2—22no—
628C2—82no—
629C2—82no—
630C2#2145947311yescharacteristic
631C2C22×#1451979611yescharacteristic
632C2#2146060311yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
Fitting, prime radicalC61×#3100C2
socleC22×C61C23×D4
hypercenter#3100D61
hypocenterC61#61882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21459681C2semidirect
2. C2×#16840482C2semidirect
3. #21458819C2semidirect
4. #21458805C2semidirect
5. #21458503C2semidirect
6. #21458466C2semidirect
7. #21456781C2semidirect
8. #21456769C2semidirect
9. #21456741C2semidirect
10. #21455896C2semidirect
11. #21455827C2semidirect
12. #21454899C2semidirect
14. #16840498C22semidirect
15. #16840487C22semidirect
20. #16840482C22semidirect
22. #16840469C22semidirect
37. #16840142C22semidirect
38. #16840134C22semidirect
40. #16840114C22semidirect
41. #16840113C22semidirect
43. #16840068C22semidirect
46. #16839983C22semidirect
48. #16839973C22semidirect
50. #16839967C22semidirect
54. C8×Dic122C22semidirect
55. #16839886C22semidirect
56. #16839856C22semidirect
57. #16839853C22semidirect
385. C61#61882semidirect

Maximal subgroups

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

subgroupindex
#214596812
C2×#168404822
#214588192
#214588052
#214585032
#214584662
#214567812
#214567692
#214567412
#214558962
#214558272
#214548992
C61×#31002
#6188261