Finite Group Databasebeta

← #30571042

Subgroups — #30571042

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#21459686C211yesmaximal, characteristic, complemented
2C2×#16840486C211yesmaximal, characteristic, complemented
3#21458814C214yesmaximal, complemented
4#21458809C214yesmaximal, complemented
5#21458508C214yesmaximal, complemented
6#21458473C214yesmaximal, complemented
7#21456789C214yesmaximal, complemented
8#21456744C211yesmaximal, characteristic, complemented
9#21455900C214yesmaximal, complemented
10#21455823C211yesmaximal, characteristic, complemented
11#21455820C211yesmaximal, characteristic
12#21454899C211yesmaximal, characteristic, complemented
13C61×#3209C211yesmaximal, characteristic, complemented
14#16840497C2212yescomplemented
15#16840491C2212yescomplemented
16#16840486—22no—
17#16840486—22no—
18#16840486C2214yes—
19#16840486C2214yes—
20#16840486C2212yescomplemented
21#16840485—22no—
22#16840473C2212yescomplemented
23#16840450—22no—
24#16840361—22no—
25#16840338—21no—
26#16840312C2211yescharacteristic
27#16840298C2214yes—
28C2×#14519772—22no—
29C2×#14519766C2212yes—
30C2×#14519757—21no—
31#16840242—21no—
32C2×Q8×Dic61C2212yes—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244C2211yescharacteristic
35C2×#14519735—21no—
36#16840147C2218yes—
37#16840136C2212yescomplemented
38#16840118C2218yes—
39#16840112C2212yescomplemented
40#16840090C2212yes—
41#16840076C2212yes—
42#16840071C2218yescomplemented
43#16840049C2212yes—
44#16840043C2212yes—
45#16839986C2218yes—
46#16839986C2218yescomplemented
47#16839980C2218yes—
48#16839964C2212yes—
49#16839964C2214yescomplemented
50#16839958C2212yes—
51#16839912C2212yes—
52#16839901C2214yes—
53#16839898C2212yes—
54C8×Dic122C2214yescomplemented
55#16839886C2214yescomplemented
56Q16×Dic61C2218yescomplemented
57C4×#11758201C2214yes—
58#16839847C2218yescomplemented
59C4×D488C2214yes—
60#16839815C2211yescharacteristic
61#16839812C2214yes—
62C61×#533C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#494C2212yescomplemented
65C61×#487C2212yescomplemented
66C61×#462C2214yescomplemented
67C61×#451C2214yes—
68C61×#431C2211yescharacteristic
69C2×#11758222—21no—
70C22×Dic122D411yescharacteristic
71C22×D244C2311yescharacteristic
72C22×C4×D61—21no—
73C2×#11758201C2314yes—
74C2×#11758201C2318yes—
75C2×D488C2314yes—
76#14519777—22no—
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519772—24no—
80#14519772—28no—
81#14519772—28no—
82#14519772—24no—
83#14519768—24no—
84#14519768C2314yes—
85#14519766—24no—
86#14519766C2318yes—
87#14519766C2314yes—
88#14519765—22no—
89#14519765—21no—
90#14519762—24no—
91#14519761—24no—
92#14519757—22no—
93#14519757—22no—
94#14519757—24no—
95#14519757—24no—
96#14519753—24no—
97#14519751—21no—
98#14519751C2312yes—
99Q8×Dic61—24no—
100Q8×Dic61C2318yes—
101Q8×Dic61C2314yes—
102C4×Dic122—22no—
103C4×Dic122D412yes—
104C4×Dic122D412yes—
105C4×Dic122C2312yes—
106C4×Dic122C2312yes—
107C4×D244—21no—
108C4×D244—21no—
109C4×D244—21no—
110C4×D244—22no—
111C4×D244C2312yes—
112C4×D244C2312yes—
113C4×D244C2312yes—
114#14519739C2314yes—
115#14519737—24no—
116#14519735—22no—
117#14519735—24no—
118#14519735—24no—
119#14519735—24no—
120C2×C4⋊Dic61—22no—
121C2×C4⋊Dic61D411yescharacteristic
122C2×C4⋊Dic61C2311yescharacteristic
123C2×#11758196D412yes—
124C4⋊C4×D61—22no—
125C2×#11758198—21no—
126C2×#11758198—22no—
127C2×#11758198—22no—
128C2×C4×Dic61—22no—
129C2×C4×Dic61C2312yes—
130#14519711C2314yes—
131#14519708C2314yes—
132C4⋊#92418C2312yes—
133#14519706C2318yes—
134#14519701C2314yes—
135#14519701C2318yes—
136#14519699C2314yes—
137#14519699C2314yes—
138#14519697C2314yes—
139C22⋊#92418C2312yes—
140C4×#92418C2312yes—
141C8×Dic61C2318yes—
142C22×Q8×C61C2312yes—
143C2×Q16×C61C2318yes—
144C4⋊Q8×C61—21no—
145C4⋊Q8×C61—21no—
146C4⋊Q8×C61C2312yes—
147C4⋊Q8×C61C2312yes—
148C4⋊Q8×C61C2311yescharacteristic
149C4⋊Q8×C61C2311yescharacteristic
150#125×C61—21no—
151C22⋊Q8×C61—21no—
152C22⋊Q8×C61C2312yes—
153C4×Q8×C61—21no—
154C4×Q8×C61C2312yes—
155#117×C61—21no—
156C2×C4⋊C4×C61—21no—
157C2×C4⋊C4×C61—21no—
158C2×C42×C61C2311yescharacteristic
159C4⋊C8×C61C2312yes—
160#103×C61C2318yes—
161C22⋊C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C23×D61—21no—
164C2×Q8×C61—22no—
165C2×Q8×C61—21no—
166C2×Q8×C61—44no—
167C2×Q8×C61C2414yes—
168C2×Q8×C61C2412yes—
169C22×C4×C61—22no—
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2411yescharacteristic
173C2×C22⋊D61—22no—
174C22×Dic61—21no—
175C22×Dic61C2×D412yes—
176C22×Dic61C2×D412yes—
177#11758222—44no—
178C2×D244—21no—
179C2×D244—21no—
180C2×D244—21no—
181C2×D244—42no—
182C2×D244C2412yes—
183C2×D244C2412yes—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—24no—
188C2×C4×D61—44no—
189C2×Dic122—24no—
190C2×Dic122—21no—
191C2×Dic122C2×D412yes—
192C2×Dic122C2×D412yes—
193Q16×C61—48no—
194C2×C8×C61C2414yes—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61C2412yes—
205C22⋊C4×C61—22no—
206C42×C61—22no—
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2411yescharacteristic
210C42×C61C2411yescharacteristic
211#11758201—48no—
212#11758201—44no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—28no—
218#11758198—28no—
219#11758198—24no—
220#11758198—24no—
221#11758198—24no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61—28no—
225C4⋊Dic61—24no—
226C4⋊Dic61—24no—
227C4⋊Dic61C2×D412yes—
228C4⋊Dic61C2×D412yes—
229C4⋊Dic61C2412yes—
230C4⋊Dic61C2412yes—
231#11758196—24no—
232#11758196C2×D414yes—
233#11758196C2×D414yes—
234C4×Dic61—28no—
235C4×Dic61—24no—
236C4×Dic61C2×D414yes—
237C4×Dic61C2414yes—
238C2×#92418C2414yes—
239D488—44no—
240C23×C61C22×D411yescharacteristic
241C22×D61—22no—
242C22×D61—22no—
243C22×D61—42no—
244C22×D61—42no—
245C22×D61—42no—
246Q8×C61—24no—
247Q8×C61—28no—
248Q8×C61—44no—
249Q8×C61—81no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—24no—
253C2×C4×C61—22no—
254C2×C4×C61—41no—
255C2×C4×C61—44no—
256C2×C4×C61#13514yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2511yescharacteristic
263C22⋊D61—44no—
264C2×Dic61—24no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C22×D414yes—
270C2×Dic61C22×D414yes—
271D244—22no—
272D244—24no—
273D244—22no—
274D244—42no—
275D244—42no—
276D244—81no—
277C4×D61—44no—
278C4×D61—44no—
279C4×D61—44no—
280C4×D61—84no—
281Dic122—24no—
282Dic122—24no—
283Dic122—42no—
284C8×C61—24no—
285#92418—24no—
286#61882—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61C2×#13512yes—
290C22×C61C23×D411yescharacteristic
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—84no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61—44no—
303C4×C61—42no—
304C4×C61C2×#13512yes—
305C4×C61C2×#13512yes—
306C4×C61#58512yes—
307Dic61—24no—
308Dic61—24no—
309Dic61—42no—
310Dic61—42no—
311#3507—611no—
312C2×#556—611no—
313#3394—614no—
314#3393—614no—
315#3364—614no—
316#3360—614no—
317#3211—614no—
318#3209D6111yescharacteristic, complemented
319#3206—611no—
320#3108—614no—
321#3101—611no—
322#3100—611no—
323#2982—611no—
324C2×C61—22no—
325C2×C61#348711yescharacteristic
326C2×C61C22×#13511yescharacteristic
327C2×C61#358611yescharacteristic
328D61—82no—
329D61—82no—
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#533C2×D6111yescharacteristic
343#532—611no—
344C2×C4⋊Q8C2×D6111yescharacteristic
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×D4—611no—
353C2×#117—1221no—
354#495—618no—
355#494C2×D6112yescomplemented
356#492—612no—
357#488—618no—
358#487—612no—
359#487C2×D6112yescomplemented
360#483—612no—
361#480—612no—
362#479—618no—
363#476—612no—
364#474—612no—
365#464—618no—
366#464—618no—
367#463—618no—
368#462C2×D6114yescomplemented
369#461—612no—
370#461—614no—
371#459—612no—
372#451C2×D6114yes—
373#450—612no—
374#448—614no—
375#447—612no—
376C8×Q8—614no—
377C8×Q8—614no—
378C4×Q16—618no—
379C4×SD16—614no—
380C4×SD16—618no—
381C4×D8—614no—
382#431—611no—
383#431—614no—
384#431C2×D6111yescharacteristic
385C61#6188211yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C22×Q8—611no—
388C22×Q8C22×D6112yes—
389C22×D4—611no—
390C23×C4—1221no—
391C2×Q16C22×D6118yes—
392C2×SD16—618no—
393C2×SD16—614no—
394C2×D8—614no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—611no—
398C4⋊Q8—1222no—
399C4⋊Q8C22×D6112yes—
400C4⋊Q8C22×D6112yes—
401C4⋊Q8C22×D6111yescharacteristic
402C4⋊Q8C22×D6111yescharacteristic
403#127—611no—
404#126—1224no—
405#126—1228no—
406#126—1228no—
407#126—1224no—
408#125—21no—
409#125—614no—
410#125—1224no—
411#124—618no—
412#124—614no—
413#124—1222no—
414#124—1224no—
415#124—1221no—
416#123—1224no—
417#123—1224no—
418C22⋊Q8—21no—
419C22⋊Q8—1224no—
420C22⋊Q8—1224no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1224no—
424C22⋊Q8C22×D6112yes—
425#121—612no—
426#121—1221no—
427C4×Q8—21no—
428C4×Q8—612no—
429C4×Q8—612no—
430C4×Q8—612no—
431C4×Q8—612no—
432C4×Q8—618no—
433C4×Q8—614no—
434C4×Q8—1222no—
435C4×Q8—1224no—
436C4×Q8C22×D6112yes—
437C4×D4—614no—
438C4×D4—612no—
439C4×D4—612no—
440C4×D4—612no—
441C4×D4—1222no—
442C4×D4—1224no—
443C4×D4—1221no—
444C4×D4—1221no—
445C4×D4—1221no—
446#117—21no—
447#117—1224no—
448#117—1224no—
449#117—1224no—
450#117—1222no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—1222no—
457C2×C4⋊C4—1222no—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1221no—
461C2×C42—612no—
462C2×C42—1222no—
463C2×C42C22×D6111yescharacteristic
464#107—614no—
465#106—614no—
466C4⋊C8—618no—
467C4⋊C8—612no—
468C4⋊C8C22×D6112yes—
469#103—614no—
470#103—618no—
471#103C22×D6118yes—
472#102—614no—
473#102—614no—
474#102—614no—
475C22⋊C8—612no—
476C22⋊C8C22×D6112yes—
477C4×C8—618no—
478C4×C8—612no—
479C4×C8C22×D6112yes—
480C24—1221no—
481#41—2444no—
482C2×Q8—22no—
483C2×Q8—21no—
484C2×Q8—44no—
485C2×Q8—612no—
486C2×Q8—612no—
487C2×Q8—1221no—
488C2×Q8—1224no—
489C2×Q8C23×D6114yes—
490C2×Q8C23×D6112yes—
491C2×D4—612no—
492C2×D4—612no—
493C2×D4—1222no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—2442no—
498C22×C4—22no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—1224no—
502C22×C4—1221no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—2444no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—48no—
511SD16—2448no—
512SD16—2444no—
513D8—2444no—
514C2×C8—614no—
515C2×C8C23×D6114yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—612no—
526C4⋊C4—614no—
527C4⋊C4—614no—
528C4⋊C4—612no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1224no—
532C4⋊C4—1228no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4C23×D6112yes—
538C22⋊C4—22no—
539C22⋊C4—1228no—
540C22⋊C4—1228no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C22⋊C4—1224no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C42—22no—
549C42—614no—
550C42—614no—
551C42—1228no—
552C42—1224no—
553C42D4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C42C23×D6111yescharacteristic
556C42C23×D6111yescharacteristic
557C23—1222no—
558C23—1222no—
559C23—2442no—
560C23—2442no—
561C23—2442no—
562C23C2×D4×D6111yescharacteristic
563Q8—24no—
564Q8—28no—
565Q8—44no—
566Q8—81no—
567Q8—1224no—
568Q8—1224no—
569Q8—2442no—
570D4—1222no—
571D4—1224no—
572D4—1222no—
573D4—4881no—
574D4—2442no—
575D4—2444no—
576D4—2442no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—614no—
584C2×C4—614no—
585C2×C4—1224no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—4884no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2444no—
592C2×C4—2444no—
593C2×C4—2444no—
594C2×C4#1451980014yes—
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—4884no—
606C22—2442no—
607C22—2442no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22C2×#1451980012yes—
613C22C22×D4×D6111yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—1224no—
621C4—1224no—
622C4—2442no—
623C4—2442no—
624C4C2×#1451980012yes—
625C4C2×#1451980012yes—
626C4#1684059712yes—
627C2—22no—
628C2—4882no—
629C2—4882no—
630C2#2145947511yescharacteristic
631C2C22×#1451980011yescharacteristic
632C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3209C2
socleC22×C61C23×D4
prime radical#21455820C2
hypercenter#3209D61
hypocenterC61#61882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. #21459686C2semidirect
2. C2×#16840486C2semidirect
3. #21458814C2semidirect
4. #21458809C2semidirect
5. #21458508C2semidirect
6. #21458473C2semidirect
7. #21456789C2semidirect
8. #21456744C2semidirect
9. #21455900C2semidirect
10. #21455823C2semidirect
12. #21454899C2semidirect
13. C61×#3209C2semidirect
14. #16840497C22semidirect
15. #16840491C22semidirect
20. #16840486C22semidirect
22. #16840473C22semidirect
37. #16840136C22semidirect
39. #16840112C22semidirect
42. #16840071C22semidirect
46. #16839986C22semidirect
49. #16839964C22semidirect
54. C8×Dic122C22semidirect
55. #16839886C22semidirect
56. Q16×Dic61C22semidirect
58. #16839847C22semidirect
64. C61×#494C22semidirect
65. C61×#487C22semidirect
66. C61×#462C22semidirect
318. #3209D61semidirect
355. #494C2×D61semidirect
359. #487C2×D61semidirect
368. #462C2×D61semidirect
385. C61#61882semidirect

Maximal subgroups

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

subgroupindex
#214596862
C2×#168404862
#214588142
#214588092
#214585082
#214584732
#214567892
#214567442
#214559002
#214558232
#214558202
#214548992
C61×#32092
#6188261