Finite Group Databasebeta

← #30571040

Subgroups — #30571040

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#21459684C211yesmaximal, characteristic, complemented
2C2×#16840485C211yesmaximal, characteristic, complemented
3#21458817C214yesmaximal, complemented
4#21458808C214yesmaximal, complemented
5#21458506C214yesmaximal, complemented
6#21458470C214yesmaximal, complemented
7#21456786C214yesmaximal, complemented
8#21456769C211yesmaximal, characteristic, complemented
9#21456746C211yesmaximal, characteristic, complemented
10#21455901C214yesmaximal, complemented
11#21455828C211yesmaximal, characteristic, complemented
12#21455820C211yesmaximal, characteristic
13C61×#2982C211yesmaximal, characteristic, complemented
14#16840497C2212yescomplemented
15#16840490C2212yescomplemented
16#16840485—22no—
17#16840485—22no—
18#16840485C2214yes—
19#16840485C2214yes—
20#16840485C2212yescomplemented
21#16840484—22no—
22#16840472C2212yescomplemented
23#16840453—22no—
24#16840359—22no—
25#16840336—21no—
26#16840319C2211yescharacteristic
27#16840312C2211yescharacteristic
28C2×#14519777C2211yescharacteristic
29#16840296C2214yes—
30C2×#14519772—22no—
31C2×#14519765C2212yes—
32C2×#14519753—21no—
33#16840243—21no—
34C2×C4×Dic122C2212yes—
35C2×C4×D244C2211yescharacteristic
36C2×#14519735—21no—
37#16840146C2218yes—
38#16840142C2212yescomplemented
39#16840135C2212yescomplemented
40#16840117C2218yes—
41#16840113C2212yescomplemented
42#16840113C2212yescomplemented
43#16840093C2212yes—
44#16840076C2212yes—
45#16840070C2218yescomplemented
46#16840053C2212yes—
47#16840043C2212yes—
48#16839985C2218yes—
49#16839985C2218yescomplemented
50#16839978C2218yes—
51#16839973C2214yescomplemented
52#16839968C2212yes—
53#16839968C2214yescomplemented
54#16839958C2212yes—
55#16839920C2214yes—
56#16839915C2212yes—
57#16839903C2214yes—
58#16839898C2212yes—
59C4×Dic244C2218yescomplemented
60C4×#11758190C2214yes—
61C4×#11758190C2218yescomplemented
62C4×D488C2214yes—
63C2×C4×Q8×C61C2211yescharacteristic
64C8×Q8×C61C2214yescomplemented
65C8×Q8×C61C2214yescomplemented
66C61×#431C2214yes—
67C61×#431C2211yescharacteristic
68C61×#431C2211yescharacteristic
69C2×#11758222—21no—
70C22×Dic122C2312yes—
71C22×D244C2311yescharacteristic
72C22×C4×D61—21no—
73C2×Dic244C2318yes—
74C2×#11758190C2314yes—
75C2×#11758190C2318yes—
76C2×D488C2314yes—
77#14519777—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519777C2312yes—
81#14519777C2312yes—
82#14519777C2311yescharacteristic
83#14519777C2311yescharacteristic
84#14519775—22no—
85#14519774C2311yescharacteristic
86#14519772—24no—
87#14519772—28no—
88#14519772—28no—
89#14519771—24no—
90#14519769—21no—
91#14519769C2314yes—
92#14519767—24no—
93#14519766—22no—
94#14519765—21no—
95#14519765—24no—
96#14519765C2318yes—
97#14519765C2314yes—
98#14519762—24no—
99#14519760—24no—
100#14519756—21no—
101#14519756C2312yes—
102#14519755—24no—
103#14519753—22no—
104#14519753—22no—
105#14519753—24no—
106#14519753—24no—
107#14519751—21no—
108#14519751C2312yes—
109C4×Dic122—24no—
110C4×Dic122—21no—
111C4×Dic122C2318yes—
112C4×Dic122C2312yes—
113C4×Dic122C2314yes—
114#14519742—22no—
115C4×D244—21no—
116C4×D244—21no—
117C4×D244—21no—
118C4×D244C2312yes—
119C4×D244C2312yes—
120C4×D244C2314yes—
121C4×D244C2312yes—
122#14519739—22no—
123C4×C22⋊D61—24no—
124#14519736—21no—
125#14519735—22no—
126#14519735—24no—
127#14519735—24no—
128#14519734—24no—
129C2×C4⋊Dic61—21no—
130C2×C4⋊Dic61—21no—
131C2×C4⋊Dic61C2311yescharacteristic
132C2×#11758196—22no—
133C4⋊C4×D61—22no—
134C2×#11758198—22no—
135C2×#11758198—21no—
136C2×#11758198—22no—
137C2×C4×Dic61—22no—
138#14519711C2314yes—
139#14519709C2314yes—
140#14519702C2314yes—
141#14519702C2318yes—
142#14519702C2318yes—
143#14519699C2314yes—
144#14519699C2314yes—
145#14519699C2314yes—
146C22×Q8×C61D411yescharacteristic
147C4×Q8×C61D412yes—
148C4×Q8×C61D412yes—
149C4×Q8×C61C2312yes—
150C4×Q8×C61C2312yes—
151C2×C4⋊C4×C61D411yescharacteristic
152C2×C4⋊C4×C61D412yes—
153C2×C42×C61C2311yescharacteristic
154C2×C42×C61C2312yes—
155C4⋊C8×C61C2312yes—
156C4⋊C8×C61C2312yes—
157C4⋊C8×C61C2318yes—
158C22⋊C8×C61C2312yes—
159C22⋊C8×C61C2312yes—
160C4×C8×C61C2312yes—
161C4×C8×C61C2312yes—
162C4×C8×C61C2318yes—
163C23×D61—21no—
164C2×Q8×C61—24no—
165C2×Q8×C61C2×D412yes—
166C2×Q8×C61C2×D412yes—
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2411yescharacteristic
172C2×C22⋊D61—22no—
173C22×Dic61—21no—
174C22×Dic61—22no—
175#11758222—44no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244—21no—
179C2×D244—42no—
180C2×D244C2412yes—
181C2×D244C2412yes—
182C2×C4×D61—22no—
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—24no—
186C2×C4×D61—44no—
187C2×Dic122—22no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—44no—
191C2×Dic122C2414yes—
192C2×Dic122C2412yes—
193C2×C8×C61C2414yes—
194C2×C8×C61C2414yes—
195C4⋊C4×C61C2×D412yes—
196C4⋊C4×C61C2×D412yes—
197C4⋊C4×C61C2×D414yes—
198C4⋊C4×C61C2×D414yes—
199C42×C61C2×D414yes—
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2411yescharacteristic
203C42×C61C2411yescharacteristic
204C42×C61C2414yes—
205C22⋊Dic61—22no—
206#11758198—28no—
207#11758198—22no—
208#11758198—22no—
209#11758198—28no—
210#11758198—22no—
211#11758198—24no—
212#11758198—24no—
213#11758198—22no—
214#11758198—24no—
215C4⋊Dic61—21no—
216C4⋊Dic61—22no—
217C4⋊Dic61—22no—
218C4⋊Dic61—22no—
219C4⋊Dic61—22no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61C2412yes—
225C4⋊Dic61C2412yes—
226C4⋊Dic61C2412yes—
227#11758196—24no—
228#11758196—28no—
229#11758196—22no—
230#11758196—24no—
231#11758196—24no—
232#11758196—22no—
233C4×Dic61—28no—
234C4×Dic61—24no—
235C4×Dic61—22no—
236Dic244—48no—
237D488—44no—
238#11758190—44no—
239#11758190—48no—
240C23×C61C22×D411yescharacteristic
241C22×D61—22no—
242C22×D61—22no—
243C22×D61—42no—
244C22×D61—42no—
245C22×D61—42no—
246Q8×C61—24no—
247Q8×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61#13514yes—
253C2×C4×C61C22×D414yes—
254C2×C4×C61C22×D414yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2511yescharacteristic
261C22⋊D61—44no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—24no—
266C2×Dic61—22no—
267C2×Dic61—44no—
268C2×Dic61—42no—
269D244—24no—
270D244—22no—
271D244—22no—
272D244—42no—
273D244—42no—
274D244—81no—
275C4×D61—44no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61—84no—
279Dic122—24no—
280Dic122—28no—
281Dic122—44no—
282Dic122—42no—
283Dic122—81no—
284C8×C61—24no—
285C8×C61—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—24no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—24no—
303C4×C61C2×#13512yes—
304C4×C61C2×#13512yes—
305C4×C61#58512yes—
306Dic61—42no—
307Dic61—44no—
308Dic61—42no—
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#3209—611no—
319#3206—611no—
320#3108—614no—
321#3101—611no—
322#3100—611no—
323#2982D6111yescharacteristic, complemented
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#533—611no—
343#532—611no—
344C2×C4⋊Q8—611no—
345#529—614no—
346C2×#126—1222no—
347C2×#124—612no—
348C2×C22⋊Q8—1221no—
349#520—1221no—
350C2×C4×Q8—612no—
351C2×C4×Q8C2×D6111yescharacteristic
352C2×C4×D4—611no—
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#480—612no—
362#479—618no—
363#476—612no—
364#474—612no—
365#464—618no—
366#464—618no—
367#463—618no—
368#462—614no—
369#461—612no—
370#461—614no—
371#459—612no—
372#451—614no—
373#450—612no—
374#448—614no—
375#447—612no—
376C8×Q8C2×D6114yescomplemented
377C8×Q8C2×D6114yescomplemented
378C4×Q16—618no—
379C4×SD16—614no—
380C4×SD16—618no—
381C4×D8—614no—
382#431C2×D6111yescharacteristic
383#431C2×D6111yescharacteristic
384#431C2×D6114yes—
385C61#6188211yescharacteristic, complemented, Sylow
386C2×#41—1221no—
387C22×Q8—612no—
388C22×Q8D24411yescharacteristic
389C22×D4—611no—
390C23×C4—1221no—
391C2×Q16—618no—
392C2×SD16—618no—
393C2×SD16—614no—
394C2×D8—614no—
395C4⋊Q8—611no—
396C4⋊Q8—611no—
397C4⋊Q8—612no—
398C4⋊Q8—612no—
399C4⋊Q8—611no—
400C4⋊Q8—1222no—
401C4⋊Q8—1221no—
402C4⋊Q8—1221no—
403#127—611no—
404#126—1228no—
405#126—1228no—
406#126—1224no—
407#126—1224no—
408#125—614no—
409#125—1224no—
410#125—1221no—
411#124—618no—
412#124—614no—
413#124—1222no—
414#124—1224no—
415#124—1221no—
416#123—1224no—
417#123—1224no—
418C22⋊Q8—612no—
419C22⋊Q8—1224no—
420C22⋊Q8—1224no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1221no—
424C22⋊Q8—1224no—
425#121—612no—
426#121—1221no—
427C4×Q8—618no—
428C4×Q8—614no—
429C4×Q8—612no—
430C4×Q8—1222no—
431C4×Q8—1224no—
432C4×Q8—1221no—
433C4×Q8D24412yes—
434C4×Q8D24412yes—
435C4×Q8C22×D6112yes—
436C4×Q8C22×D6112yes—
437C4×D4—612no—
438C4×D4—612no—
439C4×D4—614no—
440C4×D4—612no—
441C4×D4—1224no—
442C4×D4—1222no—
443C4×D4—1221no—
444C4×D4—1221no—
445C4×D4—1221no—
446#117—1221no—
447#117—1224no—
448#117—1224no—
449#117—1222no—
450#117—1224no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1222no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1222no—
456C2×C4⋊C4D24411yescharacteristic
457C2×C4⋊C4D24412yes—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1222no—
461C2×C42—1222no—
462C2×C42C22×D6111yescharacteristic
463C2×C42C22×D6112yes—
464#107—614no—
465#106—614no—
466C4⋊C8C22×D6112yes—
467C4⋊C8C22×D6112yes—
468C4⋊C8C22×D6118yes—
469#103—614no—
470#103—618no—
471#103—618no—
472#102—614no—
473#102—614no—
474#102—614no—
475C22⋊C8C22×D6112yes—
476C22⋊C8C22×D6112yes—
477C4×C8C22×D6112yes—
478C4×C8C22×D6112yes—
479C4×C8C22×D6118yes—
480C24—1221no—
481#41—2444no—
482C2×Q8—24no—
483C2×Q8—614no—
484C2×Q8—612no—
485C2×Q8—1221no—
486C2×Q8—1222no—
487C2×Q8—1221no—
488C2×Q8—2444no—
489C2×Q8C2×D24412yes—
490C2×Q8C2×D24412yes—
491C2×D4—612no—
492C2×D4—612no—
493C2×D4—1222no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—2442no—
498C22×C4—1221no—
499C22×C4—1222no—
500C22×C4—1224no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—2444no—
505C22×C4C2×D24412yes—
506C22×C4C2×D24412yes—
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—2448no—
511SD16—2448no—
512SD16—2444no—
513D8—2444no—
514C2×C8C23×D6114yes—
515C2×C8C23×D6114yes—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1224no—
522C4⋊C4—1228no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1224no—
529C4⋊C4—1222no—
530C4⋊C4—1224no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4C2×D24412yes—
535C4⋊C4C2×D24412yes—
536C4⋊C4C2×D24414yes—
537C4⋊C4C2×D24414yes—
538C22⋊C4—1222no—
539C22⋊C4—1228no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1228no—
543C22⋊C4—1222no—
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C22⋊C4—1222no—
547C22⋊C4—1224no—
548C42—1228no—
549C42—1224no—
550C42—1222no—
551C42C2×D24414yes—
552C42C2×D24411yescharacteristic
553C42C2×D24411yescharacteristic
554C42C23×D6111yescharacteristic
555C42C23×D6111yescharacteristic
556C42C23×D6114yes—
557C23—1222no—
558C23—1222no—
559C23—2442no—
560C23—2442no—
561C23—2442no—
562C23C22×D24411yescharacteristic
563Q8—24no—
564Q8—24no—
565Q8—1224no—
566Q8—1228no—
567Q8—4881no—
568Q8—2444no—
569Q8—2442no—
570D4—1222no—
571D4—1224no—
572D4—1222no—
573D4—4881no—
574D4—2442no—
575D4—2444no—
576D4—2442no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1224no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—4884no—
587C2×C4—2444no—
588C2×C4—2442no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4#1451979914yes—
593C2×C4C22×D24414yes—
594C2×C4C22×D24411yescharacteristic
595C2×C4C22×D24414yes—
596C2×C4C22×D24411yescharacteristic
597C2×C4C22×D24411yescharacteristic
598C2×C4C22×D24411yescharacteristic
599C2×C4C22×D24411yescharacteristic
600C2×C4C24×D6111yescharacteristic
601C8—24no—
602C8—24no—
603C22—21no—
604C22—21no—
605C22—4884no—
606C22—2442no—
607C22—2442no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22C2×#1451979912yes—
613C22C23×D24411yescharacteristic
614C4—21no—
615C4—24no—
616C4—24no—
617C4—21no—
618C4—21no—
619C4—2444no—
620C4—2442no—
621C4—2442no—
622C4—2442no—
623C4—2442no—
624C4C2×#1451979912yes—
625C4C2×#1451979912yes—
626C4#1684059912yes—
627C2—22no—
628C2—4882no—
629C2—4882no—
630C2#2145947811yescharacteristic
631C2C22×#1451979911yescharacteristic
632C2#2146060711yescharacteristic
subgroupG/H
centerC22C23×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2982C2
socleC22×C61C23×D4
prime radical#21455820C2
hypercenter#2982D61
hypocenterC61#61882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. #21459684C2semidirect
2. C2×#16840485C2semidirect
3. #21458817C2semidirect
4. #21458808C2semidirect
5. #21458506C2semidirect
6. #21458470C2semidirect
7. #21456786C2semidirect
8. #21456769C2semidirect
9. #21456746C2semidirect
10. #21455901C2semidirect
11. #21455828C2semidirect
13. C61×#2982C2semidirect
14. #16840497C22semidirect
15. #16840490C22semidirect
20. #16840485C22semidirect
22. #16840472C22semidirect
38. #16840142C22semidirect
39. #16840135C22semidirect
41. #16840113C22semidirect
42. #16840113C22semidirect
45. #16840070C22semidirect
49. #16839985C22semidirect
51. #16839973C22semidirect
53. #16839968C22semidirect
59. C4×Dic244C22semidirect
61. C4×#11758190C22semidirect
64. C8×Q8×C61C22semidirect
65. C8×Q8×C61C22semidirect
323. #2982D61semidirect
376. C8×Q8C2×D61semidirect
377. C8×Q8C2×D61semidirect
385. C61#61882semidirect

Maximal subgroups

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

subgroupindex
#214596842
C2×#168404852
#214588172
#214588082
#214585062
#214584702
#214567862
#214567692
#214567462
#214559012
#214558282
#214558202
C61×#29822
#6188261