Finite Group Databasebeta

← #30571067

Subgroups — #30571067

Order 15616.

1814 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic, complemented
2C2×#16840497C211yesmaximal, characteristic, complemented
3#21458832C214yesmaximal, complemented
4#21458817C214yesmaximal, complemented
5#21458517C214yesmaximal, complemented
6#21458481C214yesmaximal, complemented
7#21456786C214yesmaximal, complemented
8#21456762C211yesmaximal, characteristic, complemented
9#21456739C211yesmaximal, characteristic
10#21455868C214yesmaximal, complemented
11#21455859C211yesmaximal, characteristic, complemented
12#21455851C211yesmaximal, characteristic, complemented
13C61×#2982C211yesmaximal, characteristic, complemented
14#16840497—22no—
15#16840497—22no—
16#16840497C2214yes—
17#16840497C2214yes—
18#16840497C2212yescomplemented
19#16840490C2212yescomplemented
20#16840485C2212yescomplemented
21#16840484—22no—
22#16840472C2212yescomplemented
23#16840453—22no—
24#16840359—22no—
25#16840336—21no—
26#16840319C2211yescharacteristic
27#16840312C2211yescharacteristic
28C2×#14519774C2211yescharacteristic
29#16840296C2214yes—
30C2×#14519766—21no—
31C2×#14519765C2212yes—
32C2×#14519762—22no—
33#16840243—21no—
34C2×C4×Dic122C2211yescharacteristic
35C2×C4×D244C2212yes—
36C2×#14519739—21no—
37#16840146C2218yescomplemented
38#16840139C2212yes—
39#16840135C2212yes—
40#16840117C2218yescomplemented
41#16840109C2212yes—
42#16840109C2212yes—
43#16840098C2212yes—
44#16840081C2212yes—
45#16840063C2218yescomplemented
46#16840058C2212yescomplemented
47#16840048C2212yescomplemented
48#16839985C2218yescomplemented
49#16839978C2218yescomplemented
50#16839978C2218yescomplemented
51#16839973C2212yescomplemented
52#16839963C2214yes—
53#16839963C2212yescomplemented
54#16839958C2214yes—
55#16839920C2212yes—
56#16839915C2214yes—
57#16839903C2212yes—
58#16839898C2214yes—
59C4×Dic244C2214yescomplemented
60C4×#11758190C2214yescomplemented
61C4×#11758190C2218yescomplemented
62C4×D488C2218yescomplemented
63C2×C4×Q8×C61C2211yescharacteristic
64C8×Q8×C61C2214yescomplemented
65C8×Q8×C61C2214yescomplemented
66C61×#431C2211yescharacteristic
67C61×#431C2211yescharacteristic
68C61×#431C2214yes—
69C2×#11758222—21no—
70C22×Dic122C2311yescharacteristic
71C22×D244—21no—
72C22×D244—21no—
73C22×D244C2312yes—
74C22×C4×D61—22no—
75C2×Dic244C2314yes—
76C2×#11758190C2318yes—
77C2×#11758190C2314yes—
78C2×D488C2318yes—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519775—22no—
83#14519774—21no—
84#14519774—21no—
85#14519774C2312yes—
86#14519774C2312yes—
87#14519774C2311yescharacteristic
88#14519772—24no—
89#14519771—24no—
90#14519769—21no—
91#14519769C2314yes—
92#14519767—24no—
93#14519766—22no—
94#14519766—24no—
95#14519766—24no—
96#14519765—21no—
97#14519765—24no—
98#14519765C2314yes—
99#14519765C2318yes—
100#14519762—24no—
101#14519762—28no—
102#14519762—28no—
103#14519760—24no—
104#14519756—21no—
105#14519756C2312yes—
106#14519755—24no—
107#14519753—22no—
108#14519753—22no—
109#14519751—21no—
110#14519751C2312yes—
111C4×Dic122—21no—
112C4×Dic122—21no—
113C4×Dic122—21no—
114C4×Dic122C2312yes—
115C4×Dic122C2312yes—
116C4×Dic122C2312yes—
117C4×Dic122C2314yes—
118#14519742—22no—
119C4×D244—21no—
120C4×D244—24no—
121C4×D244C2312yes—
122C4×D244C2314yes—
123C4×D244C2318yes—
124#14519739—24no—
125#14519739—24no—
126#14519739—22no—
127C4×C22⋊D61—24no—
128#14519736—21no—
129#14519735—22no—
130#14519734—24no—
131C2×C4⋊Dic61C2311yescharacteristic
132C2×#11758196—21no—
133C4⋊C4×D61—22no—
134C2×#11758198—22no—
135C2×#11758198—22no—
136C2×#11758198—22no—
137C2×C4×Dic61—21no—
138#14519711C2314yes—
139#14519709C2314yes—
140#14519702C2314yes—
141#14519702C2314yes—
142#14519702C2314yes—
143#14519699C2318yes—
144#14519699C2318yes—
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×C61C2312yes—
154C2×C42×C61C2311yescharacteristic
155C4⋊C8×C61C2312yes—
156C4⋊C8×C61C2312yes—
157C4⋊C8×C61C2318yes—
158C22⋊C8×C61C2312yes—
159C22⋊C8×C61C2312yes—
160C4×C8×C61C2318yes—
161C4×C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C23×D61—22no—
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—21no—
175#11758222—44no—
176C2×D244—22no—
177C2×D244—22no—
178C2×D244—22no—
179C2×D244—21no—
180C2×D244—22no—
181C2×D244—22no—
182C2×D244—21no—
183C2×D244—44no—
184C2×D244—44no—
185C2×D244—82no—
186C2×D244C2414yes—
187C2×D244C2412yes—
188C2×C4×D61—22no—
189C2×C4×D61—28no—
190C2×C4×D61—24no—
191C2×C4×D61—84no—
192C2×Dic122—21no—
193C2×Dic122—21no—
194C2×Dic122—21no—
195C2×Dic122—42no—
196C2×Dic122C2412yes—
197C2×Dic122C2412yes—
198C2×C8×C61C2414yes—
199C2×C8×C61C2414yes—
200C4⋊C4×C61C2×D412yes—
201C4⋊C4×C61C2×D412yes—
202C4⋊C4×C61C2×D414yes—
203C4⋊C4×C61C2×D414yes—
204C42×C61C2×D414yes—
205C42×C61C2×D411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C42×C61C2414yes—
208C42×C61C2411yescharacteristic
209C42×C61C2411yescharacteristic
210C22⋊Dic61—22no—
211#11758198—28no—
212#11758198—28no—
213#11758198—28no—
214#11758198—22no—
215#11758198—24no—
216#11758198—24no—
217#11758198—22no—
218#11758198—24no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61C2412yes—
225C4⋊Dic61C2412yes—
226C4⋊Dic61C2412yes—
227#11758196—24no—
228#11758196—22no—
229#11758196—22no—
230#11758196—22no—
231#11758196—24no—
232#11758196—24no—
233#11758196—22no—
234C4×Dic61—22no—
235C4×Dic61—22no—
236C4×Dic61—24no—
237C4×Dic61—22no—
238Dic244—44no—
239D488—48no—
240#11758190—48no—
241#11758190—44no—
242C23×C61C22×D411yescharacteristic
243C22×D61—22no—
244C22×D61—24no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61—44no—
248Q8×C61—24no—
249Q8×C61—24no—
250C2×C4×C61—24no—
251C2×C4×C61—22no—
252C2×C4×C61—42no—
253C2×C4×C61—41no—
254C2×C4×C61#13514yes—
255C2×C4×C61C22×D414yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D414yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2511yescharacteristic
263C22⋊D61—44no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271D244—24no—
272D244—28no—
273D244—44no—
274D244—44no—
275D244—44no—
276D244—42no—
277D244—81no—
278D244—82no—
279D244—82no—
280C4×D61—44no—
281C4×D61—84no—
282C4×D61—88no—
283Dic122—24no—
284Dic122—22no—
285Dic122—22no—
286Dic122—42no—
287Dic122—42no—
288Dic122—81no—
289C8×C61—24no—
290C8×C61—24no—
291#61885—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61C2×#13512yes—
295C22×C61C23×D411yescharacteristic
296C2×D61—44no—
297C2×D61—42no—
298C2×D61—44no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—44no—
302C2×D61—88no—
303C4×C61—21no—
304C4×C61—24no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—24no—
308C4×C61C2×#13512yes—
309C4×C61C2×#13512yes—
310C4×C61#58512yes—
311Dic61—42no—
312Dic61—42no—
313Dic61—42no—
314Dic61—42no—
315Dic61—42no—
316#3507—611no—
317C2×#559—611no—
318#3396—614no—
319#3394—614no—
320#3365—614no—
321#3361—614no—
322#3211—614no—
323#3208—611no—
324#3205—611no—
325#3105—614no—
326#3104—611no—
327#3103—611no—
328#2982D6111yescharacteristic, complemented
329C2×C61—22no—
330C2×C61#348711yescharacteristic
331C2×C61C22×#13511yescharacteristic
332C2×C61#358511yescharacteristic
333D61—82no—
334D61—84no—
335#559—614no—
336#559—614no—
337#559—612no—
338#559—1222no—
339#559—1222no—
340#557—612no—
341#556—612no—
342#556—1222no—
343#554—612no—
344#552—1222no—
345#539—1222no—
346#536—1221no—
347#533—611no—
348#532—611no—
349C2×#127—611no—
350#529—614no—
351C2×#124—612no—
352C2×#124—1221no—
353C2×#123—1222no—
354#520—1221no—
355C2×C4×Q8—611no—
356C2×C4×Q8C2×D6111yescharacteristic
357C2×C4×D4—612no—
358C2×C4×D4—1221no—
359#495—618no—
360#493—612no—
361#492—612no—
362#488—618no—
363#486—612no—
364#486—612no—
365#484—612no—
366#481—612no—
367#478—618no—
368#477—612no—
369#475—612no—
370#464—618no—
371#463—618no—
372#463—618no—
373#462—612no—
374#460—614no—
375#460—612no—
376#459—614no—
377#451—612no—
378#450—614no—
379#448—612no—
380#447—614no—
381C8×Q8C2×D6114yescomplemented
382C8×Q8C2×D6114yescomplemented
383C4×Q16—614no—
384C4×SD16—614no—
385C4×SD16—618no—
386C4×D8—618no—
387#431C2×D6111yescharacteristic
388#431C2×D6111yescharacteristic
389#431C2×D6114yes—
390C61#6188511yescharacteristic, complemented, Sylow
391C2×#41—1221no—
392C22×Q8—611no—
393C22×Q8D24411yescharacteristic
394C22×D4—612no—
395C22×D4—1221no—
396C22×D4—1221no—
397C23×C4—1222no—
398C2×Q16—614no—
399C2×SD16—614no—
400C2×SD16—618no—
401C2×D8—618no—
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404C4⋊Q8—611no—
405C4⋊Q8—1222no—
406#127—612no—
407#127—612no—
408#127—611no—
409#127—1221no—
410#127—1221no—
411#126—1224no—
412#126—1224no—
413#125—614no—
414#125—1224no—
415#125—1221no—
416#124—618no—
417#124—614no—
418#124—1224no—
419#124—1224no—
420#124—1222no—
421#124—1224no—
422#124—1221no—
423#123—1224no—
424#123—1228no—
425#123—1228no—
426#123—1224no—
427C22⋊Q8—612no—
428C22⋊Q8—1222no—
429C22⋊Q8—1222no—
430C22⋊Q8—1221no—
431C22⋊Q8—1224no—
432#121—612no—
433#121—1221no—
434C4×Q8—612no—
435C4×Q8—612no—
436C4×Q8—614no—
437C4×Q8—612no—
438C4×Q8—1222no—
439C4×Q8—1221no—
440C4×Q8—1221no—
441C4×Q8—1221no—
442C4×Q8D24412yes—
443C4×Q8D24412yes—
444C4×Q8C22×D6112yes—
445C4×Q8C22×D6112yes—
446C4×D4—618no—
447C4×D4—614no—
448C4×D4—612no—
449C4×D4—1224no—
450C4×D4—1224no—
451C4×D4—1224no—
452C4×D4—1222no—
453C4×D4—1224no—
454C4×D4—1221no—
455#117—1221no—
456#117—1222no—
457#117—1224no—
458C2×C4⋊C4—611no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—1222no—
461C2×C4⋊C4D24411yescharacteristic
462C2×C4⋊C4D24412yes—
463C2×C22⋊C4—1222no—
464C2×C22⋊C4—1222no—
465C2×C22⋊C4—1222no—
466C2×C42—1221no—
467C2×C42C22×D6112yes—
468C2×C42C22×D6111yescharacteristic
469#107—614no—
470#106—614no—
471C4⋊C8C22×D6112yes—
472C4⋊C8C22×D6112yes—
473C4⋊C8C22×D6118yes—
474#103—614no—
475#103—614no—
476#103—614no—
477#102—614no—
478#102—618no—
479#102—618no—
480C22⋊C8C22×D6112yes—
481C22⋊C8C22×D6112yes—
482C4×C8C22×D6118yes—
483C4×C8C22×D6112yes—
484C4×C8C22×D6112yes—
485C24—1222no—
486#41—2444no—
487C2×Q8—24no—
488C2×Q8—612no—
489C2×Q8—612no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—2442no—
494C2×Q8C2×D24412yes—
495C2×Q8C2×D24412yes—
496C2×D4—614no—
497C2×D4—612no—
498C2×D4—1222no—
499C2×D4—1221no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1221no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—1222no—
506C2×D4—4882no—
507C2×D4—2444no—
508C2×D4—2444no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1224no—
512C22×C4—1228no—
513C22×C4—1222no—
514C22×C4—4884no—
515C22×C4C2×D24412yes—
516C22×C4C2×D24412yes—
517C22×C4C2×D24411yescharacteristic
518C22×C4C2×D24411yescharacteristic
519C22×C4C23×D6111yescharacteristic
520Q16—2444no—
521SD16—2444no—
522SD16—2448no—
523D8—2448no—
524C2×C8C23×D6114yes—
525C2×C8C23×D6114yes—
526C4⋊C4—612no—
527C4⋊C4—612no—
528C4⋊C4—612no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1224no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1224no—
536C4⋊C4—1222no—
537C4⋊C4—1224no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4C2×D24412yes—
542C4⋊C4C2×D24412yes—
543C4⋊C4C2×D24414yes—
544C4⋊C4C2×D24414yes—
545C22⋊C4—1222no—
546C22⋊C4—1228no—
547C22⋊C4—1228no—
548C22⋊C4—1228no—
549C22⋊C4—1222no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1222no—
553C22⋊C4—1224no—
554C42—1222no—
555C42—1222no—
556C42—1224no—
557C42—1222no—
558C42C2×D24414yes—
559C42C2×D24411yescharacteristic
560C42C2×D24411yescharacteristic
561C42C23×D6114yes—
562C42C23×D6111yescharacteristic
563C42C23×D6111yescharacteristic
564C23—1222no—
565C23—1224no—
566C23—2444no—
567C23—2444no—
568C23—2444no—
569C23C22×D24411yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—1224no—
573Q8—1222no—
574Q8—1222no—
575Q8—4881no—
576Q8—2442no—
577Q8—2442no—
578D4—1224no—
579D4—1228no—
580D4—4881no—
581D4—4882no—
582D4—4882no—
583D4—2444no—
584D4—2444no—
585D4—2444no—
586D4—2444no—
587D4—2442no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—4884no—
598C2×C4—4888no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2444no—
602C2×C4#1451979914yes—
603C2×C4C22×D24414yes—
604C2×C4C22×D24411yescharacteristic
605C2×C4C22×D24414yes—
606C2×C4C22×D24411yescharacteristic
607C2×C4C22×D24411yescharacteristic
608C2×C4C22×D24411yescharacteristic
609C2×C4C22×D24411yescharacteristic
610C2×C4C24×D6111yescharacteristic
611C8—24no—
612C8—24no—
613C22—21no—
614C22—21no—
615C22—4888no—
616C22—2444no—
617C22—2442no—
618C22—2442no—
619C22—2444no—
620C22—2444no—
621C22—2442no—
622C22C2×#1451979912yes—
623C22C23×D24411yescharacteristic
624C4—21no—
625C4—24no—
626C4—24no—
627C4—21no—
628C4—21no—
629C4—2442no—
630C4—2442no—
631C4—2442no—
632C4—2442no—
633C4—2442no—
634C4C2×#1451979912yes—
635C4C2×#1451979912yes—
636C4#1684059912yes—
637C2—22no—
638C2—4884no—
639C2—4882no—
640C2#2145947811yescharacteristic
641C2C22×#1451979911yescharacteristic
642C2#2146060011yescharacteristic
subgroupG/H
centerC22C23×D244
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#2982C2
socleC22×C61C23×D4
prime radical#21456739C2
hypercenter#2982D61
hypocenterC61#61885
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×#16840497C2semidirect
3. #21458832C2semidirect
4. #21458817C2semidirect
5. #21458517C2semidirect
6. #21458481C2semidirect
7. #21456786C2semidirect
8. #21456762C2semidirect
10. #21455868C2semidirect
11. #21455859C2semidirect
12. #21455851C2semidirect
13. C61×#2982C2semidirect
18. #16840497C22semidirect
19. #16840490C22semidirect
20. #16840485C22semidirect
22. #16840472C22semidirect
37. #16840146C22semidirect
40. #16840117C22semidirect
45. #16840063C22semidirect
46. #16840058C22semidirect
47. #16840048C22semidirect
48. #16839985C22semidirect
49. #16839978C22semidirect
50. #16839978C22semidirect
51. #16839973C22semidirect
53. #16839963C22semidirect
59. C4×Dic244C22semidirect
60. C4×#11758190C22semidirect
61. C4×#11758190C22semidirect
62. C4×D488C22semidirect
64. C8×Q8×C61C22semidirect
65. C8×Q8×C61C22semidirect
328. #2982D61semidirect
381. C8×Q8C2×D61semidirect
382. C8×Q8C2×D61semidirect
390. C61#61885semidirect

Maximal subgroups

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

subgroupindex
#214596842
C2×#168404972
#214588322
#214588172
#214585172
#214584812
#214567862
#214567622
#214567392
#214558682
#214558592
#214558512
C61×#29822
#6188561