Finite Group Databasebeta

← #30611500

Subgroups — #30611500

Order 15616.

1028 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462657C211yesmaximal, characteristic, complemented
2#21462390C211yesmaximal, characteristic, complemented
3#21461295C212yesmaximal, complemented
4#21461289C211yesmaximal, characteristic, complemented
5#21461284C211yesmaximal, characteristic, complemented
6#21461278C212yesmaximal, complemented
7#21461274C211yesmaximal, characteristic, complemented
8#21461270C212yesmaximal, complemented
9#21461269C211yesmaximal, characteristic, complemented
10C2×#16840617C212yesmaximal, complemented
11#21459771C211yesmaximal, characteristic
12C2×#14519864—21no—
13#16840812—21no—
14#16840791—22no—
15D4×#92420—21no—
16#16840784C2211yescharacteristic, complemented
17#16840783C2212yescomplemented
18#16840727—21no—
19#16840726—22no—
20C4×#11758213C2211yescharacteristic, complemented
21C4×#11758213C2211yescharacteristic, complemented
22C4×#11758217C2212yescomplemented
23#16840721C2211yescharacteristic, complemented
24C2×#14519832C2212yescomplemented
25C2×#14519832C2212yescomplemented
26#16840619C2211yescharacteristic, complemented
27#16840619C2211yescharacteristic, complemented
28#16840619C2212yescomplemented
29#16840617C2211yescharacteristic
30#16840617C2212yes—
31#16840617C2211yescharacteristic
32#16840617C2212yescomplemented
33#16840617C2212yescomplemented
34#16840617C2212yescomplemented
35#16840617C2212yescomplemented
36#16840498C411yescharacteristic
37#16840482C411yescharacteristic
38#16840476—21no—
39#16840443—22no—
40D4×Dic122C411yescharacteristic
41#16840418C412yes—
42#16840406—21no—
43#16840406C412yes—
44#16840379—21no—
45#16840347—22no—
46#16840310C2211yescharacteristic
47#16840294C2211yescharacteristic
48#16840293C2212yes—
49C2×#14519764C2212yes—
50#16840237—21no—
51C4×#11758224C2211yescharacteristic
52D42×C61C411yescharacteristic
53#14519864—28no—
54C2×#11758213—21no—
55C2×#11758213—21no—
56C2×#11758217—22no—
57C2×D61⋊C8—21no—
58C22×#92420—22no—
59C22×#92420—22no—
60C4⋊#92420—22no—
61C4⋊#92420—21no—
62#14519834—21no—
63C22⋊#92420—22no—
64#14519832—22no—
65#14519832—22no—
66#14519832—21no—
67#14519832C2311yescharacteristic
68#14519832C2311yescharacteristic
69#14519832C2312yes—
70#14519832C2312yes—
71#14519832C2311yescharacteristic
72#14519832C2311yescharacteristic
73C4×#92420—22no—
74C4×#92420—21no—
75C4×#92420—21no—
76C2×#11758224C2×C412yes—
77C2×#11758224D411yescharacteristic
78C22×Dic122C2×C412yes—
79#14519777C2×C411yescharacteristic
80#14519776—21no—
81#14519770—22no—
82#14519770—22no—
83#14519770—21no—
84#14519768C2311yescharacteristic
85#14519764—21no—
86#14519764—22no—
87#14519764—22no—
88#14519764C2×C412yes—
89#14519764C2×C412yes—
90#14519764C2311yescharacteristic
91#14519764C2312yes—
92#14519763—22no—
93#14519759—21no—
94#14519758—22no—
95#14519758—22no—
96#14519758—22no—
97#14519757C2×C411yescharacteristic
98#14519756—21no—
99#14519756C2×C412yes—
100#14519754—22no—
101#14519754—22no—
102#14519749—22no—
103#14519749—22no—
104#14519748—21no—
105#14519748—21no—
106#14519748C2×C411yescharacteristic
107#14519746—22no—
108C4×Dic122C2×C411yescharacteristic
109C4×Dic122C2×C411yescharacteristic
110#14519742C2×C412yes—
111#14519742D411yescharacteristic
112D4×Dic61—21no—
113D4×Dic61—22no—
114D4×Dic61—22no—
115D4×Dic61—21no—
116D4×Dic61—21no—
117D4×Dic61—21no—
118D4×Dic61C2×C412yes—
119D4×Dic61D411yescharacteristic
120C4×C22⋊D61—21no—
121C4×C22⋊D61C2×C412yes—
122#14519737D412yes—
123#14519736—21no—
124#14519736—21no—
125#14519735D411yescharacteristic
126#14519733—22no—
127#14519733D412yes—
128C2×C22⋊Dic61—22no—
129C2×C22⋊Dic61—22no—
130C2×C22⋊Dic61—22no—
131C2×C4×Dic61C2312yes—
132C42×D61C2311yescharacteristic
133C22×D4×C61—21no—
134C22×D4×C61C2×C412yes—
135#127×C61C2×C411yescharacteristic
136#121×C61—21no—
137#121×C61C2×C412yes—
138#120×C61—22no—
139C4×D4×C61C2×C411yescharacteristic
140C4×D4×C61C2×C411yescharacteristic
141C24×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—44no—
147C2×D4×C61—82no—
148C2×D4×C61C22⋊C411yescharacteristic
149C2×D4×C61C22×C412yes—
150C2×D4×C61C22×C411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61C22×C412yes—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161#11758224—24no—
162#11758224—82no—
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×Dic122—22no—
167C2×Dic122—82no—
168C2×Dic122C22⋊C411yescharacteristic
169C2×Dic122C22×C411yescharacteristic
170C2×Dic122C22×C412yes—
171#11758217—28no—
172C2×#92420—28no—
173C2×#92420—22no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—22no—
179C2×#92420—44no—
180#11758213—24no—
181#11758213—42no—
182D61⋊C8—24no—
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61C22⋊C412yes—
188C42×C61C22×C411yescharacteristic
189C22⋊Dic61—22no—
190C22⋊Dic61—21no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61C22⋊C412yes—
201#11758198—21no—
202#11758198C22⋊C412yes—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C22⋊C411yescharacteristic
208C4⋊Dic61C22×C411yescharacteristic
209#11758196—22no—
210#11758196—21no—
211#11758196—22no—
212#11758196—21no—
213#11758196C22⋊C412yes—
214#11758196C22⋊C412yes—
215C4×Dic61—21no—
216C4×Dic61—22no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D412yes—
222C4×Dic61C2411yescharacteristic
223C23×C61—21no—
224C23×C61—22no—
225C23×C61—44no—
226C23×C61—82no—
227C23×C61—82no—
228C23×C61C2×C22⋊C412yes—
229C22×D61C2×C22⋊C411yescharacteristic
230D4×C61—42no—
231D4×C61—41no—
232D4×C61—44no—
233D4×C61—42no—
234D4×C61—81no—
235D4×C61—82no—
236D4×C61#10414yes—
237C2×C4×C61—21no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C23×C411yescharacteristic
244C22⋊D61—24no—
245C22⋊D61—82no—
246C22⋊D61—81no—
247C2×Dic61—24no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—42no—
252C2×Dic61—82no—
253C2×Dic61—82no—
254C2×Dic61—82no—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C22×D411yescharacteristic
259C2×Dic61C22×D411yescharacteristic
260C4×D61—22no—
261C4×D61—41no—
262C4×D61—41no—
263Dic122—22no—
264Dic122—41no—
265Dic122—42no—
266Dic122—82no—
267#92420—22no—
268#92420—22no—
269#92420—24no—
270#92420—22no—
271#92420—41no—
272#92420—42no—
273#17652—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—42no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—41no—
282C22×C61—164no—
283C22×C61C2×#10414yes—
284C22×C61C22×C22⋊C411yescharacteristic
285C2×D61—21no—
286C2×D61—21no—
287C2×D61—21no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61C2×#10412yes—
293Dic61—21no—
294Dic61—21no—
295Dic61—22no—
296Dic61—21no—
297Dic61—22no—
298Dic61—41no—
299Dic61—42no—
300Dic61—41no—
301#3514—611no—
302#2970—611no—
303#2875—611no—
304#1532—612no—
305#1530—611no—
306#1529—611no—
307#1527—612no—
308#1525—611no—
309#1524—611no—
310#1524—612no—
311C2×#329—612no—
312C2×C61—24no—
313C2×C61—82no—
314C2×C61—81no—
315C2×C61#288811yescharacteristic
316C2×C61C22×#10411yescharacteristic
317C2×C61#290111yescharacteristic
318D61—41no—
319C2×#131—1221no—
320#559—611no—
321#556—611no—
322#555—1221no—
323#551—1222no—
324D4×Q8—611no—
325#548—612no—
326#546—612no—
327#546—1221no—
328D42#692311yescharacteristic
329#542—1221no—
330#538—1222no—
331#532—611no—
332#529—612no—
333#529—611no—
334C2×#124—612no—
335#518—1221no—
336C4×#41—611no—
337#446—1221no—
338#435—1222no—
339C8×D4—1221no—
340#432—612no—
341#432—611no—
342#405—1222no—
343#405—1221no—
344C4×M16—612no—
345C4×M16—611no—
346C4×M16—611no—
347C2×#97—612no—
348C2×#97—612no—
349C2×#97—611no—
350#330—611no—
351#330—612no—
352#330—611no—
353#329—611no—
354#329—612no—
355#329—611no—
356#329—612no—
357#329—612no—
358#329—612no—
359#329—612no—
360C61#1765211yescharacteristic, complemented, Sylow
361C2×#41—611no—
362C2×#41—612no—
363C22×Q8—612no—
364C22×D4—21no—
365C22×D4C2×#692312yes—
366#131—1228no—
367C2×M16—1222no—
368C2×M16—1221no—
369C2×M16—1221no—
370C22×C8—1222no—
371C22×C8—1221no—
372C22×C8—1222no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375#127C2×#692311yescharacteristic
376#126—1222no—
377#126—1221no—
378#126—1222no—
379#125—611no—
380#124—611no—
381#124—612no—
382#124—612no—
383#124—612no—
384#124—1222no—
385#124—1222no—
386#124—1222no—
387#124—1221no—
388#123—1221no—
389#123—1222no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—612no—
393C22⋊Q8—611no—
394C22⋊Q8—1222no—
395C22⋊Q8—1221no—
396C22⋊Q8—1222no—
397#121—21no—
398#121—611no—
399#121—1222no—
400#121—1221no—
401#121—1221no—
402#121—1222no—
403#121C2×#692312yes—
404#120—22no—
405#120—1222no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—612no—
409C4×Q8—611no—
410C4×D4—611no—
411C4×D4—612no—
412C4×D4—612no—
413C4×D4—612no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4—1222no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1222no—
420C4×D4—1221no—
421C4×D4C2×#692311yescharacteristic
422C4×D4C2×#692311yescharacteristic
423#117—611no—
424#117—612no—
425#117—1222no—
426#117—1221no—
427#117—1221no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1222no—
430C2×C22⋊C4—1222no—
431C2×C42—612no—
432C2×C42—611no—
433C4⋊C8—1221no—
434C4⋊C8—1222no—
435C22⋊C8—1222no—
436C22⋊C8—1221no—
437#97—611no—
438#97—611no—
439#97—612no—
440#97—611no—
441#97—612no—
442#97—611no—
443#97—1222no—
444#97—1221no—
445#97—1222no—
446C4×C8—1222no—
447C4×C8—1221no—
448C4×C8—1221no—
449C24—22no—
450#41—1224no—
451#41—4882no—
452C2×Q8—611no—
453C2×Q8—611no—
454C2×Q8—612no—
455C2×Q8—1222no—
456C2×Q8—4882no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—44no—
462C2×D4—82no—
463C2×D4—612no—
464C2×D4—1221no—
465C2×D4—1222no—
466C2×D4C22⋊#692311yescharacteristic
467C2×D4C22×#692312yes—
468C2×D4C22×#692311yescharacteristic
469C22×C4—21no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4C22×#692312yes—
479M16—1224no—
480M16—1228no—
481M16—2442no—
482C2×C8—1222no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1224no—
487C2×C8—1228no—
488C2×C8—1221no—
489C2×C8—1222no—
490C2×C8—2444no—
491C4⋊C4—612no—
492C4⋊C4—612no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4C22⋊#692311yescharacteristic
504C4⋊C4C22×#692311yescharacteristic
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—612no—
508C22⋊C4—612no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1222no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4C22⋊#692312yes—
522C42—612no—
523C42—611no—
524C42—611no—
525C42—612no—
526C42—611no—
527C42—612no—
528C42—1222no—
529C42—1221no—
530C42C22×#692311yescharacteristic
531C23—21no—
532C23—22no—
533C23—44no—
534C23—82no—
535C23—82no—
536C23—611no—
537C23C2×C22⋊#692312yes—
538Q8—1222no—
539Q8—4882no—
540Q8—2441no—
541Q8—2442no—
542D4—41no—
543D4—42no—
544D4—44no—
545D4—42no—
546D4—81no—
547D4—82no—
548D4—1224no—
549D4—4881no—
550D4—4882no—
551D4#1451984114yes—
552C2×C4—21no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—612no—
556C2×C4—611no—
557C2×C4—612no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—1224no—
561C2×C4—1222no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—4882no—
566C2×C4—4882no—
567C2×C4—4882no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2442no—
571C2×C4C2×C22⋊#692312yes—
572C2×C4C2×C22⋊#692311yescharacteristic
573C2×C4C2×C22⋊#692311yescharacteristic
574C2×C4C23×#692311yescharacteristic
575C8—1222no—
576C8—1222no—
577C8—1224no—
578C8—1222no—
579C8—2442no—
580C8—2441no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—42no—
588C22—41no—
589C22—164no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22C2×#1451984114yes—
594C22C22×C22⋊#692311yescharacteristic
595C4—21no—
596C4—22no—
597C4—21no—
598C4—41no—
599C4—1221no—
600C4—1221no—
601C4—1222no—
602C4—1221no—
603C4—1222no—
604C4—2442no—
605C4—2441no—
606C4—2441no—
607C4C2×#1451984112yes—
608C2—24no—
609C2—81no—
610C2—82no—
611C2—2441no—
612C2#2146242911yescharacteristic
613C2C22×#1451984111yescharacteristic
614C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingD42×C61C4
socleC22×C61C22×C22⋊C4
prime radical#21459771C2
cosocleC4×Dic61C24
hypercenterD42#6923
hypocenterC61#17652
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #21462657C2semidirect
2. #21462390C2semidirect
3. #21461295C2semidirect
4. #21461289C2semidirect
5. #21461284C2semidirect
6. #21461278C2semidirect
7. #21461274C2semidirect
8. #21461270C2semidirect
9. #21461269C2semidirect
10. C2×#16840617C2semidirect
16. #16840784C22semidirect
17. #16840783C22semidirect
20. C4×#11758213C22semidirect
21. C4×#11758213C22semidirect
22. C4×#11758217C22semidirect
23. #16840721C22semidirect
24. C2×#14519832C22semidirect
25. C2×#14519832C22semidirect
26. #16840619C22semidirect
27. #16840619C22semidirect
28. #16840619C22semidirect
32. #16840617C22semidirect
33. #16840617C22semidirect
34. #16840617C22semidirect
35. #16840617C22semidirect
360. C61#17652semidirect

Maximal subgroups

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

subgroupindex
#214626572
#214623902
#214612952
#214612892
#214612842
#214612782
#214612742
#214612702
#214612692
C2×#168406172
#214597712
#1765261