Finite Group Databasebeta

← #30610636

Subgroups — #30610636

Order 15616.

984 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462738C211yesmaximal, characteristic, complemented
2#41×#92420C211yesmaximal, characteristic, complemented
3#21461178C212yesmaximal, complemented
4#21461176C212yesmaximal, complemented
5#21461172C212yesmaximal, complemented
6#21461164C212yesmaximal, complemented
7#21461120C211yesmaximal, characteristic, complemented
8#21461114C211yesmaximal, characteristic, complemented
9#21461110C211yesmaximal, characteristic, complemented
10C2×#16840610C211yesmaximal, characteristic, complemented
11#21459236C211yesmaximal, characteristic
12#16840813—21no—
13Q8×#92420—21no—
14#16840797—21no—
15#16840793—22no—
16D4×#92420—21no—
17D4×#92420—22no—
18#16840783C2212yescomplemented
19#16840778C2211yescharacteristic
20#16840778C2211yescharacteristic
21#16840778C2212yescomplemented
22#16840765C2212yescomplemented
23C2×C4⋊#92420C2211yescharacteristic
24C2×C4×#92420C2211yescharacteristic
25C2×C4×#92420C2212yescomplemented
26Q8⋊#92420C2211yescharacteristic, complemented
27Q8⋊#92420C2211yescharacteristic, complemented
28Q8⋊#92420C2212yescomplemented
29#16840610C2212yes—
30#16840610C2212yes—
31#16840610C2212yes—
32#16840610C2212yes—
33D4⋊#92420C2211yescharacteristic, complemented
34D4⋊#92420C2211yescharacteristic, complemented
35D4⋊#92420C2212yescomplemented
36#16840393C411yescharacteristic
37#16840385—21no—
38#16840385C412yes—
39#16840365—21no—
40#16840365C412yes—
41#16840349—22no—
42#16840326C411yescharacteristic
43#16840326C411yescharacteristic
44#16840307—21no—
45#16840307C2212yes—
46#16840298—21no—
47#16840298C2212yes—
48#16840272—22no—
49#41×Dic61C2211yescharacteristic
50#41×Dic61C2211yescharacteristic
51C2×#14519739C2211yescharacteristic
52C61×#537C411yescharacteristic
53C2×#11758217—22no—
54C22×#92420—21no—
55C22×#92420—22no—
56C22×#92420—21no—
57#14519844—22no—
58C4⋊#92420—21no—
59C4⋊#92420—21no—
60C4⋊#92420—22no—
61C4⋊#92420C2312yes—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2311yescharacteristic
64C22⋊#92420—21no—
65C22⋊#92420—21no—
66C22⋊#92420—22no—
67C22⋊#92420—22no—
68#14519832—22no—
69C4×#92420—21no—
70C4×#92420—22no—
71C4×#92420—21no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C2×#11758224—22no—
76C22×D244C2×C411yescharacteristic
77C22×C4×D61—21no—
78#14519776—21no—
79#14519776C2×C412yes—
80#14519773—21no—
81#14519773C2×C412yes—
82#14519772—22no—
83#14519772—22no—
84#14519768—21no—
85#14519768C2×C412yes—
86#14519766—21no—
87#14519766C2×C412yes—
88#14519763—22no—
89#14519763—22no—
90#14519761—22no—
91#14519761—22no—
92#14519759—22no—
93#14519759—22no—
94#14519757—22no—
95#14519756—22no—
96#14519751—21no—
97#14519751—21no—
98#14519751C2×C412yes—
99#14519751C2×C412yes—
100#14519749—22no—
101#14519749—22no—
102#14519748—22no—
103#14519745—22no—
104#14519745—22no—
105Q8×Dic61C2×C411yescharacteristic
106Q8×Dic61D411yescharacteristic
107Q8×Dic61D411yescharacteristic
108Q8×Dic61C2311yescharacteristic
109D4×Dic61—21no—
110D4×Dic61—21no—
111D4×Dic61C2×C411yescharacteristic
112D4×Dic61D412yes—
113D4×Dic61D412yes—
114D4×Dic61D411yescharacteristic
115D4×Dic61D411yescharacteristic
116D4×Dic61C2311yescharacteristic
117#14519739—21no—
118#14519739—21no—
119#14519739C2×C412yes—
120#14519739C2312yes—
121#14519737—22no—
122#14519737—22no—
123#14519736—21no—
124#14519736C2×C411yescharacteristic
125#14519736C2×C411yescharacteristic
126#14519736C2×C412yes—
127#14519735—22no—
128C2×#11758198—21no—
129C2×C4×Dic61—21no—
130C2×C4×Dic61C2311yescharacteristic
131C2×C4×Dic61C2312yes—
132C2×#41×C61C2×C411yescharacteristic
133C2×#41×C61C2×C411yescharacteristic
134C22⋊Q8×C61—21no—
135C22⋊Q8×C61C2×C412yes—
136#121×C61—22no—
137#121×C61—21no—
138#121×C61C2×C412yes—
139C2×C4⋊C4×C61C2×C411yescharacteristic
140C23×D61—21no—
141#41×C61—24no—
142#41×C61—24no—
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61C22⋊C412yes—
148C2×D4×C61C22⋊C411yescharacteristic
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C22×C411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D412yes—
164#11758224—82no—
165#11758224—82no—
166C2×D244—24no—
167C2×D244—21no—
168C2×D244C22×C412yes—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—81no—
172C2×C4×D61—81no—
173C2×Dic122—22no—
174#11758217—82no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—22no—
178C2×#92420—21no—
179C2×#92420—21no—
180C2×#92420—22no—
181C2×#92420—42no—
182C2×#92420—44no—
183C2×#92420—81no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C22×C412yes—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊Dic61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192C22⋊Dic61C22⋊C412yes—
193C22⋊Dic61C22⋊C412yes—
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22⋊C411yescharacteristic
206C4⋊Dic61C22⋊C412yes—
207C4⋊Dic61C22⋊C412yes—
208C4⋊Dic61C22×C411yescharacteristic
209C4⋊Dic61C22×C411yescharacteristic
210#11758196—22no—
211#11758196—22no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61C22×C411yescharacteristic
216C4×Dic61C22×C412yes—
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221C4×Dic61C2411yescharacteristic
222C23×C61—21no—
223C23×C61C2×C22⋊C412yes—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61—22no—
226C22×D61—42no—
227C22×D61—81no—
228C22×D61—81no—
229Q8×C61—22no—
230Q8×C61—22no—
231D4×C61—24no—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—42no—
235D4×C61—82no—
236C2×C4×C61—24no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—42no—
240C2×C4×C61—81no—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61#13114yes—
244C2×C4×C61C23×C411yescharacteristic
245C22⋊D61—82no—
246C22⋊D61—82no—
247C22⋊D61—82no—
248C22⋊D61—82no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—42no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—81no—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C22×D411yescharacteristic
265C2×Dic61C22×D411yescharacteristic
266D244—24no—
267D244—42no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—82no—
271Dic122—82no—
272#92420—22no—
273#92420—41no—
274#92420—42no—
275#92420—42no—
276#92420—41no—
277#92420—41no—
278#17483—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61#40812yes—
288C22×C61C22×C22⋊C411yescharacteristic
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—84no—
293C4×C61—22no—
294C4×C61—41no—
295C4×C61—42no—
296C4×C61#40812yes—
297C4×C61C2×#13112yes—
298Dic61—22no—
299Dic61—21no—
300Dic61—21no—
301Dic61—21no—
302Dic61—22no—
303Dic61—41no—
304Dic61—41no—
305Dic61#57611yescharacteristic
306Dic61#57711yescharacteristic
307#3460—611no—
308#2989—611no—
309C8×#41—611no—
310#1496—612no—
311#1495—612no—
312#1494—612no—
313#1492—612no—
314#1478—611no—
315#1477—611no—
316#1476—611no—
317C2×D4⋊C8—611no—
318C2×C61—22no—
319C2×C61—42no—
320C2×C61—81no—
321C2×C61C2×#40811yescharacteristic
322C2×C61#289711yescharacteristic
323C2×C61#290111yescharacteristic
324D61—82no—
325#544—611no—
326#542—612no—
327#542—1221no—
328#540—612no—
329#540—1221no—
330#538—1222no—
331#537#692311yescharacteristic
332#535—611no—
333#535—611no—
334#532—612no—
335#532—1221no—
336#529—612no—
337#529—1221no—
338#525—1222no—
339C4×#41—611no—
340C4×#41—611no—
341C2×C4×D4—611no—
342#446—1221no—
343C8×Q8—1221no—
344#436—1221no—
345#435—1222no—
346C8×D4—1222no—
347C8×D4—1221no—
348#432—612no—
349#431—611no—
350#431—612no—
351#431—611no—
352#423—612no—
353C2×C4⋊C8—611no—
354C2×C4×C8—611no—
355C2×C4×C8—612no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—612no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—612no—
362D4⋊C8—612no—
363D4⋊C8—612no—
364D4⋊C8—612no—
365D4⋊C8—612no—
366C61#1748311yescharacteristic, complemented, Sylow
367C2×#41—1222no—
368C2×#41C2×#692311yescharacteristic
369C2×#41C2×#692311yescharacteristic
370C22×D4—611no—
371C23×C4—1221no—
372C2×M16—1222no—
373C22×C8—1221no—
374C22×C8—1221no—
375C22×C8—1222no—
376C4⋊Q8—612no—
377C4⋊Q8—1221no—
378#127—612no—
379#127—1221no—
380#126—1222no—
381#126—1222no—
382#125—612no—
383#125—1221no—
384#124—612no—
385#124—1222no—
386#124—1222no—
387#124—1221no—
388#123—1222no—
389#123—1222no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—21no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8C2×#692312yes—
396#121—22no—
397#121—21no—
398#121—612no—
399#121—612no—
400#121—1222no—
401#121—1222no—
402#121—1221no—
403#121—1221no—
404#121—1222no—
405#121C2×#692312yes—
406#120—1222no—
407#120—1222no—
408C4×Q8—611no—
409C4×Q8—611no—
410C4×Q8—611no—
411C4×Q8—611no—
412C4×D4—612no—
413C4×D4—611no—
414C4×D4—612no—
415C4×D4—611no—
416C4×D4—611no—
417C4×D4—611no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—1222no—
421C4×D4—1221no—
422C4×D4—1221no—
423C4×D4—1222no—
424C4×D4—1221no—
425C4×D4—1221no—
426#117—611no—
427#117—612no—
428#117—611no—
429#117—1222no—
430#117—1221no—
431C2×C4⋊C4C2×#692311yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C42—611no—
434C2×C42—612no—
435C2×C42—1221no—
436C4⋊C8—612no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—1221no—
440C4⋊C8—1221no—
441C4⋊C8—1222no—
442C4⋊C8—1222no—
443C22⋊C8—1221no—
444C22⋊C8—1222no—
445C22⋊C8—1221no—
446C22⋊C8—1222no—
447#97—1222no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—612no—
451C4×C8—1221no—
452C4×C8—1222no—
453C4×C8—1221no—
454C24—1221no—
455#41—24no—
456#41—24no—
457#41—4882no—
458#41—4882no—
459C2×Q8—1222no—
460C2×Q8C22⋊#692311yescharacteristic
461C2×Q8C22×#692311yescharacteristic
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—612no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1221no—
470C2×D4—1224no—
471C2×D4C22⋊#692312yes—
472C2×D4C22⋊#692311yescharacteristic
473C2×D4C22×#692311yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—612no—
478C22×C4—611no—
479C22×C4—612no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—4881no—
485C22×C4—4881no—
486C22×C4C22×#692312yes—
487C22×C4C22×#692311yescharacteristic
488M16—4882no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1222no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1222no—
495C2×C8—4881no—
496C2×C8—2442no—
497C2×C8—2444no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—611no—
501C4⋊C4—611no—
502C4⋊C4—612no—
503C4⋊C4—612no—
504C4⋊C4—611no—
505C4⋊C4—611no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1222no—
511C4⋊C4C22×#692312yes—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—612no—
515C22⋊C4—612no—
516C22⋊C4—612no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C22⋊C4—1221no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C42—611no—
527C42—611no—
528C42—612no—
529C42—612no—
530C42—611no—
531C42—611no—
532C42—612no—
533C42—1221no—
534C42—1221no—
535C42—1221no—
536C23—21no—
537C23—1222no—
538C23—4881no—
539C23—4881no—
540C23—2442no—
541C23C2×C22⋊#692312yes—
542C23C2×C22⋊#692311yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—4882no—
546D4—24no—
547D4—22no—
548D4—22no—
549D4—42no—
550D4—82no—
551D4—1224no—
552D4—4882no—
553D4—4882no—
554D4—4882no—
555D4—4882no—
556D4—2442no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—81no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—612no—
565C2×C4—611no—
566C2×C4—612no—
567C2×C4—611no—
568C2×C4—1221no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—4882no—
575C2×C4—4881no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2442no—
582C2×C4C2×C22⋊#692312yes—
583C2×C4C2×C22⋊#692311yescharacteristic
584C2×C4#1451986414yes—
585C2×C4C23×#692311yescharacteristic
586C8—1222no—
587C8—2441no—
588C8—2442no—
589C8—2441no—
590C8—2441no—
591C8—2442no—
592C22—21no—
593C22—21no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—4884no—
601C22—2442no—
602C22—2442no—
603C22—2442no—
604C22#1684073512yes—
605C22C22×C22⋊#692311yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—611no—
610C4—611no—
611C4—1221no—
612C4—1222no—
613C4—1221no—
614C4—1221no—
615C4—1222no—
616C4—2441no—
617C4—2441no—
618C4#1684073512yes—
619C4C2×#1451986412yes—
620C2—22no—
621C2—42no—
622C2—81no—
623C2—4882no—
624C2C2×#1684073511yescharacteristic
625C2#2146244011yescharacteristic
626C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#537C4
socleC22×C61C22×C22⋊C4
prime radical#21459236C2
cosocleC4×Dic61C24
hypercenter#537#6923
hypocenterC61#17483
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #21462738C2semidirect
2. #41×#92420C2semidirect
3. #21461178C2semidirect
4. #21461176C2semidirect
5. #21461172C2semidirect
6. #21461164C2semidirect
7. #21461120C2semidirect
8. #21461114C2semidirect
9. #21461110C2semidirect
10. C2×#16840610C2semidirect
18. #16840783C22semidirect
21. #16840778C22semidirect
22. #16840765C22semidirect
25. C2×C4×#92420C22semidirect
26. Q8⋊#92420C22semidirect
27. Q8⋊#92420C22semidirect
28. Q8⋊#92420C22semidirect
33. D4⋊#92420C22semidirect
34. D4⋊#92420C22semidirect
35. D4⋊#92420C22semidirect
366. C61#17483semidirect

Maximal subgroups

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

subgroupindex
#214627382
#41×#924202
#214611782
#214611762
#214611722
#214611642
#214611202
#214611142
#214611102
C2×#168406102
#214592362
#1748361