Finite Group Databasebeta

← #30610635

Subgroups — #30610635

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#21462740C211yesmaximal, characteristic, complemented
2#21462623C211yesmaximal, characteristic, complemented
3#21461179C212yesmaximal, complemented
4#21461175C212yesmaximal, complemented
5#21461171C212yesmaximal, complemented
6#21461165C212yesmaximal, complemented
7#21461118C211yesmaximal, characteristic, complemented
8#21461114C211yesmaximal, characteristic, complemented
9#21461110C211yesmaximal, characteristic, complemented
10C2×D4⋊#92420C211yesmaximal, characteristic, complemented
11#21459232C211yesmaximal, characteristic
12#16840812—21no—
13#16840811—21no—
14#16840796—21no—
15#16840791—22no—
16#16840790—21no—
17#16840789—22no—
18#16840784C2212yescomplemented
19#16840780C2212yescomplemented
20#16840778C2211yescharacteristic
21#16840778C2211yescharacteristic
22#16840766C2212yescomplemented
23C2×C4⋊#92420C2211yescharacteristic
24C4×D61⋊C8C2212yescomplemented
25C2×C4×#92420C2211yescharacteristic
26Q8⋊#92420C2211yescharacteristic, complemented
27Q8⋊#92420C2211yescharacteristic, complemented
28Q8⋊#92420C2212yescomplemented
29D4⋊#92420C2212yes—
30D4⋊#92420C2212yes—
31D4⋊#92420C2212yes—
32D4⋊#92420C2212yes—
33D4⋊#92420C2211yescharacteristic, complemented
34D4⋊#92420C2211yescharacteristic, complemented
35D4⋊#92420C2212yescomplemented
36#16840390C411yescharacteristic
37#16840381C412yes—
38#16840379—21no—
39#16840366C412yes—
40#16840365—21no—
41#16840347—22no—
42#16840340C411yescharacteristic
43#16840329C411yescharacteristic
44#16840310C2212yes—
45#16840307—21no—
46#16840293—21no—
47#16840291C2212yes—
48#16840268—22no—
49#41×Dic61C2211yescharacteristic
50#16840230C2211yescharacteristic
51C2×D4×Dic61C2211yescharacteristic
52C61×#535C411yescharacteristic
53C2×#11758213—22no—
54C2×D61⋊C8—22no—
55C22×#92420—21no—
56C22×#92420—21no—
57#14519844—22no—
58C4⋊#92420—21no—
59C4⋊#92420—22no—
60C4⋊#92420—21no—
61C4⋊#92420C2312yes—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2311yescharacteristic
64#14519834—22no—
65#14519834—22no—
66C22⋊#92420—21no—
67C22⋊#92420—21no—
68#14519832—22no—
69C4×#92420—21no—
70C4×#92420—22no—
71C4×#92420—21no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C2×#11758222C2×C411yescharacteristic
76C2×#11758224—22no—
77C23×Dic61—21no—
78#14519777C2×C412yes—
79#14519776—21no—
80#14519773—21no—
81#14519770—22no—
82#14519770—22no—
83#14519769C2×C412yes—
84#14519768—21no—
85#14519764—21no—
86#14519764—22no—
87#14519763—22no—
88#14519761—22no—
89#14519759—22no—
90#14519758—22no—
91#14519758—22no—
92#14519756—21no—
93#14519754—22no—
94#14519754—22no—
95#14519753C2×C412yes—
96#14519751—21no—
97#14519749—22no—
98#14519749—22no—
99#14519749—22no—
100#14519748—22no—
101#14519748C2×C412yes—
102#14519748C2×C412yes—
103#14519748C2×C412yes—
104#14519746—22no—
105Q8×Dic61C2×C411yescharacteristic
106Q8×Dic61C2311yescharacteristic
107#14519742D411yescharacteristic
108#14519742D411yescharacteristic
109D4×Dic61—21no—
110D4×Dic61—21no—
111D4×Dic61—21no—
112D4×Dic61—21no—
113D4×Dic61—22no—
114D4×Dic61C2×C412yes—
115D4×Dic61C2×C411yescharacteristic
116D4×Dic61C2312yes—
117D4×Dic61C2311yescharacteristic
118#14519739D411yescharacteristic
119#14519739D411yescharacteristic
120#14519737—22no—
121#14519737D412yes—
122#14519737D412yes—
123#14519736—21no—
124#14519736C2×C411yescharacteristic
125#14519734C2×C412yes—
126#14519733—22no—
127C2×C4⋊Dic61C2×C411yescharacteristic
128C2×C22⋊Dic61—21no—
129C2×C4×Dic61—21no—
130C2×C4×Dic61C2311yescharacteristic
131C42×D61C2312yes—
132C2×#41×C61C2×C411yescharacteristic
133C22×D4×C61C2×C411yescharacteristic
134#127×C61C2×C412yes—
135#124×C61C2×C412yes—
136#121×C61—21no—
137#121×C61—21no—
138#120×C61—22no—
139#117×C61C2×C411yescharacteristic
140C24×C61—21no—
141#41×C61—24no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—24no—
148C2×D4×C61C22×C412yes—
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61C22×C411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—21no—
157C22×Dic61—22no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—21no—
161C22×Dic61—81no—
162C22×Dic61—81no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165#11758224—82no—
166#11758224—82no—
167#11758222—24no—
168C2×D244C22⋊C411yescharacteristic
169C2×C4×D61C22×C412yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×D412yes—
172C2×Dic122—22no—
173C2×Dic122C22⋊C411yescharacteristic
174C2×#92420—21no—
175C2×#92420—22no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—22no—
180C2×#92420—42no—
181C2×#92420—81no—
182#11758213—82no—
183D61⋊C8—44no—
184C4⋊C4×C61C22⋊C411yescharacteristic
185C4⋊C4×C61C22⋊C411yescharacteristic
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61C22⋊C412yes—
189C42×C61C22×C412yes—
190C22⋊Dic61—21no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—21no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—21no—
198#11758198—22no—
199#11758198C22⋊C412yes—
200#11758198C22⋊C412yes—
201C4⋊Dic61—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C22×C412yes—
206C4⋊Dic61C22×C411yescharacteristic
207C4⋊Dic61C22×C411yescharacteristic
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196C22⋊C412yes—
212#11758196C22⋊C412yes—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221C4×Dic61C2411yescharacteristic
222C23×C61—22no—
223C23×C61—21no—
224C23×C61—42no—
225C23×C61—81no—
226C23×C61—81no—
227C23×C61C2×C22⋊C411yescharacteristic
228C22×D61C2×C22⋊C412yes—
229Q8×C61—22no—
230D4×C61—24no—
231D4×C61—22no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—82no—
235D4×C61—82no—
236C2×C4×C61—21no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61#13114yes—
242C2×C4×C61C23×C411yescharacteristic
243C22⋊D61—24no—
244C22⋊D61—82no—
245C22⋊D61—82no—
246C22⋊D61—82no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—42no—
257C2×Dic61—81no—
258C2×Dic61—82no—
259C2×Dic61—81no—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C22×D411yescharacteristic
264C2×Dic61C22×D411yescharacteristic
265D244—22no—
266C4×D61—24no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—42no—
270Dic122—22no—
271Dic122—82no—
272#92420—22no—
273#92420—42no—
274#92420—41no—
275#92420—41no—
276#92420—42no—
277#92420—41no—
278#17483—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—41no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—42no—
287C22×C61—84no—
288C22×C61#40812yes—
289C22×C61C22×C22⋊C411yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—41no—
296C4×C61#40812yes—
297C4×C61C2×#13112yes—
298Dic61—22no—
299Dic61—21no—
300Dic61—21no—
301Dic61—21no—
302Dic61—41no—
303Dic61—42no—
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—81no—
320C2×C61—82no—
321C2×C61C2×#40811yescharacteristic
322C2×C61#289711yescharacteristic
323C2×C61#290111yescharacteristic
324D61—42no—
325#544—611no—
326#542—612no—
327#542—1221no—
328#540—612no—
329#540—1221no—
330#538—1222no—
331#537—611no—
332#535—611no—
333#535#692311yescharacteristic
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—612no—
360D4⋊C8—612no—
361D4⋊C8—612no—
362D4⋊C8—612no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—612no—
366C61#1748311yescharacteristic, complemented, Sylow
367C2×#41—611no—
368C2×#41—1222no—
369C2×#41C2×#692311yescharacteristic
370C22×D4C2×#692311yescharacteristic
371C23×C4—1221no—
372C2×M16—1222no—
373C22×C8—1221no—
374C22×C8—1221no—
375C22×C8—1222no—
376C4⋊Q8—612no—
377C4⋊Q8—1221no—
378#127—1221no—
379#127C2×#692312yes—
380#126—1222no—
381#126—1222no—
382#125—612no—
383#125—1221no—
384#124—1222no—
385#124—1222no—
386#124—1221no—
387#124C2×#692312yes—
388#123—1222no—
389#123—1222no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—612no—
393C22⋊Q8—1222no—
394C22⋊Q8—1221no—
395C22⋊Q8—1222no—
396#121—21no—
397#121—21no—
398#121—612no—
399#121—612no—
400#121—612no—
401#121—1222no—
402#121—1222no—
403#121—1221no—
404#121—1222no—
405#121—1222no—
406#120—22no—
407#120—1222no—
408C4×Q8—611no—
409C4×Q8—611no—
410C4×Q8—611no—
411C4×Q8—611no—
412C4×D4—611no—
413C4×D4—611no—
414C4×D4—612no—
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—611no—
419C4×D4—611no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1221no—
425C4×D4—1221no—
426#117—612no—
427#117—611no—
428#117—1222no—
429#117—1221no—
430#117C2×#692311yescharacteristic
431C2×C4⋊C4—611no—
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—1222no—
441C4⋊C8—1221no—
442C4⋊C8—1222no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1222no—
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—21no—
455#41—24no—
456#41—1224no—
457#41—4882no—
458#41—4882no—
459C2×Q8—611no—
460C2×Q8—1222no—
461C2×Q8C22×#692311yescharacteristic
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—612no—
468C2×D4—611no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4C22×#692312yes—
473C2×D4C22×#692311yescharacteristic
474C22×C4—21no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—1221no—
481C22×C4—1222no—
482C22×C4—1221no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—4881no—
486C22×C4—4881no—
487C22×C4C22×#692311yescharacteristic
488M16—4882no—
489C2×C8—1221no—
490C2×C8—1222no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1222no—
495C2×C8—4881no—
496C2×C8—2444no—
497C2×C8—2442no—
498C4⋊C4—612no—
499C4⋊C4—612no—
500C4⋊C4—612no—
501C4⋊C4—611no—
502C4⋊C4—611no—
503C4⋊C4—1222no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4C22⋊#692311yescharacteristic
511C4⋊C4C22⋊#692311yescharacteristic
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—612no—
515C22⋊C4—612no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1221no—
520C22⋊C4—1221no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1221no—
525C22⋊C4C22⋊#692312yes—
526C42—611no—
527C42—611no—
528C42—612no—
529C42—612no—
530C42—611no—
531C42—611no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42C22×#692312yes—
536C23—22no—
537C23—21no—
538C23—42no—
539C23—81no—
540C23—81no—
541C23—612no—
542C23C2×C22⋊#692311yescharacteristic
543Q8—22no—
544Q8—1222no—
545Q8—4882no—
546D4—24no—
547D4—22no—
548D4—42no—
549D4—42no—
550D4—82no—
551D4—82no—
552D4—1224no—
553D4—1222no—
554D4—4882no—
555D4—4882no—
556D4—4882no—
557C2×C4—21no—
558C2×C4—42no—
559C2×C4—611no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—612no—
563C2×C4—611no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1224no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—4881no—
573C2×C4—4882no—
574C2×C4—4881no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4C2×C22⋊#692312yes—
582C2×C4C2×C22⋊#692312yes—
583C2×C4C2×C22⋊#692311yescharacteristic
584C2×C4#1451986414yes—
585C2×C4C23×#692311yescharacteristic
586C8—1222no—
587C8—2441no—
588C8—2442no—
589C8—2441no—
590C8—2442no—
591C8—2441no—
592C22—21no—
593C22—21no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—84no—
601C22—1222no—
602C22—1222no—
603C22—1222no—
604C22#1684073612yes—
605C22C22×C22⋊#692311yescharacteristic
606C4—22no—
607C4—22no—
608C4—41no—
609C4—611no—
610C4—611no—
611C4—1222no—
612C4—1221no—
613C4—1221no—
614C4—1221no—
615C4—2442no—
616C4—2441no—
617C4—2441no—
618C4#1684073612yes—
619C4C2×#1451986412yes—
620C2—22no—
621C2—82no—
622C2—81no—
623C2—2442no—
624C2C2×#1684073611yescharacteristic
625C2#2146244011yescharacteristic
626C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#535C4
socleC22×C61C22×C22⋊C4
prime radical#21459232C2
cosocleC4×Dic61C24
hypercenter#535#6923
hypocenterC61#17483
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #21462740C2semidirect
2. #21462623C2semidirect
3. #21461179C2semidirect
4. #21461175C2semidirect
5. #21461171C2semidirect
6. #21461165C2semidirect
7. #21461118C2semidirect
8. #21461114C2semidirect
9. #21461110C2semidirect
10. C2×D4⋊#92420C2semidirect
18. #16840784C22semidirect
19. #16840780C22semidirect
22. #16840766C22semidirect
24. C4×D61⋊C8C22semidirect
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
#214627402
#214626232
#214611792
#214611752
#214611712
#214611652
#214611182
#214611142
#214611102
C2×D4⋊#924202
#214592322
#1748361