Finite Group Databasebeta

← #30615535

Subgroups — #30615535

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840841C211yesmaximal, characteristic
2C22×#14519847C211yesmaximal, characteristic, complemented
3#21462462C211yesmaximal, characteristic, complemented
4C2×#16840759C211yesmaximal, characteristic
5#21461789C214yesmaximal, complemented
6#21461788C214yesmaximal, complemented
7#21461499C211yesmaximal, characteristic, complemented
8#21461484C211yesmaximal, characteristic, complemented
9#416×D61C211yesmaximal, characteristic
10#16840841—24no—
11#16840841—24no—
12C2×#14519865C2212yes—
13C2×#14519864—22no—
14C2×#14519864C2212yes—
15C22×#11758213—21no—
16C22×#11758213C2211yescharacteristic
17C22×#11758213C2211yescharacteristic
18C2×D61⋊M16—21no—
19C2×D61⋊M16C2211yescharacteristic
20C2×#14519856C2211yescharacteristic
21C2×#14519847—24no—
22C2×#14519847C2214yescomplemented
23#16840769C2214yescomplemented
24#16840768—22no—
25#16840767—21no—
26#16840766—21no—
27#16840759—24no—
28C2×#14519842C2212yes—
29C2×#14519841C2212yes—
30#16840632C2214yescomplemented
31(C2×#6923)⋊C8C2214yescomplemented
32C2×#41×D61C2211yescharacteristic
33C22×C8×D61C2211yescharacteristic
34#117×D61C2211yescharacteristic
35#16839759C411yescharacteristic, complemented
36#16839758C411yescharacteristic, complemented
37#16839757C2212yes—
38#16839756C411yescharacteristic, complemented
39#16839755C2212yes—
40#103×D61C412yescomplemented
41#102×D61C412yescomplemented
42C61×#416C411yescharacteristic, complemented
43#14519865—24no—
44#14519865—24no—
45#14519864—216no—
46#14519864—24no—
47#14519864—24no—
48C2×#11758213—28no—
49C2×#11758213—21no—
50C2×#11758213—22no—
51C2×#11758213—22no—
52C2×#11758213—22no—
53C2×#11758213—22no—
54C2×#11758213—21no—
55C2×#11758213D411yescharacteristic
56C2×#11758213D411yescharacteristic
57C2×#11758213D414yes—
58C2×#11758213C2312yes—
59C2×#11758213C2312yes—
60D61⋊M16—28no—
61D61⋊M16—22no—
62D61⋊M16—22no—
63C2×#11758217—22no—
64C2×#11758217D412yes—
65C2×D61⋊C8—22no—
66C2×D61⋊C8C2312yes—
67C22×#92420—21no—
68C22×#92420—22no—
69C22×#92420—21no—
70#14519856—22no—
71#14519856—22no—
72C2×C4⋊#6923—21no—
73C2×C22⋊#6923—21no—
74C2×C4×#6923C2312yes—
75#14519847—28no—
76#14519844—24no—
77C4⋊#92420—24no—
78#14519842—24no—
79#14519841—24no—
80C2×#11758226C2311yescharacteristic
81C2×#11758222—21no—
82C2×#11758222C2×C411yescharacteristic
83C2×#11758224—21no—
84C2×#11758224C2311yescharacteristic
85#41×D61—22no—
86#41×D61—22no—
87#41×D61D414yes—
88#41×D61Q814yes—
89C2×Q8×D61C2×C411yescharacteristic
90C2×D4×D61—21no—
91C2×D4×D61C2×C411yescharacteristic
92C22×C4×D61—21no—
93C22×C4×D61C2311yescharacteristic
94C22×#92418C2×C411yescharacteristic
95C2×C8×D61—22no—
96C2×C8×D61—22no—
97C2×C8×D61C2×C412yes—
98C2×C8×D61C2312yes—
99#14519736C2×C411yescharacteristic
100#14519735C2312yes—
101#14519734—21no—
102#14519733—21no—
103C4⋊C4×D61C2×C412yes—
104C22⋊C4×D61—21no—
105C42×D61—21no—
106#14519702C2×C412yes—
107#14519701C2×C412yes—
108#14519700C2×C412yes—
109#14519699C2×C412yes—
110#14519698C2×C412yes—
111#14519697C2×C412yes—
112C2×#41×C61C2×C411yescharacteristic
113C22×C8×C61C2×C411yescharacteristic
114#117×C61C2×C411yescharacteristic
115#103×C61C2×C412yes—
116#102×C61C2×C412yes—
117C23×D61—21no—
118C23×D61#4111yescharacteristic
119C22×#6923—22no—
120#41×C61—22no—
121#41×C61C4⋊C414yes—
122C2×Q8×C61C22×C411yescharacteristic
123C2×D4×C61—21no—
124C2×D4×C61C22×C411yescharacteristic
125C22×C4×C61—21no—
126C22×C4×C61C22×C411yescharacteristic
127C2×C22⋊D61—21no—
128C2×C22⋊D61—21no—
129C2×C22⋊D61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132#11758226—21no—
133#11758226—21no—
134#11758226C2×D412yes—
135#11758226C2×Q812yes—
136Q8×D61—22no—
137Q8×D61C4⋊C414yes—
138#11758224—24no—
139#11758224—24no—
140#11758224—21no—
141#11758224—21no—
142#11758224C2×D412yes—
143#11758224C2×Q812yes—
144D4×D61—24no—
145D4×D61—24no—
146D4×D61—22no—
147D4×D61C4⋊C414yes—
148#11758222—22no—
149#11758222—24no—
150#11758222—24no—
151#11758222C4⋊C414yes—
152C2×D244—21no—
153C2×D244C22×C411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—22no—
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×Q812yes—
165C2×C4×D61#4112yes—
166C2×C4×D61#4111yescharacteristic
167C2×C4×D61C2411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122C22×C411yescharacteristic
170C22⋊#6923—42no—
171#11758217—28no—
172#11758217—24no—
173C2×#92420—28no—
174C2×#92420—24no—
175C2×#92420—22no—
176C2×#92420—22no—
177C2×#92420—42no—
178C2×#92420C2×D412yes—
179C2×#92420C2×D412yes—
180C4⋊#6923—42no—
181C4×#6923—22no—
182C4×#6923—22no—
183#11758213—28no—
184#11758213—22no—
185#11758213—22no—
186#11758213—24no—
187#11758213—22no—
188#11758213—22no—
189#11758213—22no—
190#11758213—42no—
191#11758213C2×D414yes—
192D61⋊C8—28no—
193D61⋊C8—22no—
194D61⋊C8—22no—
195C2×C8×C61—22no—
196C2×C8×C61C22×C412yes—
197C4⋊C4×C61C22×C412yes—
198C22⋊C4×C61—21no—
199C42×C61—21no—
200C22⋊Dic61—21no—
201#11758198—22no—
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204C4×Dic61—21no—
205C4×Dic61—22no—
206C2×#92418—22no—
207C2×#92418C22×C412yes—
208C8×D61—24no—
209C8×D61—22no—
210C8×D61—22no—
211C23×C61—21no—
212C23×C61C4×D411yescharacteristic
213C22×D61—21no—
214C22×D61—24no—
215C22×D61—42no—
216C22×D61—82no—
217C22×D61C4×D411yescharacteristic
218C22×D61C22⋊Q812yes—
219C22×D61#12312yes—
220C22×D61C2×#4111yescharacteristic
221C2×#6923—44no—
222C2×#6923—42no—
223Q8×C61—21no—
224Q8×C61C2×C4⋊C412yes—
225D4×C61—24no—
226D4×C61—21no—
227D4×C61C2×C4⋊C412yes—
228C2×C4×C61—21no—
229C2×C4×C61—24no—
230C2×C4×C61—22no—
231C2×C4×C61C2×C4⋊C412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C22⋊D61—24no—
236C22⋊D61—22no—
237C22⋊D61—22no—
238C22⋊D61—24no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61#12012yes—
245C2×Dic61C22⋊Q812yes—
246C2×Dic61C22×D411yescharacteristic
247D244—22no—
248D244—22no—
249D244—21no—
250D244C2×C4⋊C412yes—
251C4×D61—24no—
252C4×D61—24no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—82no—
256C4×D61C2×C4⋊C412yes—
257C4×D61C4×D412yes—
258C4×D61#12012yes—
259C4×D61C22⋊Q812yes—
260C4×D61C22⋊Q812yes—
261C4×D61#12312yes—
262C4×D61C22×D411yescharacteristic
263C4×D61C22×D411yescharacteristic
264C4×D61C22×Q811yescharacteristic
265C4×D61C2×#4111yescharacteristic
266Dic122—21no—
267Dic122—24no—
268Dic122C2×C4⋊C412yes—
269#92420—22no—
270#92420—22no—
271#92420—24no—
272#92420—42no—
273C8×C61—22no—
274#92418—22no—
275#20886—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—42no—
279C22×C61#38512yes—
280C22×C61C2×C4×D411yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—84no—
288C2×D61#38512yes—
289C2×D61C2×C22⋊Q811yescharacteristic
290C2×D61C2×#12311yescharacteristic
291#6923—82no—
292C4×C61—22no—
293C4×C61—42no—
294C4×C61#38512yes—
295C4×C61C22×C4⋊C411yescharacteristic
296C4×C61C2×C4×D411yescharacteristic
297Dic61—22no—
298Dic61—42no—
299Dic61#38512yes—
300Dic61C2×#12011yescharacteristic
301Dic61C2×C22⋊Q811yescharacteristic
302C2×#568—611no—
303C22×#108—611no—
304#2905—611no—
305C2×#421—611no—
306C2×#416—611no—
307#1866—614no—
308#1866—614no—
309#1733—611no—
310#1729—611no—
311C2×C61—22no—
312C2×C61—42no—
313C2×C61C2×#38511yescharacteristic
314C2×C61#295712yes—
315D61—21no—
316D61—22no—
317D61—21no—
318D61—42no—
319C22×#41—611no—
320#568—1224no—
321#568—1224no—
322C2×#131—612no—
323C2×#131—612no—
324C2×#131—1222no—
325C22×M16—611no—
326C22×M16—611no—
327C22×M16—611no—
328C22×M16—1221no—
329C22×M16—1221no—
330C23×C8—611no—
331C2×#117—611no—
332C2×#117—611no—
333C2×#108—614no—
334C2×#108—1224no—
335#424—614no—
336#424—1222no—
337#423—1221no—
338#423—1221no—
339#421—1224no—
340C2×#104—612no—
341C2×#104—612no—
342#416—611no—
343#416—611no—
344#416—611no—
345#416—612no—
346#416—612no—
347#416#692311yescharacteristic, complemented
348C2×#103—612no—
349C2×#102—612no—
350#339—614no—
351#336—614no—
352C61#2088611yescharacteristic, complemented, Sylow
353C2×#41—614no—
354C2×#41—614no—
355C2×#41—611no—
356C2×#41—611no—
357C2×#41—611no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41C2×#692311yescharacteristic
363C22×Q8—611no—
364C22×D4—611no—
365C22×D4—1221no—
366C23×C4—611no—
367C23×C4—1221no—
368#131—1224no—
369#131—1224no—
370#131—1224no—
371#131—1224no—
372#131—12216no—
373C2×M16—612no—
374C2×M16—611no—
375C2×M16—611no—
376C2×M16—612no—
377C2×M16—614no—
378C2×M16—612no—
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1221no—
382C2×M16—1222no—
383C2×M16—1222no—
384C2×M16—1222no—
385C2×M16—1228no—
386C2×M16—1228no—
387C2×M16—1221no—
388C2×M16—1222no—
389C2×M16—1222no—
390C22×C8—612no—
391C22×C8—611no—
392C22×C8—612no—
393C22×C8—612no—
394C22×C8—1222no—
395C22×C8—1222no—
396C22×C8—1222no—
397C22×C8—1222no—
398C22×C8—1221no—
399C22×C8—1221no—
400C22×C8C2×#692311yescharacteristic
401#117—611no—
402#117—612no—
403#117—1221no—
404#117—1221no—
405#117—1222no—
406#117—1222no—
407#117C2×#692311yescharacteristic
408C2×C4⋊C4—612no—
409C2×C4⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C42—612no—
413C2×C42—1221no—
414#108—1228no—
415C4⋊C8—1224no—
416C4⋊C8—1224no—
417#104—1224no—
418#104—1224no—
419#103—612no—
420#103—612no—
421#103—612no—
422#103C2×#692312yes—
423#102—612no—
424#102—612no—
425#102—612no—
426#102C2×#692312yes—
427C24—611no—
428C24—1221no—
429#41—22no—
430#41—612no—
431#41—612no—
432#41—612no—
433#41—612no—
434#41—614no—
435#41—1224no—
436#41—1224no—
437#41—1224no—
438#41—1221no—
439#41—1224no—
440#41—1221no—
441#41—1221no—
442#41—1221no—
443#41—1222no—
444#41C4⋊#692314yes—
445C2×Q8—614no—
446C2×Q8—611no—
447C2×Q8—1221no—
448C2×Q8—1222no—
449C2×Q8C22×#692311yescharacteristic
450C2×D4—21no—
451C2×D4—614no—
452C2×D4—611no—
453C2×D4—1221no—
454C2×D4—1222no—
455C2×D4—1221no—
456C2×D4—1224no—
457C2×D4—1224no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4C22×#692311yescharacteristic
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1224no—
473C22×C4—1224no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1222no—
479C22×C4C22×#692311yescharacteristic
480M16—614no—
481M16—1222no—
482M16—1222no—
483M16—1224no—
484M16—1222no—
485M16—1224no—
486M16—1228no—
487M16—1228no—
488M16—1222no—
489M16—1222no—
490M16—2442no—
491C2×C8—22no—
492C2×C8—612no—
493C2×C8—612no—
494C2×C8—612no—
495C2×C8—1222no—
496C2×C8—1224no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1224no—
502C2×C8—1228no—
503C2×C8—1228no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—2442no—
507C2×C8C22×#692312yes—
508C4⋊C4—612no—
509C4⋊C4—1222no—
510C4⋊C4—2442no—
511C4⋊C4C22×#692312yes—
512C22⋊C4—21no—
513C22⋊C4—1222no—
514C22⋊C4—1221no—
515C22⋊C4—2442no—
516C42—21no—
517C42—1222no—
518C42—1222no—
519C42—1222no—
520C42—1221no—
521C23—21no—
522C23—611no—
523C23—611no—
524C23—612no—
525C23—612no—
526C23—1224no—
527C23—1221no—
528C23—4882no—
529C23—2442no—
530C23D4×#692311yescharacteristic
531Q8—21no—
532Q8—612no—
533Q8—1224no—
534Q8—1221no—
535Q8C2×C4⋊#692312yes—
536D4—24no—
537D4—21no—
538D4—612no—
539D4—1222no—
540D4—1222no—
541D4—1224no—
542D4—1222no—
543D4—1222no—
544D4—1224no—
545D4—1221no—
546D4C2×C4⋊#692312yes—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—611no—
560C2×C4—611no—
561C2×C4—612no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1224no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1221no—
570C2×C4—1224no—
571C2×C4—1222no—
572C2×C4—4882no—
573C2×C4—2442no—
574C2×C4—2444no—
575C2×C4C2×C4⋊#692312yes—
576C2×C4D4×#692311yescharacteristic
577C2×C4D4×#692312yes—
578C2×C4C23×#692311yescharacteristic
579C8—22no—
580C8—1222no—
581C8—1222no—
582C8—1224no—
583C8—1222no—
584C8—2442no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—612no—
589C22—611no—
590C22—611no—
591C22—1222no—
592C22—1222no—
593C22—1222no—
594C22—4884no—
595C22—2442no—
596C22—2442no—
597C22—2442no—
598C22#1684070912yes—
599C22C2×D4×#692311yescharacteristic
600C4—22no—
601C4—42no—
602C4—611no—
603C4—611no—
604C4—612no—
605C4—1222no—
606C4—4882no—
607C4—2442no—
608C4#1684070912yes—
609C4C22×C4⋊#692311yescharacteristic
610C4C2×D4×#692311yescharacteristic
611C2—22no—
612C2—42no—
613C2—1221no—
614C2—1222no—
615C2—1221no—
616C2—2442no—
617C2C2×#1684070911yescharacteristic
618C2#2146259812yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#416C4
socleC22×C61C2×C4×D4
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#416#6923
hypocenterC61#20886
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×#14519847C2semidirect
3. #21462462C2semidirect
5. #21461789C2semidirect
6. #21461788C2semidirect
7. #21461499C2semidirect
8. #21461484C2semidirect
22. C2×#14519847C22semidirect
23. #16840769C22semidirect
30. #16840632C22semidirect
31. (C2×#6923)⋊C8C22semidirect
35. #16839759C4semidirect
36. #16839758C4semidirect
38. #16839756C4semidirect
40. #103×D61C4semidirect
41. #102×D61C4semidirect
42. C61×#416C4semidirect
347. #416#6923semidirect
352. C61#20886semidirect

Maximal subgroups

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

subgroupindex
C2×#168408412
C22×#145198472
#214624622
C2×#168407592
#214617892
#214617882
#214614992
#214614842
#416×D612
#2088661