Finite Group Databasebeta

← #30611506

Subgroups — #30611506

Order 15616.

964 conjugacy classes of subgroups.

The subgroup lattice has 586 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#21461292C211yesmaximal, characteristic, complemented
5#21461284C211yesmaximal, characteristic, complemented
6#21461281C212yesmaximal, complemented
7#21461277C211yesmaximal, characteristic, complemented
8#21461270C212yesmaximal, complemented
9#21461269C211yesmaximal, characteristic, complemented
10C2×#16840619C212yesmaximal, complemented
11#21459788C211yesmaximal, characteristic
12C2×#14519864—21no—
13#16840812—21no—
14#16840791—22no—
15D4×#92420—21no—
16#16840784C2211yescharacteristic
17#16840783C2212yes—
18#16840727—21no—
19#16840726—22no—
20C4×#11758213C2211yescharacteristic
21C4×#11758213C2211yescharacteristic
22C4×#11758217C2212yes—
23#16840721C2211yescharacteristic
24C2×#14519832C2212yes—
25C2×#14519832C2212yes—
26#16840619C2211yescharacteristic
27#16840619C2212yes—
28#16840619C2212yes—
29#16840619C2212yes—
30#16840619C2211yescharacteristic
31#16840619C2212yes—
32#16840619C2212yes—
33#16840617C2211yescharacteristic
34#16840617C2211yescharacteristic
35#16840617C2212yes—
36#16840487C411yescharacteristic
37#16840469C411yescharacteristic
38#16840469C411yescharacteristic
39#16840464C412yes—
40#16840454—21no—
41#16840449—22no—
42#16840447C412yes—
43#16840411—21no—
44#16840372—21no—
45#16840356—22no—
46#16840317C2211yescharacteristic
47#16840294C2211yescharacteristic
48#16840293C2212yes—
49C2×#14519768C2212yes—
50#16840245—21no—
51C4×#11758224C2211yescharacteristic
52C61×#547C411yescharacteristic
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×#11758224D411yescharacteristic
77#14519777C2×C411yescharacteristic
78#14519775—21no—
79#14519772—22no—
80#14519770—22no—
81#14519770—21no—
82#14519770—22no—
83#14519768—21no—
84#14519768—22no—
85#14519768—22no—
86#14519768C2×C412yes—
87#14519768C2×C412yes—
88#14519768C2311yescharacteristic
89#14519768C2312yes—
90#14519767—22no—
91#14519767—22no—
92#14519764C2311yescharacteristic
93#14519761—22no—
94#14519760—22no—
95#14519759—21no—
96#14519758—22no—
97#14519757—21no—
98#14519757C2×C411yescharacteristic
99#14519756—21no—
100#14519756C2×C412yes—
101#14519754—22no—
102#14519754—22no—
103#14519753—21no—
104#14519749—22no—
105#14519748C2×C411yescharacteristic
106Q8×Dic61—21no—
107Q8×Dic61—21no—
108C4×Dic122C2×C411yescharacteristic
109C4×Dic122C2×C411yescharacteristic
110#14519742—21no—
111#14519742C2×C412yes—
112#14519742D411yescharacteristic
113D4×Dic61—21no—
114D4×Dic61—22no—
115D4×Dic61—22no—
116D4×Dic61C2×C412yes—
117D4×Dic61D411yescharacteristic
118C4×C22⋊D61—21no—
119C4×C22⋊D61C2×C412yes—
120#14519737D412yes—
121#14519736—22no—
122#14519736—21no—
123#14519735C2×C412yes—
124#14519735D411yescharacteristic
125#14519733—22no—
126#14519733D412yes—
127C2×C4⋊Dic61—22no—
128C2×C4⋊Dic61—21no—
129C2×C4⋊Dic61C2×C412yes—
130C2×#11758196—22no—
131C2×C4×Dic61C2312yes—
132C42×D61C2311yescharacteristic
133C2×#41×C61—21no—
134C4⋊Q8×C61C2×C411yescharacteristic
135#123×C61—22no—
136C22⋊Q8×C61—21no—
137#121×C61C2×C412yes—
138C4×D4×C61C2×C411yescharacteristic
139C4×D4×C61C2×C411yescharacteristic
140C2×C4⋊C4×C61C2×C412yes—
141#41×C61—44no—
142C2×Q8×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61C22⋊C411yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C412yes—
149C2×C22⋊D61—21no—
150C2×C22⋊D61C22⋊C412yes—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61C2×D412yes—
154C22×Dic61C2×D412yes—
155#11758224—24no—
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122C22⋊C411yescharacteristic
161C2×Dic122C22×C411yescharacteristic
162#11758217—28no—
163C2×#92420—28no—
164C2×#92420—22no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—22no—
170C2×#92420—44no—
171#11758213—24no—
172#11758213—42no—
173D61⋊C8—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22⋊C411yescharacteristic
177C4⋊C4×C61C22×C411yescharacteristic
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61C22⋊C412yes—
182C42×C61C22×C411yescharacteristic
183C22⋊Dic61—22no—
184C22⋊Dic61—21no—
185C22⋊Dic61—22no—
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188C22⋊Dic61C22⋊C412yes—
189#11758198—22no—
190#11758198—21no—
191#11758198C22⋊C412yes—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C412yes—
202#11758196—22no—
203#11758196—21no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—21no—
208#11758196—22no—
209#11758196C22⋊C412yes—
210#11758196C22⋊C412yes—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61C22×C412yes—
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D412yes—
219C4×Dic61C2411yescharacteristic
220C23×C61—21no—
221C23×C61C2×C22⋊C412yes—
222C22×D61C2×C22⋊C411yescharacteristic
223Q8×C61—42no—
224D4×C61—41no—
225D4×C61—44no—
226D4×C61#10414yes—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—42no—
230C2×C4×C61—44no—
231C2×C4×C61—42no—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C23×C411yescharacteristic
236C22⋊D61—24no—
237C22⋊D61—81no—
238C2×Dic61—24no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—42no—
243C2×Dic61—82no—
244C2×Dic61—82no—
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C22×D411yescharacteristic
249C2×Dic61C22×D411yescharacteristic
250C4×D61—22no—
251C4×D61—41no—
252C4×D61—41no—
253Dic122—22no—
254Dic122—41no—
255Dic122—81no—
256#92420—22no—
257#92420—22no—
258#92420—24no—
259#92420—22no—
260#92420—41no—
261#92420—42no—
262#17653—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×#10414yes—
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C4×C61—21no—
274C4×C61—22no—
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—42no—
278C4×C61C2×#10412yes—
279Dic61—21no—
280Dic61—21no—
281Dic61—22no—
282Dic61—21no—
283Dic61—22no—
284Dic61—41no—
285Dic61—42no—
286Dic61—41no—
287#3515—611no—
288#2970—611no—
289#2875—611no—
290#1532—612no—
291#1531—611no—
292#1529—611no—
293#1528—612no—
294#1526—611no—
295#1524—611no—
296#1524—612no—
297C2×#330—612no—
298C2×C61—24no—
299C2×C61—81no—
300C2×C61#288911yescharacteristic
301C2×C61C22×#10411yescharacteristic
302C2×C61#290111yescharacteristic
303D61—41no—
304C2×#131—1221no—
305#557—611no—
306#554—611no—
307#554—611no—
308#553—612no—
309#552—612no—
310#552—1221no—
311#552—1222no—
312#547—1221no—
313#547#692311yescharacteristic
314#541—1221no—
315#539—1222no—
316#533—611no—
317#529—612no—
318#529—611no—
319C2×#125—612no—
320#520—1221no—
321C4×#41—611no—
322#446—1221no—
323#435—1222no—
324C8×D4—1221no—
325#432—612no—
326#432—611no—
327#405—1222no—
328#405—1221no—
329C4×M16—612no—
330C4×M16—611no—
331C4×M16—611no—
332C2×#97—612no—
333C2×#97—612no—
334C2×#97—611no—
335#330—611no—
336#330—612no—
337#330—612no—
338#330—612no—
339#330—611no—
340#330—612no—
341#330—612no—
342#329—611no—
343#329—612no—
344#329—611no—
345C61#1765311yescharacteristic, complemented, Sylow
346C2×#41—21no—
347C2×#41—611no—
348#131—1228no—
349C2×M16—1222no—
350C2×M16—1221no—
351C2×M16—1221no—
352C22×C8—1222no—
353C22×C8—1221no—
354C22×C8—1222no—
355C4⋊Q8—611no—
356C4⋊Q8—1221no—
357C4⋊Q8C2×#692311yescharacteristic
358#126—1222no—
359#126—1221no—
360#126—1222no—
361#126—1222no—
362#125—611no—
363#125—612no—
364#125—612no—
365#125—612no—
366#125—1222no—
367#125—1222no—
368#125—1222no—
369#125—1222no—
370#125—1221no—
371#124—611no—
372#123—22no—
373#123—1222no—
374#123—1221no—
375#123—1222no—
376#123—1222no—
377C22⋊Q8—21no—
378C22⋊Q8—612no—
379C22⋊Q8—611no—
380C22⋊Q8—1222no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—1221no—
384C22⋊Q8—1222no—
385#121—611no—
386#121—1222no—
387#121C2×#692312yes—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—611no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×D4—611no—
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—612no—
399C4×D4—1222no—
400C4×D4—1221no—
401C4×D4—1222no—
402C4×D4—1221no—
403C4×D4C2×#692311yescharacteristic
404C4×D4C2×#692311yescharacteristic
405#117—611no—
406#117—612no—
407#117—612no—
408#117—1222no—
409#117—1221no—
410#117—1222no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4C2×#692312yes—
416C2×C42—612no—
417C2×C42—611no—
418C4⋊C8—1221no—
419C4⋊C8—1222no—
420C22⋊C8—1222no—
421C22⋊C8—1221no—
422#97—611no—
423#97—611no—
424#97—612no—
425#97—611no—
426#97—612no—
427#97—611no—
428#97—1222no—
429#97—1221no—
430#97—1222no—
431C4×C8—1222no—
432C4×C8—1221no—
433C4×C8—1221no—
434#41—44no—
435#41—1224no—
436C2×Q8—21no—
437C2×Q8—611no—
438C2×Q8—611no—
439C2×Q8—1221no—
440C2×D4—22no—
441C2×D4—612no—
442C2×D4—1221no—
443C2×D4C22⋊#692311yescharacteristic
444C2×D4C22×#692311yescharacteristic
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4C22×#692312yes—
455M16—1224no—
456M16—1228no—
457M16—2442no—
458C2×C8—1222no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1224no—
463C2×C8—1228no—
464C2×C8—1221no—
465C2×C8—1222no—
466C2×C8—2444no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—611no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—611no—
473C4⋊C4—612no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—1221no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4C22⋊#692311yescharacteristic
489C4⋊C4C22×#692311yescharacteristic
490C4⋊C4C22×#692312yes—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—612no—
494C22⋊C4—612no—
495C22⋊C4—1222no—
496C22⋊C4—1221no—
497C22⋊C4—1221no—
498C22⋊C4—1221no—
499C22⋊C4—1221no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4C22⋊#692312yes—
503C42—612no—
504C42—611no—
505C42—611no—
506C42—612no—
507C42—611no—
508C42—612no—
509C42—1222no—
510C42—1222no—
511C42—1221no—
512C42C22×#692311yescharacteristic
513C23—21no—
514C23—611no—
515C23C2×C22⋊#692312yes—
516Q8—42no—
517Q8—1222no—
518Q8—4881no—
519Q8—2441no—
520D4—41no—
521D4—44no—
522D4—1224no—
523D4—4881no—
524D4#1451984114yes—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—611no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1221no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2442no—
545C2×C4C2×C22⋊#692312yes—
546C2×C4C2×C22⋊#692311yescharacteristic
547C2×C4C2×C22⋊#692311yescharacteristic
548C2×C4C23×#692311yescharacteristic
549C8—1222no—
550C8—1222no—
551C8—1224no—
552C8—1222no—
553C8—2442no—
554C8—2441no—
555C22—22no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22C2×#1451984114yes—
564C22C22×C22⋊#692311yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—41no—
569C4—42no—
570C4—1221no—
571C4—1221no—
572C4—1222no—
573C4—1221no—
574C4—1222no—
575C4—2441no—
576C4—2441no—
577C4—2442no—
578C4C2×#1451984112yes—
579C2—24no—
580C2—81no—
581C2—2441no—
582C2#2146243111yescharacteristic
583C2C22×#1451984111yescharacteristic
584C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#547C4
socleC22×C61C22×C22⋊C4
prime radical#21459788C2
cosocleC4×Dic61C24
hypercenter#547#6923
hypocenterC61#17653
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #21462657C2semidirect
2. #21462390C2semidirect
3. #21461295C2semidirect
4. #21461292C2semidirect
5. #21461284C2semidirect
6. #21461281C2semidirect
7. #21461277C2semidirect
8. #21461270C2semidirect
9. #21461269C2semidirect
10. C2×#16840619C2semidirect
345. C61#17653semidirect

Maximal subgroups

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

subgroupindex
#214626572
#214623902
#214612952
#214612922
#214612842
#214612812
#214612772
#214612702
#214612692
C2×#168406192
#214597882
#1765361