Finite Group Databasebeta

← #30611695

Subgroups — #30611695

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462662C211yesmaximal, characteristic, complemented
2#21462660C211yesmaximal, characteristic, complemented
3#21461310C211yesmaximal, characteristic, complemented
4#21461303C212yesmaximal, complemented
5#21461294C212yesmaximal, complemented
6#21461284C212yesmaximal, complemented
7#21461278C211yesmaximal, characteristic, complemented
8#21461278C211yesmaximal, characteristic, complemented
9#21461276C211yesmaximal, characteristic, complemented
10#21461269C212yesmaximal, complemented
11#21459710C211yesmaximal, characteristic
12#16840812—21no—
13#16840811—21no—
14#16840796—21no—
15#16840795—22no—
16#16840794—22no—
17#16840790—21no—
18#16840785C2212yescomplemented
19#16840784C2212yescomplemented
20#16840783C2211yescharacteristic
21#16840783C2211yescharacteristic
22C4×#11758213C2212yescomplemented
23C4×#11758217C2211yescharacteristic
24C4×#11758217C2211yescharacteristic
25#16840721C2212yescomplemented
26#16840619C2211yescharacteristic, complemented
27#16840619C2212yescomplemented
28#16840619C2211yescharacteristic, complemented
29#16840619C2211yescharacteristic, complemented
30#16840619C2211yescharacteristic, complemented
31#16840619C2212yescomplemented
32#16840617C2212yes—
33#16840617C2212yes—
34#16840617C2212yes—
35#16840617C2212yes—
36#16840495—21no—
37#16840495—21no—
38#16840482—21no—
39#16840482—21no—
40#16840438—22no—
41#16840438—22no—
42#16840377C412yes—
43#16840377C412yes—
44#16840338C411yescharacteristic
45#16840329C411yescharacteristic
46#16840315C2211yescharacteristic
47#16840294C2212yes—
48#16840294C2212yes—
49C2×#14519764C2211yescharacteristic
50#16840239C411yescharacteristic
51#16840230C2211yescharacteristic
52C61×#518C411yescharacteristic
53C2×#11758213—22no—
54C2×#11758217—21no—
55C2×#11758217—21no—
56C2×D61⋊C8—22no—
57#14519844—22no—
58C4⋊#92420—21no—
59C4⋊#92420—21no—
60C4⋊#92420—22no—
61#14519834—22no—
62#14519834—22no—
63C22⋊#92420—21no—
64C22⋊#92420—21no—
65#14519832—22no—
66#14519832C2311yescharacteristic
67#14519832C2312yes—
68#14519832C2311yescharacteristic
69#14519832C2311yescharacteristic
70#14519832C2311yescharacteristic
71#14519832C2312yes—
72C4×#92420—21no—
73C4×#92420—21no—
74C4×#92420—22no—
75C2×#11758222C2×C411yescharacteristic
76C22×Dic122C2×C411yescharacteristic
77#14519777—21no—
78#14519776C2×C411yescharacteristic
79#14519776C2×C411yescharacteristic
80#14519773—21no—
81#14519770—22no—
82#14519770—22no—
83#14519770—22no—
84#14519770—22no—
85#14519768C2311yescharacteristic
86#14519768C2311yescharacteristic
87#14519765—21no—
88#14519764—21no—
89#14519764—21no—
90#14519764—21no—
91#14519764C2×C412yes—
92#14519764C2312yes—
93#14519763—22no—
94#14519763—22no—
95#14519761—22no—
96#14519760—22no—
97#14519760—22no—
98#14519759—22no—
99#14519759—22no—
100#14519757C2×C412yes—
101#14519757C2×C412yes—
102#14519756—22no—
103#14519756—22no—
104#14519754—21no—
105#14519754—21no—
106#14519748C2×C412yes—
107#14519748C2×C412yes—
108#14519746—22no—
109C4×Dic122—21no—
110C4×Dic122—21no—
111C4×Dic122C2×C412yes—
112C4×Dic122C2×C412yes—
113#14519742—21no—
114#14519742—21no—
115#14519742D411yescharacteristic
116#14519742D411yescharacteristic
117#14519739D411yescharacteristic
118#14519739D411yescharacteristic
119C4×C22⋊D61—22no—
120C4×C22⋊D61—22no—
121#14519737D412yes—
122#14519737D412yes—
123#14519736—22no—
124#14519736—22no—
125#14519736C2×C411yescharacteristic
126#14519734C2×C412yes—
127#14519733—22no—
128C2×C22⋊Dic61—21no—
129C2×C22⋊Dic61—21no—
130C2×C4×Dic61C2311yescharacteristic
131C42×D61C2312yes—
132C22×D4×C61C2×C411yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61C2×C412yes—
136C4×D4×C61C2×C412yes—
137#117×C61C2×C411yescharacteristic
138#117×C61C2×C411yescharacteristic
139C2×C22⋊C4×C61—22no—
140C24×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—42no—
143C2×D4×C61C22×C412yes—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61C22×C411yescharacteristic
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61C22⋊C412yes—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758222—42no—
154C2×D244C22⋊C411yescharacteristic
155C2×C4×D61C22×C412yes—
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—42no—
162C2×Dic122C22⋊C411yescharacteristic
163C2×Dic122C22×C412yes—
164#11758217—42no—
165#11758217—42no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—22no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—22no—
172#11758213—44no—
173D61⋊C8—44no—
174C4⋊C4×C61C22⋊C411yescharacteristic
175C4⋊C4×C61C22⋊C411yescharacteristic
176C4⋊C4×C61C22×C411yescharacteristic
177C4⋊C4×C61C22×C411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61C22⋊C412yes—
182C42×C61—21no—
183C42×C61C22×C412yes—
184C22⋊Dic61—21no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198C22⋊C412yes—
194#11758198C22⋊C412yes—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201#11758196—22no—
202#11758196—22no—
203#11758196—21no—
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—21no—
212C4×Dic61C22×C411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2411yescharacteristic
218C23×C61—22no—
219C23×C61—42no—
220C23×C61—81no—
221C23×C61—81no—
222C23×C61C2×C22⋊C411yescharacteristic
223C22×D61C2×C22⋊C412yes—
224D4×C61—42no—
225D4×C61—42no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—82no—
230C2×C4×C61—82no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C22⋊D61—42no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C22×D411yescharacteristic
248C2×Dic61C22×D411yescharacteristic
249D244—22no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61—42no—
253Dic122—22no—
254Dic122—42no—
255Dic122—42no—
256Dic122—81no—
257Dic122—81no—
258#92420—22no—
259#92420—22no—
260#92420—42no—
261#92420—42no—
262#92420—41no—
263#92420—41no—
264#17691—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—162no—
272C22×C61C22×C22⋊C411yescharacteristic
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C4×C61—21no—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61#41312yes—
282Dic61—22no—
283Dic61—21no—
284Dic61—21no—
285Dic61—21no—
286Dic61—21no—
287Dic61—42no—
288Dic61—41no—
289Dic61—41no—
290#3510—611no—
291#2971—611no—
292#2970—611no—
293#1537—611no—
294#1535—612no—
295#1532—612no—
296#1529—612no—
297#1527—611no—
298#1527—611no—
299#1526—611no—
300#1524—612no—
301C2×C61—41no—
302C2×C61—82no—
303C2×C61C2×#41311yescharacteristic
304C2×C61#290111yescharacteristic
305C2×C61#290111yescharacteristic
306D61—42no—
307#559—1221no—
308#559—1221no—
309#556—1221no—
310#556—1221no—
311#551—1222no—
312#551—1222no—
313#542—612no—
314#542—612no—
315#536—611no—
316#535—611no—
317#533—611no—
318#529—612no—
319#529—612no—
320C2×#124—611no—
321#519—611no—
322#518#692311yescharacteristic
323C4×#41—611no—
324#446—1221no—
325C8×Q8—1221no—
326#436—1222no—
327#436—1221no—
328#435—1222no—
329C8×D4—1221no—
330#432—611no—
331#432—611no—
332#432—612no—
333#432—612no—
334C4×M16—611no—
335C4×M16—611no—
336C4×M16—612no—
337C2×#97—612no—
338#330—612no—
339#330—612no—
340#330—611no—
341#330—611no—
342#330—611no—
343#330—611no—
344#329—612no—
345#329—612no—
346#329—612no—
347#329—612no—
348C61#1769111yescharacteristic, complemented, Sylow
349C2×#41—611no—
350C22×Q8—611no—
351C22×D4C2×#692311yescharacteristic
352C2×M16—1221no—
353C2×M16—1221no—
354C2×M16—1222no—
355C22×C8—1222no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8—1221no—
359#127—1221no—
360#126—1222no—
361#126—1222no—
362#126—1222no—
363#126—1222no—
364#125—611no—
365#125—611no—
366#124—612no—
367#124—612no—
368#124—1221no—
369#124—1221no—
370#124—1221no—
371#124—1222no—
372#124—1222no—
373#124—1221no—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378#123—1222no—
379C22⋊Q8—612no—
380C22⋊Q8—612no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385#121—612no—
386#121—612no—
387#120—1222no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—612no—
401C4×D4—612no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4C2×#692312yes—
405C4×D4C2×#692312yes—
406#117—612no—
407#117—611no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117C2×#692311yescharacteristic
412#117C2×#692311yescharacteristic
413C2×C22⋊C4—22no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—611no—
417C2×C42—612no—
418C4⋊C8—1221no—
419C4⋊C8—1222no—
420C4⋊C8—1221no—
421C4⋊C8—1222no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8—1222no—
425C22⋊C8—1222no—
426#97—611no—
427#97—612no—
428#97—611no—
429#97—611no—
430#97—611no—
431#97—612no—
432#97—1222no—
433C4×C8—1221no—
434C4×C8—1222no—
435C4×C8—1221no—
436C24—21no—
437#41—2442no—
438C2×Q8—611no—
439C2×Q8—612no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—2442no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—612no—
447C2×D4—611no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4C22×#692312yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—1221no—
459C22×C4C22×#692311yescharacteristic
460M16—2444no—
461M16—2442no—
462M16—2442no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1222no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—2444no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4C22⋊#692311yescharacteristic
485C4⋊C4C22⋊#692311yescharacteristic
486C4⋊C4C22×#692311yescharacteristic
487C4⋊C4C22×#692311yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—612no—
492C22⋊C4—612no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1221no—
502C22⋊C4C22⋊#692312yes—
503C42—21no—
504C42—612no—
505C42—611no—
506C42—611no—
507C42—612no—
508C42—611no—
509C42—611no—
510C42—1221no—
511C42—1222no—
512C42—1222no—
513C42C22×#692312yes—
514C23—22no—
515C23—42no—
516C23—81no—
517C23—81no—
518C23—612no—
519C23C2×C22⋊#692311yescharacteristic
520Q8—1222no—
521Q8—4881no—
522Q8—4881no—
523Q8—2442no—
524Q8—2442no—
525D4—42no—
526D4—42no—
527D4—1222no—
528D4—4882no—
529D4—4882no—
530D4—2442no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—612no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2442no—
550C2×C4C2×C22⋊#692312yes—
551C2×C4C2×C22⋊#692312yes—
552C2×C4C2×C22⋊#692311yescharacteristic
553C2×C4C23×#692311yescharacteristic
554C8—1222no—
555C8—1222no—
556C8—2442no—
557C8—2442no—
558C8—2441no—
559C8—2441no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—162no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22C22×C22⋊#692311yescharacteristic
571C4—21no—
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—1222no—
577C4—1221no—
578C4—1221no—
579C4—1221no—
580C4—1221no—
581C4—2442no—
582C4—2441no—
583C4—2441no—
584C4#1684074812yes—
585C2—41no—
586C2—82no—
587C2—2442no—
588C2C2×#1684074811yescharacteristic
589C2#2146244911yescharacteristic
590C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#518C4
socleC22×C61C22×C22⋊C4
prime radical#16840329C4
cosocleC4×Dic61C24
hypercenter#518#6923
hypocenterC61#17691
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #21462662C2semidirect
2. #21462660C2semidirect
3. #21461310C2semidirect
4. #21461303C2semidirect
5. #21461294C2semidirect
6. #21461284C2semidirect
7. #21461278C2semidirect
8. #21461278C2semidirect
9. #21461276C2semidirect
10. #21461269C2semidirect
18. #16840785C22semidirect
19. #16840784C22semidirect
22. C4×#11758213C22semidirect
25. #16840721C22semidirect
26. #16840619C22semidirect
27. #16840619C22semidirect
28. #16840619C22semidirect
29. #16840619C22semidirect
30. #16840619C22semidirect
31. #16840619C22semidirect
348. C61#17691semidirect

Maximal subgroups

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

subgroupindex
#214626622
#214626602
#214613102
#214613032
#214612942
#214612842
#214612782
#214612762
#214612692
#214597102
#1769161