Finite Group Databasebeta

← D61×#1680 #29662393

Subgroups — D61×#1680 #29662393

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#487×D61C211yesmaximal, characteristic
2#475×D61C211yesmaximal, characteristic
3#462×D61C211yesmaximal, characteristic
4C8×Q8×D61C211yesmaximal, characteristic
5#21445252C211yesmaximal, characteristic, complemented
6#21445251C211yesmaximal, characteristic, complemented
7#21445250C211yesmaximal, characteristic, complemented
8#21445249C211yesmaximal, characteristic, complemented
9#21445248C211yesmaximal, characteristic, complemented
10#21445247C211yesmaximal, characteristic, complemented
11#21445246C211yesmaximal, characteristic, complemented
12#334×D61C211yesmaximal, characteristic
13#332×D61C211yesmaximal, characteristic
14#332×D61C211yesmaximal, characteristic
15C61×#1680C211yesmaximal, characteristic, complemented
16C2×Q16×D61—21no—
17C4⋊Q8×D61C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×Q8×D61C2211yescharacteristic
20#16840114C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840111—21no—
24#16840111—21no—
25#16840048C2211yescharacteristic
26#16840047C2211yescharacteristic
27#16840046—21no—
28#16840045—21no—
29#16840044C2211yescharacteristic
30#16839973C2211yescharacteristic
31#16839972C2211yescharacteristic
32#16839971—21no—
33#16839970—21no—
34#16839969C2211yescharacteristic
35Q8×#92418C2211yescharacteristic
36C8×Dic122—21no—
37C8×Dic122C2211yescharacteristic
38#16839886—21no—
39#16839886C2211yescharacteristic
40#106×D61—21no—
41C4⋊C8×D61—21no—
42C4⋊C8×D61C2211yescharacteristic
43C4⋊C8×D61C2211yescharacteristic
44C4⋊C8×D61C2211yescharacteristic
45#103×D61—21no—
46#103×D61—21no—
47#103×D61—21no—
48#103×D61—21no—
49C4×C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic
52#16839512C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839509C2211yescharacteristic
55#16839509C2211yescharacteristic
56#16839508C2211yescharacteristic
57#16839508C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#475C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61—21no—
69C2×Q8×D61D411yescharacteristic
70C2×Dic244—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73Q16×D61—44no—
74C2×C8×D61—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61D411yescharacteristic
78C2×C8×D61D411yescharacteristic
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519776—21no—
82#14519776—21no—
83#14519776—21no—
84#14519776—21no—
85#14519775—21no—
86#14519775—21no—
87Q8×Dic61D411yescharacteristic
88C4×Dic122—21no—
89C4×Dic122C2311yescharacteristic
90#14519742—21no—
91#14519742D411yescharacteristic
92C4⋊C4×D61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61D411yescharacteristic
98C42×D61—21no—
99C42×D61C2311yescharacteristic
100#14519709—21no—
101#14519708—21no—
102#14519708—21no—
103C4⋊#92418—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519702—21no—
113#14519702—21no—
114#14519702—21no—
115#14519702—21no—
116#14519701—21no—
117#14519701—21no—
118#14519701—21no—
119#14519701—21no—
120#14519700—21no—
121#14519700—21no—
122#14519700—21no—
123#14519700—21no—
124C4×#92418—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×Q16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61—21no—
142#103×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148Q8×D61—22no—
149Q8×D61—42no—
150Q8×D61—42no—
151Q8×D61—42no—
152Q8×D61—42no—
153Q8×D61SD1614yes—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61#4111yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—41no—
165C2×Dic122—41no—
166C2×Dic122C2×D411yescharacteristic
167Q16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D411yescharacteristic
179C42×C61—21no—
180C42×C61C2411yescharacteristic
181#11758202—42no—
182#11758202—42no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C2×D411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61#4111yescharacteristic
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207Dic244—42no—
208C8×D61—22no—
209C8×D61—22no—
210C8×D61—42no—
211C8×D61—42no—
212C8×D61—42no—
213C22×D61#12111yescharacteristic
214Q8×C61—21no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61C2×SD1612yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61#12111yescharacteristic
232C2×Dic61#12111yescharacteristic
233C2×Dic61#12111yescharacteristic
234C4×D61—24no—
235C4×D61—22no—
236C4×D61—42no—
237C4×D61—82no—
238C4×D61—82no—
239C4×D61C2×SD1612yes—
240C4×D61#13512yes—
241C4×D61#13512yes—
242C4×D61#13712yes—
243Dic122—21no—
244Dic122—24no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—82no—
250Dic122—82no—
251Dic122C2×SD1612yes—
252C8×C61—21no—
253C8×C61—21no—
254C8×C61—41no—
255C8×C61—41no—
256C8×C61—41no—
257#92418—21no—
258#92418—21no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262C2×#1680—611nomaximal, Sylow
263C22×C61C2×#12111yescharacteristic
264C2×D61#46412yes—
265C2×D61#46512yes—
266C2×D61#47312yes—
267C4×C61—22no—
268C4×C61—21no—
269C4×C61—41no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61C22×SD1611yescharacteristic
273C4×C61C2×#13511yescharacteristic
274C4×C61C2×#13511yescharacteristic
275C4×C61C2×#13711yescharacteristic
276Dic61—22no—
277Dic61—21no—
278Dic61—41no—
279Dic61—81no—
280Dic61—81no—
281Dic61#46412yes—
282Dic61#46512yes—
283C2×#487—611no—
284C2×#475—611no—
285C2×#462—611no—
286C2×C8×Q8—611no—
287#1680—611no—
288#1680—611no—
289#1680—611no—
290#1680—611no—
291#1680—611no—
292#1680—611no—
293#1680—611no—
294#1680D6111yescharacteristic, direct product factor
295C2×#334—611no—
296C2×#332—611no—
297C2×#332—611no—
298C2×C61C2×#46411yescharacteristic
299C2×C61C2×#46511yescharacteristic
300C2×C61C2×#47311yescharacteristic
301D61#168014yesdirect product factor
302C22×Q16—1221no—
303C2×C4⋊Q8—611no—
304C2×C4⋊Q8—611no—
305C2×C4×Q8—611no—
306#487—611no—
307#487—611no—
308#487—611no—
309#487—1221no—
310#487—1221no—
311#487C2×D6111yescharacteristic
312#475—611no—
313#475—611no—
314#475—611no—
315#475—1221no—
316#475—1221no—
317#475C2×D6111yescharacteristic
318#462—611no—
319#462—611no—
320#462—611no—
321#462—1221no—
322#462—1221no—
323#462C2×D6111yescharacteristic
324C8×Q8—611no—
325C8×Q8—611no—
326C8×Q8—611no—
327C8×Q8—1221no—
328C8×Q8—1221no—
329C8×Q8C2×D6111yescharacteristic
330C2×#106—1221no—
331C2×C4⋊C8—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8—611no—
334C2×C4⋊C8—1221no—
335C2×#103—1221no—
336C2×#103—1221no—
337C2×#103—1221no—
338C2×#103—1221no—
339C2×C4×C8—611no—
340C2×C4×C8—1221no—
341#334—611no—
342#334—611no—
343#334—611no—
344#334C2×D6111yescharacteristic
345#332—611no—
346#332—611no—
347#332—611no—
348#332—611no—
349#332—611no—
350#332—611no—
351#332C2×D6111yescharacteristic
352#332C2×D6111yescharacteristic
353C61C2×#168011yescharacteristic, complemented, Sylow
354C22×Q8—611no—
355C22×Q8—1221no—
356C22×Q8—1221no—
357C2×Q16—21no—
358C2×Q16—1221no—
359C2×Q16—1221no—
360C2×Q16—1221no—
361C2×Q16—2444no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—1221no—
365C22×C8—1221no—
366C22×C8—1221no—
367C4⋊Q8—611no—
368C4⋊Q8—611no—
369C4⋊Q8—1221no—
370C4⋊Q8—1221no—
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8—1221no—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377C4×Q8—611no—
378C4×Q8—611no—
379C4×Q8—611no—
380C4×Q8—1221no—
381C4×Q8—1221no—
382C4×Q8C22×D6111yescharacteristic
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C42—611no—
390C2×C42—1221no—
391#106—21no—
392#106—1221no—
393#106—1221no—
394#106—1221no—
395C4⋊C8—21no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—1221no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8C22×D6111yescharacteristic
406C4⋊C8C22×D6111yescharacteristic
407C4⋊C8C22×D6111yescharacteristic
408#103—21no—
409#103—21no—
410#103—21no—
411#103—21no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424C4×C8—21no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—1221no—
429C4×C8—1221no—
430C4×C8—1221no—
431C4×C8C22×D6111yescharacteristic
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—614no—
435C2×Q8—611no—
436C2×Q8—1221no—
437C2×Q8—1222no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—2441no—
441C2×Q8—2441no—
442C2×Q8—2442no—
443C2×Q8—2442no—
444C2×Q8—2442no—
445C2×Q8—2442no—
446C2×Q8D4×D6111yescharacteristic
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—2441no—
453C22×C4—2441no—
454Q16—42no—
455Q16—2442no—
456Q16—2442no—
457Q16—2442no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—2442no—
469C2×C8—2442no—
470C2×C8—2442no—
471C2×C8D4×D6111yescharacteristic
472C2×C8D4×D6111yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—611no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4D4×D6111yescharacteristic
492C42—21no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42C23×D6111yescharacteristic
500C23—611no—
501Q8—21no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—612no—
507Q8—1224no—
508Q8—1221no—
509Q8—4882no—
510Q8—4882no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8SD16×D6112yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—611no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—4882no—
532C2×C4—4882no—
533C2×C4—2442no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4#41×D6111yescharacteristic
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—1221no—
545C8—1221no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C22—612no—
550C22—612no—
551C22—612no—
552C22#121×D6111yescharacteristic
553C4—22no—
554C4—21no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—612no—
559C4—612no—
560C4—1222no—
561C4—1221no—
562C4—4881no—
563C4—4881no—
564C4—2441no—
565C4C2×SD16×D6111yescharacteristic
566C4#135×D6111yescharacteristic
567C4#135×D6111yescharacteristic
568C4#137×D6111yescharacteristic
569C2—614no—
570C2#464×D6111yescharacteristic
571C2#465×D6111yescharacteristic
572C2#473×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1680C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1680D61
hypocenterC61C2×#1680
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1680

Complements

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

NH ≅ G/Ntype
5. #21445252C2semidirect
6. #21445251C2semidirect
7. #21445250C2semidirect
8. #21445249C2semidirect
9. #21445248C2semidirect
10. #21445247C2semidirect
11. #21445246C2semidirect
15. C61×#1680C2semidirect
294. #1680D61direct (N × H)
301. D61#1680direct (N × H)
353. C61C2×#1680semidirect

Maximal subgroups

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

subgroupindex
#487×D612
#475×D612
#462×D612
C8×Q8×D612
#214452522
#214452512
#214452502
#214452492
#214452482
#214452472
#214452462
#334×D612
#332×D612
C61×#16802
C2×#168061