Finite Group Databasebeta

← D61×#2013 #29707067

Subgroups — D61×#2013 #29707067

Order 15616.

2254 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×SD16×D61C212yesmaximal
2C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
3C2×#103×D61C212yesmaximal, complemented
4#390×D61C211yesmaximal, characteristic
5#21446904C211yesmaximal, characteristic, complemented
6#21446903C212yesmaximal, complemented
7#21446902C211yesmaximal, characteristic, complemented
8#21446901C211yesmaximal, characteristic, complemented
9#21446900C212yesmaximal, complemented
10#336×D61C211yesmaximal, characteristic, complemented
11C61×#2013C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2212yes—
13C22×D4×D61C2211yescharacteristic
14C22×#11758190C2212yes—
15C22×#11758200C2212yes—
16C22×#11758201C2212yes—
17C2×SD16×D61—28no—
18C2×SD16×D61—28no—
19C2×SD16×D61—28no—
20C22×C8×D61C2212yes—
21C2×#14519777C2211yescharacteristic, complemented
22C2×#14519776—22no—
23C2×#14519775C2212yescomplemented
24C4⋊Q8×D61—28no—
25C2×C4⋊C4×D61—22no—
26C2×C4⋊C4×D61C2211yescharacteristic
27C2×C22⋊C4×D61—22no—
28C2×C42×D61C2211yescharacteristic
29C2×#14519702C2212yescomplemented
30C2×#14519701C2212yescomplemented
31C2×#14519700C2212yescomplemented
32#103×D61—28no—
33#16839665C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839662—22no—
37#16839517C2211yescharacteristic, complemented
38#16839516C2212yescomplemented
39C22×SD16×C61C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic, complemented
41C2×#103×C61C2212yescomplemented
42C61×#390C2211yescharacteristic
43C61×#336C2211yescharacteristic, complemented
44C24×D61—21no—
45C22×Dic122—22no—
46C22×Dic122C2312yes—
47C2×Q8×D61—24no—
48C2×Q8×D61—24no—
49C2×Q8×D61—24no—
50C2×Q8×D61—24no—
51C2×Q8×D61—24no—
52C2×Q8×D61D418yes—
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C2×D4×D61—24no—
56C2×D4×D61—24no—
57C2×D4×D61—22no—
58C2×D4×D61D414yes—
59C22×C4×D61—22no—
60C22×C4×D61—21no—
61C22×C4×D61D412yes—
62C22×C4×D61C2311yescharacteristic
63C2×#11758190—24no—
64C2×#11758190—24no—
65C2×#11758190—24no—
66C2×#11758200—24no—
67C2×#11758200—24no—
68C2×#11758200—24no—
69C2×#11758201—24no—
70C2×#11758201—24no—
71C2×#11758201—24no—
72SD16×D61—264no—
73C22×#92418C2312yes—
74C2×C8×D61—24no—
75C2×C8×D61—24no—
76C2×C8×D61D418yes—
77#14519777—24no—
78#14519776—216no—
79#14519775—24no—
80#14519775—24no—
81C2×C4⋊Dic61—22no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—24no—
84C2×#11758196—21no—
85C4⋊C4×D61—216no—
86C4⋊C4×D61—24no—
87C2×C22⋊Dic61—22no—
88C2×#11758198—22no—
89C2×#11758198—22no—
90C22⋊C4×D61—48no—
91C2×C4×Dic61D412yes—
92C2×C4×Dic61C2311yescharacteristic
93C42×D61—24no—
94#14519702—24no—
95#14519701—24no—
96#14519700—24no—
97C22×Q8×C61C2312yes—
98C22×D4×C61C2311yescharacteristic
99C2×SD16×C61—24no—
100C2×SD16×C61—24no—
101C2×SD16×C61—24no—
102C22×C8×C61C2312yes—
103C4⋊Q8×C61—24no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—22no—
107C2×C42×C61C2311yescharacteristic
108#103×C61—24no—
109C24×C61—21no—
110C23×D61—21no—
111C23×D61—24no—
112C23×D61—42no—
113C23×D61—42no—
114C23×D61—44no—
115C23×D61—44no—
116C23×D61C2×D411yescharacteristic
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61C2×D414yes—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61C2×D412yes—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—24no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61—42no—
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61C2×D412yes—
137C22×Dic61C2×D411yescharacteristic
138Q8×D61—216no—
139Q8×D61—216no—
140Q8×D61—48no—
141Q8×D61—48no—
142D4×D61—216no—
143D4×D61—48no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244C2×D412yes—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—28no—
151C2×C4×D61—44no—
152C2×C4×D61—44no—
153C2×C4×D61—42no—
154C2×C4×D61—44no—
155C2×C4×D61SD1618yescomplemented
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D414yes—
158C2×C4×D61C2×D414yes—
159C2×C4×D61#4112yes—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—28no—
166C2×Dic122—44no—
167C2×Dic122—44no—
168C2×Dic122C2×D414yes—
169SD16×C61—216no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61C2×D414yes—
173C4⋊C4×C61—28no—
174C4⋊C4×C61—22no—
175C22⋊C4×C61—44no—
176C42×C61—22no—
177C22⋊Dic61—44no—
178#11758201—216no—
179#11758200—216no—
180#11758198—44no—
181#11758198—44no—
182C4⋊Dic61—28no—
183C4⋊Dic61—22no—
184#11758196—216no—
185#11758196—42no—
186C4×Dic61—22no—
187C4×Dic61—24no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418C2×D414yes—
191#11758190—216no—
192C8×D61—216no—
193C23×C61—22no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—48no—
205C22×D61—416no—
206C22×D61—88no—
207C22×D61—84no—
208C22×D61—84no—
209C22×D61#12012yes—
210C22×D61#12112yes—
211C22×D61#12712yes—
212C22×D61C2×SD1614yes—
213C22×D61#13714yes—
214Q8×C61—24no—
215Q8×C61—24no—
216Q8×C61—42no—
217Q8×C61—42no—
218D4×C61—24no—
219D4×C61—42no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61C2×SD1614yescomplemented
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D412yes—
230C2×C4×C61C22×D412yes—
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—48no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61#12012yes—
241C2×Dic61#12112yes—
242C2×Dic61#12712yes—
243C2×Dic61C2×SD1614yescomplemented
244D244—24no—
245D244—42no—
246C4×D61—216no—
247C4×D61—416no—
248C4×D61—84no—
249C4×D61#12018yes—
250C4×D61#12118yes—
251Dic122—24no—
252Dic122—24no—
253Dic122—42no—
254Dic122—42no—
255Dic122—416no—
256C8×C61—24no—
257#92418—24no—
258C2×#2013—611nomaximal, Sylow
259C22×C61—42no—
260C22×C61—44no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—82no—
264C22×C61C2×#12011yescharacteristic
265C22×C61C2×#12111yescharacteristic
266C22×C61C2×#12711yescharacteristic
267C22×C61C22×SD1612yes—
268C22×C61C2×#13712yes—
269C2×D61—42no—
270C2×D61—44no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—82no—
274C2×D61—816no—
275C2×D61#39214yes—
276C2×D61#44818yes—
277C2×D61#46118yes—
278C2×D61#49214yes—
279C2×D61#49714yes—
280C4×C61—24no—
281C4×C61—44no—
282C4×C61—81no—
283C4×C61C2×#12012yes—
284C4×C61C2×#12112yes—
285Dic61—24no—
286Dic61—44no—
287Dic61—81no—
288Dic61#39214yes—
289C23×SD16—612no—
290C22×C4⋊Q8—611no—
291C22×#103—612no—
292C2×#390—611no—
293#2013—612no—
294#2013—611no—
295#2013—611no—
296#2013—612no—
297#2013—611no—
298#2013D6111yescharacteristic, direct product factor
299C2×#336—611no—
300C2×C61—82no—
301C2×C61C2×#39211yescharacteristic
302C2×C61C2×#44812yes—
303C2×C61C2×#46112yes—
304C2×C61C2×#49211yescharacteristic
305C2×C61C2×#49711yescharacteristic
306D61—82no—
307D61#201318yesdirect product factor
308C23×Q8—612no—
309C23×D4—611no—
310C22×SD16—612no—
311C22×SD16—612no—
312C22×SD16—612no—
313C22×SD16—1228no—
314C22×SD16—1228no—
315C22×SD16—1228no—
316C22×SD16C2×D6112yes—
317C23×C8—612no—
318C2×C4⋊Q8—612no—
319C2×C4⋊Q8—611no—
320C2×C4⋊Q8—1222no—
321C2×C4⋊Q8—1228no—
322C2×C4⋊Q8C2×D6111yescharacteristic, complemented
323C22×C4⋊C4—611no—
324C22×C4⋊C4—1222no—
325C22×C22⋊C4—1222no—
326C22×C42—611no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#103—1228no—
331C2×#103C2×D6112yescomplemented
332#390—611no—
333#390—611no—
334#390—611no—
335#390—1222no—
336#390C2×D6111yescharacteristic
337#336—612no—
338#336—611no—
339#336C2×D6111yescharacteristic, complemented
340C61C2×#201311yescharacteristic, complemented, Sylow
341C25—1221no—
342C22×Q8—618no—
343C22×Q8—612no—
344C22×Q8—1222no—
345C22×Q8—1224no—
346C22×Q8—1224no—
347C22×Q8—1224no—
348C22×Q8—1224no—
349C22×Q8—1224no—
350C22×Q8C22×D6112yes—
351C22×D4—614no—
352C22×D4—611no—
353C22×D4—1224no—
354C22×D4—1224no—
355C22×D4—1222no—
356C22×D4—1221no—
357C22×D4C22×D6111yescharacteristic
358C23×C4—611no—
359C23×C4—612no—
360C23×C4—1222no—
361C23×C4—1221no—
362C2×SD16—24no—
363C2×SD16—24no—
364C2×SD16—24no—
365C2×SD16—1224no—
366C2×SD16—1224no—
367C2×SD16—1224no—
368C2×SD16—1224no—
369C2×SD16—12264no—
370C2×SD16—1224no—
371C2×SD16—1224no—
372C2×SD16—1224no—
373C2×SD16—1224no—
374C2×SD16—1224no—
375C22×C8—612no—
376C22×C8—618no—
377C22×C8—1224no—
378C22×C8—1224no—
379C22×C8C22×D6112yes—
380C4⋊Q8—24no—
381C4⋊Q8—1224no—
382C4⋊Q8—12216no—
383C4⋊Q8—1224no—
384C4⋊Q8—1224no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—12216no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—22no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—2448no—
398C2×C42—612no—
399C2×C42—611no—
400C2×C42—1224no—
401C2×C42C22×D6111yescharacteristic
402#103—24no—
403#103—1224no—
404#103—1224no—
405#103—1224no—
406C24—21no—
407C24—611no—
408C24—1224no—
409C24—1221no—
410C24—2444no—
411C24—2442no—
412C24—2442no—
413C24—2444no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—614no—
420C2×Q8—1228no—
421C2×Q8—12216no—
422C2×Q8—12216no—
423C2×Q8—1222no—
424C2×Q8—1222no—
425C2×Q8—1222no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—2444no—
429C2×Q8—2444no—
430C2×Q8—2448no—
431C2×Q8—2448no—
432C2×Q8D4×D6114yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—612no—
437C2×D4—12216no—
438C2×D4—1224no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—1221no—
442C2×D4—2448no—
443C2×D4—2442no—
444C2×D4—2442no—
445C2×D4D4×D6112yes—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—611no—
449C22×C4—614no—
450C22×C4—614no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—618no—
455C22×C4—1224no—
456C22×C4—1224no—
457C22×C4—1228no—
458C22×C4—1222no—
459C22×C4—1221no—
460C22×C4—2444no—
461C22×C4—2442no—
462C22×C4—2444no—
463C22×C4—2444no—
464C22×C4D4×D6112yes—
465C22×C4C23×D6111yescharacteristic
466SD16—216no—
467SD16—12216no—
468SD16—12216no—
469SD16—12216no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—614no—
473C2×C8—12216no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8D4×D6114yes—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—12216no—
480C4⋊C4—1228no—
481C4⋊C4—1222no—
482C4⋊C4—2442no—
483C22⋊C4—44no—
484C22⋊C4—2444no—
485C22⋊C4—2444no—
486C22⋊C4—2444no—
487C42—22no—
488C42—1224no—
489C42—1222no—
490C23—22no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—612no—
496C23—612no—
497C23—614no—
498C23—614no—
499C23—612no—
500C23—1222no—
501C23—4884no—
502C23—4884no—
503C23—4888no—
504C23—2448no—
505C23—24416no—
506C23—2441no—
507C23—2442no—
508C23—2442no—
509C23—2441no—
510C23C2×D4×D6111yescharacteristic
511Q8—24no—
512Q8—24no—
513Q8—42no—
514Q8—42no—
515Q8—1224no—
516Q8—1224no—
517Q8—24416no—
518Q8—2442no—
519Q8—2442no—
520D4—24no—
521D4—42no—
522D4—1224no—
523D4—2448no—
524D4—2442no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—618no—
536C2×C4—618no—
537C2×C4—614no—
538C2×C4—12216no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1224no—
542C2×C4—4884no—
543C2×C4—24416no—
544C2×C4—2442no—
545C2×C4—2441no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4SD16×D6114yescomplemented
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×D4×D6112yes—
551C2×C4C2×D4×D6112yes—
552C2×C4#41×D6111yescharacteristic
553C8—24no—
554C8—1224no—
555C22—42no—
556C22—44no—
557C22—81no—
558C22—81no—
559C22—82no—
560C22—614no—
561C22—614no—
562C22—614no—
563C22—618no—
564C22—618no—
565C22—48816no—
566C22—4881no—
567C22—4881no—
568C22—4882no—
569C22—2442no—
570C22—2444no—
571C22#120×D6111yescharacteristic
572C22#121×D6111yescharacteristic
573C22#127×D6111yescharacteristic
574C22C2×SD16×D6112yes—
575C22#137×D6112yes—
576C4—24no—
577C4—44no—
578C4—81no—
579C4—614no—
580C4—1224no—
581C4—4881no—
582C4—2444no—
583C4#120×D6112yes—
584C4#121×D6112yes—
585C2—82no—
586C2—618no—
587C2—4882no—
588C2#392×D6111yescharacteristic
589C2#448×D6112yes—
590C2#461×D6112yes—
591C2#492×D6111yescharacteristic
592C2#497×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2013C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
hypercenter#2013D61
hypocenterC61C2×#2013
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊Q8×D61C2semidirect
3. C2×#103×D61C2semidirect
5. #21446904C2semidirect
6. #21446903C2semidirect
7. #21446902C2semidirect
8. #21446901C2semidirect
9. #21446900C2semidirect
10. #336×D61C2semidirect
11. C61×#2013C2semidirect
21. C2×#14519777C22semidirect
23. C2×#14519775C22semidirect
29. C2×#14519702C22semidirect
30. C2×#14519701C22semidirect
31. C2×#14519700C22semidirect
37. #16839517C22semidirect
38. #16839516C22semidirect
40. C2×C4⋊Q8×C61C22semidirect
41. C2×#103×C61C22semidirect
43. C61×#336C22semidirect
155. C2×C4×D61SD16semidirect
227. C2×C4×C61C2×SD16semidirect
243. C2×Dic61C2×SD16semidirect
298. #2013D61direct (N × H)
307. D61#2013direct (N × H)
322. C2×C4⋊Q8C2×D61semidirect
331. C2×#103C2×D61semidirect
339. #336C2×D61semidirect
340. C61C2×#2013semidirect
548. C2×C4SD16×D61semidirect

Maximal subgroups

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

subgroupindex
C22×SD16×D612
C2×C4⋊Q8×D612
C2×#103×D612
#390×D612
#214469042
#214469032
#214469022
#214469012
#214469002
#336×D612
C61×#20132
C2×#201361