Finite Group Databasebeta

← #30610753

Subgroups — #30610753

Order 15616.

1112 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#423C211yesmaximal, characteristic, complemented
2C4×D61⋊M16C211yesmaximal, characteristic, complemented
3#21461168C212yesmaximal, complemented
4#21461165C212yesmaximal, complemented
5#21461154C214yesmaximal, complemented
6#21461135C214yesmaximal, complemented
7#21459114C211yesmaximal, characteristic
8C2×D61⋊M16—21no—
9C2×D61⋊M16—21no—
10#16840767—21no—
11#16840766C2212yes—
12#16840765—21no—
13#16840765C2212yes—
14D61⋊(C4⋊C8)—21no—
15D61⋊(C4⋊C8)C2212yes—
16C4×#11758213—21no—
17C4×#11758213C2212yes—
18C4×#11758217—21no—
19C4×#11758217C2212yes—
20#16840721—21no—
21C4×D61⋊C8C2212yes—
22Q8⋊#92420C2218yescomplemented
23D4⋊#92420C2218yescomplemented
24#16840319C411yescharacteristic
25#16840317C411yescharacteristic
26#16840317C2211yescharacteristic
27#16840310C411yescharacteristic
28#16840310C2211yescharacteristic
29#16840294—21no—
30#16840294—21no—
31#16840293—21no—
32#16840293—21no—
33#16840291C412yes—
34#16840291C2212yes—
35#16840272—21no—
36#16840272—21no—
37#16840269C412yes—
38#41×Dic61C2212yes—
39C4×#11758224—21no—
40C4×#11758224—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#532C411yescharacteristic
43C2×#11758213—22no—
44C2×#11758213—22no—
45D61⋊M16—28no—
46D61⋊M16—44no—
47C2×#11758217—22no—
48C2×#11758217—22no—
49C2×D61⋊C8—22no—
50C2×D61⋊C8—22no—
51#14519844—22no—
52#14519844—22no—
53C4⋊#92420—22no—
54C4⋊#92420C2314yes—
55#14519832—22no—
56#14519832—22no—
57C4×#92420—22no—
58C4×#92420C2314yes—
59C2×#11758224—21no—
60C2×#11758224—21no—
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519777C2×C411yescharacteristic
66#14519770—22no—
67#14519770—22no—
68#14519769C2×C412yes—
69#14519768—21no—
70#14519768—21no—
71#14519764—21no—
72#14519764—21no—
73#14519759—22no—
74#14519759—22no—
75#14519758—22no—
76#14519757—21no—
77#14519757—21no—
78#14519756—21no—
79#14519756—21no—
80#14519753C2×C412yes—
81#14519753C2×C412yes—
82#14519748—21no—
83#14519748—21no—
84#14519748C2×C412yes—
85#14519748C2×C412yes—
86#14519745—22no—
87Q8×Dic61C2×C412yes—
88Q8×Dic61C2312yes—
89C4×Dic122—21no—
90C4×Dic122—21no—
91#14519742—21no—
92#14519742—21no—
93D4×Dic61—22no—
94D4×Dic61—22no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61C2×C412yes—
98D4×Dic61C2312yes—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—22no—
101#14519737—22no—
102#14519737—22no—
103#14519736—22no—
104#14519736C2×C412yes—
105#14519735—21no—
106#14519735—21no—
107#14519733—22no—
108#14519733—22no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61C2311yescharacteristic
120C42×D61C2311yescharacteristic
121C2×#41×C61C2×C412yes—
122C4⋊Q8×C61C2×C411yescharacteristic
123#127×C61C2×C411yescharacteristic
124#124×C61C2×C412yes—
125#121×C61—21no—
126#121×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61C22⋊C411yescharacteristic
131#41×C61—44no—
132C2×Q8×C61C22×C412yes—
133C2×D4×C61—22no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C22×C412yes—
137C22×C4×C61—22no—
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C2×C22⋊D61—22no—
142C2×C22⋊D61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148#11758224—44no—
149#11758224—42no—
150#11758224—42no—
151C2×C4×D61—24no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—42no—
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22×C411yescharacteristic
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169#11758217—28no—
170#11758217—44no—
171C2×#92420—24no—
172C2×#92420—24no—
173#11758213—28no—
174#11758213—44no—
175D61⋊C8—28no—
176D61⋊C8—44no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190#11758198—22no—
191#11758198—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22×C412yes—
197C4⋊Dic61C22×C412yes—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22×C411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2411yescharacteristic
217C23×C61—22no—
218C23×C61C2×C22⋊C411yescharacteristic
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—41no—
222C22×D61C2×C22⋊C411yescharacteristic
223C22×D61C2×C22⋊C411yescharacteristic
224Q8×C61—24no—
225D4×C61—24no—
226D4×C61—44no—
227D4×C61—42no—
228D4×C61—42no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—44no—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C22⋊D61—44no—
241C22⋊D61—44no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—41no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—44no—
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C22×D411yescharacteristic
256C2×Dic61C22×D411yescharacteristic
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—21no—
264C4×D61—21no—
265C4×D61—22no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61#10414yes—
269C4×D61AGL(1,ℤ/8)12yes—
270C4×D61#13712yes—
271Dic122—42no—
272Dic122—42no—
273#92420—28no—
274#92420—44no—
275#17504—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61C22×C22⋊C411yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—21no—
286C2×D61—21no—
287C2×D61—44no—
288C2×D61#41712yes—
289C2×D61C2×#10412yes—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—42no—
294C4×C61#41712yes—
295C4×C61C2×#10412yes—
296C4×C61#56812yes—
297Dic61—21no—
298Dic61—21no—
299Dic61—21no—
300Dic61—21no—
301Dic61—42no—
302Dic61—42no—
303Dic61C2×#10412yes—
304Dic61C2×AGL(1,ℤ/8)11yescharacteristic
305Dic61C2×#13711yescharacteristic
306#3444—611no—
307C2×#423—611no—
308C2×C4×M16—611no—
309#1493—612no—
310#1492—612no—
311#1489—614no—
312#1483—614no—
313C2×C61—41no—
314C2×C61—82no—
315C2×C61#287911yescharacteristic
316C2×C61C2×#41711yescharacteristic
317C2×C61C22×#10411yescharacteristic
318D61—22no—
319D61—22no—
320C22×M16—1221no—
321C22×M16—1221no—
322#533—611no—
323#533—611no—
324#533—611no—
325#532—611no—
326#532—611no—
327#532#692311yescharacteristic
328#529—612no—
329#529—612no—
330#529—1221no—
331#529—1221no—
332#529—1221no—
333#529—1221no—
334#525—612no—
335#525—1221no—
336#525—1221no—
337C4×#41—612no—
338C4×#41—1221no—
339C4×#41—1221no—
340C22×C42—611no—
341#423—612no—
342#423—612no—
343#423—1221no—
344#423—1221no—
345C2×C4⋊C8—612no—
346C2×C4⋊C8—1221no—
347C4×M16—612no—
348C4×M16—612no—
349C4×M16—1221no—
350C4×M16—1221no—
351C2×#97—1221no—
352C2×C4×C8—612no—
353Q8⋊C8—618no—
354D4⋊C8—618no—
355C61#1750411yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41C2×#692312yes—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—611no—
362C2×M16—1222no—
363C2×M16—1222no—
364C2×M16—1228no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—2444no—
368C22×C8—1222no—
369C22×C8—1222no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8C2×#692311yescharacteristic
373#127C2×#692311yescharacteristic
374#126—1222no—
375#126—1222no—
376#125—612no—
377#125—1221no—
378#125—1221no—
379#124—1221no—
380#124—1221no—
381#124C2×#692312yes—
382#123—1222no—
383#123—1222no—
384#123—1222no—
385C22⋊Q8—612no—
386C22⋊Q8—612no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391#121—21no—
392#121—21no—
393#121—612no—
394#121—612no—
395#121—1221no—
396#121—1221no—
397#120—1222no—
398C4×Q8—612no—
399C4×Q8—612no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—612no—
407C4×D4—612no—
408C4×D4—1221no—
409C4×D4—1221no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416#117—612no—
417#117—1222no—
418#117—1222no—
419#117—1222no—
420#117—1221no—
421#117—1221no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1222no—
434C2×C42C2×#692311yescharacteristic
435C4⋊C8—614no—
436C4⋊C8—1222no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439#97—1222no—
440#97—1222no—
441C4×C8—614no—
442C4×C8—1222no—
443C24—611no—
444#41—44no—
445#41—2442no—
446#41—2442no—
447#41—2444no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8C22×#692312yes—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4C22×#692312yes—
457C22×C4—22no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—1224no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—2442no—
479C22×C4C22⋊#692311yescharacteristic
480C22×C4C22⋊#692311yescharacteristic
481C22×C4C22×#692311yescharacteristic
482M16—1228no—
483M16—1228no—
484M16—2444no—
485M16—2444no—
486C2×C8—1224no—
487C2×C8—1228no—
488C2×C8—1224no—
489C2×C8—2444no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42C22⋊#692311yescharacteristic
527C42C22⋊#692311yescharacteristic
528C42C22×#692311yescharacteristic
529C42C22×#692311yescharacteristic
530C23—22no—
531C23—611no—
532C23—611no—
533C23—1222no—
534C23—1222no—
535C23—2441no—
536C23C2×C22⋊#692311yescharacteristic
537Q8—24no—
538Q8—2442no—
539Q8—2442no—
540D4—24no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—2444no—
545D4—2444no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—44no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—614no—
556C2×C4—611no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2444no—
579C2×C4—2441no—
580C2×C4C2×C22⋊#692311yescharacteristic
581C2×C4C2×C22⋊#692311yescharacteristic
582C2×C4C2×C22⋊#692311yescharacteristic
583C2×C4C2×C22⋊#692311yescharacteristic
584C2×C4C2×C22⋊#692311yescharacteristic
585C2×C4C23×#692311yescharacteristic
586C8—1228no—
587C8—2444no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—612no—
595C22—612no—
596C22—1221no—
597C22—1221no—
598C22—1221no—
599C22—1221no—
600C22—2444no—
601C22C22×C22⋊#692311yescharacteristic
602C4—21no—
603C4—21no—
604C4—21no—
605C4—42no—
606C4—611no—
607C4—611no—
608C4—612no—
609C4—1221no—
610C4—1221no—
611C4—1221no—
612C4—1221no—
613C4—2442no—
614C4—2442no—
615C4#1684075312yes—
616C4D61⋊#10412yes—
617C4#1684084012yes—
618C2—41no—
619C2—82no—
620C2—1222no—
621C2—1222no—
622C2#2146240511yescharacteristic
623C2C2×#1684075311yescharacteristic
624C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#532C4
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC4×Dic61C24
hypercenter#532#6923
hypocenterC61#17504
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. D61⋊#423C2semidirect
2. C4×D61⋊M16C2semidirect
3. #21461168C2semidirect
4. #21461165C2semidirect
5. #21461154C2semidirect
6. #21461135C2semidirect
22. Q8⋊#92420C22semidirect
23. D4⋊#92420C22semidirect
355. C61#17504semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4232
C4×D61⋊M162
#214611682
#214611652
#214611542
#214611352
#214591142
#1750461