Finite Group Databasebeta

← #30614148

Subgroups — #30614148

Order 15616.

1522 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840755C211yesmaximal, characteristic
2C22×#14519840C211yesmaximal, characteristic
3#21462379C211yesmaximal, characteristic
4#21462256C211yesmaximal, characteristic
5#21461591C214yesmaximal, complemented
6C4×#14519840C214yesmaximal, complemented
7#21461508C211yesmaximal, characteristic
8#21461504C211yesmaximal, characteristic
9#519×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C22×D61⋊C8C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13C22×C4⋊#6923C2211yescharacteristic
14#16840755C418yescomplemented
15C2×#14519840C418yescomplemented
16C2×#14519840C2212yes—
17C2×#14519840C2212yes—
18C2×#14519840C2214yes—
19#16840727—22no—
20#16840726—22no—
21#16840721C2212yes—
22C4×D61⋊C8C2212yes—
23C4⋊C4×#6923—22no—
24C4×C4⋊#6923C2214yes—
25C22⋊C4×#6923—22no—
26#16840635C2212yes—
27#16840635C2212yes—
28#16840635C2214yes—
29C22×Q8×D61C2211yescharacteristic
30#16840240C411yescharacteristic, complemented
31#16840239—22no—
32#16840239C411yescharacteristic, complemented
33#16840239C411yescharacteristic, complemented
34#16840238—22no—
35#16840238C414yescomplemented
36C4×Q8×D61—22no—
37C4×Q8×D61C2214yes—
38#117×D61—22no—
39#117×D61C2211yescharacteristic
40#117×D61C2211yescharacteristic
41C61×#519C411yescharacteristic, complemented
42C23×#6923—21no—
43C2×#11758213—21no—
44D61⋊M16—22no—
45D61⋊M16C2×C414yes—
46C2×#11758217—21no—
47C2×D61⋊C8—21no—
48C2×D61⋊C8—22no—
49C2×D61⋊C8C2×C414yes—
50C2×D61⋊C8C2311yescharacteristic
51C2×D61⋊C8C2312yes—
52C2×D61⋊C8C2311yescharacteristic
53C22×#92420—21no—
54#14519856—22no—
55#14519856C2×C414yes—
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—22no—
58C2×C4⋊#6923C2×C414yes—
59C2×C4⋊#6923C2312yes—
60C2×C4⋊#6923C2312yes—
61C2×C22⋊#6923—21no—
62C2×C4×#6923—21no—
63C2×C4×#6923—24no—
64#14519840C2×C418yes—
65#14519840C2×C418yes—
66#14519840C2×C4116yescomplemented
67#14519832—24no—
68C4×#92420—24no—
69(C2×#6923)⋊C4—24no—
70C22×Dic122—21no—
71C22×Dic122C2×C411yescharacteristic
72C2×Q8×D61—21no—
73C2×Q8×D61—21no—
74C2×Q8×D61—22no—
75C2×Q8×D61—22no—
76C2×Q8×D61C2×C414yes—
77C2×Q8×D61C2×C414yes—
78C2×Q8×D61C2×C412yes—
79C2×Q8×D61C2312yes—
80C22×C4×D61—21no—
81C22×C4×D61C2311yescharacteristic
82Q8×Dic61—22no—
83Q8×Dic61C2×C414yescomplemented
84C4×Dic122—22no—
85C4×Dic122—24no—
86C4×Dic122—24no—
87C4×Dic122C2×C414yescomplemented
88#14519742—22no—
89#14519742—24no—
90#14519742—24no—
91#14519742C2×C414yescomplemented
92#14519736—22no—
93#14519736C2×C411yescharacteristic
94#14519736C2×C411yescharacteristic
95#14519735—24no—
96#14519735C2×C412yes—
97#14519735D412yes—
98#14519734—21no—
99#14519734—24no—
100#14519734C2×C412yes—
101#14519733—21no—
102#14519733—22no—
103#14519733—22no—
104#14519733D412yes—
105C4⋊C4×D61—24no—
106C4⋊C4×D61D412yes—
107C4⋊C4×D61C2312yes—
108C22⋊C4×D61—21no—
109C22⋊C4×D61D412yes—
110C42×D61—24no—
111C42×D61—21no—
112C42×D61C2312yes—
113C22×Q8×C61C2×C411yescharacteristic
114C4×Q8×C61—22no—
115C4×Q8×C61C2×C414yescomplemented
116#117×C61—22no—
117#117×C61C2×C411yescharacteristic
118#117×C61C2×C411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C22×#6923—22no—
121C22×#6923—22no—
122C22×#6923—24no—
123C22×#6923—82no—
124C2×Q8×C61—21no—
125C2×Q8×C61—22no—
126C2×Q8×C61C4214yescomplemented
127C2×Q8×C61C22×C412yes—
128C22×C4×C61—21no—
129C22×C4×C61C22×C411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132Q8×D61—24no—
133Q8×D61—24no—
134Q8×D61—24no—
135Q8×D61—24no—
136Q8×D61—24no—
137Q8×D61—22no—
138Q8×D61—22no—
139Q8×D61C4218yescomplemented
140Q8×D61C22×C414yes—
141Q8×D61C22×C414yes—
142C2×C4×D61—22no—
143C2×C4×D61—24no—
144C2×C4×D61—24no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C22×C412yes—
149C2×C4×D61C22×C412yes—
150C2×C4×D61C2×D414yes—
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61#4112yes—
154C2×C4×D61#4112yes—
155C2×C4×D61C2411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—24no—
160C2×Dic122—24no—
161C2×Dic122—22no—
162C2×Dic122C4214yescomplemented
163C2×Dic122C22×C412yes—
164C22⋊#6923—24no—
165#11758217—24no—
166C2×#92420—24no—
167C2×#92420—21no—
168C2×#92420—22no—
169C2×#92420—21no—
170C4⋊#6923—24no—
171C4⋊#6923—22no—
172C4⋊#6923—22no—
173C4⋊#6923C22×C414yes—
174C4⋊#6923C22×C414yes—
175C4×#6923—24no—
176C4×#6923—84no—
177#11758213—24no—
178D61⋊C8—24no—
179D61⋊C8—21no—
180D61⋊C8—21no—
181D61⋊C8—22no—
182D61⋊C8C22×C412yes—
183D61⋊C8C22×C412yes—
184D61⋊C8C22×C414yes—
185C4⋊C4×C61—24no—
186C4⋊C4×C61C22⋊C412yes—
187C4⋊C4×C61C22×C412yes—
188C22⋊C4×C61—21no—
189C22⋊C4×C61C22⋊C412yes—
190C42×C61—24no—
191C42×C61—21no—
192C42×C61C22×C412yes—
193C22⋊Dic61—21no—
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198C22⋊C414yes—
197C4⋊Dic61—24no—
198C4⋊Dic61C22⋊C412yes—
199C4⋊Dic61C22×C412yes—
200#11758196—24no—
201#11758196—24no—
202#11758196—22no—
203#11758196C22⋊C414yes—
204C4×Dic61—24no—
205C4×Dic61—21no—
206C4×Dic61—24no—
207C4×Dic61—22no—
208C4×Dic61—24no—
209C4×Dic61C22×C412yes—
210C4×Dic61C2×D414yes—
211C23×C61C2×C22⋊C411yescharacteristic
212C22×D61—22no—
213C22×D61—22no—
214C22×D61C2×C22⋊C411yescharacteristic
215C22×D61C4×D412yes—
216C22×D61C4×D412yes—
217C22×D61C22×D411yescharacteristic
218C2×#6923—24no—
219C2×#6923—24no—
220C2×#6923—42no—
221C2×#6923—44no—
222C2×#6923—42no—
223C2×#6923—84no—
224Q8×C61—22no—
225Q8×C61—24no—
226Q8×C61C2×C4214yes—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61C2×C4212yes—
231C2×C4×C61C2×C22⋊C414yes—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61#11712yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61C2×C22⋊C414yes—
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C4×D412yes—
241C2×Dic61C4×D412yes—
242C2×Dic61C22×D411yescharacteristic
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—28no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61C2×C4212yes—
251C4×D61#11712yes—
252C4×D61C4×D412yes—
253C4×D61C4×D412yes—
254C4×D61C4×D412yes—
255C4×D61C4×D412yes—
256C4×D61C23×C411yescharacteristic
257C4×D61C23×C411yescharacteristic
258C4×D61C2×#4111yescharacteristic
259C4×D61C2×#4111yescharacteristic
260Dic122—22no—
261Dic122—24no—
262Dic122—24no—
263Dic122—24no—
264Dic122—28no—
265Dic122C2×C4214yes—
266#92420—22no—
267#92420—22no—
268#92420—24no—
269#20417—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61C4×C22⋊C412yes—
272C22×C61C22×C22⋊C411yescharacteristic
273C2×D61—22no—
274C2×D61—24no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61C4×C22⋊C412yes—
278C2×D61C2×C4×D411yescharacteristic
279C2×D61C2×C4×D411yescharacteristic
280C2×D61#57712yes—
281C2×D61#57812yes—
282#6923—44no—
283#6923—44no—
284C4×C61—24no—
285C4×C61—24no—
286C4×C61—42no—
287C4×C61C4×C22⋊C412yes—
288C4×C61C22×C4211yescharacteristic
289C4×C61C2×#11711yescharacteristic
290Dic61—24no—
291Dic61—24no—
292Dic61—42no—
293Dic61C4×C22⋊C412yes—
294Dic61C2×C4×D411yescharacteristic
295Dic61C2×C4×D411yescharacteristic
296C2×#519—611no—
297C2×#419—611no—
298C22×#103—611no—
299C2×#405—611no—
300#2274—611no—
301#1763—614no—
302C4×#103—614no—
303#1736—611no—
304#1735—611no—
305C2×C61—22no—
306C2×C61C2×C4×C22⋊C411yescharacteristic
307C2×C61#289812yes—
308D61—22no—
309D61#294712yes—
310D61#294911yescharacteristic
311D61#295011yescharacteristic
312C23×Q8—611no—
313C22×M16—611no—
314C23×C8—611no—
315#519—611no—
316#519—614no—
317#519—611no—
318#519—611no—
319#519—1222no—
320#519—1222no—
321#519#692311yescharacteristic, complemented
322C2×C4×Q8—614no—
323C2×C4×Q8—1222no—
324C2×#117—611no—
325C2×#117—611no—
326C2×#117—611no—
327C2×#117—1222no—
328C22×C4⋊C4—611no—
329#419—618no—
330C2×#103—618no—
331C2×#103—612no—
332C2×#103—612no—
333C2×#103—614no—
334#405—1222no—
335#405—1222no—
336C2×#97—612no—
337C2×C4×C8—612no—
338C4×C4⋊C4—614no—
339C4×C4⋊C4—1222no—
340C4×C22⋊C4—1222no—
341#340—612no—
342#340—612no—
343#340—614no—
344C61#2041711yescharacteristic, complemented, Sylow
345C22×Q8—614no—
346C22×Q8—614no—
347C22×Q8—612no—
348C22×Q8—612no—
349C22×Q8—611no—
350C22×Q8—1221no—
351C22×Q8—1221no—
352C22×Q8—1221no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8C2×#692311yescharacteristic
356C23×C4—611no—
357C23×C4—1221no—
358C23×C4—1221no—
359C2×M16—614no—
360C2×M16—1221no—
361C2×M16—1222no—
362C2×M16—1221no—
363C22×C8—614no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—612no—
367C22×C8—1221no—
368C22×C8—1222no—
369C22×C8—1221no—
370C4×Q8—22no—
371C4×Q8—614no—
372C4×Q8—614no—
373C4×Q8—614no—
374C4×Q8—1224no—
375C4×Q8—1224no—
376C4×Q8—1222no—
377C4×Q8—1224no—
378C4×Q8—1224no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×Q8C2×#692314yescomplemented
382#117—22no—
383#117—611no—
384#117—612no—
385#117—612no—
386#117—612no—
387#117—614no—
388#117—611no—
389#117—612no—
390#117—1224no—
391#117—1221no—
392#117—1222no—
393#117—1221no—
394#117—1224no—
395#117—1222no—
396#117—1222no—
397#117—1222no—
398#117C2×#692311yescharacteristic
399#117C2×#692311yescharacteristic
400C2×C4⋊C4—612no—
401C2×C4⋊C4—614no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—612no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—1224no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1221no—
408C2×C22⋊C4—612no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—612no—
412C2×C42—1221no—
413C2×C42—1224no—
414C2×C42—1221no—
415C2×C42—1224no—
416#103—618no—
417#103—618no—
418#103—6116no—
419#97—1224no—
420C4×C8—1224no—
421#95—1224no—
422C24—611no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—614no—
426C2×Q8—614no—
427C2×Q8—618no—
428C2×Q8—614no—
429C2×Q8—612no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—1224no—
433C2×Q8—1224no—
434C2×Q8—1224no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×Q8—1224no—
438C2×Q8—1224no—
439C2×Q8—1224no—
440C2×Q8—1224no—
441C2×Q8—1221no—
442C2×Q8—1222no—
443C2×Q8C4×#692314yescomplemented
444C2×Q8C22×#692312yes—
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—614no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1224no—
459C22×C4—1224no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1224no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—4882no—
466C22×C4C22×#692311yescharacteristic
467M16—1224no—
468M16—1224no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—614no—
472C2×C8—1221no—
473C2×C8—1222no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1224no—
478C2×C8—1224no—
479C2×C8—1222no—
480C4⋊C4—24no—
481C4⋊C4—614no—
482C4⋊C4—612no—
483C4⋊C4—614no—
484C4⋊C4—614no—
485C4⋊C4—612no—
486C4⋊C4—1222no—
487C4⋊C4—1224no—
488C4⋊C4—1224no—
489C4⋊C4—1222no—
490C4⋊C4—1224no—
491C4⋊C4—1222no—
492C4⋊C4—1224no—
493C4⋊C4C22⋊#692312yes—
494C4⋊C4C22×#692312yes—
495C22⋊C4—21no—
496C22⋊C4—612no—
497C22⋊C4—614no—
498C22⋊C4—1222no—
499C22⋊C4—1224no—
500C22⋊C4—1221no—
501C22⋊C4C22⋊#692312yes—
502C42—24no—
503C42—21no—
504C42—614no—
505C42—612no—
506C42—1222no—
507C42—1224no—
508C42—1224no—
509C42—1224no—
510C42—1221no—
511C42—1224no—
512C42—4884no—
513C42C22×#692312yes—
514C23—611no—
515C23—611no—
516C23—612no—
517C23—612no—
518C23—1222no—
519C23—1222no—
520C23C2×C22⋊#692311yescharacteristic
521Q8—22no—
522Q8—24no—
523Q8—614no—
524Q8—1224no—
525Q8—1224no—
526Q8—1228no—
527Q8—1222no—
528Q8—1224no—
529Q8C2×C4×#692314yes—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—612no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—614no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1228no—
553C2×C4—1222no—
554C2×C4—1224no—
555C2×C4—1224no—
556C2×C4—1224no—
557C2×C4—1222no—
558C2×C4—4884no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2444no—
563C2×C4—2442no—
564C2×C4C2×C4×#692312yes—
565C2×C4C2×C22⋊#692314yes—
566C2×C4C2×C22⋊#692311yescharacteristic
567C2×C4#1451985612yes—
568C2×C4C23×#692311yescharacteristic
569C8—1222no—
570C8—1222no—
571C8—1224no—
572C22—22no—
573C22—612no—
574C22—612no—
575C22—611no—
576C22—611no—
577C22—612no—
578C22—1222no—
579C22—1224no—
580C22—1222no—
581C22—1222no—
582C22C4×C22⋊#692312yes—
583C22C22×C22⋊#692311yescharacteristic
584C4—24no—
585C4—24no—
586C4—42no—
587C4—611no—
588C4—611no—
589C4—612no—
590C4—1224no—
591C4—1224no—
592C4—2444no—
593C4—2444no—
594C4—2442no—
595C4C4×C22⋊#692312yes—
596C4C22×C4×#692311yescharacteristic
597C4C2×#1451985611yescharacteristic
598C2—22no—
599C2—611no—
600C2—612no—
601C2—611no—
602C2—1222no—
603C2C2×C4×C22⋊#692311yescharacteristic
604C2#2146244112yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC4×C61C22×C42
FrattiniC2×C4C23×#6923
FittingC61×#519C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#519#6923
hypocenterC61#20417
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
5. #21461591C2semidirect
6. C4×#14519840C2semidirect
14. #16840755C4semidirect
15. C2×#14519840C4semidirect
30. #16840240C4semidirect
32. #16840239C4semidirect
33. #16840239C4semidirect
35. #16840238C4semidirect
41. C61×#519C4semidirect
66. #14519840C2×C4semidirect
83. Q8×Dic61C2×C4semidirect
87. C4×Dic122C2×C4semidirect
91. #14519742C2×C4semidirect
115. C4×Q8×C61C2×C4semidirect
126. C2×Q8×C61C42semidirect
139. Q8×D61C42semidirect
162. C2×Dic122C42semidirect
321. #519#6923semidirect
344. C61#20417semidirect
381. C4×Q8C2×#6923semidirect
443. C2×Q8C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407552
C22×#145198402
#214623792
#214622562
#214615912
C4×#145198402
#214615082
#214615042
#519×D612
#2041761