Finite Group Databasebeta

← #30616892

Subgroups — #30616892

Order 15616.

1342 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519864C211yesmaximal, characteristic, complemented
2C2×#16840807C212yesmaximal, complemented
3#21462535C211yesmaximal, characteristic, complemented
4#21461808C212yesmaximal, complemented
5#21461800C211yesmaximal, characteristic, complemented
6#21461786C212yesmaximal, complemented
7#21461771C211yesmaximal, characteristic, complemented
8#21461660C211yesmaximal, characteristic, complemented
9#21461598C212yesmaximal, complemented
10#21461558C211yesmaximal, characteristic, complemented
11#21455038C211yesmaximal, characteristic
12C2×#14519865C2211yescharacteristic, complemented
13C2×#14519864—28no—
14C2×#14519864—21no—
15C2×#14519864—21no—
16C2×#14519864C2212yes—
17C2×#14519864C2212yes—
18C22×#11758213C2211yescharacteristic
19C22×#11758217—21no—
20C22×D61⋊C8C2211yescharacteristic
21C23×#92420—21no—
22#16840807—28no—
23#16840792—21no—
24#16840791—21no—
25#16840790—21no—
26#16840789—21no—
27C2×#14519848C2212yescomplemented
28#16840769C2211yescharacteristic, complemented
29C2×#14519842C2212yescomplemented
30C2×#14519841C2212yes—
31C2×#14519841C2212yes—
32#16840737C2211yescharacteristic, complemented
33#16840733C2211yescharacteristic
34#16840727C2211yescharacteristic, complemented
35C2×#14519831C2212yescomplemented
36C4×C22⋊#6923C2211yescharacteristic
37#16840638C2212yescomplemented
38#16840632C2212yescomplemented
39(C2×#6923)⋊C8C2212yescomplemented
40C22×#11758224C2211yescharacteristic
41C2×#14519798C2212yes—
42#16840275C2211yescharacteristic
43#16839921C412yescomplemented
44#16839917C411yescharacteristic, complemented
45#16839908C412yescomplemented
46#16839903C411yescharacteristic, complemented
47#16839896C411yescharacteristic, complemented
48#16839755C2212yes—
49C22⋊C8×D61C2211yescharacteristic
50C61×#450C411yescharacteristic, complemented
51#14519865—24no—
52#14519864—28no—
53#14519864—28no—
54#14519864—24no—
55#14519864—24no—
56#14519864—24no—
57#14519864D418yescomplemented
58C2×#11758213—21no—
59C2×#11758213—24no—
60C2×#11758213—21no—
61C2×#11758213C2311yescharacteristic
62C2×#11758213C2311yescharacteristic
63C2×#11758213C2312yes—
64C2×#11758217—28no—
65C2×#11758217—22no—
66C2×#11758217—22no—
67C2×D61⋊C8—24no—
68C2×D61⋊C8—21no—
69C2×D61⋊C8D411yescharacteristic
70C2×D61⋊C8D411yescharacteristic
71C2×D61⋊C8C2311yescharacteristic
72C2×D61⋊C8C2311yescharacteristic
73C22×#92420—28no—
74C22×#92420—21no—
75C22×#92420—22no—
76C22×#92420—22no—
77C22×#92420D411yescharacteristic
78C22×#92420D411yescharacteristic
79C2×C22⋊#6923—21no—
80C2×C4×#6923C2312yes—
81#14519848—24no—
82#14519844—21no—
83C4⋊#92420—21no—
84#14519842—24no—
85#14519841—24no—
86#14519841—24no—
87#14519834—21no—
88#14519834—21no—
89#14519834—21no—
90C22⋊#92420—21no—
91#14519832—21no—
92#14519831—24no—
93C4×#92420—21no—
94(C2×#6923)⋊C4—21no—
95C2×#11758226C2311yescharacteristic
96C2×#11758224—24no—
97C2×#11758224—24no—
98C2×#11758224—21no—
99C2×#11758224—21no—
100C2×#11758224C2×C412yes—
101C2×#11758224C2312yes—
102C22×Dic122C2×C411yescharacteristic
103C22×C22⋊D61—21no—
104C23×Dic61—21no—
105C22×C4×D61C2311yescharacteristic
106#14519798—24no—
107#14519798—24no—
108C2×#11758202C2×C412yes—
109C2×#11758190C2×C412yes—
110C2×#11758199C2×C412yes—
111C2×C8×D61C2312yes—
112#14519762—21no—
113#14519753C2×C411yescharacteristic
114#14519752C2×C411yescharacteristic
115#14519751C2×C411yescharacteristic
116#14519747—21no—
117#14519739—21no—
118#14519737—21no—
119#14519735C2311yescharacteristic
120#14519702C2×C412yes—
121#14519700C2×C412yes—
122#14519698C2×C412yes—
123C22⋊#92418C2×C411yescharacteristic
124#14519689C2×C412yes—
125C22×D4×C61C2×C411yescharacteristic
126C2×SD16×C61C2×C412yes—
127C22⋊Q8×C61C2×C411yescharacteristic
128#102×C61C2×C412yes—
129C22⋊C8×C61C2×C411yescharacteristic
130C24×C61—21no—
131C23×D61#4111yescharacteristic
132C22×#6923—22no—
133C2×Q8×C61C22×C411yescharacteristic
134C2×D4×C61—24no—
135C2×D4×C61—21no—
136C2×D4×C61C22×C412yes—
137C22×C4×C61C22×C411yescharacteristic
138C2×C22⋊D61—28no—
139C2×C22⋊D61—22no—
140C2×C22⋊D61—22no—
141C2×C22⋊D61—21no—
142C22×Dic61—24no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61C2×D411yescharacteristic
148#11758226—22no—
149#11758226—22no—
150#11758224—28no—
151#11758224—24no—
152#11758224—24no—
153#11758224—24no—
154#11758224—22no—
155#11758224—22no—
156#11758224—24no—
157#11758224C2×D414yes—
158#11758224#4114yes—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C22×C411yescharacteristic
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—21no—
165C2×C4×D61C22×C411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61#4111yescharacteristic
168C2×C4×D61C2411yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—24no—
171C2×Dic122C22×C412yes—
172C22⋊#6923—42no—
173#11758217—28no—
174#11758217—28no—
175C2×#92420—24no—
176C2×#92420—28no—
177C2×#92420—22no—
178C2×#92420—22no—
179C2×#92420—28no—
180C2×#92420—22no—
181C2×#92420—21no—
182C2×#92420—21no—
183C2×#92420C2×D411yescharacteristic
184C2×#92420C2×D411yescharacteristic
185C2×#92420C2×D411yescharacteristic
186C2×#92420C2×D411yescharacteristic
187C4×#6923—22no—
188C4×#6923—22no—
189#11758213—24no—
190#11758213—22no—
191#11758213—22no—
192#11758213—22no—
193#11758213—42no—
194#11758213C2×D414yes—
195D61⋊C8—22no—
196D61⋊C8—22no—
197D61⋊C8—22no—
198D61⋊C8—42no—
199D61⋊C8C2×D414yes—
200SD16×C61—24no—
201C2×C8×C61C22×C412yes—
202C4⋊C4×C61—21no—
203C4⋊C4×C61C22×C411yescharacteristic
204C22⋊C4×C61—21no—
205#11758202—24no—
206#11758199—24no—
207#11758198—21no—
208#11758198—21no—
209#11758198—21no—
210#11758198—21no—
211C4⋊Dic61C22×C411yescharacteristic
212#11758196—21no—
213C4×Dic61—21no—
214C2×#92418C22×C412yes—
215#11758190—24no—
216C8×D61—22no—
217C8×D61—22no—
218C23×C61—24no—
219C23×C61—22no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61C4×D411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—21no—
226C22×D61—41no—
227C22×D61C4×D411yescharacteristic
228C22×D61C2×#4111yescharacteristic
229C2×#6923—42no—
230C2×#6923—42no—
231C2×#6923—42no—
232Q8×C61—22no—
233D4×C61—24no—
234D4×C61—22no—
235D4×C61C4×D414yes—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61C4×D411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C22⋊D61—28no—
242C22⋊D61—24no—
243C22⋊D61—24no—
244C22⋊D61—42no—
245C2×Dic61—24no—
246C2×Dic61—28no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—24no—
250C2×Dic61—21no—
251C2×Dic61—24no—
252C2×Dic61—21no—
253C2×Dic61C4×D411yescharacteristic
254C2×Dic61C22×D411yescharacteristic
255D244—22no—
256D244—42no—
257D244—42no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—21no—
261C4×D61—21no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61C22×D412yes—
265C4×D61C2×#4112yes—
266Dic122—24no—
267Dic122—22no—
268Dic122C4×D414yes—
269#92420—21no—
270#92420—21no—
271#92420—24no—
272#92420—41no—
273#92420—41no—
274#92420#12012yes—
275#92420#12012yes—
276C8×C61—22no—
277#92418—22no—
278#21467—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—24no—
282C22×C61—48no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61#44412yescomplemented
286C22×C61C2×C4×D411yescharacteristic
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—44no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61#52512yes—
297#6923—82no—
298C4×C61—21no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61C2×C4×D412yes—
302Dic61—24no—
303Dic61—21no—
304Dic61—41no—
305Dic61C2×#12011yescharacteristic
306Dic61#52511yescharacteristic
307C22×#131—611no—
308#3002—611no—
309C2×#444—612no—
310#2929—611no—
311#1876—612no—
312#1871—611no—
313#1866—612no—
314#1855—611no—
315#1789—611no—
316#1765—612no—
317#1752—611no—
318C2×C61—22no—
319C2×C61—44no—
320C2×C61C4×#12011yescharacteristic
321C2×C61C2×#44411yescharacteristic
322C2×C61#295511yescharacteristic
323D61—22no—
324D61—41no—
325D61—81no—
326C22×#41—611no—
327C2×#135—612no—
328C2×#131—612no—
329C2×#131—612no—
330C2×#131—611no—
331C2×#131—1221no—
332C2×#131—1221no—
333C2×#131—1228no—
334C22×M16—611no—
335C22×M16—1221no—
336C23×C8—611no—
337C23×C8—1221no—
338#525—611no—
339#452—612no—
340#451—611no—
341#450#692311yescharacteristic, complemented
342#449—612no—
343#448—611no—
344#447—611no—
345#444—1228no—
346#435—1221no—
347#435—1221no—
348C8×D4—1221no—
349C8×D4—1221no—
350C2×#108—612no—
351#424—611no—
352C2×#104—612no—
353C2×#104—612no—
354C2×#104—612no—
355#416—612no—
356#408—611no—
357#407—611no—
358C2×C22⋊C8—611no—
359#405—611no—
360C2×#97—612no—
361C4×C22⋊C4—611no—
362#341—612no—
363#339—612no—
364#336—612no—
365C61#2146711yescharacteristic, complemented, Sylow
366C2×#41—611no—
367C2×#41—612no—
368C2×#41—612no—
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41—1224no—
372C2×#41—1224no—
373C22×Q8—611no—
374C22×D4—1221no—
375C22×D4C2×#692311yescharacteristic
376C23×C4—611no—
377C23×C4—1221no—
378#135—1224no—
379#135—1224no—
380C2×Q16—612no—
381C2×SD16—612no—
382C2×SD16C2×#692312yes—
383C2×D8—612no—
384#131—618no—
385#131—1224no—
386#131—1224no—
387#131—1224no—
388#131—1224no—
389#131—1228no—
390#131—1228no—
391C2×M16—612no—
392C2×M16—611no—
393C2×M16—611no—
394C2×M16—1222no—
395C2×M16—1222no—
396C2×M16—1221no—
397C2×M16—1224no—
398C2×M16—1221no—
399C2×M16—1228no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8—612no—
407C22×C8—1222no—
408C22×C8—1222no—
409C22×C8—1224no—
410C22×C8—1221no—
411C22×C8—1221no—
412C22×C8—1228no—
413#123—1221no—
414C22⋊Q8—611no—
415C22⋊Q8C2×#692311yescharacteristic
416#121—611no—
417#121—611no—
418#120—1221no—
419C4×D4—1221no—
420C4×D4—1221no—
421#117—611no—
422C2×C22⋊C4—1221no—
423C2×C42—612no—
424#108—1224no—
425C4⋊C8—1221no—
426C4⋊C8—1221no—
427#104—1224no—
428#104—1224no—
429#104—1224no—
430#103—612no—
431#103—612no—
432#102—612no—
433#102C2×#692312yes—
434C22⋊C8—611no—
435C22⋊C8—612no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440C22⋊C8C2×#692311yescharacteristic
441#97—1224no—
442#97—1221no—
443C4×C8—1221no—
444#95—1221no—
445C24—21no—
446C24—611no—
447#41—614no—
448#41—614no—
449#41—1222no—
450#41—1222no—
451#41—1222no—
452#41—1222no—
453#41—1224no—
454#41—1224no—
455#41—1224no—
456#41—1228no—
457#41—1224no—
458C2×Q8—612no—
459C2×Q8—1224no—
460C2×Q8—1221no—
461C2×Q8C22×#692311yescharacteristic
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—611no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1228no—
470C2×D4—1221no—
471C2×D4C22×#692312yes—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4C22×#692311yescharacteristic
487Q16—1224no—
488SD16—24no—
489SD16—1224no—
490D8—1224no—
491M16—614no—
492M16—1228no—
493M16—1222no—
494M16—1224no—
495M16—1228no—
496M16—1222no—
497M16—1222no—
498M16—2442no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—614no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—612no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—1228no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—1224no—
515C2×C8—1228no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—2442no—
519C2×C8C22×#692312yes—
520C4⋊C4—21no—
521C4⋊C4—611no—
522C4⋊C4—1221no—
523C4⋊C4C22×#692311yescharacteristic
524C22⋊C4—21no—
525C22⋊C4—1221no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C22⋊C4—1221no—
529C22⋊C4—2442no—
530C42—1221no—
531C42—1222no—
532C42—1222no—
533C23—24no—
534C23—22no—
535C23—42no—
536C23—42no—
537C23—611no—
538C23—611no—
539C23—1222no—
540C23—1222no—
541C23—1221no—
542C23—2441no—
543C23D4×#692311yescharacteristic
544Q8—22no—
545Q8—614no—
546Q8—1224no—
547Q8—1222no—
548D4—24no—
549D4—22no—
550D4—1224no—
551D4—1224no—
552D4—1228no—
553D4—1222no—
554D4—2442no—
555D4—2442no—
556D4—2442no—
557D4D4×#692314yes—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—611no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—611no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1221no—
569C2×C4—1224no—
570C2×C4—1221no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1228no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4D4×#692311yescharacteristic
583C2×C4C23×#692311yescharacteristic
584C8—22no—
585C8—612no—
586C8—612no—
587C8—1221no—
588C8—1221no—
589C8—1224no—
590C8—1222no—
591C8—2441no—
592C8—2441no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—48no—
597C22—42no—
598C22—42no—
599C22—612no—
600C22—1221no—
601C22—1221no—
602C22—1221no—
603C22—1221no—
604C22—1221no—
605C22—2444no—
606C22—2441no—
607C22—2441no—
608C22—2441no—
609C22#1684080712yescomplemented
610C22C2×D4×#692311yescharacteristic
611C4—21no—
612C4—41no—
613C4—41no—
614C4—611no—
615C4—611no—
616C4—1221no—
617C4—1224no—
618C4—4882no—
619C4—2441no—
620C4C2×D4×#692312yes—
621C2—22no—
622C2—44no—
623C2—1222no—
624C2—4881no—
625C2—2441no—
626C2#120×#692311yescharacteristic
627C2C2×#1684080711yescharacteristic
628C2#2146259111yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#450C4
socleC22×C61C2×C4×D4
prime radical#21455038C2
cosocleC2×C4×D61C24
hypercenter#450#6923
hypocenterC61#21467
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14519864C2semidirect
2. C2×#16840807C2semidirect
3. #21462535C2semidirect
4. #21461808C2semidirect
5. #21461800C2semidirect
6. #21461786C2semidirect
7. #21461771C2semidirect
8. #21461660C2semidirect
9. #21461598C2semidirect
10. #21461558C2semidirect
12. C2×#14519865C22semidirect
27. C2×#14519848C22semidirect
28. #16840769C22semidirect
29. C2×#14519842C22semidirect
32. #16840737C22semidirect
34. #16840727C22semidirect
35. C2×#14519831C22semidirect
37. #16840638C22semidirect
38. #16840632C22semidirect
39. (C2×#6923)⋊C8C22semidirect
43. #16839921C4semidirect
44. #16839917C4semidirect
45. #16839908C4semidirect
46. #16839903C4semidirect
47. #16839896C4semidirect
50. C61×#450C4semidirect
57. #14519864D4semidirect
285. C22×C61#444semidirect
341. #450#6923semidirect
365. C61#21467semidirect
609. C22#16840807semidirect

Maximal subgroups

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

subgroupindex
C22×#145198642
C2×#168408072
#214625352
#214618082
#214618002
#214617862
#214617712
#214616602
#214615982
#214615582
#214550382
#2146761