Finite Group Databasebeta

← #30610803

Subgroups — #30610803

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462631C211yesmaximal, characteristic, complemented
2#21462464C211yesmaximal, characteristic, complemented
3#21461174C212yesmaximal, complemented
4#21461169C211yesmaximal, characteristic, complemented
5#21461156C212yesmaximal, complemented
6#21461145C211yesmaximal, characteristic, complemented
7#21461134C212yesmaximal, complemented
8#21461129C211yesmaximal, characteristic, complemented
9#21461116C212yesmaximal, complemented
10D61⋊(D4⋊C8)C211yesmaximal, characteristic, complemented
11#21459376C211yesmaximal, characteristic
12C2×#14519864—21no—
13#16840812—21no—
14#16840797—21no—
15#16840794—22no—
16#16840781C2211yescharacteristic
17#16840778C2212yescomplemented
18#16840769—21no—
19#16840768—22no—
20#16840766C2211yescharacteristic
21#16840765C2212yescomplemented
22D61⋊(C4⋊C8)C2211yescharacteristic
23C2×C4⋊#92420C2212yescomplemented
24C4×#11758217C2212yescomplemented
25C4×D61⋊C8C2211yescharacteristic
26#16840612C2212yescomplemented
27#16840612C2212yescomplemented
28#16840612C2212yescomplemented
29#16840612C2212yescomplemented
30#16840610C2211yescharacteristic
31#16840610C2211yescharacteristic
32#16840610C2212yes—
33D4⋊#92420C2211yescharacteristic
34D4⋊#92420C2211yescharacteristic
35D4⋊#92420C2212yes—
36#16840482C411yescharacteristic
37#16840473C411yescharacteristic
38#16840459—21no—
39#16840451—21no—
40#16840451C412yes—
41#16840442—22no—
42#16840417C411yescharacteristic
43#16840409C412yes—
44#16840317C2211yescharacteristic
45#16840293C2212yes—
46#16840291—22no—
47#16840272—21no—
48#16840230C2212yes—
49C4×#11758224C2211yescharacteristic
50C4×D4×D61C2211yescharacteristic
51#117×D61—21no—
52C61×#555C411yescharacteristic
53#14519864—44no—
54C2×#11758213—21no—
55C2×#11758217—22no—
56C2×#11758217—22no—
57C2×D61⋊C8—21no—
58C2×D61⋊C8—21no—
59C22×#92420—22no—
60#14519844—21no—
61#14519844—22no—
62C4⋊#92420—22no—
63C4⋊#92420—21no—
64C4⋊#92420—22no—
65C4⋊#92420—21no—
66C4⋊#92420C2311yescharacteristic
67C4⋊#92420C2312yes—
68C4⋊#92420C2311yescharacteristic
69#14519834—21no—
70C22⋊#92420—22no—
71#14519832—22no—
72C4×#92420—21no—
73C4×#92420C2311yescharacteristic
74C4×#92420C2311yescharacteristic
75C4×#92420C2312yes—
76C2×#11758222C2×C412yes—
77C2×#11758224D411yescharacteristic
78C2×D4×D61C2×C411yescharacteristic
79C22×C4×D61—21no—
80#14519777C2×C411yescharacteristic
81#14519770—22no—
82#14519770—22no—
83#14519770—22no—
84#14519769—22no—
85#14519768—21no—
86#14519768C2×C412yes—
87#14519764C2×C412yes—
88#14519759—22no—
89#14519759—22no—
90#14519759—22no—
91#14519758—22no—
92#14519757—21no—
93#14519757C2×C412yes—
94#14519756—21no—
95#14519756C2×C412yes—
96#14519753—21no—
97#14519753—22no—
98#14519748—21no—
99#14519748—22no—
100#14519745—22no—
101Q8×Dic61—21no—
102C4×Dic122C2×C411yescharacteristic
103#14519742C2×C412yes—
104#14519742D411yescharacteristic
105#14519742C2312yes—
106C4×D244C2×C411yescharacteristic
107D4×Dic61—22no—
108D4×Dic61D411yescharacteristic
109D4×Dic61C2311yescharacteristic
110#14519739C2×C412yes—
111#14519739C2311yescharacteristic
112C4×C22⋊D61—21no—
113C4×C22⋊D61C2×C412yes—
114#14519737—22no—
115#14519737—22no—
116#14519737—21no—
117#14519737D412yes—
118#14519736—21no—
119#14519735—21no—
120#14519735D411yescharacteristic
121#14519734—22no—
122#14519734—22no—
123#14519733—22no—
124#14519733D412yes—
125C4⋊C4×D61—21no—
126C4⋊C4×D61—21no—
127C4⋊C4×D61C2×C411yescharacteristic
128C22⋊C4×D61—21no—
129C22⋊C4×D61—22no—
130C2×C4×Dic61C2312yes—
131C42×D61—22no—
132C42×D61C2311yescharacteristic
133C4⋊Q8×C61C2×C411yescharacteristic
134#124×C61—22no—
135#121×C61—21no—
136#121×C61C2×C412yes—
137C4×D4×C61C2×C411yescharacteristic
138C4×D4×C61C2×C411yescharacteristic
139#117×C61—21no—
140#117×C61C2×C412yes—
141C23×D61—21no—
142C2×Q8×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61C22⋊C411yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61C22×C412yes—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61C22⋊C412yes—
151C22×Dic61—21no—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154#11758224—22no—
155#11758224—22no—
156D4×D61—22no—
157D4×D61—41no—
158#11758222—24no—
159#11758222—42no—
160C2×D244C22×C411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×C4×D61—21no—
165C2×C4×D61—22no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61C22×C411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×Dic122C22⋊C411yescharacteristic
172C2×Dic122C22×C412yes—
173#11758217—44no—
174#11758217—44no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—22no—
178C2×#92420—21no—
179C2×#92420—22no—
180C2×#92420—21no—
181C2×#92420—44no—
182#11758213—42no—
183D61⋊C8—42no—
184D61⋊C8—42no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C22⋊C411yescharacteristic
187C4⋊C4×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61C22⋊C412yes—
193C42×C61—22no—
194C42×C61C22×C411yescharacteristic
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—22no—
198C22⋊Dic61—21no—
199C22⋊Dic61C22⋊C412yes—
200#11758198—22no—
201#11758198—21no—
202#11758198—22no—
203#11758198—21no—
204#11758198C22⋊C412yes—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—22no—
208C4⋊Dic61—22no—
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22×C411yescharacteristic
211#11758196—22no—
212#11758196—22no—
213#11758196—21no—
214#11758196—22no—
215#11758196—22no—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61C22×C412yes—
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D412yes—
223C4×Dic61C2×D412yes—
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2411yescharacteristic
226C23×C61—21no—
227C23×C61C2×C22⋊C412yes—
228C22×D61—21no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—81no—
232C22×D61C2×C22⋊C411yescharacteristic
233Q8×C61—81no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—22no—
237D4×C61—82no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C22⋊D61—24no—
248C22⋊D61—42no—
249C22⋊D61—44no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C22×D411yescharacteristic
261C2×Dic61C22×D411yescharacteristic
262D244—21no—
263D244#13112yescomplemented
264C4×D61—21no—
265C4×D61—21no—
266C4×D61—41no—
267C4×D61—44no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—82no—
273C4×D61AGL(1,ℤ/8)12yes—
274C4×D61#13712yes—
275Dic122—22no—
276Dic122—22no—
277Dic122—22no—
278#92420—22no—
279#92420—41no—
280#92420—42no—
281#92420—41no—
282#92420—41no—
283#92420—42no—
284#17512—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61C22×C22⋊C411yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—22no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—84no—
298C2×D61#41712yes—
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—41no—
302C4×C61—42no—
303C4×C61#41712yes—
304C4×C61C2×#13111yescharacteristic
305C4×C61#56811yescharacteristic
306Dic61—21no—
307Dic61—22no—
308Dic61—21no—
309Dic61—22no—
310Dic61—42no—
311Dic61—41no—
312Dic61C2×AGL(1,ℤ/8)11yescharacteristic
313Dic61C2×#13711yescharacteristic
314#3475—611no—
315#2965—611no—
316#2906—611no—
317#1495—612no—
318#1493—611no—
319#1489—612no—
320#1486—611no—
321#1483—612no—
322#1481—611no—
323#1477—612no—
324C2×D4⋊C8—611no—
325C2×C61—42no—
326C2×C61—81no—
327C2×C61#288011yescharacteristic
328C2×C61C2×#41711yescharacteristic
329C2×C61#290111yescharacteristic
330D61—22no—
331D61—42no—
332C2×#131—1221no—
333#556—611no—
334#555#692311yescharacteristic
335#554—611no—
336#553—1221no—
337#552—612no—
338#552—1221no—
339#551—1222no—
340#547—611no—
341#546—612no—
342#533—611no—
343#529—612no—
344#529—1222no—
345#525—1221no—
346C4×#41—611no—
347C4×#41—612no—
348C2×C4×D4—611no—
349C2×#117—1221no—
350#446—1221no—
351#436—1221no—
352#435—1222no—
353#431—612no—
354#431—611no—
355#424—1222no—
356#424—1221no—
357#423—612no—
358#423—611no—
359C2×C4⋊C8—612no—
360C2×C4⋊C8—611no—
361C4×M16—612no—
362C2×C4×C8—611no—
363Q8⋊C8—612no—
364Q8⋊C8—612no—
365Q8⋊C8—612no—
366Q8⋊C8—612no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—612no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8—612no—
373C61#1751211yescharacteristic, complemented, Sylow
374C2×#41—611no—
375C2×#41—612no—
376C22×D4—611no—
377C23×C4—1221no—
378#131—2444no—
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1221no—
382C22×C8—1222no—
383C22×C8—1221no—
384C22×C8—1221no—
385C4⋊Q8—611no—
386C4⋊Q8C2×#692311yescharacteristic
387#126—1222no—
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1222no—
392#125—1221no—
393#124—22no—
394#124—612no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—612no—
401C22⋊Q8—1221no—
402C22⋊Q8—1222no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405#121—21no—
406#121—1222no—
407#121—1221no—
408#121C2×#692312yes—
409#120—1222no—
410C4×Q8—611no—
411C4×Q8—611no—
412C4×Q8—612no—
413C4×Q8—612no—
414C4×Q8—1221no—
415C4×D4—611no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—611no—
419C4×D4—611no—
420C4×D4—612no—
421C4×D4—611no—
422C4×D4—1221no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4C2×#692311yescharacteristic
428C4×D4C2×#692311yescharacteristic
429#117—21no—
430#117—611no—
431#117—612no—
432#117—1222no—
433#117—1222no—
434#117—1222no—
435#117—1221no—
436#117—1221no—
437#117C2×#692312yes—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1221no—
443C2×C42—612no—
444C2×C42—611no—
445C2×C42—1222no—
446C4⋊C8—611no—
447C4⋊C8—612no—
448C4⋊C8—611no—
449C4⋊C8—1222no—
450C4⋊C8—1221no—
451C4⋊C8—1222no—
452C4⋊C8—1221no—
453C4⋊C8—1221no—
454C4⋊C8—1222no—
455C22⋊C8—1222no—
456C22⋊C8—1221no—
457#97—1222no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8—612no—
461C4×C8—1221no—
462C24—1221no—
463#41—1224no—
464#41—1222no—
465#41—1222no—
466#41—2442no—
467C2×Q8—21no—
468C2×Q8—611no—
469C2×Q8—612no—
470C2×D4—22no—
471C2×D4—612no—
472C2×D4—611no—
473C2×D4—1221no—
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—2441no—
477C2×D4C22⋊#692311yescharacteristic
478C2×D4C22×#692311yescharacteristic
479C22×C4—21no—
480C22×C4—611no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1222no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4C22×#692312yes—
494M16—2444no—
495M16—2442no—
496M16—2444no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1222no—
500C2×C8—1221no—
501C2×C8—1222no—
502C2×C8—1221no—
503C2×C8—2442no—
504C2×C8—2442no—
505C2×C8—2444no—
506C4⋊C4—21no—
507C4⋊C4—611no—
508C4⋊C4—612no—
509C4⋊C4—612no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4C22⋊#692311yescharacteristic
521C4⋊C4C22×#692311yescharacteristic
522C4⋊C4C22×#692312yes—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—21no—
526C22⋊C4—612no—
527C22⋊C4—612no—
528C22⋊C4—1221no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1221no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1221no—
536C22⋊C4C22⋊#692312yes—
537C42—22no—
538C42—611no—
539C42—612no—
540C42—612no—
541C42—611no—
542C42—611no—
543C42—612no—
544C42—1222no—
545C42—1222no—
546C42C22×#692311yescharacteristic
547C23—21no—
548C23—611no—
549C23—1221no—
550C23—4881no—
551C23—2442no—
552C23—2442no—
553C23C2×C22⋊#692312yes—
554Q8—81no—
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558D4—22no—
559D4—21no—
560D4—21no—
561D4—82no—
562D4—612no—
563D4—1224no—
564D4—1221no—
565D4—2442no—
566D4—2444no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—611no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—611no—
577C2×C4—612no—
578C2×C4—611no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1221no—
585C2×C4—4882no—
586C2×C4—2441no—
587C2×C4—2444no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2442no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4C2×C22⋊#692312yes—
596C2×C4C2×C22⋊#692311yescharacteristic
597C2×C4C2×C22⋊#692311yescharacteristic
598C2×C4C23×#692311yescharacteristic
599C8—1222no—
600C8—2441no—
601C8—2441no—
602C8—2442no—
603C8—2441no—
604C8—2442no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—612no—
612C22—1221no—
613C22—1221no—
614C22—1222no—
615C22—4884no—
616C22—2441no—
617C22—2441no—
618C22C22×C22⋊#692311yescharacteristic
619C4—22no—
620C4—21no—
621C4—41no—
622C4—42no—
623C4—611no—
624C4—611no—
625C4—1221no—
626C4—1222no—
627C4—1221no—
628C4—1222no—
629C4—2442no—
630C4—2441no—
631C4#1684075312yes—
632C4D61⋊#13111yescharacteristic
633C4#1684084111yescharacteristic
634C2—42no—
635C2—81no—
636C2—1222no—
637C2—2442no—
638C2#2146240911yescharacteristic
639C2C2×#1684075311yescharacteristic
640C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#555C4
socleC22×C61C22×C22⋊C4
prime radical#21459376C2
cosocleC4×Dic61C24
hypercenter#555#6923
hypocenterC61#17512
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #21462631C2semidirect
2. #21462464C2semidirect
3. #21461174C2semidirect
4. #21461169C2semidirect
5. #21461156C2semidirect
6. #21461145C2semidirect
7. #21461134C2semidirect
8. #21461129C2semidirect
9. #21461116C2semidirect
10. D61⋊(D4⋊C8)C2semidirect
17. #16840778C22semidirect
21. #16840765C22semidirect
23. C2×C4⋊#92420C22semidirect
24. C4×#11758217C22semidirect
26. #16840612C22semidirect
27. #16840612C22semidirect
28. #16840612C22semidirect
29. #16840612C22semidirect
263. D244#131semidirect
373. C61#17512semidirect

Maximal subgroups

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

subgroupindex
#214626312
#214624642
#214611742
#214611692
#214611562
#214611452
#214611342
#214611292
#214611162
D61⋊(D4⋊C8)2
#214593762
#1751261