Finite Group Databasebeta

← #30610778

Subgroups — #30610778

Order 15616.

980 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462624C211yesmaximal, characteristic, complemented
2#21462515C211yesmaximal, characteristic, complemented
3#21461172C212yesmaximal, complemented
4#21461168C211yesmaximal, characteristic, complemented
5#21461150C212yesmaximal, complemented
6#21461145C211yesmaximal, characteristic, complemented
7#21461132C212yesmaximal, complemented
8#21461128C211yesmaximal, characteristic, complemented
9#21461110C212yesmaximal, complemented
10D61⋊(D4⋊C8)C211yesmaximal, characteristic, complemented
11#21459382C211yesmaximal, characteristic
12#16840811—21no—
13#16840789—22no—
14D4×#92420—21no—
15#16840786—22no—
16#16840783C2212yescomplemented
17#16840781C2211yescharacteristic
18#16840779C2212yescomplemented
19#16840778C2212yescomplemented
20#16840766C2211yescharacteristic
21D61⋊(C4⋊C8)C2211yescharacteristic
22#16840736—21no—
23#16840727—21no—
24C4×D61⋊C8C2211yescharacteristic
25C2×C4×#92420C2212yescomplemented
26Q8⋊#92420C2212yescomplemented
27Q8⋊#92420C2212yescomplemented
28Q8⋊#92420C2212yescomplemented
29Q8⋊#92420C2212yescomplemented
30#16840610C2211yescharacteristic
31#16840610C2211yescharacteristic
32#16840610C2212yes—
33D4⋊#92420C2211yescharacteristic
34D4⋊#92420C2211yescharacteristic
35D4⋊#92420C2212yes—
36#16840486C411yescharacteristic
37#16840477C411yescharacteristic
38#16840469C411yescharacteristic
39#16840464—21no—
40#16840447—21no—
41#16840446C412yes—
42#16840435—22no—
43#16840400C412yes—
44#16840317C2211yescharacteristic
45#16840298C2212yes—
46#16840289—22no—
47#16840272—21no—
48#41×Dic61C2212yes—
49C4×#11758224C2211yescharacteristic
50C4×D4×D61C2211yescharacteristic
51C2×#14519735—21no—
52C61×#547C411yescharacteristic
53C2×#11758213—21no—
54C2×D61⋊C8—21no—
55C2×D61⋊C8—21no—
56C22×#92420—22no—
57#14519844—21no—
58#14519844—22no—
59C4⋊#92420—21no—
60C4⋊#92420—21no—
61C4⋊#92420C2311yescharacteristic
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2312yes—
64#14519834—21no—
65C22⋊#92420—22no—
66C22⋊#92420—22no—
67C22⋊#92420—22no—
68#14519832—22no—
69C4×#92420—22no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2312yes—
74C4×#92420C2311yescharacteristic
75C2×#11758224D411yescharacteristic
76C2×D4×D61C2×C411yescharacteristic
77C22×C4×D61—21no—
78#14519777C2×C411yescharacteristic
79#14519772—22no—
80#14519772—22no—
81#14519770—22no—
82#14519768—21no—
83#14519768C2×C412yes—
84#14519767—22no—
85#14519766C2×C412yes—
86#14519763—22no—
87#14519761—22no—
88#14519760—22no—
89#14519759—22no—
90#14519757—21no—
91#14519757—21no—
92#14519756—21no—
93#14519754—22no—
94#14519753C2×C412yes—
95#14519751C2×C412yes—
96#14519749—22no—
97#14519748—21no—
98#14519745—22no—
99Q8×Dic61C2×C412yes—
100Q8×Dic61C2312yes—
101C4×Dic122C2×C411yescharacteristic
102#14519742—21no—
103#14519742D411yescharacteristic
104C4×D244C2×C411yescharacteristic
105D4×Dic61—22no—
106D4×Dic61—22no—
107D4×Dic61C2×C412yes—
108D4×Dic61D411yescharacteristic
109D4×Dic61C2311yescharacteristic
110#14519739C2311yescharacteristic
111C4×C22⋊D61—21no—
112C4×C22⋊D61C2×C412yes—
113#14519737—22no—
114#14519737—21no—
115#14519737D412yes—
116#14519736—22no—
117#14519736C2×C412yes—
118#14519735—21no—
119#14519735—21no—
120#14519735—22no—
121#14519735—21no—
122#14519735—22no—
123#14519735D411yescharacteristic
124#14519733D412yes—
125C2×C4⋊Dic61—21no—
126C4⋊C4×D61C2×C411yescharacteristic
127C2×#11758198—22no—
128C22⋊C4×D61—21no—
129C2×C4×Dic61—22no—
130C2×C4×Dic61C2312yes—
131C42×D61C2311yescharacteristic
132C2×#41×C61C2×C412yes—
133C4⋊Q8×C61C2×C411yescharacteristic
134#123×C61—22no—
135C22⋊Q8×C61C2×C412yes—
136#121×C61—21no—
137C4×D4×C61C2×C411yescharacteristic
138C4×D4×C61C2×C411yescharacteristic
139C2×C4⋊C4×C61—21no—
140C23×D61—21no—
141#41×C61—24no—
142#41×C61—42no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—22no—
145C2×D4×C61C22⋊C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61C22×C412yes—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61C22⋊C412yes—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61C2×D412yes—
156C22×Dic61C2×D412yes—
157#11758224—22no—
158#11758224—22no—
159D4×D61—22no—
160D4×D61—41no—
161C2×D244C22×C411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61C22×C411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×Dic122—21no—
171C2×Dic122C22⋊C411yescharacteristic
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—22no—
176C2×#92420—22no—
177C2×#92420—21no—
178C2×#92420—44no—
179#11758213—42no—
180D61⋊C8—42no—
181D61⋊C8—42no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C22⋊C411yescharacteristic
186C4⋊C4×C61C22×C411yescharacteristic
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61C22⋊C412yes—
190C42×C61C22×C411yescharacteristic
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—21no—
194C22⋊Dic61C22⋊C412yes—
195#11758198—22no—
196#11758198—21no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—21no—
201#11758198C22⋊C412yes—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61C22⋊C411yescharacteristic
207C4⋊Dic61C22×C411yescharacteristic
208C4⋊Dic61C22×C412yes—
209#11758196—21no—
210#11758196—22no—
211#11758196—22no—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2411yescharacteristic
223C23×C61—21no—
224C23×C61C2×C22⋊C412yes—
225C22×D61—21no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—81no—
229C22×D61C2×C22⋊C411yescharacteristic
230Q8×C61—22no—
231Q8×C61—22no—
232D4×C61—21no—
233D4×C61—22no—
234D4×C61—44no—
235D4×C61#13112yes—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—44no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C23×C411yescharacteristic
246C22⋊D61—24no—
247C22⋊D61—42no—
248C22⋊D61—82no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—82no—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C22×D411yescharacteristic
261C2×Dic61C22×D411yescharacteristic
262D244—21no—
263D244—21no—
264C4×D61—21no—
265C4×D61—21no—
266C4×D61—41no—
267C4×D61—42no—
268C4×D61—42no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61#13512yes—
272C4×D61#13512yes—
273Dic122—22no—
274Dic122—81no—
275#92420—22no—
276#92420—41no—
277#92420—42no—
278#92420—41no—
279#92420—41no—
280#92420—42no—
281#17508—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61C22×C22⋊C411yescharacteristic
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—84no—
295C2×D61#41612yes—
296C4×C61—22no—
297C4×C61—21no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61#41612yes—
301C4×C61C2×#13111yescharacteristic
302C4×C61#56811yescharacteristic
303Dic61—21no—
304Dic61—22no—
305Dic61—21no—
306Dic61—22no—
307Dic61—41no—
308Dic61—42no—
309Dic61C2×#13511yescharacteristic
310Dic61C2×#13511yescharacteristic
311#3475—611no—
312C8×#41—611no—
313#2925—611no—
314#1494—612no—
315#1493—611no—
316#1488—612no—
317#1486—611no—
318#1482—612no—
319#1481—611no—
320#1476—612no—
321C2×D4⋊C8—611no—
322C2×C61—42no—
323C2×C61—42no—
324C2×C61#288011yescharacteristic
325C2×C61C2×#41611yescharacteristic
326C2×C61#290111yescharacteristic
327D61—22no—
328D61—81no—
329#556—611no—
330#555—611no—
331#554—611no—
332#553—1221no—
333#552—612no—
334#552—1221no—
335#551—1222no—
336#547#692311yescharacteristic
337#546—612no—
338#533—611no—
339#529—612no—
340#529—1222no—
341#525—1221no—
342C4×#41—611no—
343C4×#41—612no—
344C2×C4×D4—611no—
345C2×#117—1221no—
346C8×Q8—1221no—
347C8×D4—1221no—
348C8×D4—1222no—
349#433—1222no—
350#432—612no—
351#431—612no—
352#431—611no—
353#431—612no—
354#423—611no—
355C2×C4⋊C8—611no—
356#408—1221no—
357#405—1221no—
358C2×C4×C8—612no—
359C2×C4×C8—611no—
360Q8⋊C8—612no—
361Q8⋊C8—612no—
362Q8⋊C8—612no—
363Q8⋊C8—612no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—612no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—612no—
370C61#1750811yescharacteristic, complemented, Sylow
371C2×#41—611no—
372C2×#41C2×#692312yes—
373C22×D4—611no—
374C23×C4—1221no—
375C2×M16—1221no—
376C22×C8—1222no—
377C22×C8—1221no—
378C22×C8—1221no—
379C4⋊Q8—611no—
380C4⋊Q8C2×#692311yescharacteristic
381#126—1222no—
382#126—1222no—
383#126—1222no—
384#125—612no—
385#125—1221no—
386#125—1222no—
387#124—612no—
388#124—1222no—
389#123—22no—
390#123—1222no—
391#123—1222no—
392#123—1222no—
393C22⋊Q8—612no—
394C22⋊Q8—1222no—
395C22⋊Q8—1221no—
396C22⋊Q8—1221no—
397C22⋊Q8—1221no—
398C22⋊Q8C2×#692312yes—
399#121—21no—
400#121—612no—
401#121—1222no—
402#121—1221no—
403#120—1222no—
404C4×Q8—611no—
405C4×Q8—611no—
406C4×Q8—612no—
407C4×Q8—612no—
408C4×Q8—1221no—
409C4×D4—611no—
410C4×D4—612no—
411C4×D4—612no—
412C4×D4—611no—
413C4×D4—612no—
414C4×D4—611no—
415C4×D4—611no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1222no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4C2×#692311yescharacteristic
422C4×D4C2×#692311yescharacteristic
423#117—611no—
424#117—612no—
425#117—612no—
426#117—1222no—
427#117—1222no—
428#117—1222no—
429#117—1221no—
430#117—1221no—
431#117—1221no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1222no—
437C2×C42—612no—
438C2×C42—611no—
439C2×C42—1222no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—612no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8—1222no—
447C22⋊C8—1222no—
448C22⋊C8—1222no—
449C22⋊C8—1222no—
450C22⋊C8—1221no—
451#97—1222no—
452C4×C8—611no—
453C4×C8—612no—
454C4×C8—611no—
455C4×C8—1222no—
456C4×C8—1221no—
457C4×C8—1222no—
458C24—1221no—
459#41—24no—
460#41—42no—
461#41—1222no—
462#41—1222no—
463C2×Q8—611no—
464C2×Q8—1221no—
465C2×Q8C22×#692312yes—
466C2×D4—22no—
467C2×D4—612no—
468C2×D4—611no—
469C2×D4—1221no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—2441no—
473C2×D4C22⋊#692311yescharacteristic
474C2×D4C22×#692311yescharacteristic
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4C22×#692312yes—
490M16—2442no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1222no—
494C2×C8—1221no—
495C2×C8—1222no—
496C2×C8—1221no—
497C2×C8—2442no—
498C2×C8—2444no—
499C2×C8—2442no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—611no—
504C4⋊C4—612no—
505C4⋊C4—612no—
506C4⋊C4—611no—
507C4⋊C4—612no—
508C4⋊C4—1221no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1221no—
515C4⋊C4C22⋊#692311yescharacteristic
516C4⋊C4C22×#692311yescharacteristic
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—612no—
520C22⋊C4—612no—
521C22⋊C4—1222no—
522C22⋊C4—1221no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1221no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1221no—
530C22⋊C4C22⋊#692312yes—
531C42—612no—
532C42—612no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—612no—
537C42—1222no—
538C42—1222no—
539C42—1222no—
540C42C22×#692311yescharacteristic
541C23—21no—
542C23—611no—
543C23—1221no—
544C23—4881no—
545C23—2442no—
546C23—2442no—
547C23C2×C22⋊#692312yes—
548Q8—22no—
549Q8—22no—
550Q8—1222no—
551Q8—4881no—
552D4—22no—
553D4—21no—
554D4—44no—
555D4—1224no—
556D4—1221no—
557D4—1221no—
558D4—4882no—
559D4—2442no—
560D4#1451986412yes—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—611no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—611no—
572C2×C4—612no—
573C2×C4—611no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—4882no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2442no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4C2×C22⋊#692312yes—
590C2×C4C2×C22⋊#692311yescharacteristic
591C2×C4C2×C22⋊#692311yescharacteristic
592C2×C4C23×#692311yescharacteristic
593C8—1222no—
594C8—2441no—
595C8—2441no—
596C8—2442no—
597C8—2441no—
598C8—2442no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—612no—
606C22—1221no—
607C22—1221no—
608C22—4884no—
609C22—2441no—
610C22—2441no—
611C22—2441no—
612C22C22×C22⋊#692311yescharacteristic
613C4—22no—
614C4—21no—
615C4—42no—
616C4—41no—
617C4—611no—
618C4—611no—
619C4—1221no—
620C4—1222no—
621C4—1221no—
622C4—1222no—
623C4—2441no—
624C4—2442no—
625C4#1684075212yes—
626C4C2×#1451986411yescharacteristic
627C4#1684084011yescharacteristic
628C2—42no—
629C2—42no—
630C2—1222no—
631C2—4881no—
632C2#2146240711yescharacteristic
633C2C2×#1684075211yescharacteristic
634C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#547C4
socleC22×C61C22×C22⋊C4
prime radical#21459382C2
cosocleC4×Dic61C24
hypercenter#547#6923
hypocenterC61#17508
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #21462624C2semidirect
2. #21462515C2semidirect
3. #21461172C2semidirect
4. #21461168C2semidirect
5. #21461150C2semidirect
6. #21461145C2semidirect
7. #21461132C2semidirect
8. #21461128C2semidirect
9. #21461110C2semidirect
10. D61⋊(D4⋊C8)C2semidirect
16. #16840783C22semidirect
18. #16840779C22semidirect
19. #16840778C22semidirect
25. C2×C4×#92420C22semidirect
26. Q8⋊#92420C22semidirect
27. Q8⋊#92420C22semidirect
28. Q8⋊#92420C22semidirect
29. Q8⋊#92420C22semidirect
370. C61#17508semidirect

Maximal subgroups

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

subgroupindex
#214626242
#214625152
#214611722
#214611682
#214611502
#214611452
#214611322
#214611282
#214611102
D61⋊(D4⋊C8)2
#214593822
#1750861