Finite Group Databasebeta

← #30610868

Subgroups — #30610868

Order 15616.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462747C211yesmaximal, characteristic, complemented
2#21462503C211yesmaximal, characteristic, complemented
3#21461181C212yesmaximal, complemented
4#21461168C211yesmaximal, characteristic, complemented
5#21461155C211yesmaximal, characteristic, complemented
6#21461150C212yesmaximal, complemented
7#21461142C211yesmaximal, characteristic, complemented
8#21461127C212yesmaximal, complemented
9#21461114C212yesmaximal, complemented
10#21461111C211yesmaximal, characteristic, complemented
11#21459431C211yesmaximal, characteristic
12#16840812—21no—
13#16840791—22no—
14D4×#92420—21no—
15#16840786—22no—
16#16840783C2212yes—
17#16840781C2211yescharacteristic
18#16840779C2212yes—
19#16840778C2212yes—
20#16840769—21no—
21#16840766C2211yescharacteristic
22C2×C4⋊#92420C2212yes—
23#16840736—21no—
24C4×#11758213C2211yescharacteristic
25C4×D61⋊C8C2211yescharacteristic
26#16840612C2211yescharacteristic
27#16840612C2211yescharacteristic
28#16840612C2212yes—
29Q8⋊#92420C2212yes—
30Q8⋊#92420C2212yes—
31Q8⋊#92420C2212yes—
32Q8⋊#92420C2212yes—
33D4⋊#92420C2211yescharacteristic
34D4⋊#92420C2211yescharacteristic
35D4⋊#92420C2212yes—
36#16840487C411yescharacteristic
37#16840483C412yes—
38#16840482C411yescharacteristic
39#16840449—22no—
40#16840447—21no—
41D4×Dic122C411yescharacteristic
42#16840418—21no—
43#16840411C412yes—
44#16840317C2211yescharacteristic
45#16840314C2212yes—
46#16840293—21no—
47#16840289—22no—
48#41×Dic61C2212yes—
49C4×#11758224C2211yescharacteristic
50C4×#11758224C2211yescharacteristic
51C2×#14519742—21no—
52C61×#547C411yescharacteristic
53C2×#11758213—21no—
54C2×#11758213—21no—
55C2×D61⋊C8—21no—
56C22×#92420—22no—
57#14519844—22no—
58C4⋊#92420—22no—
59C4⋊#92420—21no—
60C4⋊#92420—22no—
61C4⋊#92420C2311yescharacteristic
62C4⋊#92420C2312yes—
63C4⋊#92420C2311yescharacteristic
64#14519834—21no—
65C22⋊#92420—22no—
66C22⋊#92420—22no—
67C22⋊#92420—22no—
68#14519832—21no—
69#14519832—22no—
70C4×#92420—21no—
71C4×#92420—21no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C2×#11758224C2×C411yescharacteristic
76C2×#11758224D411yescharacteristic
77C22×Dic122—21no—
78#14519777C2×C411yescharacteristic
79#14519776C2×C412yes—
80#14519775C2×C412yes—
81#14519772—22no—
82#14519770—22no—
83#14519770—22no—
84#14519768—21no—
85#14519768—21no—
86#14519767—22no—
87#14519767—22no—
88#14519764—21no—
89#14519763—22no—
90#14519760—22no—
91#14519758—22no—
92#14519757—21no—
93#14519756—21no—
94#14519756C2×C412yes—
95#14519754—22no—
96#14519754—22no—
97#14519753C2×C412yes—
98#14519749—22no—
99Q8×Dic61C2×C412yes—
100Q8×Dic61C2312yes—
101C4×Dic122C2×C411yescharacteristic
102C4×Dic122C2×C411yescharacteristic
103#14519742—21no—
104#14519742—21no—
105#14519742—22no—
106#14519742—22no—
107#14519742—21no—
108#14519742D411yescharacteristic
109#14519742C2311yescharacteristic
110D4×Dic61—22no—
111D4×Dic61—22no—
112D4×Dic61C2×C412yes—
113D4×Dic61D411yescharacteristic
114D4×Dic61C2311yescharacteristic
115C4×C22⋊D61—21no—
116C4×C22⋊D61C2×C412yes—
117#14519737—21no—
118#14519737D412yes—
119#14519736—22no—
120#14519736C2×C412yes—
121#14519735—21no—
122#14519735C2×C411yescharacteristic
123#14519735D411yescharacteristic
124#14519733—22no—
125#14519733—21no—
126#14519733D412yes—
127C2×#11758196—22no—
128C2×C4×Dic61—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—
140#41×C61—24no—
141#41×C61—42no—
142C2×Q8×C61C22×C412yes—
143C2×D4×C61—22no—
144C2×D4×C61C22⋊C411yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61C22×C412yes—
149C2×C22⋊D61—21no—
150C2×C22⋊D61C22⋊C412yes—
151C22×Dic61—22no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D412yes—
155C22×Dic61C2×D412yes—
156#11758224—22no—
157#11758224—41no—
158#11758224—42no—
159C2×C4×D61C22×C411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×Dic122—21no—
163C2×Dic122—22no—
164C2×Dic122—44no—
165C2×Dic122C22⋊C411yescharacteristic
166C2×Dic122C22×C411yescharacteristic
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—22no—
172C2×#92420—22no—
173C2×#92420—82no—
174#11758213—42no—
175#11758213—81no—
176D61⋊C8—42no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22⋊C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61C22⋊C412yes—
185C42×C61C22×C411yescharacteristic
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—21no—
190C22⋊Dic61C22⋊C412yes—
191#11758198—22no—
192#11758198—21no—
193#11758198C22⋊C412yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199C4⋊Dic61C22×C412yes—
200#11758196—22no—
201#11758196—21no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—21no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—22no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—22no—
213C4×Dic61—21no—
214C4×Dic61—22no—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C4×Dic61C2411yescharacteristic
221C23×C61—21no—
222C23×C61C2×C22⋊C412yes—
223C22×D61C2×C22⋊C411yescharacteristic
224Q8×C61—22no—
225Q8×C61—22no—
226D4×C61—21no—
227D4×C61—22no—
228D4×C61—44no—
229D4×C61#13112yes—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—44no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C22×D411yescharacteristic
256C2×Dic61C22×D411yescharacteristic
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—22no—
260C4×D61—21no—
261C4×D61—21no—
262Dic122—22no—
263Dic122—21no—
264Dic122—21no—
265Dic122—44no—
266Dic122—42no—
267#92420—41no—
268#92420—41no—
269#92420—42no—
270#92420—42no—
271#92420—41no—
272#92420—41no—
273#17523—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C4×C61—22no—
285C4×C61—21no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61#42112yes—
289C4×C61C2×#13111yescharacteristic
290C4×C61#56811yescharacteristic
291Dic61—21no—
292Dic61—22no—
293Dic61—21no—
294Dic61—22no—
295Dic61—41no—
296Dic61—41no—
297Dic61—42no—
298Dic61#57711yescharacteristic
299Dic61#57811yescharacteristic
300#3482—611no—
301#2990—611no—
302#2922—611no—
303#1497—612no—
304#1493—611no—
305#1489—611no—
306#1488—612no—
307#1485—611no—
308#1481—612no—
309#1477—612no—
310#1476—611no—
311C2×C61—42no—
312C2×C61—42no—
313C2×C61#288011yescharacteristic
314C2×C61#289811yescharacteristic
315C2×C61C2×#42111yescharacteristic
316D61—41no—
317#557—611no—
318#556—612no—
319#556—611no—
320#552—1221no—
321#552—1222no—
322D4×Q8—611no—
323#548—1221no—
324#547—612no—
325#547#692311yescharacteristic
326#533—611no—
327#533—612no—
328#529—1221no—
329#529—1222no—
330C4×#41—611no—
331C4×#41—612no—
332C4×#41—611no—
333C2×C4×Q8—1221no—
334#446—1221no—
335#435—1222no—
336C8×D4—1221no—
337#433—1222no—
338#432—612no—
339#431—612no—
340#431—612no—
341#431—611no—
342#424—1221no—
343#423—611no—
344C2×C4⋊C8—612no—
345#408—1221no—
346C4×M16—611no—
347C2×C4×C8—611no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—612no—
351Q8⋊C8—612no—
352Q8⋊C8—611no—
353Q8⋊C8—611no—
354Q8⋊C8—612no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—612no—
358C61#1752311yescharacteristic, complemented, Sylow
359C2×#41—611no—
360C2×#41—611no—
361C2×#41C2×#692312yes—
362C22×Q8—1221no—
363C2×M16—1221no—
364C2×M16—1221no—
365C22×C8—1222no—
366C22×C8—1221no—
367C4⋊Q8—612no—
368C4⋊Q8—612no—
369C4⋊Q8—611no—
370C4⋊Q8C2×#692311yescharacteristic
371#126—1222no—
372#126—1222no—
373#126—1222no—
374#125—1222no—
375#125—1221no—
376#125—1222no—
377#125—1221no—
378#124—1222no—
379#124—1221no—
380#123—22no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—612no—
384C22⋊Q8—612no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8C2×#692312yes—
390#121—21no—
391#121—1222no—
392C4×Q8—611no—
393C4×Q8—611no—
394C4×Q8—612no—
395C4×Q8—612no—
396C4×Q8—611no—
397C4×Q8—611no—
398C4×Q8—1221no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×D4—611no—
404C4×D4—612no—
405C4×D4—612no—
406C4×D4—611no—
407C4×D4—612no—
408C4×D4—1221no—
409C4×D4—1221no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4C2×#692311yescharacteristic
413C4×D4C2×#692311yescharacteristic
414#117—611no—
415#117—612no—
416#117—612no—
417#117—611no—
418#117—1221no—
419#117—1222no—
420#117—1222no—
421#117—1221no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—1222no—
424C2×C42—612no—
425C2×C42—611no—
426C2×C42—1221no—
427C2×C42—1222no—
428C4⋊C8—611no—
429C4⋊C8—612no—
430C4⋊C8—611no—
431C4⋊C8—1222no—
432C4⋊C8—1221no—
433C4⋊C8—1222no—
434C4⋊C8—1222no—
435C22⋊C8—1222no—
436C22⋊C8—1222no—
437C22⋊C8—1222no—
438C22⋊C8—1221no—
439#97—1221no—
440#97—1222no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—612no—
444C4×C8—1221no—
445C4×C8—1221no—
446#41—24no—
447#41—42no—
448#41—1222no—
449#41—2442no—
450#41—2441no—
451C2×Q8—611no—
452C2×Q8—611no—
453C2×Q8—1222no—
454C2×Q8—1221no—
455C2×Q8—2444no—
456C2×Q8C22×#692312yes—
457C2×D4—22no—
458C2×D4—612no—
459C2×D4—1221no—
460C2×D4C22⋊#692311yescharacteristic
461C2×D4C22×#692311yescharacteristic
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4C22×#692312yes—
473M16—4881no—
474M16—2442no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1222no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1222no—
481C2×C8—4882no—
482C2×C8—2442no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—612no—
487C4⋊C4—612no—
488C4⋊C4—611no—
489C4⋊C4—611no—
490C4⋊C4—612no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1221no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4C22⋊#692311yescharacteristic
502C4⋊C4C22×#692311yescharacteristic
503C22⋊C4—22no—
504C22⋊C4—21no—
505C22⋊C4—612no—
506C22⋊C4—612no—
507C22⋊C4—1221no—
508C22⋊C4—1222no—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4C22⋊#692312yes—
514C42—611no—
515C42—611no—
516C42—612no—
517C42—612no—
518C42—611no—
519C42—612no—
520C42—1221no—
521C42—1221no—
522C42—1222no—
523C42—1222no—
524C42—1222no—
525C42—1221no—
526C42C22×#692311yescharacteristic
527C23—21no—
528C23—611no—
529C23C2×C22⋊#692312yes—
530Q8—22no—
531Q8—22no—
532Q8—1222no—
533Q8—1221no—
534Q8—1221no—
535Q8—2444no—
536Q8—2442no—
537D4—22no—
538D4—21no—
539D4—44no—
540D4—2442no—
541D4—2442no—
542D4#1451986412yes—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—611no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1221no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2444no—
569C2×C4C2×C22⋊#692312yes—
570C2×C4C2×C22⋊#692311yescharacteristic
571C2×C4C2×C22⋊#692311yescharacteristic
572C2×C4C23×#692311yescharacteristic
573C8—2441no—
574C8—2441no—
575C8—2442no—
576C8—2441no—
577C8—2441no—
578C8—2442no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—1221no—
586C22—1221no—
587C22—1221no—
588C22C22×C22⋊#692311yescharacteristic
589C4—22no—
590C4—21no—
591C4—42no—
592C4—41no—
593C4—611no—
594C4—611no—
595C4—1221no—
596C4—1222no—
597C4—1221no—
598C4—1222no—
599C4—2441no—
600C4—2441no—
601C4—2442no—
602C4#1684075912yes—
603C4C2×#1451986411yescharacteristic
604C4#1684084011yescharacteristic
605C2—42no—
606C2—42no—
607C2—2441no—
608C2#2146240711yescharacteristic
609C2#2146244111yescharacteristic
610C2C2×#1684075911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#547C4
socleC22×C61C22×C22⋊C4
prime radical#21459431C2
cosocleC4×Dic61C24
hypercenter#547#6923
hypocenterC61#17523
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. #21462747C2semidirect
2. #21462503C2semidirect
3. #21461181C2semidirect
4. #21461168C2semidirect
5. #21461155C2semidirect
6. #21461150C2semidirect
7. #21461142C2semidirect
8. #21461127C2semidirect
9. #21461114C2semidirect
10. #21461111C2semidirect
358. C61#17523semidirect

Maximal subgroups

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

subgroupindex
#214627472
#214625032
#214611812
#214611682
#214611552
#214611502
#214611422
#214611272
#214611142
#214611112
#214594312
#1752361