Finite Group Databasebeta

← #30610848

Subgroups — #30610848

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462749C211yesmaximal, characteristic, complemented
2C4×#14519865C211yesmaximal, characteristic, complemented
3#21461171C212yesmaximal, complemented
4#21461168C211yesmaximal, characteristic, complemented
5#21461162C212yesmaximal, complemented
6#21461155C211yesmaximal, characteristic, complemented
7#21461142C211yesmaximal, characteristic, complemented
8#21461135C212yesmaximal, complemented
9#21461111C211yesmaximal, characteristic, complemented
10D61⋊(Q8⋊C8)C212yesmaximal, complemented
11#21459438C211yesmaximal, characteristic
12C2×#14519865—21no—
13#16840813—21no—
14#16840792—22no—
15#16840790—21no—
16#16840784C2212yes—
17#16840781C2211yescharacteristic
18#16840766C2211yescharacteristic
19D61⋊(C4⋊C8)C2212yes—
20#16840727—22no—
21#16840727—21no—
22C4×#11758213C2212yes—
23C4×#11758213C2211yescharacteristic
24C4×D61⋊C8C2212yes—
25C4×D61⋊C8C2211yescharacteristic
26#16840612C2211yescharacteristic
27#16840612C2212yes—
28#16840612C2211yescharacteristic
29Q8⋊#92420C2212yes—
30Q8⋊#92420C2212yes—
31Q8⋊#92420C2212yes—
32Q8⋊#92420C2212yes—
33D4⋊#92420C2211yescharacteristic
34D4⋊#92420C2211yescharacteristic
35D4⋊#92420C2212yes—
36#16840490C411yescharacteristic
37#16840480—21no—
38#16840453C412yes—
39#16840448—22no—
40#16840427C411yescharacteristic
41#16840420C412yes—
42#16840417C411yescharacteristic
43#16840414—21no—
44#16840317C2211yescharacteristic
45#16840314—21no—
46#16840295—22no—
47#16840291C2212yes—
48C4×#11758226C2211yescharacteristic
49#16840230—21no—
50C4×#11758224C2211yescharacteristic
51C4×Q8×D61C2212yes—
52C61×#556C411yescharacteristic
53#14519865—28no—
54C2×#11758213—22no—
55C2×#11758213—21no—
56C2×#11758213—21no—
57C2×D61⋊C8—22no—
58C2×D61⋊C8—21no—
59C2×D61⋊C8—22no—
60#14519844—22no—
61C4⋊#92420—21no—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2312yes—
64C4⋊#92420C2311yescharacteristic
65#14519834—21no—
66#14519834—22no—
67#14519832—21no—
68#14519832—22no—
69#14519832—22no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420—21no—
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C4×#92420C2311yescharacteristic
76C2×#11758226D411yescharacteristic
77C2×#11758222—21no—
78C2×#11758224C2×C411yescharacteristic
79C2×Q8×D61C2×C412yes—
80#14519777C2×C411yescharacteristic
81#14519776—21no—
82#14519775—21no—
83#14519771—22no—
84#14519770—22no—
85#14519769C2×C412yes—
86#14519768—21no—
87#14519767—22no—
88#14519767—22no—
89#14519763—22no—
90#14519762—22no—
91#14519759—22no—
92#14519759—22no—
93#14519757—21no—
94#14519756—21no—
95#14519755—22no—
96#14519754—22no—
97#14519753C2×C412yes—
98#14519753C2×C412yes—
99#14519750—22no—
100#14519748C2×C412yes—
101Q8×Dic61D411yescharacteristic
102Q8×Dic61C2312yes—
103C4×Dic122—22no—
104C4×Dic122C2×C412yes—
105C4×Dic122C2×C411yescharacteristic
106#14519742—21no—
107#14519742—22no—
108#14519742—21no—
109#14519742C2311yescharacteristic
110C4×D244C2×C412yes—
111C4×D244C2×C411yescharacteristic
112D4×Dic61C2311yescharacteristic
113#14519739—21no—
114#14519739D411yescharacteristic
115#14519739D412yes—
116C4×C22⋊D61—21no—
117#14519737—21no—
118#14519737—22no—
119#14519737—22no—
120#14519735—21no—
121#14519735C2×C411yescharacteristic
122#14519735D412yes—
123#14519735D411yescharacteristic
124#14519734—22no—
125#14519733—21no—
126#14519733—22no—
127C4⋊C4×D61—22no—
128C4⋊C4×D61C2×C412yes—
129C2×C4×Dic61—21no—
130C42×D61—22no—
131C42×D61C2312yes—
132C42×D61C2311yescharacteristic
133C4⋊Q8×C61C2×C411yescharacteristic
134#126×C61—22no—
135#124×C61C2×C412yes—
136C22⋊Q8×C61—21no—
137C4×Q8×C61C2×C412yes—
138C4×Q8×C61C2×C411yescharacteristic
139C4×D4×C61C2×C411yescharacteristic
140#117×C61—21no—
141C2×Q8×C61C22⋊C411yescharacteristic
142C2×Q8×C61C22×C412yes—
143C2×D4×C61C22×C411yescharacteristic
144C22×C4×C61—21no—
145C2×C22⋊D61—21no—
146C2×C22⋊D61—22no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149#11758226—24no—
150Q8×D61—24no—
151Q8×D61—42no—
152#11758224—22no—
153#11758224—41no—
154#11758222—44no—
155C2×D244C22⋊C412yes—
156C2×D244C22⋊C411yescharacteristic
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61C22×C412yes—
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—22no—
167C2×Dic122C22×C411yescharacteristic
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—22no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—22no—
174#11758213—28no—
175#11758213—24no—
176#11758213—81no—
177D61⋊C8—28no—
178D61⋊C8—24no—
179D61⋊C8—82no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22⋊C412yes—
185C4⋊C4×C61C22×C411yescharacteristic
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C42×C61—22no—
190C42×C61C22×C411yescharacteristic
191C42×C61C22×C412yes—
192C22⋊Dic61—22no—
193C22⋊Dic61—21no—
194#11758198—22no—
195#11758198—22no—
196#11758198—21no—
197#11758198C22⋊C412yes—
198#11758198C22⋊C412yes—
199#11758198C22⋊C412yes—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22⋊C412yes—
203C4⋊Dic61C22⋊C411yescharacteristic
204C4⋊Dic61C22×C412yes—
205C4⋊Dic61C22×C411yescharacteristic
206#11758196—22no—
207#11758196—21no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—21no—
214C4×Dic61—22no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—22no—
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D412yes—
223C4×Dic61C2411yescharacteristic
224C23×C61—21no—
225C22×D61C2×C22⋊C411yescharacteristic
226C22×D61C2×C22⋊C412yes—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—22no—
230D4×C61—21no—
231D4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—42no—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—44no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—42no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C22×D411yescharacteristic
251C2×Dic61C22×D411yescharacteristic
252D244—24no—
253D244#10414yes—
254C4×D61—24no—
255C4×D61—21no—
256C4×D61—22no—
257C4×D61—21no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—21no—
262C4×D61—22no—
263C4×D61—21no—
264C4×D61—44no—
265C4×D61—44no—
266Dic122—21no—
267Dic122—44no—
268Dic122—42no—
269Dic122#13112yescomplemented
270#92420—22no—
271#92420—24no—
272#92420—22no—
273#92420—41no—
274#92420—42no—
275#92420—41no—
276#17519—611nomaximal, Sylow
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—21no—
286C2×D61C2×#10414yes—
287C4×C61—22no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61C2×#10412yes—
293C4×C61C2×#13111yescharacteristic
294C4×C61#56811yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—21no—
298Dic61—21no—
299Dic61—41no—
300Dic61—42no—
301Dic61#57711yescharacteristic
302Dic61#57811yescharacteristic
303#3482—611no—
304#2990—611no—
305C4×#131—611no—
306#1494—612no—
307#1493—611no—
308#1491—612no—
309#1489—611no—
310#1485—611no—
311#1483—612no—
312#1476—611no—
313C2×Q8⋊C8—612no—
314C2×C61—81no—
315C2×C61#288011yescharacteristic
316C2×C61#289811yescharacteristic
317C2×C61C22×#10411yescharacteristic
318D61—24no—
319D61—41no—
320C2×#131—1221no—
321#557—611no—
322#556—1221no—
323#556#692311yescharacteristic
324#552—612no—
325#552—1222no—
326D4×Q8—611no—
327#548—612no—
328#547—611no—
329#547—1221no—
330#533—611no—
331#533—1221no—
332#529—612no—
333#529—1222no—
334C4×#41—611no—
335C4×#41—611no—
336C4×#41—1221no—
337C2×C4×Q8—612no—
338#446—1221no—
339#435—1222no—
340C8×D4—1221no—
341#432—612no—
342#431—611no—
343#423—611no—
344C2×C4⋊C8—612no—
345#405—1222no—
346#405—1221no—
347C4×M16—612no—
348C4×M16—611no—
349C2×C4×C8—612no—
350C2×C4×C8—611no—
351Q8⋊C8—612no—
352Q8⋊C8—612no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355Q8⋊C8—611no—
356Q8⋊C8—612no—
357Q8⋊C8—611no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—612no—
361C61#1751911yescharacteristic, complemented, Sylow
362C2×#41—611no—
363C2×#41—611no—
364C2×#41—1221no—
365C22×Q8—612no—
366#131—1228no—
367C2×M16—1222no—
368C2×M16—1221no—
369C2×M16—1221no—
370C22×C8—1222no—
371C22×C8—1221no—
372C22×C8—1222no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375C4⋊Q8—1221no—
376C4⋊Q8C2×#692311yescharacteristic
377#126—22no—
378#126—1222no—
379#126—1222no—
380#125—612no—
381#125—1221no—
382#125—1222no—
383#125—1222no—
384#124—1222no—
385#124C2×#692312yes—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—21no—
390C22⋊Q8—612no—
391C22⋊Q8—612no—
392C22⋊Q8—1222no—
393C22⋊Q8—1222no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396#121—612no—
397#121—1222no—
398C4×Q8—611no—
399C4×Q8—612no—
400C4×Q8—611no—
401C4×Q8—612no—
402C4×Q8—611no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×Q8C2×#692312yes—
408C4×Q8C2×#692311yescharacteristic
409C4×D4—611no—
410C4×D4—612no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—612no—
414C4×D4—1222no—
415C4×D4—1221no—
416C4×D4—1222no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4C2×#692311yescharacteristic
420#117—21no—
421#117—612no—
422#117—611no—
423#117—611no—
424#117—1222no—
425#117—1221no—
426#117—1222no—
427#117—1221no—
428C2×C4⋊C4—612no—
429C2×C4⋊C4—1222no—
430C2×C42—612no—
431C2×C42—611no—
432C2×C42—1222no—
433C2×C42—1221no—
434C4⋊C8—611no—
435C4⋊C8—612no—
436C4⋊C8—611no—
437C4⋊C8—1221no—
438C4⋊C8—1222no—
439C22⋊C8—1221no—
440C22⋊C8—1222no—
441#97—1221no—
442#97—1222no—
443#97—1222no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—612no—
447C4×C8—1221no—
448C4×C8—1222no—
449C4×C8—1221no—
450#41—1224no—
451#41—1222no—
452#41—2444no—
453#41—2441no—
454C2×Q8—611no—
455C2×Q8—1222no—
456C2×Q8—1221no—
457C2×Q8—1224no—
458C2×Q8—2442no—
459C2×Q8C22⋊#692311yescharacteristic
460C2×Q8C22×#692312yes—
461C2×D4—611no—
462C2×D4—612no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4C22×#692311yescharacteristic
466C22×C4—21no—
467C22×C4—612no—
468C22×C4—611no—
469C22×C4—612no—
470C22×C4—611no—
471C22×C4—612no—
472C22×C4—611no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1221no—
477M16—1228no—
478M16—1224no—
479M16—4881no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1222no—
483C2×C8—1228no—
484C2×C8—1224no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1222no—
488C2×C8—4882no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—612no—
493C4⋊C4—611no—
494C4⋊C4—612no—
495C4⋊C4—611no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1221no—
506C4⋊C4C22⋊#692311yescharacteristic
507C4⋊C4C22⋊#692312yes—
508C4⋊C4C22×#692311yescharacteristic
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—22no—
512C22⋊C4—612no—
513C22⋊C4—612no—
514C22⋊C4—612no—
515C22⋊C4—1221no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1221no—
519C22⋊C4—1222no—
520C42—22no—
521C42—611no—
522C42—612no—
523C42—611no—
524C42—612no—
525C42—611no—
526C42—1221no—
527C42—1221no—
528C42—1222no—
529C42—1221no—
530C42—1222no—
531C42C22×#692311yescharacteristic
532C42C22×#692312yes—
533C23—21no—
534C23—612no—
535C23—611no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—612no—
540Q8—1221no—
541Q8—2444no—
542Q8—2442no—
543D4—21no—
544D4—21no—
545D4—614no—
546D4—1224no—
547D4—2442no—
548D4—2444no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—612no—
555C2×C4—611no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1224no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1222no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2442no—
574C2×C4C2×C22⋊#692312yes—
575C2×C4C2×C22⋊#692311yescharacteristic
576C2×C4C2×C22⋊#692312yes—
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C23×#692311yescharacteristic
579C8—1222no—
580C8—1224no—
581C8—1222no—
582C8—2441no—
583C8—2441no—
584C8—2442no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—614no—
589C22—1222no—
590C22—1222no—
591C22—1221no—
592C22—1221no—
593C22—1221no—
594C22C22×C22⋊#692311yescharacteristic
595C4—22no—
596C4—21no—
597C4—22no—
598C4—42no—
599C4—41no—
600C4—611no—
601C4—611no—
602C4—1221no—
603C4—1222no—
604C4—1222no—
605C4—1221no—
606C4—2441no—
607C4—2442no—
608C4D61⋊#10412yes—
609C4D61⋊#13111yescharacteristic
610C4#1684084111yescharacteristic
611C2—81no—
612C2—1224no—
613C2—2441no—
614C2#2146241011yescharacteristic
615C2#2146244111yescharacteristic
616C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#556C4
socleC22×C61C22×C22⋊C4
prime radical#21459438C2
cosocleC4×Dic61C24
hypercenter#556#6923
hypocenterC61#17519
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #21462749C2semidirect
2. C4×#14519865C2semidirect
3. #21461171C2semidirect
4. #21461168C2semidirect
5. #21461162C2semidirect
6. #21461155C2semidirect
7. #21461142C2semidirect
8. #21461135C2semidirect
9. #21461111C2semidirect
10. D61⋊(Q8⋊C8)C2semidirect
269. Dic122#131semidirect
361. C61#17519semidirect

Maximal subgroups

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

subgroupindex
#214627492
C4×#145198652
#214611712
#214611682
#214611622
#214611552
#214611422
#214611352
#214611112
D61⋊(Q8⋊C8)2
#214594382
#1751961