Finite Group Databasebeta

← #30616736

Subgroups — #30616736

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462500C211yesmaximal, characteristic, complemented
2#117×#6923C211yesmaximal, characteristic, complemented
3#21462006C211yesmaximal, characteristic, complemented
4#21462005C211yesmaximal, characteristic, complemented
5#21461955C211yesmaximal, characteristic, complemented
6#21461952C211yesmaximal, characteristic, complemented
7#21461838C212yesmaximal, complemented
8#21461836C212yesmaximal, complemented
9#21461583C212yesmaximal, complemented
10C4×#14519842C212yesmaximal, complemented
11#21459210C211yesmaximal, characteristic
12C2×#14519856C2211yescharacteristic
13C22×C4×#6923C2211yescharacteristic
14#16840787—22no—
15#16840786—22no—
16#16840767C2211yescharacteristic
17#16840766C2211yescharacteristic
18C2×#14519842C2212yes—
19C2×#14519842C2212yes—
20C2×#14519842C2212yes—
21C2×#14519842C2212yes—
22#16840733C2212yes—
23C4×#11758213C2212yes—
24C4⋊C4×#6923—22no—
25C22⋊C4×#6923—22no—
26#16840698C2211yescharacteristic
27#16840698C2211yescharacteristic
28C42×#6923C2211yescharacteristic
29C42×#6923C2211yescharacteristic
30#16840632C2212yes—
31#16840632C2212yes—
32#16840632C2212yes—
33#16840632C2212yes—
34#16840385C412yescomplemented
35#16840384C412yescomplemented
36#16840373C411yescharacteristic, complemented
37#16840368C411yescharacteristic, complemented
38#16840367C411yescharacteristic, complemented
39#16840360—22no—
40#16840351—22no—
41#16840318C2211yescharacteristic
42#16840313C2211yescharacteristic
43#16840297—22no—
44#16840295—22no—
45#16840275C2212yes—
46C4×#11758226C2212yes—
47#117×D61C2211yescharacteristic
48C61×#541C411yescharacteristic, complemented
49C23×#6923—21no—
50C2×#11758213C2312yes—
51C2×#11758213C2312yes—
52#14519856—22no—
53#14519856—22no—
54C2×C4⋊#6923—21no—
55C2×C22⋊#6923—21no—
56C2×C4×#6923—21no—
57C2×C4×#6923—22no—
58C2×C4×#6923—22no—
59C2×C4×#6923—22no—
60C2×C4×#6923—22no—
61C2×C4×#6923C2311yescharacteristic
62C2×C4×#6923C2311yescharacteristic
63C2×C4×#6923C2311yescharacteristic
64C2×C4×#6923C2311yescharacteristic
65#14519844—22no—
66C4⋊#92420—22no—
67#14519842—24no—
68#14519842—24no—
69#14519842—24no—
70#14519842—24no—
71#14519834—22no—
72C22⋊#92420—22no—
73#14519832—22no—
74C4×#92420—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77C2×#11758226C2312yes—
78C22×C4×D61C2311yescharacteristic
79#14519776C2×C412yescomplemented
80#14519774C2×C411yescharacteristic, complemented
81#14519772—22no—
82#14519772—22no—
83#14519771—22no—
84#14519769C2×C411yescharacteristic, complemented
85#14519767—22no—
86#14519766C2×C412yescomplemented
87#14519765—22no—
88#14519763—22no—
89#14519762—22no—
90#14519762—22no—
91#14519762—22no—
92#14519762—22no—
93#14519759—22no—
94#14519757—22no—
95#14519755—22no—
96#14519755—22no—
97#14519753C2×C412yes—
98#14519752—22no—
99#14519752—22no—
100#14519752C2×C412yes—
101#14519751C2×C412yes—
102#14519750—22no—
103#14519747—22no—
104Q8×Dic61C2×C412yescomplemented
105#14519742—22no—
106C4×D244—22no—
107C4×D244C2×C412yescomplemented
108#14519739—22no—
109#14519739—22no—
110#14519739—22no—
111#14519739C2×C412yescomplemented
112#14519737—22no—
113#14519736C2×C411yescharacteristic
114#14519735—22no—
115#14519735D412yes—
116#14519735C2312yes—
117#14519734C2×C411yescharacteristic
118#14519734C2×C411yescharacteristic
119#14519733D412yes—
120C4⋊C4×D61D412yes—
121C22⋊C4×D61D412yes—
122C42×D61—22no—
123C42×D61C2311yescharacteristic
124C42×D61C2311yescharacteristic
125C4⋊Q8×C61C2×C411yescharacteristic, complemented
126#126×C61—22no—
127#125×C61—22no—
128#125×C61C2×C411yescharacteristic, complemented
129C22⋊Q8×C61C2×C412yes—
130C4×Q8×C61C2×C412yescomplemented
131#117×C61C2×C411yescharacteristic
132C23×D61C2×D411yescharacteristic
133C22×#6923—21no—
134C22×#6923—24no—
135C22×#6923—21no—
136C22×#6923—21no—
137C22×#6923—21no—
138C22×#6923—44no—
139C2×Q8×C61C22×C412yes—
140C22×C4×C61C22×C411yescharacteristic
141C2×C22⋊D61—22no—
142C22×Dic61C2×D411yescharacteristic
143#11758226—22no—
144#11758226—22no—
145#11758226—22no—
146#11758226—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244C22×C412yes—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2411yescharacteristic
164C2×Dic122—22no—
165C22⋊#6923—24no—
166C2×#92420—22no—
167C2×#92420—22no—
168C4⋊#6923—24no—
169C4×#6923—24no—
170C4×#6923—21no—
171C4×#6923—21no—
172C4×#6923—21no—
173C4×#6923—21no—
174C4×#6923—44no—
175C4×#6923—44no—
176C4×#6923#4112yes—
177C4×#6923#4112yes—
178C4×#6923#4112yes—
179C4×#6923#4112yes—
180#11758213—22no—
181#11758213—22no—
182#11758213—22no—
183#11758213—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61C22⋊C412yes—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—22no—
191C22⋊C4×C61C22⋊C412yes—
192C42×C61—22no—
193C42×C61C22×C411yescharacteristic
194C42×C61C22×C411yescharacteristic
195C22⋊Dic61C22⋊C412yes—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198C22⋊C412yes—
205#11758198C22⋊C412yes—
206C4⋊Dic61—22no—
207C4⋊Dic61C22⋊C412yes—
208C4⋊Dic61C22×C412yes—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C22×C411yescharacteristic
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C23×C61C2×C22⋊C411yescharacteristic
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—22no—
224C22×D61—42no—
225C22×D61C2×C22⋊C411yescharacteristic
226C22×D61C22×D411yescharacteristic
227C2×#6923—24no—
228C2×#6923—22no—
229C2×#6923—22no—
230C2×#6923—22no—
231C2×#6923—22no—
232C2×#6923—48no—
233C2×#6923—42no—
234C2×#6923—42no—
235C2×#6923—42no—
236Q8×C61—22no—
237Q8×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C23×C411yescharacteristic
246C22⋊D61—82no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61C2×C22⋊C412yes—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C22×D411yescharacteristic
254D244—22no—
255D244—22no—
256D244—42no—
257D244—42no—
258D244—82no—
259D244—82no—
260C4×D61—21no—
261C4×D61—21no—
262C4×D61—21no—
263C4×D61—21no—
264C4×D61—21no—
265C4×D61—21no—
266C4×D61—42no—
267C4×D61—42no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—82no—
271C4×D61—82no—
272C4×D61C2×#4111yescharacteristic
273C4×D61C2×#4111yescharacteristic
274C4×D61C2×#4111yescharacteristic
275C4×D61C2×#4111yescharacteristic
276Dic122—82no—
277#92420—42no—
278#92420—42no—
279#21419—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—21no—
282C22×C61C22×C22⋊C411yescharacteristic
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—21no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61#52511yescharacteristic
292C2×D61#52511yescharacteristic
293#6923—24no—
294#6923—42no—
295#6923—42no—
296#6923—42no—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—21no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61C4×#4111yescharacteristic
303C4×C61C4×#4111yescharacteristic
304Dic61—22no—
305Dic61—22no—
306Dic61—21no—
307Dic61—42no—
308Dic61#53211yescharacteristic
309Dic61#53211yescharacteristic
310#3455—611no—
311#2921—611no—
312C4×#117—611no—
313#1995—611no—
314#1994—611no—
315#1963—611no—
316#1962—611no—
317#1897—612no—
318#1896—612no—
319#1759—612no—
320C4×#104—612no—
321C2×C61—41no—
322C2×C61#229711yescharacteristic
323C2×C61#290112yes—
324D61—21no—
325D61—21no—
326D61—41no—
327D61—82no—
328#542—612no—
329#542—612no—
330#541—611no—
331#541#692311yescharacteristic, complemented
332#540—611no—
333#540—611no—
334#539—1222no—
335#538—1222no—
336#533—611no—
337#532—611no—
338#529—1222no—
339#529—1222no—
340#525—612no—
341C4×#41—612no—
342C2×#117—611no—
343C2×#117—611no—
344C22×C42—611no—
345#433—1222no—
346#433—1222no—
347#423—611no—
348#423—611no—
349C2×#104—612no—
350C2×#104—612no—
351C2×#104—612no—
352C2×#104—612no—
353#407—612no—
354C4×M16—612no—
355C4×C4⋊C4—1222no—
356C4×C22⋊C4—1222no—
357#376—611no—
358#376—611no—
359C43—611no—
360C43—611no—
361#339—612no—
362#339—612no—
363#339—612no—
364#339—612no—
365C61#2141911yescharacteristic, complemented, Sylow
366C2×#41—612no—
367C23×C4—611no—
368C23×C4—1221no—
369C2×M16—612no—
370C2×M16—612no—
371C4⋊Q8—612no—
372C4⋊Q8C2×#692311yescharacteristic, complemented
373#127—611no—
374#126—22no—
375#126—1222no—
376#126—1222no—
377#126—1222no—
378#125—22no—
379#125—611no—
380#125—1222no—
381#125C2×#692311yescharacteristic, complemented
382#124—612no—
383#124—1222no—
384#124—1222no—
385#123—1222no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389#123—1222no—
390C22⋊Q8—612no—
391C22⋊Q8—1222no—
392C22⋊Q8—1222no—
393C22⋊Q8—1222no—
394C22⋊Q8C2×#692312yes—
395#121—612no—
396#121—612no—
397#121—1222no—
398#121—1222no—
399#121—1222no—
400#120—1222no—
401C4×Q8—612no—
402C4×Q8—1222no—
403C4×Q8C2×#692312yescomplemented
404C4×D4—612no—
405C4×D4—612no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411#117—611no—
412#117—612no—
413#117—612no—
414#117—611no—
415#117—611no—
416#117—612no—
417#117—1222no—
418#117—1222no—
419#117—1222no—
420#117C2×#692311yescharacteristic
421C2×C4⋊C4—612no—
422C2×C4⋊C4—1221no—
423C2×C22⋊C4—612no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—1222no—
432C2×C42—1222no—
433C2×C42—1222no—
434C2×C42—1221no—
435C2×C42—1222no—
436C2×C42—1222no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439#104—1224no—
440#104—1224no—
441#104—1224no—
442#104—1224no—
443C22⋊C8—1222no—
444C22⋊C8—1222no—
445#97—1222no—
446C4×C8—1222no—
447#95—1222no—
448#95—1222no—
449C24—611no—
450#41—1222no—
451#41—1222no—
452#41—1222no—
453#41—1222no—
454C2×Q8—1222no—
455C2×Q8C22×#692312yes—
456C2×D4—612no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1222no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—2444no—
481C22×C4C22×#692311yescharacteristic
482M16—1222no—
483M16—1222no—
484M16—1222no—
485M16—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1224no—
500C4⋊C4—1222no—
501C4⋊C4C22⋊#692312yes—
502C4⋊C4C22×#692312yes—
503C22⋊C4—22no—
504C22⋊C4—612no—
505C22⋊C4—612no—
506C22⋊C4—612no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1224no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4C22⋊#692312yes—
517C42—22no—
518C42—612no—
519C42—612no—
520C42—611no—
521C42—611no—
522C42—612no—
523C42—612no—
524C42—612no—
525C42—612no—
526C42—1222no—
527C42—1222no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—1224no—
533C42—2444no—
534C42—2444no—
535C42C22×#692311yescharacteristic
536C42C22×#692311yescharacteristic
537C23—611no—
538C23—611no—
539C23—1221no—
540C23—1221no—
541C23—1222no—
542C23—2442no—
543C23C2×C22⋊#692311yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—4882no—
547D4—1222no—
548D4—1222no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—2442no—
553D4—2442no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—611no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1222no—
573C2×C4—1224no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—4882no—
581C2×C4—4882no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2448no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4C2×C22⋊#692312yes—
591C2×C4C2×C22⋊#692312yes—
592C2×C4C2×C22⋊#692311yescharacteristic
593C2×C4C23×#692311yescharacteristic
594C8—2442no—
595C8—2442no—
596C22—21no—
597C22—22no—
598C22—611no—
599C22—611no—
600C22—1222no—
601C22—1221no—
602C22—1222no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22C22×C22⋊#692311yescharacteristic
609C4—21no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—611no—
615C4—611no—
616C4—1224no—
617C4—1221no—
618C4—1222no—
619C4—1222no—
620C4—2442no—
621C4—2442no—
622C4—2442no—
623C4—2442no—
624C4#41×#692311yescharacteristic
625C4#41×#692311yescharacteristic
626C2—41no—
627C2—1221no—
628C2—1221no—
629C2—4882no—
630C2—2441no—
631C2#2146228011yescharacteristic
632C2#2146245012yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#541C4
socleC22×C61C22×C22⋊C4
prime radical#21459210C2
cosocleC2×C4×D61C24
hypercenter#541#6923
hypocenterC61#21419
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #21462500C2semidirect
2. #117×#6923C2semidirect
3. #21462006C2semidirect
4. #21462005C2semidirect
5. #21461955C2semidirect
6. #21461952C2semidirect
7. #21461838C2semidirect
8. #21461836C2semidirect
9. #21461583C2semidirect
10. C4×#14519842C2semidirect
34. #16840385C4semidirect
35. #16840384C4semidirect
36. #16840373C4semidirect
37. #16840368C4semidirect
38. #16840367C4semidirect
48. C61×#541C4semidirect
79. #14519776C2×C4semidirect
80. #14519774C2×C4semidirect
84. #14519769C2×C4semidirect
86. #14519766C2×C4semidirect
104. Q8×Dic61C2×C4semidirect
107. C4×D244C2×C4semidirect
111. #14519739C2×C4semidirect
125. C4⋊Q8×C61C2×C4semidirect
128. #125×C61C2×C4semidirect
130. C4×Q8×C61C2×C4semidirect
331. #541#6923semidirect
365. C61#21419semidirect
372. C4⋊Q8C2×#6923semidirect
381. #125C2×#6923semidirect
403. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625002
#117×#69232
#214620062
#214620052
#214619552
#214619522
#214618382
#214618362
#214615832
C4×#145198422
#214592102
#2141961