Finite Group Databasebeta

← #30615621

Subgroups — #30615621

Order 15616.

1682 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840840C211yesmaximal, characteristic, complemented
2C2×#16840761C211yesmaximal, characteristic, complemented
3C2×#16840760C211yesmaximal, characteristic, complemented
4#21462406C211yesmaximal, characteristic, complemented
5#21461808C218yesmaximal, complemented
6#21461511C211yesmaximal, characteristic, complemented
7#21461484C211yesmaximal, characteristic, complemented
8C22×#14519798C211yesmaximal, characteristic
9#16840840—24no—
10#16840840—24no—
11C2×#14519864—22no—
12C2×#14519864C2214yescomplemented
13C2×D61⋊M16C2211yescharacteristic
14C2×D61⋊M16C2211yescharacteristic
15C2×D61⋊M16C2211yescharacteristic
16C22×#11758217—22no—
17C2×#14519856C2211yescharacteristic
18#16840761—24no—
19#16840760—24no—
20C2×#14519842C2214yescomplemented
21C2×#14519841C2214yescomplemented
22#16840737—22no—
23#16840737C2214yescomplemented
24#16840734—22no—
25#16840638C2214yescomplemented
26(C2×#6923)⋊C8C2214yescomplemented
27C22×#11758226C2211yescharacteristic
28C22×#11758224C2211yescharacteristic
29C2×#14519798—24no—
30C2×#14519798—24no—
31C2×#14519798C414yescomplemented
32C2×#14519798C2214yes—
33C22×#11758202C411yescharacteristic, complemented
34C22×#11758190C411yescharacteristic, complemented
35C22×#11758199C411yescharacteristic, complemented
36C22×C8×D61C2211yescharacteristic
37C22×SD16×C61C411yescharacteristic, complemented
38#14519864—216no—
39#14519864—28no—
40#14519864—28no—
41C2×#11758213D412yes—
42C2×#11758213C2312yes—
43C2×#11758213C2312yes—
44D61⋊M16—22no—
45D61⋊M16—22no—
46D61⋊M16—22no—
47D61⋊M16—22no—
48D61⋊M16—22no—
49D61⋊M16D414yes—
50C2×#11758217—216no—
51C2×#11758217—21no—
52C2×#11758217—21no—
53C2×#11758217—24no—
54C2×#11758217D412yes—
55C2×D61⋊C8—21no—
56C2×D61⋊C8—21no—
57C2×D61⋊C8C2312yes—
58C22×#92420—24no—
59#14519856—22no—
60#14519856—22no—
61C2×C4⋊#6923—21no—
62C2×C22⋊#6923—21no—
63C2×C4×#6923C2312yes—
64#14519842—28no—
65#14519841—28no—
66#14519834—28no—
67C2×#11758226—21no—
68C2×#11758226—21no—
69C2×#11758226—22no—
70C2×#11758226—22no—
71C2×#11758226—22no—
72C2×#11758226—22no—
73C2×#11758226C2×C412yes—
74C2×#11758226C2312yes—
75C2×#11758224—21no—
76C2×#11758224—22no—
77C2×#11758224—22no—
78C2×#11758224—21no—
79C2×#11758224C2×C412yes—
80C2×#11758224D414yes—
81C2×#11758224D414yes—
82C2×#11758224C2312yes—
83C22×Dic122C2×C411yescharacteristic
84C22×D244—21no—
85C22×D244C2×C411yescharacteristic
86C22×C22⋊D61—21no—
87C23×Dic61—21no—
88C22×C4×D61—21no—
89C22×C4×D61C2311yescharacteristic
90#14519798—216no—
91#14519798—28no—
92#14519798—28no—
93C2×#11758202—24no—
94C2×#11758202C2×C414yes—
95C2×#11758190—24no—
96C2×#11758190C2×C414yes—
97C2×#11758199—24no—
98C2×#11758199C2×C414yes—
99C22×#92418C2×C411yescharacteristic
100C2×C8×D61—22no—
101C2×C8×D61—22no—
102C2×C8×D61C2×C412yes—
103C2×C8×D61C2312yes—
104C22×Q8×C61C2×C411yescharacteristic
105C22×D4×C61C2×C411yescharacteristic
106C2×SD16×C61—24no—
107C2×SD16×C61C2×C414yes—
108C22×C8×C61C2×C411yescharacteristic
109C24×C61—21no—
110C23×D61—21no—
111C23×D61#4111yescharacteristic
112C22×#6923—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C22×C412yes—
117C2×D4×C61—22no—
118C2×D4×C61—21no—
119C2×D4×C61C22⋊C414yes—
120C2×D4×C61C22×C412yes—
121C22×C4×C61—21no—
122C22×C4×C61C22×C411yescharacteristic
123C2×C22⋊D61—22no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C2×C22⋊D61—22no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—24no—
130C22×Dic61—24no—
131C22×Dic61—22no—
132C22×Dic61C2×D411yescharacteristic
133#11758226—24no—
134#11758226—24no—
135#11758226—22no—
136#11758226—22no—
137#11758226—22no—
138#11758226—22no—
139#11758226—42no—
140#11758226—42no—
141#11758226—42no—
142#11758226—42no—
143#11758224—24no—
144#11758224—24no—
145#11758224—24no—
146#11758224—24no—
147#11758224—24no—
148#11758224—22no—
149#11758224—22no—
150#11758224C22⋊C418yes—
151#11758224C2×D414yes—
152#11758224C2×D414yes—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244—42no—
159C2×D244—42no—
160C2×D244C22×C412yes—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61C22×C411yescharacteristic
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61#4112yes—
171C2×C4×D61#4111yescharacteristic
172C2×C4×D61C2411yescharacteristic
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122C22⋊C414yes—
176C2×Dic122C22×C412yes—
177C22⋊#6923—42no—
178#11758217—216no—
179#11758217—24no—
180#11758217—24no—
181#11758217—42no—
182C2×#92420—216no—
183C2×#92420—22no—
184C2×#92420—22no—
185C2×#92420C2×D414yes—
186C4⋊#6923—42no—
187C4×#6923—22no—
188C4×#6923—22no—
189#11758213—24no—
190#11758213—22no—
191#11758213—22no—
192#11758213—22no—
193#11758213C2×D414yes—
194D61⋊C8—24no—
195D61⋊C8—22no—
196D61⋊C8—22no—
197D61⋊C8—42no—
198SD16×C61—28no—
199C2×C8×C61—22no—
200C2×C8×C61C22×C412yes—
201#11758202—28no—
202#11758199—28no—
203C2×#92418—22no—
204C2×#92418C22×C412yes—
205#11758190—28no—
206C8×D61—24no—
207C8×D61—22no—
208C8×D61—22no—
209C23×C61—22no—
210C23×C61—24no—
211C23×C61—82no—
212C23×C61C4×D411yescharacteristic
213C22×D61—22no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—82no—
217C22×D61C4×D411yescharacteristic
218C22×D61#12112yes—
219C22×D61#12312yes—
220C22×D61C2×#4111yescharacteristic
221C2×#6923—44no—
222C2×#6923—42no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—42no—
226D4×C61—22no—
227D4×C61—24no—
228D4×C61C2×C22⋊C414yes—
229C2×C4×C61—22no—
230C2×C4×C61—42no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C22⋊D61—24no—
236C22⋊D61—24no—
237C22⋊D61—28no—
238C2×Dic61—22no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—28no—
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61#12012yes—
245C2×Dic61#12112yes—
246C2×Dic61C22×D411yescharacteristic
247D244—22no—
248D244—22no—
249D244—42no—
250D244—42no—
251D244—44no—
252D244—42no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—44no—
256C4×D61C2×C22⋊C412yes—
257C4×D61C4×D412yes—
258C4×D61#12012yes—
259C4×D61#12112yes—
260C4×D61#12112yes—
261C4×D61#12312yes—
262C4×D61C22×D411yescharacteristic
263C4×D61C22×D411yescharacteristic
264C4×D61C22×D411yescharacteristic
265C4×D61C2×#4111yescharacteristic
266Dic122—22no—
267Dic122—24no—
268Dic122C2×C22⋊C414yes—
269#92420—24no—
270#92420—24no—
271#92420—42no—
272C8×C61—22no—
273#92418—22no—
274#20919—611nomaximal, Sylow
275C22×C61—22no—
276C22×C61—24no—
277C22×C61—44no—
278C22×C61—84no—
279C22×C61#38612yes—
280C22×C61C2×C4×D411yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—44no—
286C2×D61—42no—
287C2×D61—84no—
288C2×D61#38612yes—
289C2×D61C2×#12111yescharacteristic
290C2×D61C2×#12311yescharacteristic
291#6923—82no—
292C4×C61—42no—
293C4×C61#38612yes—
294C4×C61C22×C22⋊C411yescharacteristic
295C4×C61C2×C4×D411yescharacteristic
296Dic61—24no—
297Dic61#38612yes—
298Dic61C2×#12011yescharacteristic
299Dic61C2×#12111yescharacteristic
300C22×#135—611no—
301C2×#568—611no—
302C2×#421—611no—
303C2×#421—611no—
304#2879—611no—
305#1876—618no—
306C2×#341—611no—
307#1729—611no—
308C2×C61—22no—
309C2×C61—44no—
310C2×C61C2×#38611yescharacteristic
311C2×C61#295712yes—
312D61—21no—
313D61—21no—
314D61—22no—
315D61—82no—
316C22×#41—611no—
317C22×#41—611no—
318C2×#135—614no—
319C2×#135—614no—
320C2×#135—1224no—
321C2×#135—1224no—
322C22×Q16—611no—
323C22×SD16—611no—
324C22×SD16#692311yescharacteristic, complemented
325C22×D8—611no—
326#568—1224no—
327#568—1224no—
328C2×#131—614no—
329C2×#131—1222no—
330C22×M16—611no—
331C22×M16—611no—
332C22×M16—611no—
333C22×M16—1222no—
334C23×C8—611no—
335C2×#117—611no—
336#421—1224no—
337#421—1224no—
338C2×#104—614no—
339C2×#104—614no—
340#408—614no—
341#408—1222no—
342#407—1222no—
343#341—614no—
344#336—614no—
345C61#2091911yescharacteristic, complemented, Sylow
346C2×#41—612no—
347C2×#41—612no—
348C2×#41—614no—
349C2×#41—614no—
350C2×#41—612no—
351C2×#41—612no—
352C2×#41—1221no—
353C2×#41—1222no—
354C2×#41—1221no—
355C2×#41—1222no—
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1222no—
359C2×#41—1222no—
360C2×#41—1222no—
361C2×#41—1222no—
362C22×Q8—611no—
363C22×Q8C2×#692311yescharacteristic
364C22×D4—611no—
365C22×D4—1221no—
366C22×D4—1221no—
367C22×D4C2×#692311yescharacteristic
368C23×C4—611no—
369C23×C4—1221no—
370C23×C4—1221no—
371#135—1228no—
372#135—1228no—
373#135—12216no—
374C2×Q16—614no—
375C2×Q16—1224no—
376C2×SD16—24no—
377C2×SD16—614no—
378C2×SD16—1224no—
379C2×SD16C2×#692314yes—
380C2×D8—614no—
381C2×D8—1224no—
382#131—1228no—
383#131—1228no—
384#131—12216no—
385C2×M16—612no—
386C2×M16—612no—
387C2×M16—614no—
388C2×M16—612no—
389C2×M16—612no—
390C2×M16—1224no—
391C2×M16—1221no—
392C2×M16—1222no—
393C2×M16—1222no—
394C2×M16—1222no—
395C2×M16—12216no—
396C2×M16—1221no—
397C2×M16—1222no—
398C2×M16—1222no—
399C22×C8—612no—
400C22×C8—611no—
401C22×C8—612no—
402C22×C8—612no—
403C22×C8—1222no—
404C22×C8—1222no—
405C22×C8—1224no—
406C22×C8—1221no—
407C22×C8—1221no—
408C22×C8C2×#692311yescharacteristic
409#117—1222no—
410#117—1222no—
411C2×C4⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C42—612no—
414#104—1228no—
415#104—1228no—
416C22⋊C8—1228no—
417C24—21no—
418C24—611no—
419C24—1221no—
420#41—614no—
421#41—614no—
422#41—618no—
423#41—1224no—
424#41—1222no—
425#41—1224no—
426#41—1224no—
427#41—1224no—
428#41—1224no—
429#41—1222no—
430#41—1222no—
431#41—1224no—
432#41—1222no—
433#41—1222no—
434#41—1224no—
435#41—1222no—
436#41—2442no—
437#41—2442no—
438#41—2442no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—614no—
444C2×Q8—612no—
445C2×Q8—1222no—
446C2×Q8—1221no—
447C2×Q8C22×#692312yes—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—612no—
451C2×D4—1222no—
452C2×D4—1222no—
453C2×D4—1224no—
454C2×D4—1224no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1221no—
460C2×D4—2442no—
461C2×D4—2442no—
462C2×D4C22⋊#692314yes—
463C2×D4C22×#692312yes—
464C22×C4—21no—
465C22×C4—611no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—611no—
469C22×C4—612no—
470C22×C4—611no—
471C22×C4—612no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1221no—
481C22×C4—1222no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4C22×#692311yescharacteristic
485Q16—1228no—
486SD16—28no—
487SD16—1228no—
488D8—1228no—
489M16—614no—
490M16—1224no—
491M16—1224no—
492M16—1224no—
493M16—1222no—
494M16—12216no—
495M16—1222no—
496M16—1222no—
497M16—2442no—
498C2×C8—22no—
499C2×C8—614no—
500C2×C8—612no—
501C2×C8—1224no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1224no—
508C2×C8—12216no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—2442no—
512C2×C8C22×#692312yes—
513C4⋊C4—2442no—
514C22⋊C4—2442no—
515C42—1222no—
516C42—1222no—
517C23—22no—
518C23—24no—
519C23—82no—
520C23—611no—
521C23—611no—
522C23—612no—
523C23—612no—
524C23—1222no—
525C23—4882no—
526C23—2442no—
527C23—2442no—
528C23D4×#692311yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—42no—
532Q8—614no—
533Q8—1224no—
534Q8—1222no—
535D4—24no—
536D4—22no—
537D4—1224no—
538D4—1224no—
539D4—1228no—
540D4—1222no—
541D4—1222no—
542D4—2442no—
543D4—2442no—
544D4—2444no—
545D4—2442no—
546D4C2×C22⋊#692314yes—
547C2×C4—22no—
548C2×C4—42no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—612no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1228no—
566C2×C4—1224no—
567C2×C4—1222no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2444no—
571C2×C4—2442no—
572C2×C4—2444no—
573C2×C4C2×C22⋊#692312yes—
574C2×C4D4×#692311yescharacteristic
575C2×C4D4×#692312yes—
576C2×C4C23×#692311yescharacteristic
577C8—22no—
578C8—1224no—
579C8—1224no—
580C8—1222no—
581C8—2442no—
582C22—22no—
583C22—24no—
584C22—44no—
585C22—84no—
586C22—611no—
587C22—611no—
588C22—612no—
589C22—1222no—
590C22—1222no—
591C22—4884no—
592C22—2442no—
593C22—2442no—
594C22—2444no—
595C22—2442no—
596C22#1684071012yes—
597C22C2×D4×#692311yescharacteristic
598C4—42no—
599C4—611no—
600C4—611no—
601C4—612no—
602C4—1224no—
603C4—4882no—
604C4#1684071012yes—
605C4C22×C22⋊#692311yescharacteristic
606C4C2×D4×#692311yescharacteristic
607C2—22no—
608C2—44no—
609C2—1221no—
610C2—1221no—
611C2—1222no—
612C2—4882no—
613C2C2×#1684071011yescharacteristic
614C2#2146259712yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC22×SD16×C61C4
socleC22×C61C2×C4×D4
prime radicalC22×#14519798C2
cosocleC2×C4×D61C24
hypercenterC22×SD16#6923
hypocenterC61#20919
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16840840C2semidirect
2. C2×#16840761C2semidirect
3. C2×#16840760C2semidirect
4. #21462406C2semidirect
5. #21461808C2semidirect
6. #21461511C2semidirect
7. #21461484C2semidirect
12. C2×#14519864C22semidirect
20. C2×#14519842C22semidirect
21. C2×#14519841C22semidirect
23. #16840737C22semidirect
25. #16840638C22semidirect
26. (C2×#6923)⋊C8C22semidirect
31. C2×#14519798C4semidirect
33. C22×#11758202C4semidirect
34. C22×#11758190C4semidirect
35. C22×#11758199C4semidirect
37. C22×SD16×C61C4semidirect
324. C22×SD16#6923semidirect
345. C61#20919semidirect

Maximal subgroups

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

subgroupindex
C2×#168408402
C2×#168407612
C2×#168407602
#214624062
#214618082
#214615112
#214614842
C22×#145197982
#2091961