Finite Group Databasebeta

← #29656525

Subgroups — #29656525

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457202C211yesmaximal, characteristic, complemented
2#21456269C211yesmaximal, characteristic, complemented
3#21455879C211yesmaximal, characteristic, complemented
4#21444946C212yesmaximal, complemented
5#21444940C212yesmaximal, complemented
6#21444884C212yesmaximal, complemented
7#21444878C212yesmaximal, complemented
8#21444487C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3508C211yesmaximal, characteristic, complemented
12C2×#14519803—22no—
13#16840312C2211yescharacteristic
14#16840149—22no—
15#16840138C2212yes—
16#16840134C2212yes—
17#16840086—21no—
18#16840057C2211yescharacteristic, complemented
19#16840057C2212yescomplemented
20#16840052C2211yescharacteristic, complemented
21#16840039—21no—
22#16840010—22no—
23#16839964—21no—
24#16839962C2211yescharacteristic, complemented
25#16839962C2212yescomplemented
26#16839957C2211yescharacteristic, complemented
27#16839907—21no—
28C4×#11758201—21no—
29C4×#11758199—21no—
30#16839815C2211yescharacteristic
31#16839786C2211yescharacteristic
32C4×#11758194C2211yescharacteristic
33#16839506C2212yes—
34#16839506C2212yes—
35Q8⋊#92418C2212yescomplemented
36Q8⋊#92418C2212yescomplemented
37Q8⋊#92418C2212yescomplemented
38D4⋊#92418C2212yescomplemented
39C61×#559—21no—
40C61×#559C2212yes—
41C61×#550—21no—
42C61×#550C2212yes—
43C61×#546—22no—
44C61×#546—22no—
45C61×#540—22no—
46C61×#535—21no—
47C61×#533C2211yescharacteristic
48C61×#532—21no—
49C61×#532—21no—
50C2×#127×C61C2211yescharacteristic
51C61×#518—21no—
52C4×#41×C61C2211yescharacteristic
53C2×#11758222—21no—
54#14519803—48no—
55C2×#11758200—24no—
56C2×#11758201—21no—
57C2×#11758201—22no—
58C2×#11758199—21no—
59C2×#11758199—24no—
60C2×#11758194—21no—
61C2×#11758194—21no—
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519710—22no—
70#14519710—21no—
71#14519708—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75C4⋊#92418C2311yescharacteristic
76#14519701—22no—
77#14519701—21no—
78#14519698—21no—
79#14519697—21no—
80#14519697—22no—
81#14519697—21no—
82C22⋊#92418—21no—
83#14519688—21no—
84C4×#92418—21no—
85C4×#92418C2312yes—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61D411yescharacteristic
89C22×D4×C61—21no—
90C22×D4×C61—21no—
91C22×D4×C61D412yes—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61D411yescharacteristic
94#127×C61—22no—
95#127×C61—21no—
96#127×C61—21no—
97#127×C61D412yes—
98#127×C61C2312yes—
99#125×C61—21no—
100#124×C61—22no—
101#124×C61—22no—
102#124×C61—22no—
103#123×C61—24no—
104#123×C61—24no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61D412yes—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—22no—
110#121×C61—24no—
111#121×C61—24no—
112#121×C61—22no—
113#120×C61—24no—
114C4×Q8×C61—21no—
115C4×Q8×C61D411yescharacteristic
116C4×Q8×C61C2312yes—
117C4×Q8×C61C2311yescharacteristic
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C4×D4×C61—22no—
126C4×D4×C61D411yescharacteristic
127C4×D4×C61C2311yescharacteristic
128#117×C61—21no—
129#117×C61—21no—
130#117×C61D411yescharacteristic
131C2×C22⋊C4×C61—24no—
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C24×C61—22no—
135#41×C61—41no—
136#41×C61—41no—
137#41×C61—82no—
138#41×C61—82no—
139#41×C61—84no—
140C2×Q8×C61—21no—
141C2×Q8×C61C2×D411yescharacteristic
142C2×Q8×C61C2×D412yes—
143C2×D4×C61—21no—
144C2×D4×C61—24no—
145C2×D4×C61—24no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—22no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61—44no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D414yes—
158C22×C4×C61—24no—
159C22×C4×C61—24no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C2×C22⋊D61—41no—
167#11758222—44no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×C4×D61—41no—
172C2×Dic122—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D412yes—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—24no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—22no—
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199#11758201—42no—
200#11758201—82no—
201#11758200—48no—
202#11758199—48no—
203#11758199—42no—
204#11758198—41no—
205#11758198—41no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208#11758194—24no—
209#11758194—81no—
210C2×#92418—22no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—22no—
214C23×C61—24no—
215C23×C61—21no—
216C23×C61—44no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61#12011yescharacteristic
220C22×D61—41no—
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—42no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—44no—
229D4×C61—44no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—42no—
235D4×C61—44no—
236D4×C61—82no—
237D4×C61—84no—
238D4×C61—84no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—42no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61—84no—
254C2×C4×C61—84no—
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C22⋊D61—82no—
262C2×Dic61—41no—
263D244—42no—
264D244—41no—
265D244—41no—
266D244—41no—
267D244—41no—
268C4×D61—82no—
269Dic122—42no—
270#92418—24no—
271#92418—41no—
272#92418—41no—
273#92418—42no—
274#18568—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—41no—
279C22×C61—44no—
280C22×C61—41no—
281C22×C61—44no—
282C22×C61—41no—
283C22×C61—44no—
284C22×C61—164no—
285C22×C61C2×#12011yescharacteristic
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61#45312yes—
298C4×C61#57511yescharacteristic
299C4×C61#57511yescharacteristic
300C4×C61#57611yescharacteristic
301C4×C61#57711yescharacteristic
302Dic61—81no—
303#3508D6111yescharacteristic, complemented
304#3254—611no—
305#3151—611no—
306#3106—611no—
307#1631—612no—
308#1630—612no—
309#1621—612no—
310#1620—612no—
311#1545—611no—
312#1489—611no—
313#1485—611no—
314C2×C61—41no—
315C2×C61—81no—
316C2×C61—84no—
317C2×C61#300511yescharacteristic
318C2×C61#301211yescharacteristic
319C2×C61C2×#45311yescharacteristic
320D61—161no—
321C2×AGL(1,ℤ/8)—1222no—
322#559—21no—
323#559C2×D6112yes—
324#550—21no—
325#550C2×D6112yes—
326#546—22no—
327#546—22no—
328#540—22no—
329#535—21no—
330#533C2×D6111yescharacteristic
331#532—21no—
332#532—21no—
333#532—611no—
334C2×#127C2×D6111yescharacteristic
335#518—21no—
336C4×#41C2×D6111yescharacteristic
337#496—1222no—
338#493—612no—
339#492—612no—
340#482—1221no—
341#477—611no—
342#477—612no—
343#476—611no—
344#474—1221no—
345#468—1222no—
346#461—1221no—
347#460—611no—
348#460—612no—
349#459—611no—
350#449—1221no—
351C4×SD16—1221no—
352C4×D8—1221no—
353#431—611no—
354#423—611no—
355C4×M16—611no—
356#331—612no—
357#331—612no—
358Q8⋊C8—612no—
359Q8⋊C8—612no—
360Q8⋊C8—612no—
361D4⋊C8—612no—
362C61#1856811yescharacteristic, complemented, Sylow
363C2×#41—22no—
364C2×#41—22no—
365C2×#41—1221no—
366C2×#41C22⋊D6111yescharacteristic
367C22×D4—21no—
368C22×D4—21no—
369C22×D4C22⋊D6112yes—
370AGL(1,ℤ/8)—2448no—
371C2×SD16—1222no—
372C2×SD16—1221no—
373C2×SD16—1224no—
374C2×D8—1224no—
375C2×D8—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C4⋊Q8—611no—
379C4⋊Q8C22⋊D6111yescharacteristic
380C4⋊Q8C22⋊D6111yescharacteristic
381#127—22no—
382#127—21no—
383#127—21no—
384#127—611no—
385#127C22⋊D6112yes—
386#127C22×D6112yes—
387#125—21no—
388#124—22no—
389#124—22no—
390#124—22no—
391#124—1221no—
392#123—24no—
393#123—24no—
394C22⋊Q8—21no—
395C22⋊Q8C22⋊D6112yes—
396#121—22no—
397#121—22no—
398#121—22no—
399#121—24no—
400#121—24no—
401#121—22no—
402#121—1221no—
403#121—1221no—
404#120—24no—
405C4×Q8—21no—
406C4×Q8C22⋊D6111yescharacteristic
407C4×Q8C22×D6111yescharacteristic
408C4×Q8C22×D6112yes—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4C22⋊D6111yescharacteristic
420C4×D4C22×D6111yescharacteristic
421#117—21no—
422#117—21no—
423#117C22⋊D6111yescharacteristic
424C2×C22⋊C4—24no—
425C2×C42—21no—
426C2×C42C22×D6111yescharacteristic
427#107—1221no—
428#107—1222no—
429#106—1221no—
430C4⋊C8—612no—
431C4⋊C8—611no—
432C4⋊C8—611no—
433C4⋊C8—1221no—
434#103—1221no—
435#103—1222no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1222no—
440C22⋊C8—1221no—
441#97—1221no—
442C4×C8—612no—
443C4×C8—1221no—
444C24—22no—
445#41—41no—
446#41—41no—
447#41—82no—
448#41—82no—
449#41—84no—
450#41—2444no—
451C2×Q8—21no—
452C2×Q8—1221no—
453C2×Q8C2×C22⋊D6111yescharacteristic
454C2×Q8C2×C22⋊D6112yes—
455C2×D4—21no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—81no—
467C2×D4—81no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—2441no—
472C2×D4C2×C22⋊D6111yescharacteristic
473C2×D4C2×C22⋊D6114yes—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—2441no—
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483SD16—4882no—
484SD16—2448no—
485SD16—2442no—
486D8—2448no—
487D8—2442no—
488M16—1224no—
489M16—4881no—
490C2×C8—1222no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1222no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C4⋊C4C2×C22⋊D6111yescharacteristic
505C4⋊C4C2×C22⋊D6112yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—21no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42—21no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42C2×C22⋊D6111yescharacteristic
521C42C2×C22⋊D6111yescharacteristic
522C42C2×C22⋊D6111yescharacteristic
523C42C23×D6111yescharacteristic
524C23—24no—
525C23—21no—
526C23—44no—
527C23—82no—
528C23—82no—
529C23—2441no—
530C23#1451974611yescharacteristic
531Q8—21no—
532Q8—21no—
533Q8—22no—
534Q8—22no—
535Q8—42no—
536Q8—2442no—
537D4—21no—
538D4—21no—
539D4—44no—
540D4—44no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—44no—
547D4—82no—
548D4—84no—
549D4—84no—
550D4—4882no—
551D4—2442no—
552D4—2441no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—4882no—
573C2×C4—2441no—
574C2×C4#1451974611yescharacteristic
575C2×C4#1451974611yescharacteristic
576C2×C4#1451974611yescharacteristic
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4C22×C22⋊D6111yescharacteristic
580C8—1224no—
581C8—2441no—
582C8—2441no—
583C8—2442no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—41no—
588C22—44no—
589C22—41no—
590C22—44no—
591C22—41no—
592C22—44no—
593C22—164no—
594C22—4881no—
595C22—4881no—
596C22—4881no—
597C22C2×#1451974611yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—4881no—
607C4#1683993012yes—
608C4#1684056011yescharacteristic
609C4#1684056011yescharacteristic
610C4#1684056511yescharacteristic
611C4#1684057311yescharacteristic
612C2—41no—
613C2—84no—
614C2—81no—
615C2—9761no—
616C2#2145506111yescharacteristic
617C2#2145512311yescharacteristic
618C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#18568
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21457202C2semidirect
2. #21456269C2semidirect
3. #21455879C2semidirect
4. #21444946C2semidirect
5. #21444940C2semidirect
6. #21444884C2semidirect
7. #21444878C2semidirect
8. #21444487C2semidirect
9. #21444258C2semidirect
10. #21444240C2semidirect
11. C61×#3508C2semidirect
18. #16840057C22semidirect
19. #16840057C22semidirect
20. #16840052C22semidirect
24. #16839962C22semidirect
25. #16839962C22semidirect
26. #16839957C22semidirect
35. Q8⋊#92418C22semidirect
36. Q8⋊#92418C22semidirect
37. Q8⋊#92418C22semidirect
38. D4⋊#92418C22semidirect
303. #3508D61semidirect
362. C61#18568semidirect

Maximal subgroups

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

subgroupindex
#214572022
#214562692
#214558792
#214449462
#214449402
#214448842
#214448782
#214444872
#214442582
#214442402
C61×#35082
#1856861