Finite Group Databasebeta

← #29656679

Subgroups — #29656679

Order 15616.

1124 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#21457197C211yesmaximal, characteristic, complemented
2#21456215C211yesmaximal, characteristic, complemented
3#21455879C211yesmaximal, characteristic, complemented
4#21444984C212yesmaximal, complemented
5#21444970C212yesmaximal, complemented
6#21444922C212yesmaximal, complemented
7#21444908C212yesmaximal, complemented
8#21444504C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3499C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840124—22no—
14#16840114C2212yes—
15#16840110C2212yes—
16#16840086—21no—
17#16840057C2211yescharacteristic
18#16840052C2211yescharacteristic
19#16840052C2212yes—
20#16840049—21no—
21#16840013—22no—
22#16839962C2211yescharacteristic
23#16839957C2211yescharacteristic
24#16839957C2212yes—
25#16839954—21no—
26#16839907—21no—
27C4×#11758201—21no—
28C4×#11758199—21no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839761—22no—
32C4×#11758194C2211yescharacteristic
33#16839508C2212yes—
34#16839508C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38D4⋊#92418C2212yes—
39C2×2−1+4×C61—21no—
40D4×Q8×C61—21no—
41D4×Q8×C61C2212yes—
42C61×#547—22no—
43C61×#547—21no—
44C61×#547—22no—
45C61×#547C2212yes—
46C61×#536—22no—
47C61×#536—21no—
48C61×#532—21no—
49C61×#532C2211yescharacteristic
50C2×C4⋊Q8×C61—21no—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C4×#41×C61C2211yescharacteristic
53C2×#11758222—21no—
54C2×#11758201—21no—
55C2×#11758199—21no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519710—21no—
67#14519708—22no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2311yescharacteristic
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73#14519701—21no—
74#14519700—24no—
75#14519698—22no—
76#14519698—21no—
77#14519697—22no—
78#14519697—21no—
79#14519697—21no—
80#14519697—24no—
81C22⋊#92418—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
852−1+4×C61—44no—
862−1+4×C61—44no—
87C2×#41×C61—22no—
88C2×#41×C61—24no—
89C2×#41×C61—22no—
90C2×#41×C61—21no—
91C2×#41×C61D411yescharacteristic
92C22×Q8×C61—22no—
93C22×Q8×C61—22no—
94C22×Q8×C61—21no—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—22no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61D411yescharacteristic
101C4⋊Q8×C61D412yes—
102C4⋊Q8×C61C2312yes—
103#127×C61D411yescharacteristic
104#124×C61—22no—
105#124×C61—21no—
106#123×C61—24no—
107#123×C61—24no—
108#123×C61—24no—
109C22⋊Q8×C61—24no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—24no—
112C22⋊Q8×C61—22no—
113#121×C61—22no—
114#121×C61—22no—
115#121×C61—21no—
116#121×C61D412yes—
117C4×Q8×C61D411yescharacteristic
118C4×Q8×C61C2311yescharacteristic
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124C4×D4×C61D411yescharacteristic
125C4×D4×C61C2312yes—
126C4×D4×C61C2311yescharacteristic
127#117×C61—21no—
128#117×C61D411yescharacteristic
129C2×C4⋊C4×C61—22no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—22no—
132C2×C4⋊C4×C61D412yes—
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#41×C61—22no—
136#41×C61—22no—
137#41×C61—48no—
138#41×C61—42no—
139#41×C61—42no—
140#41×C61—416no—
141#41×C61—44no—
142#41×C61—44no—
143C2×Q8×C61—24no—
144C2×Q8×C61—22no—
145C2×Q8×C61—21no—
146C2×Q8×C61—22no—
147C2×Q8×C61—48no—
148C2×Q8×C61—42no—
149C2×Q8×C61—42no—
150C2×Q8×C61—44no—
151C2×Q8×C61—44no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61—24no—
155C2×D4×C61—22no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61C2×D412yes—
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—21no—
161C22×C4×C61—24no—
162C22×C4×C61—24no—
163C22×C4×C61—22no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—22no—
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C2×C22⋊D61—41no—
171#11758222—44no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×C4×D61—41no—
176C2×Dic122—21no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C2×D412yes—
186C4⋊C4×C61C2×D414yes—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—24no—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—22no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61C2×D411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2411yescharacteristic
203#11758201—42no—
204#11758199—42no—
205#11758199—82no—
206#11758198—41no—
207#11758198—41no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210#11758194—42no—
211#11758194—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—22no—
215C2×#92418—22no—
216C23×C61—21no—
217C23×C61—22no—
218C23×C61#12011yescharacteristic
219C22×D61—41no—
220Q8×C61—21no—
221Q8×C61—44no—
222Q8×C61—42no—
223Q8×C61—48no—
224Q8×C61—42no—
225Q8×C61—42no—
226Q8×C61AGL(1,ℤ/8)12yes—
227D4×C61—24no—
228D4×C61—21no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—42no—
232D4×C61—44no—
233D4×C61—48no—
234D4×C61—42no—
235D4×C61AGL(1,ℤ/8)12yes—
236C2×C4×C61—24no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—48no—
248C2×C4×C61—44no—
249C2×C4×C61—48no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
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—42no—
271#92418—41no—
272#92418—41no—
273#92418—42no—
274#18582—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—22no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61C2×#12011yescharacteristic
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—44no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61#45412yes—
297C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C61#57711yescharacteristic
300C4×C61#57811yescharacteristic
301Dic61—81no—
302#3499D6111yescharacteristic, complemented
303#3253—611no—
304#3141—611no—
305#3106—611no—
306#1637—612no—
307#1635—612no—
308#1627—612no—
309#1625—612no—
310#1549—611no—
311#1489—611no—
312#1484—611no—
313C2×C61—41no—
314C2×C61—42no—
315C2×C61—82no—
316C2×C61#300311yescharacteristic
317C2×C61#301311yescharacteristic
318C2×C61C2×#45411yescharacteristic
319D61—161no—
320C2×2−1+4—21no—
321D4×Q8—21no—
322D4×Q8C2×D6112yes—
323#547—22no—
324#547—21no—
325#547—22no—
326#547C2×D6112yes—
327#536—22no—
328#536—21no—
329#532—21no—
330#532—611no—
331#532C2×D6111yescharacteristic
332C2×C4⋊Q8—21no—
333C2×C4⋊Q8C2×D6111yescharacteristic
334C4×#41C2×D6111yescharacteristic
335#489—1222no—
336#487—612no—
337#486—612no—
338#482—1221no—
339#477—611no—
340#476—611no—
341#476—612no—
342#476—1221no—
343#469—1222no—
344#460—611no—
345#459—611no—
346#459—612no—
347#459—1221no—
348#449—1221no—
349C4×SD16—1221no—
350C4×D8—1221no—
351#431—611no—
352#423—611no—
353#417—1222no—
354C4×M16—611no—
355#332—612no—
356#332—612no—
357Q8⋊C8—612no—
358D4⋊C8—612no—
359D4⋊C8—612no—
360D4⋊C8—612no—
361C61#1858211yescharacteristic, complemented, Sylow
3622−1+4—44no—
3632−1+4—44no—
364C2×#41—22no—
365C2×#41—24no—
366C2×#41—22no—
367C2×#41—21no—
368C2×#41—1221no—
369C2×#41C22⋊D6111yescharacteristic
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—21no—
373C2×SD16—1221no—
374C2×D8—1222no—
375C2×D8—1221no—
376C2×M16—1221no—
377C2×M16—1221no—
378C4⋊Q8—22no—
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381C4⋊Q8—21no—
382C4⋊Q8—21no—
383C4⋊Q8—611no—
384C4⋊Q8C22⋊D6111yescharacteristic
385C4⋊Q8C22⋊D6112yes—
386C4⋊Q8C22×D6112yes—
387#127—611no—
388#127C22⋊D6111yescharacteristic
389#124—22no—
390#124—21no—
391#124—1221no—
392#123—24no—
393#123—24no—
394#123—24no—
395C22⋊Q8—24no—
396C22⋊Q8—22no—
397C22⋊Q8—24no—
398C22⋊Q8—22no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121—1221no—
403#121—1221no—
404#121C22⋊D6112yes—
405C4×Q8C22⋊D6111yescharacteristic
406C4×Q8C22×D6111yescharacteristic
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4C22⋊D6111yescharacteristic
415C4×D4C22×D6111yescharacteristic
416C4×D4C22×D6112yes—
417#117—21no—
418#117C22⋊D6111yescharacteristic
419C2×C4⋊C4—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4C22⋊D6112yes—
423C2×C42—21no—
424C2×C42C22×D6111yescharacteristic
425#107—1221no—
426#106—1221no—
427#106—1222no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—612no—
431C4⋊C8—1221no—
432#103—1221no—
433#103—1224no—
434#102—1221no—
435#102—1222no—
436#102—1224no—
437#102—1221no—
438#102—1221no—
439#102—1222no—
440C22⋊C8—1221no—
441#97—1221no—
442C4×C8—612no—
443C4×C8—1221no—
444#41—22no—
445#41—22no—
446#41—48no—
447#41—42no—
448#41—42no—
449#41—416no—
450#41—44no—
451#41—44no—
452#41—2444no—
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—48no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—44no—
461C2×Q8—44no—
462C2×Q8—1221no—
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×D4—21no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—2441no—
473C2×D4C2×C22⋊D6112yes—
474C2×D4C2×C22⋊D6111yescharacteristic
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—2441no—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486SD16—2442no—
487D8—4882no—
488D8—2442no—
489M16—4881no—
490M16—2442no—
491C2×C8—1222no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1222no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4C2×C22⋊D6114yes—
506C4⋊C4C2×C22⋊D6112yes—
507C4⋊C4C2×C22⋊D6111yescharacteristic
508C4⋊C4C2×C22⋊D6111yescharacteristic
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—2441no—
517C22⋊C4—2441no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42C2×C22⋊D6111yescharacteristic
522C42C2×C22⋊D6111yescharacteristic
523C42C2×C22⋊D6111yescharacteristic
524C42C23×D6111yescharacteristic
525C23—21no—
526C23—22no—
527C23—2441no—
528C23#1451974611yescharacteristic
529Q8—21no—
530Q8—44no—
531Q8—42no—
532Q8—48no—
533Q8—42no—
534Q8—42no—
535Q8—2442no—
536Q8#1451980312yes—
537D4—24no—
538D4—21no—
539D4—22no—
540D4—22no—
541D4—42no—
542D4—44no—
543D4—48no—
544D4—42no—
545D4—4882no—
546D4—2442no—
547D4—2441no—
548D4—2441no—
549D4—2441no—
550D4—2441no—
551D4#1451980312yes—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—4882no—
572C2×C4—2441no—
573C2×C4#1451974611yescharacteristic
574C2×C4#1451974611yescharacteristic
575C2×C4#1451974611yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C8—2442no—
580C8—2441no—
581C8—2441no—
582C8—2442no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22C2×#1451974611yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—44no—
603C4—42no—
604C4—4881no—
605C4#1683993512yes—
606C4C2×#1451980311yescharacteristic
607C4C2×#1451980311yescharacteristic
608C4#1684057011yescharacteristic
609C4#1684057711yescharacteristic
610C2—41no—
611C2—42no—
612C2—82no—
613C2—9761no—
614C2#2145504611yescharacteristic
615C2#2145513411yescharacteristic
616C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3499C2
socleC22×C61C2×#120
hypercenter#3499D61
hypocenterC61#18582
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #21457197C2semidirect
2. #21456215C2semidirect
3. #21455879C2semidirect
4. #21444984C2semidirect
5. #21444970C2semidirect
6. #21444922C2semidirect
7. #21444908C2semidirect
8. #21444504C2semidirect
9. #21444258C2semidirect
10. #21444236C2semidirect
11. C61×#3499C2semidirect
302. #3499D61semidirect
361. C61#18582semidirect

Maximal subgroups

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

subgroupindex
#214571972
#214562152
#214558792
#214449842
#214449702
#214449222
#214449082
#214445042
#214442582
#214442362
C61×#34992
#1858261