Finite Group Databasebeta

← #29656767

Subgroups — #29656767

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457206C211yesmaximal, characteristic, complemented
2#21456361C211yesmaximal, characteristic, complemented
3#21455890C211yesmaximal, characteristic, complemented
4#21445103C212yesmaximal, complemented
5#21445082C212yesmaximal, complemented
6#21445068C212yesmaximal, complemented
7#21445054C212yesmaximal, complemented
8#21444504C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3499C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840168—22no—
14#16840165C2212yes—
15#16840158C2212yes—
16#16840103—21no—
17#16840054—21no—
18#16840047C2211yescharacteristic
19#16840042C2211yescharacteristic
20#16840042C2212yes—
21#16840006—22no—
22#16839972C2211yescharacteristic
23#16839967C2211yescharacteristic
24#16839967C2212yes—
25#16839959—21no—
26#16839924—21no—
27C4×#11758202—21no—
28C4×#11758200—21no—
29#16839815C2211yescharacteristic
30#16839802—22no—
31#16839786C2211yescharacteristic
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×#11758202—21no—
54C2×#11758200—21no—
55C2×#11758200—22no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—22no—
67#14519710—21no—
68#14519710—24no—
69#14519708—21no—
70#14519708—24no—
71C4⋊#92418—21no—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75#14519701—21no—
76#14519700—22no—
77#14519700—21no—
78#14519700—21no—
79#14519698—22no—
80#14519698—21no—
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×Dic122—21no—
171C2×Dic122—21no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61C2×D414yes—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—22no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2411yescharacteristic
198C22⋊Dic61—41no—
199#11758202—42no—
200#11758200—42no—
201#11758200—82no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206#11758196—41no—
207#11758196—41no—
208C4×Dic61—41no—
209#11758194—24no—
210#11758194—81no—
211C2×#92418—22no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—22no—
215C23×C61—21no—
216C23×C61—22no—
217C23×C61#12011yescharacteristic
218Q8×C61—21no—
219Q8×C61—44no—
220Q8×C61—42no—
221Q8×C61—48no—
222Q8×C61—42no—
223Q8×C61—42no—
224Q8×C61AGL(1,ℤ/8)12yes—
225D4×C61—24no—
226D4×C61—21no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—42no—
230D4×C61—44no—
231D4×C61—48no—
232D4×C61—42no—
233D4×C61AGL(1,ℤ/8)12yes—
234C2×C4×C61—24no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—24no—
244C2×C4×C61—22no—
245C2×C4×C61—48no—
246C2×C4×C61—44no—
247C2×C4×C61—48no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—42no—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×Dic61—41no—
260C2×Dic61—41no—
261Dic122—41no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265#92418—24no—
266#92418—41no—
267#92418—41no—
268#92418—42no—
269#18590—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—22no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61C2×#12011yescharacteristic
280C4×C61—21no—
281C4×C61—22no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—44no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61#45612yes—
289C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C61#57711yescharacteristic
292C4×C61#57811yescharacteristic
293Dic61—81no—
294Dic61—81no—
295#3499D6111yescharacteristic
296#3254—611no—
297#3165—611no—
298#3107—611no—
299#1654—612no—
300#1651—612no—
301#1649—612no—
302#1647—612no—
303#1549—611no—
304#1489—611no—
305#1484—611no—
306C2×C61—41no—
307C2×C61—42no—
308C2×C61—82no—
309C2×C61#300311yescharacteristic
310C2×C61#301311yescharacteristic
311C2×C61C2×#45611yescharacteristic
312C2×2−1+4—21no—
313D4×Q8—21no—
314D4×Q8C2×D6112yes—
315#547—22no—
316#547—21no—
317#547—22no—
318#547C2×D6112yes—
319#536—22no—
320#536—21no—
321#533—611no—
322#532—21no—
323#532C2×D6111yescharacteristic
324C2×C4⋊Q8—21no—
325C2×C4⋊Q8C2×D6111yescharacteristic
326C4×#41C2×D6111yescharacteristic
327#501—1222no—
328#500—612no—
329#498—612no—
330#485—1221no—
331#477—1221no—
332#475—611no—
333#474—611no—
334#474—612no—
335#468—1222no—
336#462—611no—
337#461—611no—
338#461—612no—
339#460—1221no—
340#452—1221no—
341C4×Q16—1221no—
342C4×SD16—1221no—
343#431—611no—
344#428—1222no—
345#423—611no—
346C4×M16—611no—
347#332—612no—
348#332—612no—
349Q8⋊C8—612no—
350D4⋊C8—612no—
351D4⋊C8—612no—
352D4⋊C8—612no—
353C61#1859011yescharacteristic, complemented, Sylow
3542−1+4—44no—
3552−1+4—44no—
356C2×#41—22no—
357C2×#41—24no—
358C2×#41—22no—
359C2×#41—21no—
360C2×#41C22⋊D6111yescharacteristic
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—21no—
364C2×Q16—1221no—
365C2×SD16—1222no—
366C2×SD16—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376C4⋊Q8C22⋊D6111yescharacteristic
377C4⋊Q8C22⋊D6112yes—
378C4⋊Q8C22×D6112yes—
379#127C22⋊D6111yescharacteristic
380#125—1221no—
381#124—22no—
382#124—21no—
383#123—24no—
384#123—24no—
385#123—24no—
386C22⋊Q8—24no—
387C22⋊Q8—22no—
388C22⋊Q8—24no—
389C22⋊Q8—22no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392#121—22no—
393#121—22no—
394#121—21no—
395#121C22⋊D6112yes—
396C4×Q8—1221no—
397C4×Q8—1221no—
398C4×Q8C22⋊D6111yescharacteristic
399C4×Q8C22×D6111yescharacteristic
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4C22⋊D6111yescharacteristic
406C4×D4C22×D6111yescharacteristic
407C4×D4C22×D6112yes—
408#117—21no—
409#117—1221no—
410#117C22⋊D6111yescharacteristic
411C2×C4⋊C4—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4C22⋊D6112yes—
415C2×C42—21no—
416C2×C42C22×D6111yescharacteristic
417#107—1224no—
418#107—1221no—
419#107—1222no—
420#106—1224no—
421#106—1221no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—612no—
425C4⋊C8—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1222no—
430#102—1221no—
431#102—1222no—
432C22⋊C8—1221no—
433#97—1221no—
434C4×C8—612no—
435C4×C8—1221no—
436#41—22no—
437#41—22no—
438#41—48no—
439#41—42no—
440#41—42no—
441#41—416no—
442#41—44no—
443#41—44no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—48no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—21no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×C22⋊D6112yes—
462C2×D4C2×C22⋊D6111yescharacteristic
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473Q16—2442no—
474SD16—4882no—
475SD16—2442no—
476M16—1224no—
477M16—4881no—
478C2×C8—1222no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1222no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—2441no—
495C4⋊C4—2441no—
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C4⋊C4C2×C22⋊D6112yes—
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C4⋊C4C2×C22⋊D6114yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—2441no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—2441no—
512C42C2×C22⋊D6111yescharacteristic
513C42C2×C22⋊D6111yescharacteristic
514C42C2×C22⋊D6111yescharacteristic
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—22no—
518C23#1451974611yescharacteristic
519Q8—21no—
520Q8—44no—
521Q8—42no—
522Q8—48no—
523Q8—42no—
524Q8—42no—
525Q8—2441no—
526Q8—2441no—
527Q8—2441no—
528Q8—2441no—
529Q8#1451980312yes—
530D4—24no—
531D4—21no—
532D4—22no—
533D4—22no—
534D4—42no—
535D4—44no—
536D4—48no—
537D4—42no—
538D4#1451980312yes—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4#1451974611yescharacteristic
561C2×C4#1451974611yescharacteristic
562C2×C4#1451974611yescharacteristic
563C2×C4C22×C22⋊D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C22×C22⋊D6111yescharacteristic
566C8—1224no—
567C8—2441no—
568C8—2441no—
569C8—2442no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×#1451974611yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—44no—
587C4—42no—
588C4—4881no—
589C4—4881no—
590C4#1683994512yes—
591C4C2×#1451980311yescharacteristic
592C4C2×#1451980311yescharacteristic
593C4#1684057011yescharacteristic
594C4#1684057711yescharacteristic
595C2—41no—
596C2—42no—
597C2—82no—
598C2#2145504611yescharacteristic
599C2#2145513411yescharacteristic
600C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3499C2
socleC22×C61C2×#120
hypercenter#3499D61
hypocenterC61#18590
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. #21457206C2semidirect
2. #21456361C2semidirect
3. #21455890C2semidirect
4. #21445103C2semidirect
5. #21445082C2semidirect
6. #21445068C2semidirect
7. #21445054C2semidirect
8. #21444504C2semidirect
9. #21444258C2semidirect
10. #21444236C2semidirect
353. C61#18590semidirect

Maximal subgroups

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

subgroupindex
#214572062
#214563612
#214558902
#214451032
#214450822
#214450682
#214450542
#214445042
#214442582
#214442362
C61×#34992
#1859061