Finite Group Databasebeta

← #29671171

Subgroups — #29671171

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458800C211yesmaximal, characteristic, complemented
2#21458502C211yesmaximal, characteristic, complemented
3#21458403C211yesmaximal, characteristic, complemented
4#21445439C211yesmaximal, characteristic, complemented
5#21445282C211yesmaximal, characteristic, complemented
6#21445242C211yesmaximal, characteristic, complemented
7#21445211C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445094C211yesmaximal, characteristic, complemented
11#21445012C211yesmaximal, characteristic, complemented
12#21444927C211yesmaximal, characteristic, complemented
13#21444663C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3284C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17#16840482C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840134—21no—
21#16840134C2211yescharacteristic
22#16840115—21no—
23#16840113C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840097—21no—
26#16840083—21no—
27#16840072—21no—
28#16840044C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840006—21no—
32#16839982—21no—
33#16839972C2211yescharacteristic
34#16839967—21no—
35#16839967C2211yescharacteristic
36#16839959—21no—
37#16839959C2211yescharacteristic
38Q8×#92418C2211yescharacteristic
39C8×Dic122C2211yescharacteristic
40#16839854—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839802—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839509C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#550C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#462C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#452—21no—
64C61×#449—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200D411yescharacteristic
73C2×#11758194—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519770—41no—
78#14519768—21no—
79#14519768—41no—
80#14519764—21no—
81#14519756—21no—
82Q8×Dic61—21no—
83C4×Dic122—21no—
84C4×Dic122C2311yescharacteristic
85#14519742—21no—
86#14519736—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519708—21no—
92#14519708—41no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519702—21no—
100#14519702—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700D411yescharacteristic
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×#41×C61—21no—
116C2×#41×C61—21no—
117#135×C61—44no—
118C2×Q16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61D411yescharacteristic
122C2×D8×C61—21no—
123C22×C8×C61—21no—
124C4⋊Q8×C61C2311yescharacteristic
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—81no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122C2×D411yescharacteristic
166Q16×C61—42no—
167SD16×C61—22no—
168SD16×C61—42no—
169SD16×C61—81no—
170D8×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61—42no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189C22⋊Dic61—41no—
190#11758202—42no—
191#11758200—22no—
192#11758200—42no—
193#11758200—81no—
194#11758200—81no—
195#11758200—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—41no—
208C4×Dic61—41no—
209#11758194—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—41no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61AGL(1,ℤ/8)12yes—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—82no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—161no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—82no—
245C2×C4×C61—82no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253Dic122—21no—
254Dic122—21no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—41no—
262#92418—21no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418#13512yescomplemented
268#19876—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61D4211yescharacteristic
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61C2×#13511yescharacteristic
286C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C61#57711yescharacteristic
288C4×C61#57711yescharacteristic
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293#3392—611no—
294#3364—611no—
295#3353—611no—
296#3284D6111yescharacteristic
297#1714—611no—
298#1685—611no—
299#1679—611no—
300#1674—611no—
301#1666—611no—
302#1654—611no—
303#1653—611no—
304#1641—611no—
305#1628—611no—
306#1583—611no—
307SD16⋊C8—611no—
308C2×C61—81no—
309C2×C61—81no—
310C2×C61—161no—
311C2×C61#326511yescharacteristic
312C2×C61#328411yescharacteristic
313C2×C61#328911yescharacteristic
314C2×#135—21no—
315#557—611no—
316#556—611no—
317#550C2×D6111yescharacteristic
318#547C2×D6111yescharacteristic
319#500—611no—
320#498—611no—
321#492—611no—
322#492—1221no—
323#492C2×D6111yescharacteristic
324#488—1221no—
325#487—611no—
326#486—611no—
327#484—1221no—
328#482—1221no—
329#479—1221no—
330#475—611no—
331#474—611no—
332#474—611no—
333#468—1221no—
334#465—21no—
335#464—1221no—
336#462—611no—
337#462C2×D6111yescharacteristic
338#461—611no—
339#461—1221no—
340#460—611no—
341#460—1221no—
342#459C2×D6111yescharacteristic
343#452—21no—
344#449—21no—
345C8×Q8—611no—
346C8×Q8—611no—
347C4×Q16—1221no—
348C4×SD16—611no—
349C4×SD16—1221no—
350C4×SD16C2×D6111yescharacteristic
351#436—611no—
352C8×D4C2×D6111yescharacteristic
353#428—1221no—
354#335—611no—
355#334—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#331—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364#322—611no—
365C61#1987611yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C2×#41—21no—
368#135—44no—
369C2×Q16—21no—
370C2×Q16—1221no—
371C2×SD16—21no—
372C2×SD16—21no—
373C2×SD16—611no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16C22⋊D6111yescharacteristic
379C2×D8—21no—
380C2×M16—1221no—
381C22×C8—21no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384C4⋊Q8—1221no—
385C4⋊Q8C22×D6111yescharacteristic
386#127—21no—
387#127C22×D6111yescharacteristic
388#126—2441no—
389#125—1221no—
390#125—2441no—
391#124—1221no—
392#123—41no—
393C22⋊Q8—21no—
394C22⋊Q8—1221no—
395#121—21no—
396#121—21no—
397#121—41no—
398C4×Q8—611no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C22×D6111yescharacteristic
403C4×D4—21no—
404C4×D4C22×D6111yescharacteristic
405C4×D4C22×D6111yescharacteristic
406#117—1221no—
407C2×C4⋊C4—21no—
408#107—1221no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#106—611no—
413#106—1221no—
414#106—2441no—
415#106C22⋊D6111yescharacteristic
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422#103—21no—
423#103—611no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103C22⋊D6111yescharacteristic
430#102—21no—
431#102—611no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—2441no—
436#102C22⋊D6111yescharacteristic
437C22⋊C8—21no—
438C22⋊C8—1221no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—1221no—
442C4×C8—1221no—
443C4×C8C22×D6111yescharacteristic
444#41—42no—
445#41—42no—
446#41—81no—
447#41—81no—
448C2×Q8—21no—
449C2×Q8—611no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8C2×C22⋊D6111yescharacteristic
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4D4×D6111yescharacteristic
460C2×D4C2×C22⋊D6111yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4—41no—
465Q16—42no—
466Q16—2442no—
467SD16—22no—
468SD16—42no—
469SD16—81no—
470SD16—1222no—
471SD16—4881no—
472SD16—4881no—
473SD16—4881no—
474SD16—2442no—
475D8—42no—
476M16—2442no—
477C2×C8—21no—
478C2×C8—42no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8C2×C22⋊D6111yescharacteristic
486C2×C8C2×C22⋊D6111yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—611no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—2441no—
509C22⋊C4—2441no—
510C42—21no—
511C42—1221no—
512C42—2441no—
513C42—2441no—
514C42C23×D6111yescharacteristic
515C23—21no—
516C23—21no—
517C23—41no—
518Q8—21no—
519Q8—41no—
520Q8—41no—
521Q8—1221no—
522Q8—1221no—
523Q8—2441no—
524Q8—2441no—
525Q8—2441no—
526Q8—2441no—
527Q8#1451980312yes—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—82no—
537D4—81no—
538D4—82no—
539D4—161no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—21no—
559C8—21no—
560C8—41no—
561C8—612no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22D4×C22⋊D6111yescharacteristic
577C4—22no—
578C4—21no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—4881no—
584C4—4881no—
585C4—2441no—
586C4—2441no—
587C4#135×D6111yescharacteristic
588C4C2×#1451980311yescharacteristic
589C4#1684056911yescharacteristic
590C4#1684057311yescharacteristic
591C2—81no—
592C2—81no—
593C2—161no—
594C2#2145731011yescharacteristic
595C2#2145754311yescharacteristic
596C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3284C2
socleC22×C61D42
hypercenter#3284D61
hypocenterC61#19876
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3284

Complements

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

NH ≅ G/Ntype
1. #21458800C2semidirect
2. #21458502C2semidirect
3. #21458403C2semidirect
4. #21445439C2semidirect
5. #21445282C2semidirect
6. #21445242C2semidirect
7. #21445211C2semidirect
8. #21445164C2semidirect
9. #21445103C2semidirect
10. #21445094C2semidirect
11. #21445012C2semidirect
12. #21444927C2semidirect
13. #21444663C2semidirect
14. #21444631C2semidirect
267. #92418#135semidirect
365. C61#19876semidirect

Maximal subgroups

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

subgroupindex
#214588002
#214585022
#214584032
#214454392
#214452822
#214452422
#214452112
#214451642
#214451032
#214450942
#214450122
#214449272
#214446632
#214446312
C61×#32842
#1987661