Finite Group Databasebeta

← #29670106

Subgroups — #29670106

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758200C211yesmaximal, characteristic, complemented
2SD16×Dic122C211yesmaximal, characteristic, complemented
3#21457461C211yesmaximal, characteristic
4#21445454C211yesmaximal, characteristic, complemented
5#21445316C211yesmaximal, characteristic, complemented
6#21445246C211yesmaximal, characteristic, complemented
7#21445194C211yesmaximal, characteristic, complemented
8#21445187C211yesmaximal, characteristic, complemented
9#21445082C211yesmaximal, characteristic, complemented
10#21445080C211yesmaximal, characteristic, complemented
11#21445039C211yesmaximal, characteristic, complemented
12#21444970C211yesmaximal, characteristic
13#21444663C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15C61×#3298C211yesmaximal, characteristic
16C2×#14519807—21no—
17Q8×Dic122C2211yescharacteristic
18D4×Dic122C2211yescharacteristic
19#16840160C2211yescharacteristic
20#16840159—21no—
21#16840158—21no—
22#16840158C2211yescharacteristic
23#16840158C2211yescharacteristic
24#16840141C2211yescharacteristic
25#16840114C2211yescharacteristic
26#16840052—21no—
27#16840052C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840050—21no—
30#16840047C2211yescharacteristic
31#16840045—21no—
32#16840044—21no—
33#16840044C2211yescharacteristic
34#16840018—21no—
35#16839969C2211yescharacteristic
36#16839967C2211yescharacteristic
37#16839924—21no—
38#16839917—21no—
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41SD16×Dic61—21no—
42C4×#11758200—21no—
43C4×#11758200C2211yescharacteristic
44#16839840C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56D4×Q8×C61C2211yescharacteristic
57C61×#547C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#483—21no—
60C61×#475C2211yescharacteristic
61C61×#465—21no—
62C61×#461C2211yescharacteristic
63C61×#448—21no—
64C4×SD16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C22×Dic122—21no—
68#14519807—44no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758200—21no—
73C2×#11758200D411yescharacteristic
74C2×#11758194—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519776—21no—
80#14519776—41no—
81#14519756—21no—
82#14519754—41no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86D4×Dic61—21no—
87#14519709—21no—
88#14519709—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—41no—
96#14519708D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519706—21no—
102#14519701—21no—
103#14519701—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—41no—
107#14519700D411yescharacteristic
108#14519698—21no—
109#14519698—21no—
110#14519698D411yescharacteristic
111C22⋊#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C2×#41×C61—21no—
117C22×Q8×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C22×C8×C61—21no—
121C4⋊Q8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#123×C61—41no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—21no—
127C22⋊Q8×C61—41no—
128#121×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C22×Dic61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—42no—
164C2×Dic122—41no—
165C2×Dic122—41no—
166C2×Dic122—81no—
167C2×Dic122C2×D411yescharacteristic
168SD16×C61—22no—
169SD16×C61—81no—
170C2×C8×C61—21no—
171C2×C8×C61—81no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191#11758202—42no—
192#11758202—42no—
193#11758202—81no—
194#11758200—42no—
195#11758200SD1614yescomplemented
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—21no—
203#11758196—41no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207#11758194—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C23×C61—21no—
215C23×C61—21no—
216Q8×C61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—82no—
222Q8×C61#13712yescomplemented
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—82no—
227D4×C61C2×SD1612yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61—82no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—82no—
248Dic122—21no—
249Dic122—41no—
250Dic122—41no—
251Dic122—82no—
252Dic122—81no—
253Dic122—161no—
254Dic122C2×SD1612yes—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—41no—
258#92418—21no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418C2×SD1612yes—
264#19805—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61D4211yescharacteristic
272C4×C61—22no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61C22×SD1611yescharacteristic
280C4×C61C2×#13711yescharacteristic
281C4×C61#57711yescharacteristic
282C4×C61#57711yescharacteristic
283Dic61—22no—
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287Q8×SD16—611no—
288Q8×SD16—611no—
289#3298D6111yescharacteristic
290#3276—611no—
291#1716—611no—
292#1691—611no—
293#1680—611no—
294#1671—611no—
295#1670—611no—
296#1651—611no—
297#1651—611no—
298#1645—611no—
299#1635—611no—
300#1583—611no—
301SD16⋊C8—611no—
302C2×C61—42no—
303C2×C61—81no—
304C2×C61#326611yescharacteristic
305C2×C61D4×SD1611yescharacteristic
306C2×C61#328911yescharacteristic
307C2×#137—1221no—
308Q82—611no—
309D4×Q8—611no—
310D4×Q8C2×D6111yescharacteristic
311#547C2×D6111yescharacteristic
312#498—611no—
313#498—611no—
314#498—611no—
315#498—1221no—
316#498—1221no—
317#494—611no—
318#487—611no—
319#487C2×D6111yescharacteristic
320#483—21no—
321#476—611no—
322#476—611no—
323#476—1221no—
324#476—1221no—
325#475—611no—
326#475—611no—
327#475—1221no—
328#475—1221no—
329#475C2×D6111yescharacteristic
330#470—1221no—
331#465—21no—
332#462—611no—
333#461—611no—
334#461C2×D6111yescharacteristic
335#452—1221no—
336#451—1221no—
337#448—21no—
338C8×Q8—611no—
339C8×Q8—611no—
340C4×SD16—611no—
341C4×SD16—1221no—
342C4×SD16—1221no—
343C4×SD16C2×D6111yescharacteristic
344#436—611no—
345C8×D4C2×D6111yescharacteristic
346C2×#103—21no—
347#335—611no—
348#334—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357#322—611no—
358C61#1980511yescharacteristic, complemented, Sylow
359C2×#41—21no—
360C22×Q8—21no—
361C22×Q8—1221no—
362#137—2444no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×SD16—21no—
367C2×SD16—611no—
368C2×SD16—1221no—
369C2×SD16C22⋊D6111yescharacteristic
370C2×M16—1221no—
371C22×C8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377C4⋊Q8—1221no—
378C4⋊Q8—2441no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#123—41no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—41no—
385C22⋊Q8—1221no—
386C22⋊Q8—2441no—
387#121—21no—
388C4×Q8—611no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×Q8C22×D6111yescharacteristic
392C4×D4—21no—
393C4×D4—1221no—
394C4×D4C22×D6111yescharacteristic
395C4×D4C22×D6111yescharacteristic
396C2×C4⋊C4—21no—
397#106—21no—
398#106—611no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402#106—1221no—
403#106—1221no—
404#106—1221no—
405#106—1221no—
406#106—1221no—
407#106—2441no—
408#106C22⋊D6111yescharacteristic
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415#103—21no—
416#103—21no—
417#103—21no—
418#103—21no—
419#103—611no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—2441no—
425#103C22⋊D6111yescharacteristic
426#102—611no—
427#102—1221no—
428#102—1221no—
429#102C22⋊D6111yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—1221no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8C22×D6111yescharacteristic
437#41—42no—
438#41—81no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—41no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—611no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—4881no—
448C2×Q8—2441no—
449C2×Q8—2441no—
450C2×Q8—2442no—
451C2×Q8C2×C22⋊D6111yescharacteristic
452C2×D4—41no—
453C2×D4D4×D6111yescharacteristic
454C2×D4C2×C22⋊D6111yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—2441no—
460Q16—4881no—
461Q16—2442no—
462Q16—2442no—
463SD16—22no—
464SD16—81no—
465SD16—614no—
466SD16—2442no—
467M16—2442no—
468C2×C8—21no—
469C2×C8—81no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8C2×C22⋊D6111yescharacteristic
477C2×C8C2×C22⋊D6111yescharacteristic
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—611no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C4⋊C4D4×D6111yescharacteristic
495C4⋊C4C2×C22⋊D6111yescharacteristic
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—2441no—
502C42—21no—
503C42—1221no—
504C42—2441no—
505C42C23×D6111yescharacteristic
506C23—21no—
507C23—21no—
508Q8—21no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—82no—
514Q8—612no—
515Q8—1221no—
516Q8—9761no—
517Q8—4881no—
518Q8—4882no—
519Q8—2441no—
520Q8—2441no—
521Q8#1451980712yescomplemented
522D4—21no—
523D4—21no—
524D4—21no—
525D4—82no—
526D4SD16×D6112yes—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4#1451982211yescharacteristic
547C8—21no—
548C8—21no—
549C8—41no—
550C8—612no—
551C8—1221no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C8—2441no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22D4×C22⋊D6111yescharacteristic
563C4—22no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—1222no—
571C4—4881no—
572C4—4881no—
573C4—2441no—
574C4C2×SD16×D6111yescharacteristic
575C4C2×#1451980711yescharacteristic
576C4#1684056811yescharacteristic
577C4#1684057311yescharacteristic
578C2—42no—
579C2—81no—
580C2#2145732511yescharacteristic
581C2SD16×C22⋊D6111yescharacteristic
582C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3298C2
socleC22×C61D42
prime radicalC61×#547C22
hypercenter#3298D61
hypocenterC61#19805
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3298

Complements

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

NH ≅ G/Ntype
1. Q8×#11758200C2semidirect
2. SD16×Dic122C2semidirect
4. #21445454C2semidirect
5. #21445316C2semidirect
6. #21445246C2semidirect
7. #21445194C2semidirect
8. #21445187C2semidirect
9. #21445082C2semidirect
10. #21445080C2semidirect
11. #21445039C2semidirect
13. #21444663C2semidirect
14. #21444631C2semidirect
195. #11758200SD16semidirect
222. Q8×C61#137semidirect
358. C61#19805semidirect
521. Q8#14519807semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582002
SD16×Dic1222
#214574612
#214454542
#214453162
#214452462
#214451942
#214451872
#214450822
#214450802
#214450392
#214449702
#214446632
#214446312
C61×#32982
#1980561