Finite Group Databasebeta

← #29669969

Subgroups — #29669969

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758201C211yesmaximal, characteristic, complemented
2#21457730C211yesmaximal, characteristic
3#21457686C211yesmaximal, characteristic
4#21445466C211yesmaximal, characteristic, complemented
5#21445373C211yesmaximal, characteristic, complemented
6#21445267C211yesmaximal, characteristic, complemented
7#21445251C211yesmaximal, characteristic, complemented
8#21445196C211yesmaximal, characteristic, complemented
9#21445075C211yesmaximal, characteristic, complemented
10#21445066C211yesmaximal, characteristic, complemented
11#21445040C211yesmaximal, characteristic, complemented
12#21444969C211yesmaximal, characteristic
13#21444653C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3369C211yesmaximal, characteristic, complemented
16Q8×D244C2211yescharacteristic
17#16840417C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840158—21no—
20#16840158C2211yescharacteristic
21#16840114C2211yescharacteristic
22#16840113C2211yescharacteristic
23#16840112C2211yescharacteristic
24#16840103—21no—
25#16840094—21no—
26#16840049—21no—
27#16840049C2211yescharacteristic
28#16840047—21no—
29#16840047C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840039C2211yescharacteristic
32#16840001—21no—
33#16839990—21no—
34#16839972C2211yescharacteristic
35#16839967C2211yescharacteristic
36#16839922—21no—
37#16839899—21no—
38Q8×#92418C2211yescharacteristic
39C4×#11758201—21no—
40C4×#11758201C2211yescharacteristic
41C8×D244C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839756—21no—
44#103×D61—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57C61×#556C2211yescharacteristic
58C61×#489—21no—
59C61×#487C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#475—21no—
62C61×#464—21no—
63C61×#462C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#441—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×Q8×D61—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758201D411yescharacteristic
72C22×#92418—21no—
73C2×C8×D61—21no—
74#14519777—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519759—41no—
78#14519757—21no—
79#14519757—41no—
80#14519756—21no—
81#14519748—21no—
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84C4⋊C4×D61—21no—
85#14519710—21no—
86#14519710—41no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—41no—
102#14519701D411yescharacteristic
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×Q16×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—41no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#106×C61—21no—
133#106×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C2×C22⋊D61—41no—
152Q8×D61—42no—
153Q8×D61—81no—
154#11758222—42no—
155#11758222—81no—
156C2×D244C2×D411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—41no—
163Q16×C61—42no—
164SD16×C61—22no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61—41no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758202—81no—
190#11758201—22no—
191#11758200—81no—
192#11758198—21no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—41no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—81no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C8×D61—81no—
209C23×C61—21no—
210C22×D61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—81no—
217Q8×C61—161no—
218Q8×C61#13512yes—
219D4×C61—21no—
220D4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C22⋊D61—82no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237D244—21no—
238D244#13712yes—
239C4×D61—41no—
240C4×D61—41no—
241C4×D61—82no—
242C4×D61—82no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—82no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19796—611nomaximal, Sylow
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61D4211yescharacteristic
262C2×D61—22no—
263C2×D61—41no—
264C2×D61—41no—
265C4×C61—22no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61C2×#13511yescharacteristic
274C4×C61C2×#13711yescharacteristic
275C4×C61#57711yescharacteristic
276C4×C61#57711yescharacteristic
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280Q8×SD16—611no—
281#3369D6111yescharacteristic, complemented
282#3302—611no—
283#3298—611no—
284#1718—611no—
285#1700—611no—
286#1683—611no—
287#1680—611no—
288#1671—611no—
289#1650—611no—
290#1649—611no—
291#1645—611no—
292#1635—611no—
293#1582—611no—
294SD16⋊C8—611no—
295C2×C61—81no—
296C2×C61#326611yescharacteristic
297C2×C61#328211yescharacteristic
298C2×C61#328911yescharacteristic
299D61—42no—
300Q82C2×D6111yescharacteristic
301#556C2×D6111yescharacteristic
302D4×Q8—611no—
303#547—611no—
304#500—611no—
305#498—611no—
306#498—1221no—
307#489—21no—
308#487—611no—
309#487—611no—
310#487—611no—
311#487C2×D6111yescharacteristic
312#485—1221no—
313#483—1221no—
314#476—611no—
315#476—1221no—
316#476C2×D6111yescharacteristic
317#475—21no—
318#475—611no—
319#475—611no—
320#475—1221no—
321#474—611no—
322#467—1221no—
323#465—1221no—
324#464—21no—
325#462—611no—
326#462C2×D6111yescharacteristic
327#461—611no—
328#452—1221no—
329#448—1221no—
330#446C2×D6111yescharacteristic
331C8×Q8—611no—
332#441—21no—
333C4×SD16—611no—
334C4×SD16—1221no—
335C4×SD16C2×D6111yescharacteristic
336C8×D4—611no—
337C8×D4—611no—
338#416—1221no—
339C2×#103—1221no—
340#335—611no—
341#334—611no—
342#333—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350#322—611no—
351C61#1979611yescharacteristic, complemented, Sylow
352C2×#41—1221no—
353C22×Q8—1221no—
354C2×Q16—21no—
355C2×Q16—1221no—
356C2×SD16—611no—
357C2×SD16—1221no—
358C2×SD16C22⋊D6111yescharacteristic
359C22×C8—1221no—
360C22×C8—1221no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—41no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8C22×D6111yescharacteristic
368C4⋊Q8C22×D6111yescharacteristic
369#126—41no—
370#124—21no—
371#123—2441no—
372C22⋊Q8—21no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—2441no—
376#121—1221no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22×D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—611no—
383C4×D4—1221no—
384C4×D4C22×D6111yescharacteristic
385#117—21no—
386C2×C4⋊C4—1221no—
387#107—1221no—
388#107—2441no—
389#106—21no—
390#106—611no—
391#106—1221no—
392#106—1221no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396#106C22⋊D6111yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403#103—21no—
404#103—21no—
405#103—21no—
406#103—611no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—2441no—
416#103C22⋊D6111yescharacteristic
417#102—21no—
418#102—611no—
419#102—1221no—
420#102—1221no—
421#102C22⋊D6111yescharacteristic
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—21no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8C22×D6111yescharacteristic
429#41—4881no—
430#41—2442no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—4881no—
437C2×Q8—2441no—
438C2×Q8—2442no—
439C2×Q8D4×D6111yescharacteristic
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4—611no—
442C2×D4—2441no—
443C2×D4C2×C22⋊D6111yescharacteristic
444C22×C4—21no—
445C22×C4—1221no—
446C22×C4—2441no—
447C22×C4—2441no—
448Q16—42no—
449Q16—4881no—
450SD16—22no—
451SD16—1222no—
452SD16—4881no—
453C2×C8—21no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—4881no—
461C2×C8—4881no—
462C2×C8C2×C22⋊D6111yescharacteristic
463C2×C8C2×C22⋊D6111yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4D4×D6111yescharacteristic
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—1221no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C42—21no—
490C42—21no—
491C42—41no—
492C42—41no—
493C42C23×D6111yescharacteristic
494C23—21no—
495C23—1221no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—81no—
502Q8—161no—
503Q8—4882no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508Q8#1451979712yes—
509D4—21no—
510D4—21no—
511D4—612no—
512D4—1221no—
513D4—4882no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—1221no—
524C2×C4—4882no—
525C2×C4—4882no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#1451982211yescharacteristic
533C8—21no—
534C8—21no—
535C8—41no—
536C8—1221no—
537C8—1221no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C8—2441no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—1222no—
546C22—2441no—
547C22—2441no—
548C22D4×C22⋊D6111yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—81no—
557C4—4881no—
558C4—4881no—
559C4—2441no—
560C4C2×#1451979711yescharacteristic
561C4#137×D6111yescharacteristic
562C4#1684057011yescharacteristic
563C4#1684057111yescharacteristic
564C2—81no—
565C2—2442no—
566C2#2145732311yescharacteristic
567C2#2145752711yescharacteristic
568C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3369C2
socleC22×C61D42
prime radical#16840417C22
hypercenter#3369D61
hypocenterC61#19796
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3369

Complements

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

NH ≅ G/Ntype
1. Q8×#11758201C2semidirect
4. #21445466C2semidirect
5. #21445373C2semidirect
6. #21445267C2semidirect
7. #21445251C2semidirect
8. #21445196C2semidirect
9. #21445075C2semidirect
10. #21445066C2semidirect
11. #21445040C2semidirect
13. #21444653C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3369C2semidirect
281. #3369D61semidirect
351. C61#19796semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582012
#214577302
#214576862
#214454662
#214453732
#214452672
#214452512
#214451962
#214450752
#214450662
#214450402
#214449692
#214446532
SD16⋊#924182
C61×#33692
#1979661