Finite Group Databasebeta

← #29834450

Subgroups — #29834450

Order 15616.

710 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
1C2×#16840193C211yesmaximal, characteristic, complemented
2C22×C16×D61C211yesmaximal, characteristic, complemented
3#21455402C211yesmaximal, characteristic, complemented
4#21455398C211yesmaximal, characteristic, complemented
5#21447916C211yesmaximal, characteristic, complemented
6#21447915C211yesmaximal, characteristic, complemented
7#21447912C211yesmaximal, characteristic, complemented
8#21447909C211yesmaximal, characteristic, complemented
9#21447908C211yesmaximal, characteristic, complemented
10#21447904C211yesmaximal, characteristic, complemented
11#21447903C211yesmaximal, characteristic, complemented
12#21447602C211yesmaximal, characteristic, complemented
13#21447525C211yesmaximal, characteristic, complemented
14#21447523C211yesmaximal, characteristic, complemented
15C61×#2217C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19#16840275—21no—
20#16840272—21no—
21#16840193—24no—
22C2×#14519723C2211yescharacteristic, complemented
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519718C2211yescharacteristic, complemented
25C22×#11758185C2211yescharacteristic
26C2×C16×D61—24no—
27C2×C16×D61C2211yescharacteristic
28C2×C16×D61C2211yescharacteristic
29C2×C16×D61C2211yescharacteristic
30C2×C16×D61C2211yescharacteristic
31#16840004C2211yescharacteristic
32#16840003C2211yescharacteristic
33#16840000C2211yescharacteristic
34#16839999C2211yescharacteristic
35#16839998C2211yescharacteristic
36#16839995C2211yescharacteristic
37#16839994—21no—
38#16839993—21no—
39#16839990—21no—
40#16839989—21no—
41#16839799C2211yescharacteristic
42#16839799C2211yescharacteristic
43#16839757—21no—
44#16839755—21no—
45#16839590C2211yescharacteristic, complemented
46#16839587C2211yescharacteristic, complemented
47#16839587C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839566C2211yescharacteristic, complemented
50#16839565C2211yescharacteristic, complemented
51#16839564C2211yescharacteristic, complemented
52#16839563C2211yescharacteristic, complemented
53#16839562C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C22×C16×C61C2211yescharacteristic
56C61×#467C2211yescharacteristic
57C61×#466C2211yescharacteristic
58C61×#367C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×#11758224—21no—
63C22×C4×D61D411yescharacteristic
64#14519800—22no—
65#14519800—22no—
66#14519796—22no—
67#14519796—22no—
68C2×Dic244C2311yescharacteristic
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×D488C2311yescharacteristic
72C22×#92418D411yescharacteristic
73C2×C8×D61—22no—
74C2×C8×D61—22no—
75C2×C8×D61D411yescharacteristic
76C2×C8×D61D411yescharacteristic
77C2×C8×D61C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519762—41no—
80#14519759—41no—
81#14519757—21no—
82#14519756—21no—
83#14519753—21no—
84#14519752—21no—
85#14519751—21no—
86#14519748—21no—
87#14519747—41no—
88#14519745—41no—
89D4×Dic61—41no—
90#14519739—41no—
91#14519737—41no—
92#14519737—41no—
93#14519735—21no—
94#14519735—21no—
95#14519723—22no—
96#14519722—22no—
97#14519718—22no—
98C2×#11758185—22no—
99C2×#11758185C2311yescharacteristic
100C2×#11758185C2311yescharacteristic
101C16×D61—22no—
102C16×D61—22no—
103C16×D61—22no—
104C16×D61D414yescomplemented
105#14519711C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519708—21no—
108#14519708—21no—
109#14519702—21no—
110#14519701—21no—
111#14519700—21no—
112#14519699—21no—
113#14519698—21no—
114#14519697—21no—
115C2×Q16×C61C2311yescharacteristic
116C2×D8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22⋊Q8×C61—21no—
119#121×C61—21no—
120SD32×C61—22no—
121C2×C16×C61—22no—
122C2×C16×C61C2311yescharacteristic
123C2×C16×C61C2311yescharacteristic
124#107×C61C2311yescharacteristic
125#107×C61C2311yescharacteristic
126#103×C61—21no—
127#102×C61—21no—
128C23×D61D811yescharacteristic
129C2×Q8×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—41no—
132C22×C4×C61C2×D411yescharacteristic
133C2×C22⋊D61—41no—
134C2×C22⋊D61—41no—
135C2×C22⋊D61—41no—
136C22×Dic61—41no—
137C22×Dic61D811yescharacteristic
138#11758226—42no—
139#11758226—42no—
140#11758224—42no—
141#11758224—42no—
142C2×D244—21no—
143C2×D244—41no—
144C2×D244—41no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—41no—
148C2×C4×D61D811yescharacteristic
149C2×C4×D61D811yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×Dic122—21no—
153Q16×C61—21no—
154Q16×C61—21no—
155D8×C61—21no—
156D8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2411yescharacteristic
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C22⋊C4×C61—41no—
166C22⋊Dic61—41no—
167C22⋊Dic61—41no—
168#11758201—42no—
169#11758200—42no—
170#11758198—41no—
171#11758198—41no—
172#11758198—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—41no—
179#11758196—41no—
180#11758196—41no—
181C4×Dic61—41no—
182C4×Dic61—41no—
183C2×#92418—22no—
184C2×#92418C2×D411yescharacteristic
185C2×#92418C2×D411yescharacteristic
186Dic244—21no—
187Dic244—21no—
188D488—21no—
189D488—21no—
190C8×D61—22no—
191C8×D61—21no—
192C8×D61—21no—
193C8×D61—22no—
194C8×D61C2×D412yes—
195C8×D61#4112yes—
196C16×C61—21no—
197C16×C61C2×D412yes—
198#11758185—21no—
199#11758185C2×D412yes—
200C23×C61—41no—
201C23×C61C2×D811yescharacteristic
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—41no—
206C22×D61C2×D811yescharacteristic
207C22×D61C2×D811yescharacteristic
208Q8×C61—41no—
209Q8×C61—41no—
210D4×C61—41no—
211D4×C61—41no—
212D4×C61—82no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C2×D811yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C22⋊D61—82no—
221C22⋊D61—82no—
222C22⋊D61—82no—
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—82no—
228C2×Dic61C2×D811yescharacteristic
229C2×Dic61C2×D811yescharacteristic
230D244—41no—
231D244—41no—
232D244—82no—
233D244—82no—
234C4×D61—22no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—22no—
238C4×D61—82no—
239C4×D61#12112yes—
240C4×D61#13512yes—
241Dic122—41no—
242Dic122—41no—
243C8×C61—21no—
244C8×C61C22×D411yescharacteristic
245C8×C61C2×#4111yescharacteristic
246#92418—21no—
247#92418#12112yes—
248#33600—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61#50812yescomplemented
255C22×C61C22×D811yescharacteristic
256C2×D61—28no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61#46612yes—
265C2×D61#50812yescomplemented
266C4×C61—21no—
267C4×C61—81no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C2×#12111yescharacteristic
271C4×C61C2×#13511yescharacteristic
272Dic61—21no—
273Dic61—81no—
274Dic61—81no—
275Dic61—81no—
276Dic61#46612yes—
277C2×#508—611no—
278C23×C16—611no—
279#3050—611no—
280#3050—611no—
281#2217—611no—
282#2217—611no—
283#2217—611no—
284#2217D6111yescharacteristic, complemented
285#2216—611no—
286#2216—611no—
287#2215—611no—
288#2215—611no—
289#2158—611no—
290#2138—611no—
291#2138—611no—
292C2×C61—22no—
293C2×C61—161no—
294C2×C61C2×#46611yescharacteristic
295C2×C61C2×#50811yescharacteristic
296C2×C61C2×#50811yescharacteristic
297D61—22no—
298D61—22no—
299D61—161no—
300C2×#135—611no—
301C2×#135—611no—
302C23×C8—611no—
303#525—1221no—
304#525—1221no—
305#508—1224no—
306C2×Q32—611no—
307C2×SD32—611no—
308C2×SD32C2×D6111yescharacteristic, complemented
309C2×D16—611no—
310C22×C16—611no—
311C22×C16—611no—
312C22×C16—611no—
313C22×C16—611no—
314C22×C16—611no—
315C22×C16—1224no—
316C22×C16C2×D6111yescharacteristic
317#467—611no—
318#467—611no—
319#467—611no—
320#467C2×D6111yescharacteristic
321#466—611no—
322#466—611no—
323#466—611no—
324#466C2×D6111yescharacteristic
325#465—1221no—
326#465—1221no—
327#465—1221no—
328#465—1221no—
329#427—611no—
330#427—611no—
331#416—1221no—
332#416—1221no—
333#367—611no—
334#367C2×D6111yescharacteristic, complemented
335#366—611no—
336#366—611no—
337#358—611no—
338#358—611no—
339#358—611no—
340#358C2×D6111yescharacteristic, complemented
341#357—611no—
342#357—611no—
343#357—611no—
344#357C2×D6111yescharacteristic, complemented
345C61#3360011yescharacteristic, complemented, Sylow
346C2×#41—1221no—
347C2×#41—1221no—
348C23×C4—611no—
349#135—1222no—
350#135—1222no—
351#135—1222no—
352#135—1222no—
353C2×Q16—611no—
354C2×Q16C22×D6111yescharacteristic
355C2×SD16—1221no—
356C2×SD16—1221no—
357C2×D8—611no—
358C2×D8C22×D6111yescharacteristic
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8—1222no—
365C22×C8—1222no—
366C22×C8C22×D6111yescharacteristic
367#123—2441no—
368#123—2441no—
369C22⋊Q8—21no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372C22⋊Q8—1221no—
373#121—21no—
374#121—1221no—
375#121—1221no—
376#121—1221no—
377#120—2441no—
378#120—2441no—
379C4×D4—2441no—
380C4×D4—2441no—
381C4×D4—2441no—
382C4×D4—2441no—
383#117—1221no—
384#117—1221no—
385Q32—1222no—
386SD32—22no—
387SD32—1222no—
388D16—1222no—
389C2×C16—22no—
390C2×C16—611no—
391C2×C16—614no—
392C2×C16—611no—
393C2×C16—1222no—
394C2×C16—1222no—
395C2×C16—1222no—
396C2×C16—1222no—
397C2×C16C22×D6111yescharacteristic
398C2×C16C22×D6111yescharacteristic
399#107—611no—
400#107—611no—
401#107C22×D6111yescharacteristic
402#107C22×D6111yescharacteristic
403#106—1221no—
404#106—1221no—
405#103—21no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#102—21no—
410#102—1221no—
411#102—1221no—
412#102—1221no—
413C24—611no—
414#41—2442no—
415#41—2442no—
416#41—2442no—
417#41—2442no—
418C2×Q8—21no—
419C2×Q8—1221no—
420C2×D4—21no—
421C2×D4—41no—
422C2×D4—1221no—
423C2×D4—2441no—
424C2×D4—2441no—
425C2×D4—2441no—
426C2×D4—2441no—
427C2×D4—2441no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4D4×D6111yescharacteristic
438Q16—21no—
439Q16—21no—
440Q16—1221no—
441Q16—1221no—
442SD16—2442no—
443SD16—2442no—
444D8—21no—
445D8—21no—
446D8—1221no—
447D8—1221no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—611no—
451C2×C8—612no—
452C2×C8—612no—
453C2×C8—611no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8D4×D6111yescharacteristic
460C2×C8C23×D6111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C22⋊C4—2441no—
478C22⋊C4—2441no—
479C42—2441no—
480C42—2441no—
481C16—21no—
482C16—612no—
483C16—1221no—
484C16D4×D6112yes—
485C23—41no—
486C23—611no—
487C23—611no—
488C23—1222no—
489C23—1222no—
490C23—1222no—
491C23—2441no—
492C23D8×D6111yescharacteristic
493Q8—41no—
494Q8—41no—
495Q8—2441no—
496Q8—2441no—
497D4—41no—
498D4—41no—
499D4—82no—
500D4—4882no—
501D4—4882no—
502D4—4882no—
503D4—4882no—
504D4—4882no—
505D4—2441no—
506D4—2441no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—611no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—611no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—4882no—
522C2×C4—4882no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4D8×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C8—21no—
529C8—612no—
530C8—1221no—
531C8C2×D4×D6111yescharacteristic
532C8#41×D6111yescharacteristic
533C22—21no—
534C22—21no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—612no—
539C22—612no—
540C22—1228no—
541C22—1221no—
542C22—1221no—
543C22—1221no—
544C22—1221no—
545C22—4881no—
546C22—4881no—
547C22—4881no—
548C22#1684019312yescomplemented
549C22C2×D8×D6111yescharacteristic
550C4—21no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—612no—
555C4—1221no—
556C4—4881no—
557C4—4881no—
558C4—4881no—
559C4#121×D6111yescharacteristic
560C4#135×D6111yescharacteristic
561C2—22no—
562C2—161no—
563C2—1222no—
564C2—1222no—
565C2—9761no—
566C2#466×D6111yescharacteristic
567C2#508×D6111yescharacteristic
568C2C2×#1684019311yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2217C2
socleC22×C61C22×D8
hypercenter#2217D61
hypocenterC61#33600
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. C2×#16840193C2semidirect
2. C22×C16×D61C2semidirect
3. #21455402C2semidirect
4. #21455398C2semidirect
5. #21447916C2semidirect
6. #21447915C2semidirect
7. #21447912C2semidirect
8. #21447909C2semidirect
9. #21447908C2semidirect
10. #21447904C2semidirect
11. #21447903C2semidirect
12. #21447602C2semidirect
13. #21447525C2semidirect
14. #21447523C2semidirect
15. C61×#2217C2semidirect
22. C2×#14519723C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519718C22semidirect
45. #16839590C22semidirect
46. #16839587C22semidirect
47. #16839587C22semidirect
48. #16839567C22semidirect
49. #16839566C22semidirect
50. #16839565C22semidirect
51. #16839564C22semidirect
52. #16839563C22semidirect
53. #16839562C22semidirect
54. C2×SD32×C61C22semidirect
58. C61×#367C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
104. C16×D61D4semidirect
254. C22×C61#508semidirect
265. C2×D61#508semidirect
284. #2217D61semidirect
308. C2×SD32C2×D61semidirect
334. #367C2×D61semidirect
340. #358C2×D61semidirect
344. #357C2×D61semidirect
345. C61#33600semidirect
548. C22#16840193semidirect

Maximal subgroups

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

subgroupindex
C2×#168401932
C22×C16×D612
#214554022
#214553982
#214479162
#214479152
#214479122
#214479092
#214479082
#214479042
#214479032
#214476022
#214475252
#214475232
C61×#22172
#3360061