Finite Group Databasebeta

← #29836503

Subgroups — #29836503

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3Q16×D244C211yesmaximal, characteristic, complemented
4#21457505C211yesmaximal, characteristic
5#21448090C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447911C211yesmaximal, characteristic, complemented
8#21447881C211yesmaximal, characteristic, complemented
9#21447864C211yesmaximal, characteristic, complemented
10#21447760C211yesmaximal, characteristic, complemented
11#21447754C211yesmaximal, characteristic, complemented
12#21447749C211yesmaximal, characteristic, complemented
13#21447737C211yesmaximal, characteristic, complemented
14#21447646C211yesmaximal, characteristic, complemented
15C61×#2175C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic
20Q8×D244—21no—
21#16840407—21no—
22#16840201—24no—
23C2×Dic488C2211yescharacteristic, complemented
24C2×Dic488C2211yescharacteristic, complemented
25C2×#14519723C2211yescharacteristic, complemented
26C2×#14519722C2211yescharacteristic
27C2×#14519722C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519721C2211yescharacteristic
30C2×#14519720C2211yescharacteristic
31SD32×D61—24no—
32C2×#14519716C2211yescharacteristic, complemented
33C2×C16×D61C2211yescharacteristic
34#16840146C2211yescharacteristic
35#16840142C2211yescharacteristic, complemented
36#16840004C2211yescharacteristic
37#16840004C2211yescharacteristic
38#16840004C2211yescharacteristic, complemented
39#16839999C2211yescharacteristic
40#16839977—21no—
41#16839972—21no—
42#16839919—21no—
43#16839919—21no—
44#16839914—21no—
45#16839908—21no—
46C8×D244C2211yescharacteristic
47C4⋊#11758185C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839565C2211yescharacteristic, complemented
50#16839565C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic
52#16839541C2211yescharacteristic
53#16839541C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic, complemented
56C4×D8×C61C2211yescharacteristic
57C61×#366C2211yescharacteristic, complemented
58C61×#358C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic
60C61×#346C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×Q8×D61—21no—
63C2×Q8×D61—21no—
64C2×D4×D61—21no—
65#14519796—22no—
66#14519796—41no—
67C2×Dic244—21no—
68C2×Dic244C2311yescharacteristic
69C2×Dic244C2311yescharacteristic
70C2×Dic244C2311yescharacteristic
71C2×#11758202—21no—
72C2×#11758202—21no—
73Q16×D61—22no—
74Q16×D61—22no—
75Q16×D61—41no—
76Q16×D61—41no—
77C2×#11758190—21no—
78C2×#11758200—21no—
79C2×D488C2311yescharacteristic
80C2×#11758199—21no—
81D8×D61—22no—
82D8×D61—41no—
83C2×C8×D61C2311yescharacteristic
84C2×C8×D61C2311yescharacteristic
85#14519777—21no—
86#14519777—41no—
87#14519765—21no—
88#14519761—41no—
89#14519757—21no—
90#14519757—21no—
91#14519757—21no—
92#14519757—41no—
93#14519757—41no—
94#14519752—21no—
95#14519751—41no—
96#14519747—41no—
97C4×D244—41no—
98C4×D244D411yescharacteristic
99C2×#11758198—41no—
100Dic488—22no—
101Dic488—41no—
102#14519723—22no—
103#14519722—22no—
104#14519722—41no—
105#14519721—22no—
106#14519721—22no—
107#14519720—22no—
108#14519716—22no—
109C2×#11758185C2311yescharacteristic
110C2×#11758185C2311yescharacteristic
111C16×D61—22no—
112C4⋊#92418D411yescharacteristic
113#14519700—21no—
114#14519700—21no—
115#14519700—21no—
116#14519697—21no—
117#14519689D411yescharacteristic
118#14519689D411yescharacteristic
119C2×Q16×C61C2311yescharacteristic
120C2×D8×C61C2311yescharacteristic
121C4×Q8×C61—21no—
122C4×D4×C61—21no—
123SD32×C61D414yescomplemented
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#107×C61C2311yescharacteristic
128#103×C61—21no—
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×D4×C61—21no—
134C22×C4×C61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C22×Dic61—41no—
138Q8×D61—42no—
139Q8×D61—42no—
140Q8×D61—81no—
141Q8×D61—81no—
142#11758224—42no—
143#11758224—81no—
144D4×D61—42no—
145D4×D61—81no—
146C2×D244—21no—
147C2×D244—41no—
148C2×D244D811yescharacteristic
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×Dic122—41no—
158Q16×C61—21no—
159Q16×C61C2×D412yes—
160D8×C61—21no—
161D8×C61C2×D412yes—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2411yescharacteristic
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C42×C61—41no—
169C42×C61C2×D411yescharacteristic
170#11758202—42no—
171#11758202—42no—
172#11758200—42no—
173#11758199—42no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198D811yescharacteristic
181#11758198D811yescharacteristic
182C4⋊Dic61—41no—
183C4⋊Dic61—41no—
184C4⋊Dic61—41no—
185C4⋊Dic61—41no—
186C4⋊Dic61—41no—
187C4⋊Dic61D811yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192Dic244—21no—
193Dic244—21no—
194Dic244—21no—
195Dic244—21no—
196Dic244—81no—
197D488—21no—
198D488—21no—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C22×D61—41no—
210C22×D61—82no—
211C22×D61—82no—
212C22×D61—161no—
213C22×D61C2×D811yescharacteristic
214C22×D61C2×D811yescharacteristic
215Q8×C61—22no—
216Q8×C61—41no—
217D4×C61—22no—
218D4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—161no—
223C2×C4×C61C2×D811yescharacteristic
224C2×C4×C61C2×D811yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61—82no—
227C22⋊D61—82no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—82no—
232C2×Dic61C2×D811yescharacteristic
233C2×Dic61C2×D811yescharacteristic
234D244—22no—
235D244—41no—
236D244—41no—
237D244—161no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—82no—
243C4×D61—82no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—82no—
251Dic122—82no—
252C8×C61—21no—
253C8×C61C22×D411yescharacteristic
254C8×C61H(5,𝔽2)11yescharacteristic
255#92418—21no—
256#92418—21no—
257#33832—611nomaximal, Sylow
258C22×C61—42no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61C22×D811yescharacteristic
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—164no—
271C2×D61#51012yescomplemented
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—42no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61D4211yescharacteristic
278C4×C61#57611yescharacteristic
279Dic61—21no—
280Dic61—21no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284C2×#510—611no—
285C22×SD32—611no—
286D4×Q16—611no—
287#3280—611no—
288#2251—611no—
289#2220—611no—
290#2216—611no—
291#2211—611no—
292#2208—611no—
293#2189—611no—
294#2188—611no—
295#2187—611no—
296#2185—611no—
297#2175D6111yescharacteristic, complemented
298#2168—611no—
299C2×C61—82no—
300C2×C61D4×D811yescharacteristic
301C2×C61C2×#51011yescharacteristic
302C2×C61#341611yescharacteristic
303D61—22no—
304D61—41no—
305D61—161no—
306C2×#135—611no—
307C22×Q16—611no—
308C22×Q16—611no—
309C22×D8—611no—
310D4×Q8—1221no—
311#546—1221no—
312#510—1224no—
313C2×Q32—611no—
314C2×Q32—611no—
315C2×Q32—611no—
316C2×SD32—611no—
317C2×SD32—611no—
318C2×SD32—611no—
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—1224no—
322C2×SD32C2×D6111yescharacteristic
323C2×M32—611no—
324C22×C16—611no—
325#495—611no—
326#494—611no—
327#467—611no—
328#467—611no—
329#467—611no—
330#466—611no—
331#463—1221no—
332#462—1221no—
333#451—1221no—
334#451—1221no—
335#450—1221no—
336#449—1221no—
337C4×Q16C2×D6111yescharacteristic, complemented
338C4×D8C2×D6111yescharacteristic
339C8×D4—611no—
340#366C2×D6111yescharacteristic, complemented
341C4⋊C16—611no—
342#358—611no—
343#358—611no—
344#358—611no—
345#358C2×D6111yescharacteristic, complemented
346#357—611no—
347#357C2×D6111yescharacteristic
348C22⋊C16—611no—
349C22⋊C16—611no—
350#346C2×D6111yescharacteristic, complemented
351C61#3383211yescharacteristic, complemented, Sylow
352C2×#41—1221no—
353C22×Q8—1221no—
354C22×Q8—1221no—
355C22×D4—1221no—
356#135—1222no—
357#135—2441no—
358C2×Q16—611no—
359C2×Q16—611no—
360C2×Q16—611no—
361C2×Q16—1221no—
362C2×Q16—1221no—
363C2×Q16—1222no—
364C2×Q16—1222no—
365C2×Q16—1221no—
366C2×Q16—2441no—
367C2×Q16—2441no—
368C2×Q16C22×D6111yescharacteristic
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—611no—
372C2×D8—1221no—
373C2×D8—1222no—
374C2×D8—2441no—
375C2×D8C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8—611no—
378C4⋊Q8—1221no—
379C4⋊Q8—2441no—
380#124—1221no—
381#123—2441no—
382C22⋊Q8—1221no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—2441no—
386C22⋊Q8—2441no—
387#121—1221no—
388#121—2441no—
389#120—2441no—
390C4×Q8—21no—
391C4×D4—21no—
392C4×D4—611no—
393C4×D4—2441no—
394C2×C22⋊C4—2441no—
395Q32—1222no—
396Q32—1222no—
397Q32—2441no—
398SD32—1222no—
399SD32—1222no—
400SD32—1222no—
401SD32—1222no—
402SD32—2441no—
403SD32D24414yescomplemented
404M32—1222no—
405C2×C16—611no—
406C2×C16—611no—
407C2×C16—1222no—
408C2×C16C22×D6111yescharacteristic
409C2×C16C22×D6111yescharacteristic
410#107C22×D6111yescharacteristic
411#107C22×D6111yescharacteristic
412C4⋊C8—611no—
413#103—21no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—21no—
418#102—1221no—
419C22⋊C8—611no—
420C22⋊C8—611no—
421C4×C8C22×D6111yescharacteristic
422C24—2441no—
423#41—4881no—
424#41—2442no—
425C2×Q8—21no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—4881no—
430C2×Q8—4881no—
431C2×Q8—2441no—
432C2×Q8—2441no—
433C2×Q8—2442no—
434C2×Q8—2442no—
435C2×D4—21no—
436C2×D4—611no—
437C2×D4—1221no—
438C2×D4—4881no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2442no—
442C2×D4—2441no—
443C22×C4—41no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4—2441no—
449Q16—21no—
450Q16—1221no—
451Q16—1221no—
452Q16—1221no—
453Q16—1221no—
454Q16—1221no—
455Q16—1221no—
456Q16—4881no—
457Q16—2442no—
458Q16—2442no—
459Q16C2×D24412yes—
460SD16—4881no—
461SD16—2442no—
462D8—21no—
463D8—1221no—
464D8—1221no—
465D8—2442no—
466D8C2×D24412yes—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8D4×D6111yescharacteristic
474C2×C8C23×D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—611no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C22⋊C4—41no—
485C22⋊C4—611no—
486C22⋊C4—611no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C42—41no—
494C42D4×D6111yescharacteristic
495C16—21no—
496C16—1221no—
497C16—1221no—
498C16C2×D24412yes—
499C23—41no—
500C23—611no—
501C23—611no—
502C23—9761no—
503C23—4882no—
504C23—4882no—
505C23—2441no—
506Q8—22no—
507Q8—41no—
508Q8—4882no—
509Q8—4882no—
510Q8—2441no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516D4—22no—
517D4—41no—
518D4—1222no—
519D4—9761no—
520D4—4882no—
521D4—4882no—
522D4—2441no—
523D4—2441no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—161no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4D8×D6111yescharacteristic
541C2×C4D8×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C8—21no—
544C8—1221no—
545C8—1221no—
546C8C22×D24411yescharacteristic
547C8#1451982311yescharacteristic
548C22—42no—
549C22—81no—
550C22—81no—
551C22—612no—
552C22—1221no—
553C22—1221no—
554C22—1221no—
555C22—1221no—
556C22—1221no—
557C22—9764no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22C2×D8×D6111yescharacteristic
562C4—21no—
563C4—21no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—1221no—
568C4—1221no—
569C4—4881no—
570C4—4881no—
571C4—4881no—
572C4D4×D24411yescharacteristic
573C4#1684056411yescharacteristic
574C2—82no—
575C2—1222no—
576C2—9761no—
577C2—2441no—
578C2D8×D24411yescharacteristic
579C2#510×D6111yescharacteristic
580C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2175C2
socleC22×C61C22×D8
prime radical#21457505C2
hypercenter#2175D61
hypocenterC61#33832
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
1. C2×#16840201C2semidirect
2. C2×SD32×D61C2semidirect
3. Q16×D244C2semidirect
5. #21448090C2semidirect
6. #21447933C2semidirect
7. #21447911C2semidirect
8. #21447881C2semidirect
9. #21447864C2semidirect
10. #21447760C2semidirect
11. #21447754C2semidirect
12. #21447749C2semidirect
13. #21447737C2semidirect
14. #21447646C2semidirect
15. C61×#2175C2semidirect
18. C2×Q16×D61C22semidirect
23. C2×Dic488C22semidirect
24. C2×Dic488C22semidirect
25. C2×#14519723C22semidirect
32. C2×#14519716C22semidirect
35. #16840142C22semidirect
38. #16840004C22semidirect
47. C4⋊#11758185C22semidirect
48. #16839565C22semidirect
49. #16839565C22semidirect
50. #16839565C22semidirect
53. #16839541C22semidirect
55. C4×Q16×C61C22semidirect
57. C61×#366C22semidirect
58. C61×#358C22semidirect
60. C61×#346C22semidirect
123. SD32×C61D4semidirect
271. C2×D61#510semidirect
297. #2175D61semidirect
337. C4×Q16C2×D61semidirect
340. #366C2×D61semidirect
345. #358C2×D61semidirect
350. #346C2×D61semidirect
351. C61#33832semidirect
403. SD32D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
C2×SD32×D612
Q16×D2442
#214575052
#214480902
#214479332
#214479112
#214478812
#214478642
#214477602
#214477542
#214477492
#214477372
#214476462
C61×#21752
#3383261