Finite Group Databasebeta

← #29836506

Subgroups — #29836506

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×#16840205C211yesmaximal, characteristic, complemented
2C22×#14519722C211yesmaximal, characteristic, complemented
3D4×Dic244C211yesmaximal, characteristic, complemented
4#21457506C211yesmaximal, characteristic
5#21448090C211yesmaximal, characteristic, complemented
6#21447932C211yesmaximal, characteristic, complemented
7#21447910C211yesmaximal, characteristic, complemented
8#21447882C211yesmaximal, characteristic, complemented
9#21447865C211yesmaximal, characteristic, complemented
10#21447761C211yesmaximal, characteristic, complemented
11#21447755C211yesmaximal, characteristic, complemented
12#21447750C211yesmaximal, characteristic, complemented
13#21447738C211yesmaximal, characteristic, complemented
14#21447685C211yesmaximal, characteristic, complemented
15C61×#2168C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×Dic244C2211yescharacteristic, complemented
19C22×D488C2211yescharacteristic
20D4×Dic122—21no—
21#16840408—21no—
22#16840205—24no—
23C2×Dic488C2211yescharacteristic, complemented
24C2×Dic488C2211yescharacteristic, complemented
25C2×Dic488C2211yescharacteristic, complemented
26C2×#14519722—24no—
27C2×#14519722C2211yescharacteristic
28C2×#14519722C2211yescharacteristic
29C2×#14519722C2211yescharacteristic
30C2×#14519722C2211yescharacteristic
31C2×#14519722C2211yescharacteristic
32C2×#14519722C2211yescharacteristic
33#16840146C2211yescharacteristic
34#16840142C2211yescharacteristic, complemented
35#16840003C2211yescharacteristic
36#16840003C2211yescharacteristic
37#16840003C2211yescharacteristic, complemented
38#16839998C2211yescharacteristic
39#16839978—21no—
40#16839973—21no—
41#16839920—21no—
42#16839920—21no—
43#16839915—21no—
44#16839909—21no—
45C4×Dic244C2211yescharacteristic, complemented
46C4×D488C2211yescharacteristic
47#16839588C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839567C2211yescharacteristic, complemented
50#16839567C2211yescharacteristic, complemented
51#16839567C2211yescharacteristic, complemented
52#16839564C2211yescharacteristic
53#16839564C2211yescharacteristic
54C2×M32×C61C2211yescharacteristic, complemented
55C22×C16×C61C2211yescharacteristic
56C8×D4×C61C2211yescharacteristic
57C61×C4⋊C16C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic
59C61×C22⋊C16C2211yescharacteristic, complemented
60C61×#346C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C22×Dic122—21no—
63C22×Dic122—21no—
64C22×D244—21no—
65#14519799—22no—
66#14519799—41no—
67C2×Dic244—22no—
68C2×Dic244—22no—
69C2×Dic244—21no—
70C2×Dic244—21no—
71C2×Dic244—21no—
72C2×Dic244—41no—
73C2×Dic244—41no—
74C2×Dic244C2311yescharacteristic
75C2×Dic244C2311yescharacteristic
76C2×Dic244C2311yescharacteristic
77C2×Dic244C2311yescharacteristic
78C2×#11758190—21no—
79C2×#11758190—21no—
80C2×D488—22no—
81C2×D488—21no—
82C2×D488—41no—
83C2×D488C2311yescharacteristic
84C2×D488C2311yescharacteristic
85#14519777—21no—
86#14519776—41no—
87#14519765—21no—
88#14519762—41no—
89#14519756—21no—
90#14519756—21no—
91#14519756—21no—
92#14519754—41no—
93#14519754—41no—
94#14519751—21no—
95#14519750—41no—
96#14519745—41no—
97C4×Dic122—21no—
98C4×D244—21no—
99D4×Dic61—41no—
100C22⋊C4×D61—41no—
101Dic488—22no—
102Dic488—22no—
103Dic488—41no—
104#14519722—22no—
105#14519722—22no—
106#14519722—22no—
107#14519722—22no—
108#14519722—41no—
109#14519722D414yescomplemented
110#14519711C2311yescharacteristic
111#14519711C2311yescharacteristic
112#14519702—21no—
113#14519702—21no—
114#14519702—21no—
115#14519702—21no—
116#14519699—21no—
117#14519699—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61D411yescharacteristic
121M32×C61—22no—
122C2×C16×C61—22no—
123C2×C16×C61C2311yescharacteristic
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126C2×C16×C61C2311yescharacteristic
127C4⋊C8×C61D411yescharacteristic
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×D4×C61D811yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C22×Dic61—41no—
138C22×Dic61—41no—
139#11758222—42no—
140#11758222—81no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—42no—
144C2×D244—41no—
145C2×D244—81no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—41no—
155C2×Dic122—41no—
156C2×Dic122—81no—
157C2×Dic122—81no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2411yescharacteristic
166C4⋊C4×C61D811yescharacteristic
167C22⋊C4×C61D811yescharacteristic
168C22⋊C4×C61D811yescharacteristic
169C42×C61C2×D411yescharacteristic
170C22⋊Dic61—41no—
171C22⋊Dic61—41no—
172C22⋊Dic61—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—41no—
180C4⋊Dic61—41no—
181#11758196—41no—
182#11758196—41no—
183#11758196—41no—
184#11758196—41no—
185C4×Dic61—41no—
186Dic244—21no—
187Dic244—21no—
188Dic244—21no—
189Dic244—21no—
190Dic244—21no—
191Dic244—21no—
192Dic244—21no—
193Dic244—42no—
194Dic244—42no—
195Dic244—81no—
196Dic244C2×D412yes—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—42no—
201D488C2×D412yes—
202#11758190—42no—
203#11758190—81no—
204C16×C61—21no—
205C16×C61—21no—
206C16×C61—21no—
207C16×C61C2×D412yes—
208C23×C61C2×D811yescharacteristic
209C23×C61C2×D811yescharacteristic
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—161no—
215D4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61C2×D811yescharacteristic
221C2×C4×C61C2×D811yescharacteristic
222C2×C4×C61C2×D811yescharacteristic
223C2×C4×C61C2×D811yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C22⋊D61—82no—
226C22⋊D61—161no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—82no—
234C2×Dic61—82no—
235D244—22no—
236D244—41no—
237D244—41no—
238D244—41no—
239D244—82no—
240C4×D61—82no—
241C4×D61—161no—
242Dic122—22no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—82no—
251Dic122—82no—
252C8×C61—21no—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61C22×D411yescharacteristic
256C8×C61H(5,𝔽2)11yescharacteristic
257#33832—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61#51012yescomplemented
264C22×C61C22×D811yescharacteristic
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—164no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61D4211yescharacteristic
277C4×C61#57611yescharacteristic
278Dic61—42no—
279Dic61—81no—
280Dic61—81no—
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#2175—611no—
298#2168D6111yescharacteristic, complemented
299C2×C61—22no—
300C2×C61—41no—
301C2×C61D4×D811yescharacteristic
302C2×C61C2×#51011yescharacteristic
303C2×C61#341611yescharacteristic
304D61—82no—
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—611no—
322C2×SD32—1224no—
323C2×M32C2×D6111yescharacteristic, complemented
324C22×C16C2×D6111yescharacteristic
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×Q16—611no—
338C4×D8—611no—
339C8×D4C2×D6111yescharacteristic
340#366—611no—
341C4⋊C16C2×D6111yescharacteristic, complemented
342#358—611no—
343#358—611no—
344#358—611no—
345#358—611no—
346#357—611no—
347#357—611no—
348C22⋊C16C2×D6111yescharacteristic
349C22⋊C16C2×D6111yescharacteristic, complemented
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—611no—
362C2×Q16—1222no—
363C2×Q16—1222no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×Q16—1221no—
367C2×Q16—2441no—
368C2×Q16—2441no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—611no—
372C2×D8—611no—
373C2×D8—1222no—
374C2×D8—1221no—
375C2×D8—2441no—
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
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—1221no—
391C4×D4—1221no—
392C4×D4—2441no—
393C4×D4D24411yescharacteristic
394C2×C22⋊C4—2441no—
395Q32—1222no—
396Q32—1222no—
397Q32—2441no—
398SD32—614no—
399SD32—1222no—
400SD32—1222no—
401SD32—1222no—
402SD32—1222no—
403SD32—2441no—
404M32—22no—
405C2×C16—22no—
406C2×C16C22×D6111yescharacteristic
407C2×C16C22×D6111yescharacteristic
408C2×C16C22×D6111yescharacteristic
409C2×C16C22×D6111yescharacteristic
410#107—611no—
411#107—611no—
412C4⋊C8D24411yescharacteristic
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8D24411yescharacteristic
420C22⋊C8D24411yescharacteristic
421C4×C8C22×D6111yescharacteristic
422C24—2441no—
423#41—4881no—
424#41—2442no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—4881no—
430C2×Q8—4881no—
431C2×Q8—2442no—
432C2×Q8—2442no—
433C2×Q8—2441no—
434C2×Q8—2441no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—4881no—
438C2×D4—2442no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
442C2×D4D48811yescharacteristic
443C22×C4—2441no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4—2441no—
447C22×C4C2×D24411yescharacteristic
448C22×C4C2×D24411yescharacteristic
449Q16—612no—
450Q16—1221no—
451Q16—1221no—
452Q16—1221no—
453Q16—1221no—
454Q16—1221no—
455Q16—1221no—
456Q16—1221no—
457Q16—4881no—
458Q16—2442no—
459Q16—2442no—
460SD16—4881no—
461SD16—2442no—
462D8—612no—
463D8—1221no—
464D8—1221no—
465D8—1221no—
466D8—2442no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8C2×D24411yescharacteristic
472C2×C8C2×D24411yescharacteristic
473C2×C8C2×D24411yescharacteristic
474C2×C8C23×D6111yescharacteristic
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4D48811yescharacteristic
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4D48811yescharacteristic
492C22⋊C4D48811yescharacteristic
493C42—2441no—
494C42C2×D24411yescharacteristic
495C16—21no—
496C16—21no—
497C16—21no—
498C16D4×D6112yes—
499C23—9761no—
500C23—4882no—
501C23—4882no—
502C23—2441no—
503C23—2441no—
504C23C2×D48811yescharacteristic
505C23C2×D48811yescharacteristic
506Q8—1222no—
507Q8—4882no—
508Q8—4882no—
509Q8—2441no—
510Q8—2441no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516D4—22no—
517D4—1222no—
518D4—9761no—
519D4—4882no—
520D4—4882no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—9761no—
529C2×C4—4882no—
530C2×C4—4882no—
531C2×C4—4882no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4C2×D48811yescharacteristic
539C2×C4C2×D48811yescharacteristic
540C2×C4C2×D48811yescharacteristic
541C2×C4C2×D48811yescharacteristic
542C2×C4C22×D24411yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D6111yescharacteristic
547C8#1451982311yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—9764no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—2442no—
560C22#1684020512yescomplemented
561C22C22×D48811yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—4881no—
567C4—4881no—
568C4—4881no—
569C4—4881no—
570C4—4881no—
571C4—2442no—
572C4D4×D24411yescharacteristic
573C4#1684056711yescharacteristic
574C2—22no—
575C2—41no—
576C2—9761no—
577C2—4882no—
578C2D4×D48811yescharacteristic
579C2C2×#1684020511yescharacteristic
580C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2168C2
socleC22×C61C22×D8
prime radical#21457506C2
hypercenter#2168D61
hypocenterC61#33832
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. C2×#16840205C2semidirect
2. C22×#14519722C2semidirect
3. D4×Dic244C2semidirect
5. #21448090C2semidirect
6. #21447932C2semidirect
7. #21447910C2semidirect
8. #21447882C2semidirect
9. #21447865C2semidirect
10. #21447761C2semidirect
11. #21447755C2semidirect
12. #21447750C2semidirect
13. #21447738C2semidirect
14. #21447685C2semidirect
15. C61×#2168C2semidirect
18. C22×Dic244C22semidirect
23. C2×Dic488C22semidirect
24. C2×Dic488C22semidirect
25. C2×Dic488C22semidirect
34. #16840142C22semidirect
37. #16840003C22semidirect
45. C4×Dic244C22semidirect
47. #16839588C22semidirect
48. #16839567C22semidirect
49. #16839567C22semidirect
50. #16839567C22semidirect
51. #16839567C22semidirect
54. C2×M32×C61C22semidirect
57. C61×C4⋊C16C22semidirect
59. C61×C22⋊C16C22semidirect
60. C61×#346C22semidirect
109. #14519722D4semidirect
263. C22×C61#510semidirect
298. #2168D61semidirect
323. C2×M32C2×D61semidirect
341. C4⋊C16C2×D61semidirect
349. C22⋊C16C2×D61semidirect
350. #346C2×D61semidirect
351. C61#33832semidirect
560. C22#16840205semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
C22×#145197222
D4×Dic2442
#214575062
#214480902
#214479322
#214479102
#214478822
#214478652
#214477612
#214477552
#214477502
#214477382
#214476852
C61×#21682
#3383261