Finite Group Databasebeta

← #29836703

Subgroups — #29836703

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840194C211yesmaximal, characteristic, complemented
2C22×#14519722C211yesmaximal, characteristic, complemented
3#21457562C211yesmaximal, characteristic, complemented
4#21457506C211yesmaximal, characteristic, complemented
5#21448080C211yesmaximal, characteristic, complemented
6#21447915C211yesmaximal, characteristic, complemented
7#21447915C211yesmaximal, characteristic, complemented
8#21447882C211yesmaximal, characteristic, complemented
9#21447875C211yesmaximal, characteristic, complemented
10#21447761C211yesmaximal, characteristic, complemented
11#21447755C211yesmaximal, characteristic, complemented
12#21447744C211yesmaximal, characteristic, complemented
13#21447733C211yesmaximal, characteristic, complemented
14C4×#14519722C211yesmaximal, characteristic, complemented
15D4×C16×C61C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519799C2211yescharacteristic, complemented
18C22×Dic244C2211yescharacteristic
19C22×D488C2211yescharacteristic
20#16840421—21no—
21#16840408—21no—
22#16840194—24no—
23C2×Dic488C2211yescharacteristic, complemented
24C2×Dic488C2211yescharacteristic, complemented
25C2×#14519722—24no—
26C2×#14519722C2211yescharacteristic
27C2×#14519722C2211yescharacteristic
28C2×#14519722C2211yescharacteristic
29C2×#14519722C2211yescharacteristic
30C2×#14519722C2211yescharacteristic, complemented
31C2×D976C2211yescharacteristic
32C2×D976C2211yescharacteristic
33#16840146C2211yescharacteristic
34#16840146C2211yescharacteristic, complemented
35#16840003C2211yescharacteristic
36#16840003C2211yescharacteristic, complemented
37#16839998C2211yescharacteristic
38#16839998C2211yescharacteristic
39#16839985—21no—
40#16839978—21no—
41#16839926—21no—
42#16839915—21no—
43#16839909—21no—
44#16839903—21no—
45C4×Dic244C2211yescharacteristic, complemented
46C4×D488C2211yescharacteristic
47#16839590C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839567C2211yescharacteristic, complemented
50#16839567C2211yescharacteristic, complemented
51#16839564C2211yescharacteristic
52#16839564C2211yescharacteristic
53#16839564C2211yescharacteristic, complemented
54C22×C16×C61C2211yescharacteristic
55C22×C16×C61C2211yescharacteristic, complemented
56C8×D4×C61C2211yescharacteristic
57C61×C4⋊C16C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic
59C61×C22⋊C16C2211yescharacteristic, complemented
60C4×C16×C61C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C2×#11758222—21no—
63C22×Dic122—21no—
64C22×D244—21no—
65#14519799—22no—
66#14519799—22no—
67#14519799—41no—
68#14519799—41no—
69C2×Dic244—22no—
70C2×Dic244—21no—
71C2×Dic244—41no—
72C2×Dic244C2311yescharacteristic
73C2×Dic244C2311yescharacteristic
74C2×Dic244C2311yescharacteristic
75C2×#11758190—21no—
76C2×#11758190—21no—
77C2×#11758190—21no—
78C2×#11758190—21no—
79C2×D488—22no—
80C2×D488—21no—
81C2×D488—41no—
82C2×D488C2311yescharacteristic
83C2×D488C2311yescharacteristic
84C2×D488C2311yescharacteristic
85#14519765—21no—
86#14519765—21no—
87#14519763—41no—
88#14519762—41no—
89#14519756—21no—
90#14519756—21no—
91#14519755—41no—
92#14519751—21no—
93#14519751—21no—
94#14519750—41no—
95#14519749—41no—
96#14519745—41no—
97C4×Dic122—21no—
98C4×D244—21no—
99#14519737—41no—
100C22⋊C4×D61—41no—
101Dic488—22no—
102Dic488—41no—
103#14519722—22no—
104#14519722—22no—
105#14519722—22no—
106#14519722—22no—
107#14519722D414yescomplemented
108D976—22no—
109D976—41no—
110#14519711C2311yescharacteristic
111#14519711C2311yescharacteristic
112#14519702—21no—
113#14519702—21no—
114#14519702—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61D411yescharacteristic
121C2×C16×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—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C22×Dic61—41no—
140#11758222—42no—
141#11758222—42no—
142#11758222—81no—
143#11758222—81no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—42no—
148C2×D244—41no—
149C2×D244—81no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—42no—
157C2×Dic122—41no—
158C2×Dic122—81no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2411yescharacteristic
167C4⋊C4×C61D811yescharacteristic
168C22⋊C4×C61D811yescharacteristic
169C22⋊C4×C61D811yescharacteristic
170C42×C61C2×D411yescharacteristic
171C22⋊Dic61—41no—
172C22⋊Dic61—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182#11758196—41no—
183#11758196—41no—
184#11758196—41no—
185C4×Dic61—41no—
186Dic244—21no—
187Dic244—21no—
188Dic244—21no—
189Dic244—21no—
190Dic244—21no—
191Dic244—42no—
192Dic244C2×D412yes—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—42no—
199D488C2×D412yes—
200#11758190—42no—
201#11758190—42no—
202#11758190—81no—
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—41no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—161no—
216D4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61C2×D811yescharacteristic
222C2×C4×C61C2×D811yescharacteristic
223C2×C4×C61C2×D811yescharacteristic
224C2×C4×C61C2×D811yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61—82no—
227C22⋊D61—82no—
228C22⋊D61—82no—
229C22⋊D61—161no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—82no—
236D244—22no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242D244—82no—
243C4×D61—82no—
244C4×D61—82no—
245C4×D61—161no—
246Dic122—22no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—82no—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61C22×D411yescharacteristic
257C8×C61H(5,𝔽2)11yescharacteristic
258#33848—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61#50812yescomplemented
265C22×C61C22×D811yescharacteristic
266C2×D61—42no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—164no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61D4211yescharacteristic
281C4×C61#57611yescharacteristic
282Dic61—42no—
283Dic61—81no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287C2×#508—611no—
288C22×SD32—611no—
289#3285—611no—
290#3280—611no—
291#2249—611no—
292#2217—611no—
293#2217—611no—
294#2211—611no—
295#2210—611no—
296#2189—611no—
297#2188—611no—
298#2186—611no—
299#2184—611no—
300C4×SD32—611no—
301D4×C16D6111yescharacteristic, complemented
302C2×C61—22no—
303C2×C61—41no—
304C2×C61D4×D811yescharacteristic
305C2×C61C2×#50811yescharacteristic
306C2×C61#341611yescharacteristic
307D61—82no—
308D61—161no—
309D61—161no—
310C2×#135—611no—
311C2×#135—611no—
312C22×Q16—611no—
313C22×D8—611no—
314#548—1221no—
315#546—1221no—
316#508—1224no—
317C2×Q32—611no—
318C2×Q32—611no—
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32—611no—
324C2×SD32—1224no—
325C2×D16—611no—
326C2×D16—611no—
327C22×C16C2×D6111yescharacteristic
328C22×C16C2×D6111yescharacteristic, complemented
329#495—611no—
330#495—611no—
331#467—611no—
332#467—611no—
333#466—611no—
334#466—611no—
335#464—1221no—
336#463—1221no—
337#452—1221no—
338#450—1221no—
339#449—1221no—
340#448—1221no—
341C4×Q16—611no—
342C4×D8—611no—
343C8×D4C2×D6111yescharacteristic
344#367—611no—
345C4⋊C16C2×D6111yescharacteristic, complemented
346#358—611no—
347#358—611no—
348#358—611no—
349#357—611no—
350#357—611no—
351#357—611no—
352C22⋊C16C2×D6111yescharacteristic
353C22⋊C16C2×D6111yescharacteristic, complemented
354C4×C16C2×D6111yescharacteristic, complemented
355C61#3384811yescharacteristic, complemented, Sylow
356C2×#41—1221no—
357C2×#41—1221no—
358C22×Q8—1221no—
359C22×D4—1221no—
360#135—1222no—
361#135—1222no—
362#135—2441no—
363#135—2441no—
364C2×Q16—611no—
365C2×Q16—611no—
366C2×Q16—611no—
367C2×Q16—1222no—
368C2×Q16—1221no—
369C2×Q16—2441no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—611no—
375C2×D8—611no—
376C2×D8—611no—
377C2×D8—1222no—
378C2×D8—1221no—
379C2×D8—2441no—
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382#124—1221no—
383#124—1221no—
384#124—2441no—
385#123—2441no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—1221no—
390#121—1221no—
391#121—2441no—
392#121—2441no—
393#120—2441no—
394C4×Q8—1221no—
395C4×D4—1221no—
396C4×D4—2441no—
397C4×D4D24411yescharacteristic
398C2×C22⋊C4—2441no—
399Q32—1222no—
400Q32—2441no—
401SD32—614no—
402SD32—1222no—
403SD32—1222no—
404SD32—1222no—
405SD32—1222no—
406D16—1222no—
407D16—2441no—
408C2×C16—22no—
409C2×C16—22no—
410C2×C16C22×D6111yescharacteristic
411C2×C16C22×D6111yescharacteristic
412C2×C16C22×D6111yescharacteristic
413C2×C16C22×D6111yescharacteristic
414#107—611no—
415#107—611no—
416C4⋊C8D24411yescharacteristic
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—1221no—
421#102—1221no—
422#102—1221no—
423C22⋊C8D24411yescharacteristic
424C22⋊C8D24411yescharacteristic
425C4×C8C22×D6111yescharacteristic
426C24—2441no—
427#41—4881no—
428#41—4881no—
429#41—2442no—
430#41—2442no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2441no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—4881no—
441C2×D4—2442no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4D48811yescharacteristic
448C22×C4—2441no—
449C22×C4—2441no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4C2×D24411yescharacteristic
453C22×C4C2×D24411yescharacteristic
454Q16—612no—
455Q16—1221no—
456Q16—1221no—
457Q16—1221no—
458Q16—1221no—
459Q16—1221no—
460Q16—2442no—
461SD16—4881no—
462SD16—4881no—
463SD16—2442no—
464SD16—2442no—
465D8—612no—
466D8—1221no—
467D8—1221no—
468D8—1221no—
469D8—1221no—
470D8—1221no—
471D8—2442no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8C2×D24411yescharacteristic
477C2×C8C2×D24411yescharacteristic
478C2×C8C2×D24411yescharacteristic
479C2×C8C23×D6111yescharacteristic
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4D48811yescharacteristic
486C22⋊C4—2441no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4D48811yescharacteristic
496C22⋊C4D48811yescharacteristic
497C42—2441no—
498C42C2×D24411yescharacteristic
499C16—21no—
500C16—21no—
501C16—21no—
502C16D4×D6112yes—
503C23—9761no—
504C23—4882no—
505C23—4882no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23C2×D48811yescharacteristic
510C23C2×D48811yescharacteristic
511Q8—1222no—
512Q8—4882no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518D4—22no—
519D4—1222no—
520D4—9761no—
521D4—4882no—
522D4—4882no—
523D4—4882no—
524D4—4882no—
525D4—2441no—
526D4—2441no—
527D4—2441no—
528D4—2441no—
529D4—2441no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—9761no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—4882no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C2×D48811yescharacteristic
544C2×C4C2×D48811yescharacteristic
545C2×C4C2×D48811yescharacteristic
546C2×C4C2×D48811yescharacteristic
547C2×C4C22×D24411yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8C2×D4×D6111yescharacteristic
552C8#1451982311yescharacteristic
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—9764no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—4881no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22—2442no—
568C22#1684019412yescomplemented
569C22C22×D48811yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—4881no—
575C4—4881no—
576C4—4881no—
577C4—4881no—
578C4—2442no—
579C4D4×D24411yescharacteristic
580C4#1684056711yescharacteristic
581C2—22no—
582C2—41no—
583C2—9761no—
584C2—9761no—
585C2—4882no—
586C2D4×D48811yescharacteristic
587C2C2×#1684019411yescharacteristic
588C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingD4×C16×C61C2
socleC22×C61C22×D8
hypercenterD4×C16D61
hypocenterC61#33848
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. C2×#16840194C2semidirect
2. C22×#14519722C2semidirect
3. #21457562C2semidirect
4. #21457506C2semidirect
5. #21448080C2semidirect
6. #21447915C2semidirect
7. #21447915C2semidirect
8. #21447882C2semidirect
9. #21447875C2semidirect
10. #21447761C2semidirect
11. #21447755C2semidirect
12. #21447744C2semidirect
13. #21447733C2semidirect
14. C4×#14519722C2semidirect
15. D4×C16×C61C2semidirect
17. C2×#14519799C22semidirect
23. C2×Dic488C22semidirect
24. C2×Dic488C22semidirect
30. C2×#14519722C22semidirect
34. #16840146C22semidirect
36. #16840003C22semidirect
45. C4×Dic244C22semidirect
47. #16839590C22semidirect
48. #16839567C22semidirect
49. #16839567C22semidirect
50. #16839567C22semidirect
53. #16839564C22semidirect
55. C22×C16×C61C22semidirect
57. C61×C4⋊C16C22semidirect
59. C61×C22⋊C16C22semidirect
60. C4×C16×C61C22semidirect
107. #14519722D4semidirect
264. C22×C61#508semidirect
301. D4×C16D61semidirect
328. C22×C16C2×D61semidirect
345. C4⋊C16C2×D61semidirect
353. C22⋊C16C2×D61semidirect
354. C4×C16C2×D61semidirect
355. C61#33848semidirect
568. C22#16840194semidirect

Maximal subgroups

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

subgroupindex
C2×#168401942
C22×#145197222
#214575622
#214575062
#214480802
#214479152
#214478822
#214478752
#214477612
#214477552
#214477442
#214477332
C4×#145197222
D4×C16×C612
#3384861