Finite Group Databasebeta

← #29836704

Subgroups — #29836704

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×#16840191C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457559C211yesmaximal, characteristic, complemented
4#21457505C211yesmaximal, characteristic, complemented
5#21448078C211yesmaximal, characteristic, complemented
6#21447918C211yesmaximal, characteristic, complemented
7#21447914C211yesmaximal, characteristic, complemented
8#21447882C211yesmaximal, characteristic, complemented
9#21447756C211yesmaximal, characteristic, complemented
10#21447754C211yesmaximal, characteristic, complemented
11#21447743C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13C4×#14519721C211yesmaximal, characteristic, complemented
14#21447638C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×#14519796C2211yescharacteristic, complemented
18C2×Q16×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840420—21no—
21#16840407—21no—
22#16840191—24no—
23C2×Dic488C2211yescharacteristic, complemented
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519721C2211yescharacteristic, complemented
28C2×#14519720C2211yescharacteristic
29SD32×D61—24no—
30C2×#14519718C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32C2×C16×D61C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic
34#16840004C2211yescharacteristic
35#16840000C2211yescharacteristic, complemented
36#16839999C2211yescharacteristic
37#16839995C2211yescharacteristic
38#16839981—21no—
39#16839977—21no—
40#16839921—21no—
41#16839914—21no—
42#16839908—21no—
43#16839899—21no—
44C4×Dic244C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic
50#16839563C2211yescharacteristic
51#16839541C2211yescharacteristic
52#16839541C2211yescharacteristic, complemented
53C4×#11758185C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×#11758224—21no—
63C2×Q8×D61—21no—
64C2×D4×D61—21no—
65#14519796—22no—
66#14519796—22no—
67#14519796—41no—
68#14519796—41no—
69C2×Dic244C2311yescharacteristic
70C2×Dic244C2311yescharacteristic
71C2×#11758202—21no—
72Q16×D61—22no—
73Q16×D61—41no—
74C2×#11758190—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×D488C2311yescharacteristic
78C2×#11758199—21no—
79D8×D61—22no—
80D8×D61—41no—
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83#14519765—21no—
84#14519763—41no—
85#14519761—41no—
86#14519757—21no—
87#14519754—41no—
88#14519753—21no—
89#14519752—21no—
90#14519751—41no—
91#14519750—41no—
92#14519748—21no—
93#14519747—41no—
94C4×Dic122—21no—
95C4×D244D411yescharacteristic
96#14519737—41no—
97C2×#11758198—41no—
98Dic488—22no—
99#14519723—41no—
100#14519722—22no—
101#14519721—22no—
102#14519721D414yescomplemented
103#14519720—22no—
104#14519718—41no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519702—21no—
112#14519701—21no—
113#14519700—21no—
114#14519698—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#124×C61—21no—
123C4×D4×C61—21no—
124SD32×C61—22no—
125D16×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C22×C4×C61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C22×Dic61—41no—
141C22×Dic61—41no—
142Q8×D61—42no—
143Q8×D61—81no—
144#11758224—42no—
145#11758224—42no—
146#11758224—81no—
147#11758224—81no—
148D4×D61—42no—
149D4×D61—81no—
150C2×D244—21no—
151C2×D244—41no—
152C2×D244D811yescharacteristic
153C2×C4×D61—41no—
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—41no—
159Q16×C61—21no—
160Q16×C61—21no—
161SD16×C61—81no—
162D8×C61—21no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61C2×D412yes—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2411yescharacteristic
168C4⋊C4×C61—21no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C42×C61C2×D411yescharacteristic
173C22⋊Dic61—41no—
174#11758202—42no—
175#11758200—42no—
176#11758200—42no—
177#11758199—42no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198D811yescharacteristic
184#11758198D811yescharacteristic
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187C4⋊Dic61D811yescharacteristic
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—41no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195Dic244—21no—
196Dic244C2×D412yes—
197D488—21no—
198D488—21no—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C22×D61—41no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—161no—
214C22×D61C2×D811yescharacteristic
215C22×D61C2×D811yescharacteristic
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—22no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—161no—
225C2×C4×C61C2×D811yescharacteristic
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C22⋊D61—82no—
229C22⋊D61—82no—
230C22⋊D61—82no—
231C22⋊D61—82no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—82no—
236C2×Dic61—82no—
237C2×Dic61C2×D811yescharacteristic
238C2×Dic61C2×D811yescharacteristic
239D244—22no—
240D244—41no—
241D244—41no—
242D244—161no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—82no—
248Dic122—22no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—82no—
253C8×C61—21no—
254C8×C61C22×D411yescharacteristic
255C8×C61H(5,𝔽2)11yescharacteristic
256#92418—21no—
257#92418—21no—
258#33848—611nomaximal, Sylow
259C22×C61—42no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61C22×D811yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—164no—
275C2×D61#50812yescomplemented
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61D4211yescharacteristic
281C4×C61#57611yescharacteristic
282Dic61—21no—
283Dic61—21no—
284Dic61—42no—
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#2210D6111yescharacteristic, complemented
296#2189—611no—
297#2188—611no—
298#2186—611no—
299#2184—611no—
300C4×SD32—611no—
301D4×C16—611no—
302C2×C61—82no—
303C2×C61—161no—
304C2×C61D4×D811yescharacteristic
305C2×C61C2×#50811yescharacteristic
306C2×C61#341611yescharacteristic
307D61—22no—
308D61—41no—
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—1224no—
324C2×SD32C2×D6111yescharacteristic
325C2×D16—611no—
326C2×D16C2×D6111yescharacteristic
327C22×C16—611no—
328C22×C16—611no—
329#495—611no—
330#495C2×D6111yescharacteristic, complemented
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×D8C2×D6111yescharacteristic
343C8×D4—611no—
344#367—611no—
345C4⋊C16C2×D6111yescharacteristic, complemented
346#358—611no—
347#358—611no—
348#358C2×D6111yescharacteristic, complemented
349#357—611no—
350#357—611no—
351#357C2×D6111yescharacteristic, complemented
352C22⋊C16—611no—
353C22⋊C16—611no—
354C4×C16—611no—
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—1221no—
367C2×Q16—1222no—
368C2×Q16—2441no—
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—21no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—611no—
375C2×D8—1221no—
376C2×D8—1222no—
377C2×D8—2441no—
378C2×D8C22×D6111yescharacteristic
379C2×D8C22×D6111yescharacteristic
380C22×C8—611no—
381C22×C8—611no—
382#124—21no—
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—21no—
396C4×D4—611no—
397C4×D4—2441no—
398C2×C22⋊C4—2441no—
399Q32—1222no—
400Q32—2441no—
401SD32—22no—
402SD32—614no—
403SD32—1222no—
404SD32—1222no—
405SD32—1222no—
406D16—22no—
407D16—2441no—
408C2×C16—611no—
409C2×C16—611no—
410C2×C16—1222no—
411C2×C16—1222no—
412C2×C16C22×D6111yescharacteristic
413C2×C16C22×D6111yescharacteristic
414#107—611no—
415#107C22×D6111yescharacteristic
416C4⋊C8—611no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—1221no—
422#102—1221no—
423C22⋊C8—611no—
424C22⋊C8—611no—
425C4×C8C22×D6111yescharacteristic
426C24—2441no—
427#41—4881no—
428#41—4881no—
429#41—2442no—
430#41—2442no—
431C2×Q8—21no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—2441no—
436C2×Q8—2442no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—611no—
440C2×D4—1221no—
441C2×D4—4881no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4—2442no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4—2441no—
448C22×C4—41no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4—2441no—
454Q16—21no—
455Q16—21no—
456Q16—612no—
457Q16—1221no—
458Q16—1221no—
459Q16—1221no—
460Q16—2442no—
461SD16—81no—
462SD16—4881no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—21no—
468D8—1221no—
469D8—1221no—
470D8—2442no—
471D8D4×D6112yes—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8D4×D6111yescharacteristic
479C2×C8C23×D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—611no—
482C4⋊C4—1221no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—611no—
490C22⋊C4—611no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C42—2441no—
498C42D4×D6111yescharacteristic
499C16—21no—
500C16—21no—
501C16—612no—
502C16—1221no—
503C23—41no—
504C23—41no—
505C23—611no—
506C23—611no—
507C23—9761no—
508C23—4882no—
509C23—4882no—
510C23—2441no—
511Q8—41no—
512Q8—41no—
513Q8—1222no—
514Q8—4882no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518D4—22no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—1222no—
523D4—9761no—
524D4—4882no—
525D4—4882no—
526D4—4882no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—161no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4D8×D6111yescharacteristic
546C2×C4D8×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C8—21no—
549C8—1221no—
550C8—1221no—
551C8C2×D4×D6111yescharacteristic
552C8#1451982311yescharacteristic
553C22—42no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—612no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—9764no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—21no—
572C4—81no—
573C4—81no—
574C4—1221no—
575C4—1221no—
576C4—4881no—
577C4—4881no—
578C4—2442no—
579C4#1684039611yescharacteristic
580C4#1684056611yescharacteristic
581C2—82no—
582C2—161no—
583C2—1222no—
584C2—9761no—
585C2—2441no—
586C2#2145749111yescharacteristic
587C2#508×D6111yescharacteristic
588C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
hypercenter#2210D61
hypocenterC61#33848
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#16840191C2semidirect
2. C2×SD32×D61C2semidirect
3. #21457559C2semidirect
4. #21457505C2semidirect
5. #21448078C2semidirect
6. #21447918C2semidirect
7. #21447914C2semidirect
8. #21447882C2semidirect
9. #21447756C2semidirect
10. #21447754C2semidirect
11. #21447743C2semidirect
12. #21447729C2semidirect
13. C4×#14519721C2semidirect
14. #21447638C2semidirect
15. C61×#2210C2semidirect
17. C2×#14519796C22semidirect
23. C2×Dic488C22semidirect
24. C2×#14519723C22semidirect
27. C2×#14519721C22semidirect
32. C2×C16×D61C22semidirect
35. #16840000C22semidirect
44. C4×Dic244C22semidirect
46. #16839589C22semidirect
47. #16839567C22semidirect
48. #16839565C22semidirect
52. #16839541C22semidirect
53. C4×#11758185C22semidirect
56. C61×#495C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
102. #14519721D4semidirect
275. C2×D61#508semidirect
295. #2210D61semidirect
330. #495C2×D61semidirect
345. C4⋊C16C2×D61semidirect
348. #358C2×D61semidirect
351. #357C2×D61semidirect
355. C61#33848semidirect

Maximal subgroups

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

subgroupindex
C2×#168401912
C2×SD32×D612
#214575592
#214575052
#214480782
#214479182
#214479142
#214478822
#214477562
#214477542
#214477432
#214477292
C4×#145197212
#214476382
C61×#22102
#3384861