Finite Group Databasebeta

← #29836693

Subgroups — #29836693

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×#16840195C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457561C211yesmaximal, characteristic, complemented
4#21457503C211yesmaximal, characteristic, complemented
5#21448078C211yesmaximal, characteristic, complemented
6#21447918C211yesmaximal, characteristic, complemented
7#21447914C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447762C211yesmaximal, characteristic, complemented
10#21447751C211yesmaximal, characteristic, complemented
11#21447741C211yesmaximal, characteristic, complemented
12#21447732C211yesmaximal, characteristic, complemented
13C4×#14519720C211yesmaximal, characteristic, complemented
14#21447638C211yesmaximal, characteristic, complemented
15C61×#2211C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519800C2211yescharacteristic, complemented
18C2×Q16×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840424—21no—
21#16840405—21no—
22#16840195—24no—
23C2×#14519723C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519720C2211yescharacteristic
27C2×#14519720C2211yescharacteristic, complemented
28SD32×D61—24no—
29C2×D976C2211yescharacteristic
30C2×#14519718C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32C2×C16×D61C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic, complemented
34#16840004C2211yescharacteristic, complemented
35#16840000C2211yescharacteristic
36#16839999C2211yescharacteristic
37#16839995C2211yescharacteristic
38#16839984—21no—
39#16839974—21no—
40#16839927—21no—
41#16839911—21no—
42#16839906—21no—
43#16839902—21no—
44C4×D488C2211yescharacteristic
45C8×D244C2211yescharacteristic
46#16839589C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic
51#16839541C2211yescharacteristic
52#16839541C2211yescharacteristic, complemented
53C4×#11758185C2211yescharacteristic, complemented
54C2×Q32×C61C2211yescharacteristic, complemented
55C2×SD32×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic
57C4×Q16×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic
61C2×#11758226—21no—
62C2×#11758226—21no—
63C2×Q8×D61—21no—
64C2×D4×D61—21no—
65#14519800—22no—
66#14519800—22no—
67#14519800—41no—
68#14519800—41no—
69C2×Dic244C2311yescharacteristic
70C2×#11758202—21no—
71Q16×D61—22no—
72Q16×D61—41no—
73C2×#11758190—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×D488C2311yescharacteristic
77C2×D488C2311yescharacteristic
78C2×#11758199—21no—
79D8×D61—22no—
80D8×D61—41no—
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83#14519765—21no—
84#14519765—41no—
85#14519759—41no—
86#14519757—21no—
87#14519757—41no—
88#14519753—21no—
89#14519752—21no—
90#14519752—41no—
91#14519750—41no—
92#14519748—21no—
93#14519747—41no—
94C4×D244—21no—
95C4×D244—41no—
96C4×D244D411yescharacteristic
97C22⋊C4×D61—41no—
98#14519723—41no—
99#14519722—22no—
100#14519721—22no—
101#14519720—22no—
102#14519720D414yescomplemented
103D976—22no—
104#14519718—41no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519701—21no—
112#14519700—21no—
113#14519699—21no—
114#14519698—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×Q16×C61C2311yescharacteristic
120C2×SD16×C61—21no—
121C2×D8×C61C2311yescharacteristic
122#124×C61—21no—
123C4×Q8×C61—21no—
124Q32×C61—22no—
125SD32×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#103×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137#11758226—42no—
138#11758226—42no—
139#11758226—81no—
140#11758226—81no—
141Q8×D61—42no—
142Q8×D61—81no—
143D4×D61—42no—
144D4×D61—81no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—41no—
148C2×D244—41no—
149C2×D244—41no—
150C2×D244D811yescharacteristic
151C2×C4×D61—41no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—41no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61C2×D412yes—
163SD16×C61—81no—
164D8×C61—21no—
165D8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C42×C61—41no—
173C42×C61C2×D411yescharacteristic
174C22⋊Dic61—41no—
175#11758202—42no—
176#11758201—42no—
177#11758201—42no—
178#11758199—42no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198D811yescharacteristic
186#11758198D811yescharacteristic
187C4⋊Dic61—21no—
188C4⋊Dic61—41no—
189C4⋊Dic61D811yescharacteristic
190#11758196—41no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488C2×D412yes—
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—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—161no—
214C22×D61C2×D811yescharacteristic
215C22×D61C2×D811yescharacteristic
216Q8×C61—22no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61C2×D811yescharacteristic
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C22⋊D61—82no—
229C22⋊D61—161no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61C2×D811yescharacteristic
233C2×Dic61C2×D811yescharacteristic
234D244—22no—
235D244—22no—
236D244—41no—
237D244—41no—
238D244—41no—
239D244—82no—
240D244—82no—
241D244—82no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—82no—
247C4×D61—82no—
248C4×D61—82no—
249C4×D61—161no—
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—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61C22×D811yescharacteristic
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—42no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—164no—
275C2×D61#50812yescomplemented
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—42no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61D4211yescharacteristic
282C4×C61#57611yescharacteristic
283Dic61—21no—
284Dic61—21no—
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#2211D6111yescharacteristic, complemented
295#2210—611no—
296#2189—611no—
297#2188—611no—
298#2186—611no—
299#2184—611no—
300C4×SD32—611no—
301D4×C16—611no—
302C2×C61—161no—
303C2×C61D4×D811yescharacteristic
304C2×C61C2×#50811yescharacteristic
305C2×C61#341611yescharacteristic
306D61—22no—
307D61—41no—
308D61—82no—
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×Q32C2×D6111yescharacteristic, complemented
319C2×SD32—611no—
320C2×SD32—611no—
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32—1224no—
324C2×SD32C2×D6111yescharacteristic
325C2×D16—611no—
326C2×D16—611no—
327C22×C16—611no—
328C22×C16—611no—
329#495—611no—
330#495C2×D6111yescharacteristic
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×Q16C2×D6111yescharacteristic, complemented
342C4×D8—611no—
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
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—1221no—
366C2×Q16—1222no—
367C2×Q16—2441no—
368C2×Q16C22×D6111yescharacteristic
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—21no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×D8—611no—
375C2×D8—611no—
376C2×D8—1221no—
377C2×D8—1222no—
378C2×D8—2441no—
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—21no—
395C4×D4—611no—
396C4×D4—1221no—
397C4×D4—2441no—
398C2×C22⋊C4—2441no—
399Q32—22no—
400Q32—2441no—
401SD32—22no—
402SD32—614no—
403SD32—1222no—
404SD32—1222no—
405SD32—1222no—
406D16—1222no—
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—21no—
418#103—1221no—
419#103—1221no—
420#102—1221no—
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—21no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—2441no—
436C2×Q8—2442no—
437C2×D4—21no—
438C2×D4—611no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—4881no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2442no—
447C2×D4—2441no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—2441no—
451C22×C4—2441no—
452C22×C4—2441no—
453C22×C4—2441no—
454Q16—21no—
455Q16—21no—
456Q16—21no—
457Q16—1221no—
458Q16—1221no—
459Q16—2442no—
460Q16D4×D6112yes—
461SD16—81no—
462SD16—4881no—
463SD16—2442no—
464SD16—2442no—
465D8—21no—
466D8—21no—
467D8—612no—
468D8—1221no—
469D8—1221no—
470D8—1221no—
471D8—2442no—
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—41no—
482C4⋊C4—611no—
483C4⋊C4—1221no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—611no—
489C22⋊C4—611no—
490C22⋊C4—2441no—
491C22⋊C4—2441no—
492C22⋊C4—2441no—
493C22⋊C4—2441no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C42—41no—
498C42D4×D6111yescharacteristic
499C16—21no—
500C16—21no—
501C16—612no—
502C16—1221no—
503C23—41no—
504C23—611no—
505C23—611no—
506C23—9761no—
507C23—4882no—
508C23—4882no—
509C23—2441no—
510C23—2441no—
511Q8—22no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—4882no—
516Q8—2441no—
517Q8—2441no—
518D4—41no—
519D4—41no—
520D4—1222no—
521D4—1222no—
522D4—9761no—
523D4—4882no—
524D4—4882no—
525D4—4882no—
526D4—4882no—
527D4—2441no—
528D4—2441no—
529D4—2441no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—9761no—
540C2×C4—4882no—
541C2×C4—4882no—
542C2×C4—4882no—
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—81no—
554C22—81no—
555C22—81no—
556C22—612no—
557C22—1221no—
558C22—1221no—
559C22—1221no—
560C22—1221no—
561C22—1221no—
562C22—9764no—
563C22—4881no—
564C22—4881no—
565C22—4881no—
566C22—4881no—
567C22—4881no—
568C22—2442no—
569C22C2×D8×D6111yescharacteristic
570C4—21no—
571C4—21no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—1221no—
576C4—1221no—
577C4—4881no—
578C4—4881no—
579C4#1684039611yescharacteristic
580C4#1684056611yescharacteristic
581C2—161no—
582C2—1222no—
583C2—9761no—
584C2—4882no—
585C2—2441no—
586C2#2145749111yescharacteristic
587C2#508×D6111yescharacteristic
588C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2211C2
socleC22×C61C22×D8
hypercenter#2211D61
hypocenterC61#33848
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2211

Complements

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

NH ≅ G/Ntype
1. C2×#16840195C2semidirect
2. C2×SD32×D61C2semidirect
3. #21457561C2semidirect
4. #21457503C2semidirect
5. #21448078C2semidirect
6. #21447918C2semidirect
7. #21447914C2semidirect
8. #21447875C2semidirect
9. #21447762C2semidirect
10. #21447751C2semidirect
11. #21447741C2semidirect
12. #21447732C2semidirect
13. C4×#14519720C2semidirect
14. #21447638C2semidirect
15. C61×#2211C2semidirect
17. C2×#14519800C22semidirect
23. C2×#14519723C22semidirect
27. C2×#14519720C22semidirect
32. C2×C16×D61C22semidirect
33. #16840146C22semidirect
34. #16840004C22semidirect
46. #16839589C22semidirect
47. #16839567C22semidirect
48. #16839566C22semidirect
49. #16839564C22semidirect
52. #16839541C22semidirect
53. C4×#11758185C22semidirect
54. C2×Q32×C61C22semidirect
57. C4×Q16×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
102. #14519720D4semidirect
275. C2×D61#508semidirect
294. #2211D61semidirect
318. C2×Q32C2×D61semidirect
341. C4×Q16C2×D61semidirect
345. C4⋊C16C2×D61semidirect
348. #358C2×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×#168401952
C2×SD32×D612
#214575612
#214575032
#214480782
#214479182
#214479142
#214478752
#214477622
#214477512
#214477412
#214477322
C4×#145197202
#214476382
C61×#22112
#3384861