Finite Group Databasebeta

← #29835452

Subgroups — #29835452

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C211yesmaximal, characteristic, complemented
2C2×#16840201C211yesmaximal, characteristic, complemented
3C2×#16840199C211yesmaximal, characteristic, complemented
4C2×#16840196C211yesmaximal, characteristic, complemented
5#21456244C211yesmaximal, characteristic, complemented
6#495×D61C211yesmaximal, characteristic, complemented
7#21448089C211yesmaximal, characteristic, complemented
8#21448087C211yesmaximal, characteristic, complemented
9#21448084C211yesmaximal, characteristic, complemented
10#21448082C211yesmaximal, characteristic, complemented
11#21448080C211yesmaximal, characteristic, complemented
12#21448078C211yesmaximal, characteristic, complemented
13#21448078C211yesmaximal, characteristic, complemented
14C4×#14519716C211yesmaximal, characteristic, complemented
15C61×#2249C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519798—21no—
18C2×#14519796C2211yescharacteristic
19C2×Q16×D61C2211yescharacteristic
20C2×SD16×D61—21no—
21C2×D8×D61C2211yescharacteristic
22#16840311—21no—
23#124×D61—21no—
24#16840204—24no—
25#16840201—24no—
26#16840199—24no—
27#16840196—24no—
28C2×Dic488C2211yescharacteristic, complemented
29C2×#14519723C2211yescharacteristic, complemented
30C2×#14519723C2211yescharacteristic, complemented
31C2×#14519722C2211yescharacteristic
32C2×#14519722C2211yescharacteristic, complemented
33C2×#14519721C2211yescharacteristic
34C2×#14519721C2211yescharacteristic, complemented
35C2×#14519720C2211yescharacteristic
36C2×#14519720C2211yescharacteristic, complemented
37C2×D976C2211yescharacteristic
38C2×#14519718C2211yescharacteristic
39C2×#14519718C2211yescharacteristic
40C2×#14519716C2211yescharacteristic
41C2×#14519716C2211yescharacteristic
42#16840147C2211yescharacteristic
43#16840146C2211yescharacteristic
44#16840146C2211yescharacteristic, complemented
45#16840145C2211yescharacteristic
46#16840144—21no—
47#16840144—21no—
48#16840143—21no—
49#16840140C2211yescharacteristic, complemented
50#16840137C2211yescharacteristic, complemented
51#16840133—21no—
52C4×C8×D61C2211yescharacteristic
53#16839537C2211yescharacteristic, complemented
54#16839537C2211yescharacteristic, complemented
55C4×#11758185C2211yescharacteristic, complemented
56C2×Q32×C61C2211yescharacteristic, complemented
57C2×SD32×C61C2211yescharacteristic
58C2×SD32×C61C2211yescharacteristic, complemented
59C2×D16×C61C2211yescharacteristic
60C61×#495C2211yescharacteristic
61C61×#495C2211yescharacteristic, complemented
62C4×C16×C61C2211yescharacteristic, complemented
63C2×#11758226—21no—
64C2×#11758224—21no—
65C2×Q8×D61—21no—
66C2×D4×D61—21no—
67#14519800—22no—
68#14519800—22no—
69#14519798—44no—
70#14519796—22no—
71#14519796—22no—
72C2×Dic244C2311yescharacteristic
73C2×Dic244C2311yescharacteristic
74C2×#11758202—21no—
75C2×#11758202—21no—
76Q16×D61—22no—
77Q16×D61—22no—
78C2×#11758190—21no—
79C2×#11758190—21no—
80C2×#11758200—21no—
81C2×#11758200—21no—
82C2×#11758201—21no—
83C2×#11758201—21no—
84SD16×D61—44no—
85C2×D488C2311yescharacteristic
86C2×D488C2311yescharacteristic
87C2×#11758199—21no—
88C2×#11758199—21no—
89D8×D61—22no—
90D8×D61—22no—
91C2×C8×D61D411yescharacteristic
92C2×C8×D61C2311yescharacteristic
93#14519775—21no—
94#14519773—21no—
95#14519766—21no—
96#14519765—21no—
97#14519765—21no—
98#14519764—21no—
99#14519763—42no—
100#14519750—42no—
101#14519737—42no—
102C22⋊C4×D61—42no—
103C42×D61D411yescharacteristic
104Dic488—22no—
105#14519723—22no—
106#14519723—22no—
107#14519722—22no—
108#14519722—22no—
109#14519721—22no—
110#14519721—22no—
111#14519720—22no—
112#14519720—22no—
113D976—22no—
114#14519718—22no—
115#14519718—22no—
116#14519716D414yescomplemented
117#14519716D414yescomplemented
118C2×#11758185C2311yescharacteristic
119C2×#11758185C2311yescharacteristic
120C4×#92418D411yescharacteristic
121C8×Dic61D411yescharacteristic
122C8×Dic61C2311yescharacteristic
123C2×Q16×C61C2311yescharacteristic
124C2×Q16×C61C2311yescharacteristic
125C2×SD16×C61—21no—
126C2×SD16×C61—21no—
127C2×D8×C61C2311yescharacteristic
128C2×D8×C61C2311yescharacteristic
129#124×C61—21no—
130#124×C61—21no—
131Q32×C61—22no—
132SD32×C61—22no—
133SD32×C61—22no—
134D16×C61—22no—
135C2×C16×C61C2311yescharacteristic
136C2×C16×C61C2311yescharacteristic
137C4×C8×C61C2311yescharacteristic
138C23×D61—41no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—42no—
146C22×Dic61—41no—
147#11758226—42no—
148#11758226—42no—
149Q8×D61—42no—
150Q8×D61—42no—
151#11758224—42no—
152#11758224—42no—
153D4×D61—42no—
154D4×D61—42no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61D812yes—
161C2×C4×D61C2×D411yescharacteristic
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—41no—
165Q16×C61—21no—
166Q16×C61—21no—
167Q16×C61—21no—
168Q16×C61—21no—
169SD16×C61—42no—
170SD16×C61—42no—
171D8×C61—21no—
172D8×C61—21no—
173D8×C61—21no—
174D8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2411yescharacteristic
177C22⋊C4×C61—42no—
178C22⋊C4×C61—42no—
179C42×C61C2×D411yescharacteristic
180C22⋊Dic61—42no—
181#11758202—42no—
182#11758202—42no—
183#11758201—42no—
184#11758201—42no—
185#11758200—42no—
186#11758200—42no—
187#11758199—42no—
188#11758199—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758196—42no—
193C4×Dic61D812yes—
194C4×Dic61C2×D411yescharacteristic
195C2×#92418C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197Dic244—21no—
198Dic244—21no—
199Dic244—21no—
200Dic244—21no—
201D488—21no—
202D488—21no—
203D488—21no—
204D488—21no—
205#11758190—42no—
206#11758190—42no—
207C8×D61D814yes—
208C8×D61C2×D412yes—
209C8×D61C2×D412yes—
210C16×C61C2×D412yes—
211C16×C61C2×D412yes—
212#11758185C2×D412yes—
213#11758185C2×D412yes—
214C23×C61—41no—
215C23×C61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—84no—
219C22×D61—82no—
220C22×D61C2×D811yescharacteristic
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C2×D812yes—
232C2×C4×C61C22×D411yescharacteristic
233C22⋊D61—84no—
234C22⋊D61—82no—
235C22⋊D61—82no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—82no—
239C2×Dic61C2×D812yes—
240C2×Dic61C2×D811yescharacteristic
241D244—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—82no—
246C4×D61—42no—
247C4×D61—82no—
248C4×D61—82no—
249C4×D61#12712yes—
250C4×D61C2×D812yes—
251Dic122—41no—
252Dic122—41no—
253Dic122—41no—
254Dic122—41no—
255Dic122—82no—
256C8×C61C2×D812yes—
257C8×C61C22×D411yescharacteristic
258C8×C61C22×D411yescharacteristic
259#92418#12712yes—
260#92418C2×D812yes—
261#33719—611nomaximal, Sylow
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61C22×D811yescharacteristic
267C2×D61—22no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—82no—
271C2×D61—82no—
272C2×D61—164no—
273C2×D61#49312yes—
274C4×C61—22no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61C2×#12711yescharacteristic
278C4×C61C22×D811yescharacteristic
279Dic61—22no—
280Dic61—81no—
281Dic61—81no—
282Dic61#49312yes—
283C2×#510—611no—
284C2×#510—611no—
285C2×#509—611no—
286C2×#509—611no—
287#3147—611no—
288C2×#495—611no—
289#2251—611no—
290#2251—611no—
291#2250—611no—
292#2250—611no—
293#2249—611no—
294#2249—611no—
295#2249—611no—
296#2249D6111yescharacteristic, complemented
297C4×M32—611no—
298C2×C61—161no—
299C2×C61—161no—
300C2×C61C2×#49311yescharacteristic
301C2×C61#341412yes—
302D61—22no—
303D61—161no—
304D61—161no—
305C2×#135—611no—
306C2×#135—611no—
307C2×#135—1221no—
308C22×Q16—611no—
309C22×SD16—1221no—
310C22×D8—611no—
311#532—1221no—
312C2×#124—1221no—
313#510—1224no—
314#510—1224no—
315#509—1224no—
316#509—1224no—
317C2×Q32—611no—
318C2×Q32—611no—
319C2×Q32—611no—
320C2×Q32C2×D6111yescharacteristic, complemented
321C2×SD32—611no—
322C2×SD32—611no—
323C2×SD32—611no—
324C2×SD32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32C2×D6111yescharacteristic
328C2×SD32C2×D6111yescharacteristic, complemented
329C2×D16—611no—
330C2×D16—611no—
331C2×D16—611no—
332C2×D16C2×D6111yescharacteristic
333C2×M32—611no—
334C2×M32—611no—
335#495—611no—
336#495—611no—
337#495—611no—
338#495—611no—
339#495—1221no—
340#495—1221no—
341#495—1221no—
342#495C2×D6111yescharacteristic
343#495C2×D6111yescharacteristic, complemented
344#494—611no—
345#493—611no—
346#492—1221no—
347C2×C4×C8—611no—
348#346—611no—
349#346—611no—
350C4×C16—611no—
351C4×C16C2×D6111yescharacteristic, complemented
352C61#3371911yescharacteristic, complemented, Sylow
353C2×#41—1221no—
354C2×#41—1221no—
355C22×Q8—1221no—
356C22×D4—1221no—
357#135—1222no—
358#135—1222no—
359#135—1222no—
360#135—1222no—
361#135—2444no—
362C2×Q16—611no—
363C2×Q16—611no—
364C2×Q16—1222no—
365C2×Q16—1222no—
366C2×Q16—1221no—
367C2×Q16—1221no—
368C2×Q16C22×D6111yescharacteristic
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—2444no—
379C2×D8—611no—
380C2×D8—611no—
381C2×D8—1222no—
382C2×D8—1222no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×D8C22×D6111yescharacteristic
386C2×D8C22×D6111yescharacteristic
387C22×C8—611no—
388C22×C8—611no—
389C4⋊Q8—1221no—
390#127—1221no—
391#124—21no—
392#124—21no—
393#124—1221no—
394#124—1221no—
395#124—1221no—
396#124—1221no—
397#124—2442no—
398#121—2442no—
399C4×D4—2442no—
400C2×C22⋊C4—2442no—
401C2×C42—611no—
402Q32—22no—
403Q32—1222no—
404Q32—1222no—
405Q32—1222no—
406SD32—22no—
407SD32—22no—
408SD32—1222no—
409SD32—1222no—
410SD32—1222no—
411SD32—1222no—
412SD32—1222no—
413SD32—1222no—
414D16—22no—
415D16—1222no—
416D16—1222no—
417D16—1222no—
418M32—614no—
419M32—614no—
420C2×C16—611no—
421C2×C16—611no—
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8C22×D6111yescharacteristic
428C24—2441no—
429#41—2442no—
430#41—2442no—
431#41—2442no—
432#41—2442no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—2441no—
438C2×Q8—2442no—
439C2×Q8—2442no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—2441no—
445C2×D4—2442no—
446C2×D4—2442no—
447C2×D4—2442no—
448C2×D4—2441no—
449C2×D4—2441no—
450C22×C4—611no—
451C22×C4—612no—
452C22×C4—2441no—
453C22×C4—2441no—
454C22×C4—2441no—
455Q16—21no—
456Q16—21no—
457Q16—21no—
458Q16—21no—
459Q16—1221no—
460Q16—1221no—
461Q16—1221no—
462Q16—1221no—
463Q16—2442no—
464Q16—2442no—
465SD16—42no—
466SD16—42no—
467SD16—2442no—
468SD16—2442no—
469SD16—2442no—
470SD16—2442no—
471SD16—2442no—
472SD16—2442no—
473D8—21no—
474D8—21no—
475D8—21no—
476D8—21no—
477D8—1221no—
478D8—1221no—
479D8—1221no—
480D8—1221no—
481D8—2442no—
482D8—2442no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—614no—
488C2×C8D4×D6111yescharacteristic
489C2×C8C23×D6111yescharacteristic
490C4⋊C4—2442no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—2442no—
494C22⋊C4—2442no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C42—611no—
498C42—612no—
499C42D4×D6111yescharacteristic
500C16—612no—
501C16—612no—
502C16D4×D6112yes—
503C16D4×D6112yes—
504C23—41no—
505C23—41no—
506C23—611no—
507C23—4882no—
508C23—4884no—
509C23—2441no—
510C23—2441no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—4882no—
516Q8—2441no—
517Q8—2441no—
518Q8—2441no—
519Q8—2441no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—4882no—
525D4—4882no—
526D4—4884no—
527D4—4882no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—611no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—4882no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—2442no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4D8×D6112yes—
545C2×C4C2×D4×D6111yescharacteristic
546C8—612no—
547C8—612no—
548C8D8×D6112yes—
549C8C2×D4×D6111yescharacteristic
550C8C2×D4×D6111yescharacteristic
551C22—82no—
552C22—82no—
553C22—81no—
554C22—81no—
555C22—612no—
556C22—1222no—
557C22—9764no—
558C22—4882no—
559C22—4882no—
560C22—4881no—
561C22—4881no—
562C22C2×D8×D6111yescharacteristic
563C4—22no—
564C4—81no—
565C4—81no—
566C4—612no—
567C4—1222no—
568C4—4881no—
569C4—4881no—
570C4#127×D6111yescharacteristic
571C4C2×D8×D6111yescharacteristic
572C2—161no—
573C2—161no—
574C2—1222no—
575C2—9761no—
576C2—9761no—
577C2#493×D6111yescharacteristic
578C2#2145895212yes—
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2249C2
socleC22×C61C22×D8
hypercenter#2249D61
hypocenterC61#33719
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
2. C2×#16840201C2semidirect
3. C2×#16840199C2semidirect
4. C2×#16840196C2semidirect
5. #21456244C2semidirect
6. #495×D61C2semidirect
7. #21448089C2semidirect
8. #21448087C2semidirect
9. #21448084C2semidirect
10. #21448082C2semidirect
11. #21448080C2semidirect
12. #21448078C2semidirect
13. #21448078C2semidirect
14. C4×#14519716C2semidirect
15. C61×#2249C2semidirect
28. C2×Dic488C22semidirect
29. C2×#14519723C22semidirect
30. C2×#14519723C22semidirect
32. C2×#14519722C22semidirect
34. C2×#14519721C22semidirect
36. C2×#14519720C22semidirect
44. #16840146C22semidirect
49. #16840140C22semidirect
50. #16840137C22semidirect
53. #16839537C22semidirect
54. #16839537C22semidirect
55. C4×#11758185C22semidirect
56. C2×Q32×C61C22semidirect
58. C2×SD32×C61C22semidirect
61. C61×#495C22semidirect
62. C4×C16×C61C22semidirect
116. #14519716D4semidirect
117. #14519716D4semidirect
296. #2249D61semidirect
320. C2×Q32C2×D61semidirect
328. C2×SD32C2×D61semidirect
343. #495C2×D61semidirect
351. C4×C16C2×D61semidirect
352. C61#33719semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
C2×#168402012
C2×#168401992
C2×#168401962
#214562442
#495×D612
#214480892
#214480872
#214480842
#214480822
#214480802
#214480782
C4×#145197162
C61×#22492
#3371961