Finite Group Databasebeta

← #29834603

Subgroups — #29834603

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C211yesmaximal, characteristic, complemented
2#21458920C211yesmaximal, characteristic, complemented
3#21455541C211yesmaximal, characteristic
4#473×D61C211yesmaximal, characteristic, complemented
5#21447952C211yesmaximal, characteristic, complemented
6#21447951C211yesmaximal, characteristic, complemented
7#21447950C211yesmaximal, characteristic, complemented
8#21447949C211yesmaximal, characteristic, complemented
9#21447943C211yesmaximal, characteristic, complemented
10#21447942C211yesmaximal, characteristic, complemented
11#21447941C211yesmaximal, characteristic, complemented
12#21447612C211yesmaximal, characteristic, complemented
13#21447567C211yesmaximal, characteristic, complemented
14#362×D61C211yesmaximal, characteristic, complemented
15C61×#2222C211yesmaximal, characteristic, complemented
16#16840561—21no—
17#137×D61—21no—
18C2×#14519800C2211yescharacteristic
19C2×Q16×D61C2211yescharacteristic
20#131×D61C2211yescharacteristic
21#16840204—21no—
22#16840204—21no—
23#16840204—21no—
24#16840204—21no—
25C2×#14519723C2211yescharacteristic, complemented
26C2×#14519722C2211yescharacteristic, complemented
27C2×#14519720C2211yescharacteristic, complemented
28#16840180—21no—
29#16840180C2211yescharacteristic
30#16840179C2211yescharacteristic
31#16840178—21no—
32#16840178C2211yescharacteristic
33C2×#14519716—21no—
34C2×#14519716C2211yescharacteristic
35M32×D61—21no—
36M32×D61C2211yescharacteristic
37#16840038C2211yescharacteristic
38#16840037C2211yescharacteristic
39#16840036—21no—
40#16840035—21no—
41#16840034C2211yescharacteristic
42#16840034C2211yescharacteristic
43#16840033C2211yescharacteristic
44#16840032C2211yescharacteristic
45#16840028—21no—
46#16840027—21no—
47#108×D61C2211yescharacteristic
48#108×D61C2211yescharacteristic
49#16839745—21no—
50#101×D61—21no—
51#16839592C2211yescharacteristic, complemented
52#16839591C2211yescharacteristic, complemented
53#16839591C2211yescharacteristic, complemented
54#16839579C2211yescharacteristic, complemented
55#16839578C2211yescharacteristic, complemented
56#16839578C2211yescharacteristic, complemented
57#16839577C2211yescharacteristic, complemented
58#16839576C2211yescharacteristic, complemented
59#16839574C2211yescharacteristic, complemented
60C2×Q32×C61C2211yescharacteristic, complemented
61C61×#504C2211yescharacteristic
62C61×#473C2211yescharacteristic
63C61×#473C2211yescharacteristic
64C61×#368C2211yescharacteristic, complemented
65C61×#362C2211yescharacteristic, complemented
66C61×#362C2211yescharacteristic, complemented
67C2×#11758226—21no—
68#41×D61D411yescharacteristic
69C2×Q8×D61—21no—
70#14519810—21no—
71#14519809—41no—
72#14519809—41no—
73#14519808—21no—
74#14519808—21no—
75#14519807—21no—
76#14519806—41no—
77#14519805—21no—
78#14519804—41no—
79#14519802—21no—
80#14519800—21no—
81#14519800—21no—
82#14519800—21no—
83#14519800—21no—
84#14519800—41no—
85#14519798—41no—
86C2×Dic244C2311yescharacteristic
87C2×#11758202—21no—
88Q16×D61—21no—
89Q16×D61—21no—
90Q16×D61—21no—
91Q16×D61—21no—
92Q16×D61—41no—
93C2×#11758201—21no—
94SD16×D61—41no—
95C2×D488C2311yescharacteristic
96#14519785—21no—
97#14519785D411yescharacteristic
98#14519784D411yescharacteristic
99#14519783—21no—
100#14519783C2311yescharacteristic
101M16×D61—21no—
102M16×D61—21no—
103M16×D61—21no—
104M16×D61D411yescharacteristic
105C2×C8×D61—21no—
106C2×C8×D61C2311yescharacteristic
107#14519723—21no—
108#14519723—21no—
109#14519722—21no—
110#14519722—21no—
111#14519720—21no—
112#14519720—21no—
113#14519717—21no—
114#14519717C2311yescharacteristic
115#14519716—21no—
116#14519716—21no—
117#14519716—21no—
118#14519716D411yescharacteristic
119#14519716D411yescharacteristic
120#14519716D411yescharacteristic
121#14519716D411yescharacteristic
122C2×#11758185—21no—
123C2×#11758185C2311yescharacteristic
124C16×D61—21no—
125C16×D61—21no—
126C16×D61—21no—
127#14519713C2311yescharacteristic
128#14519713C2311yescharacteristic
129#14519712—21no—
130#14519712—21no—
131#14519696—21no—
132#14519696—21no—
133#14519695—21no—
134#14519695—21no—
135#14519693—21no—
136#14519693—21no—
137#137×C61—21no—
138#137×C61—21no—
139C2×Q16×C61C2311yescharacteristic
140C2×Q16×C61C2311yescharacteristic
141#131×C61C2311yescharacteristic
142Q32×C61—21no—
143Q32×C61—21no—
144M32×C61—21no—
145M32×C61C2311yescharacteristic
146C2×C16×C61—21no—
147C2×C16×C61C2311yescharacteristic
148#108×C61C2311yescharacteristic
149#108×C61C2311yescharacteristic
150#101×C61—21no—
151#101×C61—21no—
152#41×C61C2×D411yescharacteristic
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155#11758226—41no—
156#11758226—41no—
157#11758226—41no—
158#11758226—41no—
159#11758226D811yescharacteristic
160Q8×D61—41no—
161Q8×D61—41no—
162Q8×D61—41no—
163Q8×D61—41no—
164Q8×D61D811yescharacteristic
165#11758224—21no—
166#11758224D811yescharacteristic
167D4×D61—21no—
168D4×D61D811yescharacteristic
169#11758222—21no—
170#11758222C2×D411yescharacteristic
171C2×D244—21no—
172C2×D244—41no—
173C2×C4×D61—21no—
174C2×C4×D61—41no—
175C2×C4×D61—41no—
176C2×C4×D61C2×D411yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—41no—
179Q16×C61—21no—
180Q16×C61—21no—
181Q16×C61—21no—
182Q16×C61—21no—
183Q16×C61—41no—
184Q16×C61—41no—
185SD16×C61—41no—
186SD16×C61—41no—
187M16×C61—21no—
188M16×C61—21no—
189M16×C61—21no—
190M16×C61C2×D411yescharacteristic
191C2×C8×C61—21no—
192C2×C8×C61C2411yescharacteristic
193#11758202—41no—
194#11758202—41no—
195#11758202—41no—
196#11758202—41no—
197#11758202—41no—
198#11758201—41no—
199#11758201—41no—
200#11758201—41no—
201#11758201—41no—
202#11758201—41no—
203#11758200—41no—
204#11758199—41no—
205#11758194—21no—
206#11758194—21no—
207#11758194—21no—
208#11758194C2×D411yescharacteristic
209C2×#92418—21no—
210C2×#92418C2×D411yescharacteristic
211Dic244—21no—
212Dic244—21no—
213Dic244—41no—
214D488—21no—
215D488—21no—
216D488—41no—
217#11758190—41no—
218#11758190—41no—
219#11758189—21no—
220#11758189—21no—
221#11758189—21no—
222#11758189—41no—
223#11758189—41no—
224C8×D61—21no—
225C8×D61—21no—
226C8×D61—21no—
227C8×D61—41no—
228C8×D61—41no—
229C8×D61C2×D411yescharacteristic
230C8×D61C2×D411yescharacteristic
231C8×D61#4111yescharacteristic
232C8×D61#4111yescharacteristic
233C16×C61—21no—
234C16×C61C2×D411yescharacteristic
235C16×C61C2×D411yescharacteristic
236#11758185—21no—
237#11758185C2×D411yescharacteristic
238#11758185C2×D411yescharacteristic
239C22×D61—21no—
240C22×D61—41no—
241C22×D61#13511yescharacteristic
242Q8×C61—41no—
243Q8×C61—41no—
244Q8×C61—41no—
245Q8×C61—41no—
246Q8×C61C2×D811yescharacteristic
247D4×C61—21no—
248D4×C61C2×D811yescharacteristic
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C22×D411yescharacteristic
253C22⋊D61—21no—
254C22⋊D61—21no—
255C22⋊D61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61#13511yescharacteristic
259D244—21no—
260D244—41no—
261D244—41no—
262D244—81no—
263D244—81no—
264D244C2×D811yescharacteristic
265C4×D61—21no—
266C4×D61—21no—
267C4×D61—21no—
268C4×D61—81no—
269C4×D61—81no—
270C4×D61—81no—
271C4×D61—81no—
272C4×D61#12111yescharacteristic
273C4×D61#12111yescharacteristic
274C4×D61C2×D811yescharacteristic
275C4×D61C2×D811yescharacteristic
276Dic122—21no—
277Dic122—41no—
278Dic122—41no—
279Dic122—81no—
280Dic122—81no—
281Dic122C2×D811yescharacteristic
282C8×C61—21no—
283C8×C61—41no—
284C8×C61—41no—
285C8×C61C22×D411yescharacteristic
286C8×C61C2×#4111yescharacteristic
287#92418—21no—
288#92418—41no—
289#92418—41no—
290#92418#12111yescharacteristic
291#92418#12111yescharacteristic
292#33619—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61C2×#13511yescharacteristic
295C2×D61—21no—
296C2×D61—41no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—161no—
300C2×D61#46611yescharacteristic
301C2×D61#46611yescharacteristic
302C4×C61—21no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61C2×#12111yescharacteristic
306C4×C61C22×D811yescharacteristic
307Dic61—21no—
308Dic61—81no—
309Dic61#46611yescharacteristic
310Dic61#46611yescharacteristic
311C2×#510—611no—
312#3407—611no—
313#3067—611no—
314C2×#473—611no—
315#2223—611no—
316#2223—611no—
317#2223—611no—
318#2223—611no—
319#2222—611no—
320#2222—611no—
321#2222—611no—
322#2222D6111yescharacteristic, complemented
323#2161—611no—
324#2148—611no—
325C2×#362—611no—
326C2×C61—21no—
327C2×C61—41no—
328C2×C61C2×#46611yescharacteristic
329D61—21no—
330D61—21no—
331D61—41no—
332D61—161no—
333#575—1221no—
334C2×#137—1221no—
335C2×#135—611no—
336C22×Q16—611no—
337C2×#131—611no—
338#510—1221no—
339#510—1221no—
340#510—1221no—
341#510—1221no—
342C2×Q32—611no—
343C2×Q32C2×D6111yescharacteristic, complemented
344C2×SD32—611no—
345C2×SD32—611no—
346#504—611no—
347#504—611no—
348#504—611no—
349#504—1221no—
350#504—1221no—
351#504C2×D6111yescharacteristic
352C2×M32—611no—
353C2×M32—611no—
354C2×M32—1221no—
355C2×M32—1221no—
356#473—611no—
357#473—611no—
358#473—611no—
359#473—611no—
360#473—1221no—
361#473—1221no—
362#473C2×D6111yescharacteristic
363#473C2×D6111yescharacteristic
364#472—611no—
365#472—611no—
366#471—1221no—
367#471—1221no—
368C2×#108—611no—
369C2×#108—611no—
370#413—1221no—
371C2×#101—1221no—
372#368—611no—
373#368—611no—
374#368—611no—
375#368C2×D6111yescharacteristic, complemented
376#362—611no—
377#362—611no—
378#362—611no—
379#362—611no—
380#362C2×D6111yescharacteristic, complemented
381#362C2×D6111yescharacteristic, complemented
382#361—611no—
383#361—611no—
384C61#3361911yescharacteristic, complemented, Sylow
385C2×#41—611no—
386C2×#41—1221no—
387C22×Q8—1221no—
388#137—21no—
389#137—21no—
390#137—1221no—
391#137—1221no—
392#137—1221no—
393#137—1221no—
394#137—2441no—
395#137—2441no—
396#137—2441no—
397AGL(1,ℤ/8)—1221no—
398AGL(1,ℤ/8)—1221no—
399AGL(1,ℤ/8)—2441no—
400#135—1221no—
401#135—1221no—
402#135—1221no—
403#135—1221no—
404#135—2441no—
405#135—2441no—
406C2×Q16—611no—
407C2×Q16—1221no—
408C2×Q16—1221no—
409C2×Q16—1221no—
410C2×Q16—1221no—
411C2×Q16—1221no—
412C2×Q16—2441no—
413C2×Q16C22×D6111yescharacteristic
414C2×Q16C22×D6111yescharacteristic
415C2×SD16—1221no—
416C2×SD16—2441no—
417C2×D8—611no—
418#131—611no—
419#131—611no—
420#131—611no—
421#131—1221no—
422#131—1221no—
423#131C22×D6111yescharacteristic
424C2×M16—611no—
425C2×M16—1221no—
426C2×M16—1221no—
427C2×M16—1221no—
428C22×C8—611no—
429C22×C8—1221no—
430Q32—21no—
431Q32—21no—
432Q32—1221no—
433Q32—1221no—
434SD32—1221no—
435SD32—1221no—
436SD32—1221no—
437SD32—1221no—
438M32—21no—
439M32—611no—
440M32—611no—
441M32—611no—
442M32—611no—
443M32—611no—
444M32—1221no—
445M32—1221no—
446M32—1221no—
447M32—1221no—
448M32C22×D6111yescharacteristic
449C2×C16—21no—
450C2×C16—611no—
451C2×C16—1221no—
452C2×C16—1221no—
453C2×C16—1221no—
454C2×C16—1221no—
455C2×C16C22×D6111yescharacteristic
456#108—611no—
457#108—611no—
458#108—1221no—
459#108—1221no—
460#108C22×D6111yescharacteristic
461#108C22×D6111yescharacteristic
462#101—21no—
463#101—21no—
464#101—1221no—
465#101—1221no—
466#101—1221no—
467#101—1221no—
468#100—1221no—
469#100—1221no—
470#41—611no—
471#41—611no—
472#41—611no—
473#41—1221no—
474#41—1221no—
475#41—2441no—
476#41—2441no—
477#41—2441no—
478#41—2441no—
479#41D4×D6111yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—611no—
483C2×Q8—1221no—
484C2×Q8—2441no—
485C2×Q8—2441no—
486C2×Q8—2441no—
487C2×Q8—2441no—
488C2×Q8—2441no—
489C2×D4—611no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—2441no—
493C22×C4—611no—
494C22×C4—1221no—
495C22×C4—2441no—
496C22×C4—2441no—
497Q16—21no—
498Q16—21no—
499Q16—21no—
500Q16—21no—
501Q16—41no—
502Q16—41no—
503Q16—1221no—
504Q16—1221no—
505Q16—2441no—
506Q16—2441no—
507Q16—2441no—
508Q16—2441no—
509Q16—2441no—
510Q16—2441no—
511SD16—41no—
512SD16—41no—
513SD16—2441no—
514SD16—2441no—
515SD16—2441no—
516SD16—2441no—
517SD16—2441no—
518SD16—2441no—
519SD16—2441no—
520SD16—2441no—
521D8—1221no—
522D8—1221no—
523D8—2441no—
524D8—2441no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—611no—
529M16—1221no—
530M16—1221no—
531M16—1221no—
532M16—1221no—
533M16—1221no—
534M16—1221no—
535M16—2441no—
536M16—2441no—
537M16D4×D6111yescharacteristic
538C2×C8—21no—
539C2×C8—611no—
540C2×C8—611no—
541C2×C8—611no—
542C2×C8—611no—
543C2×C8—611no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—2441no—
549C2×C8—2441no—
550C2×C8C23×D6111yescharacteristic
551C16—21no—
552C16—611no—
553C16—611no—
554C16—1221no—
555C16D4×D6111yescharacteristic
556C16D4×D6111yescharacteristic
557C23—611no—
558C23—1221no—
559C23—2441no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—611no—
565Q8—1221no—
566Q8—4881no—
567Q8—4881no—
568Q8—2441no—
569Q8—2441no—
570Q8D8×D6111yescharacteristic
571D4—21no—
572D4—611no—
573D4—1221no—
574D4—1221no—
575D4—1221no—
576D4—1221no—
577D4—4881no—
578D4—4881no—
579D4—2441no—
580D4—2441no—
581D4D8×D6111yescharacteristic
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—4881no—
595C2×C4—4881no—
596C2×C4—4881no—
597C2×C4—4881no—
598C2×C4—2441no—
599C2×C4C2×D4×D6111yescharacteristic
600C8—21no—
601C8—41no—
602C8—41no—
603C8—611no—
604C8—611no—
605C8—1221no—
606C8—2441no—
607C8—2441no—
608C8C2×D4×D6111yescharacteristic
609C8#41×D6111yescharacteristic
610C22—21no—
611C22—611no—
612C22—611no—
613C22—1221no—
614C22—9761no—
615C22—4881no—
616C22—4881no—
617C22—2441no—
618C22#135×D6111yescharacteristic
619C4—21no—
620C4—81no—
621C4—81no—
622C4—611no—
623C4—611no—
624C4—1221no—
625C4—4881no—
626C4#121×D6111yescharacteristic
627C4C2×D8×D6111yescharacteristic
628C2—21no—
629C2—41no—
630C2—1221no—
631C2—1221no—
632C2—9761no—
633C2—2441no—
634C2#466×D6111yescharacteristic
subgroupG/H
centerC2#466×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2222C2
socleC2×C61C2×#466
prime radical#21455541C2
hypercenter#2222D61
hypocenterC61#33619
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #2222

Complements

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

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
2. #21458920C2semidirect
4. #473×D61C2semidirect
5. #21447952C2semidirect
6. #21447951C2semidirect
7. #21447950C2semidirect
8. #21447949C2semidirect
9. #21447943C2semidirect
10. #21447942C2semidirect
11. #21447941C2semidirect
12. #21447612C2semidirect
13. #21447567C2semidirect
14. #362×D61C2semidirect
15. C61×#2222C2semidirect
25. C2×#14519723C22semidirect
26. C2×#14519722C22semidirect
27. C2×#14519720C22semidirect
51. #16839592C22semidirect
52. #16839591C22semidirect
53. #16839591C22semidirect
54. #16839579C22semidirect
55. #16839578C22semidirect
56. #16839578C22semidirect
57. #16839577C22semidirect
58. #16839576C22semidirect
59. #16839574C22semidirect
60. C2×Q32×C61C22semidirect
64. C61×#368C22semidirect
65. C61×#362C22semidirect
66. C61×#362C22semidirect
322. #2222D61semidirect
343. C2×Q32C2×D61semidirect
375. #368C2×D61semidirect
380. #362C2×D61semidirect
381. #362C2×D61semidirect
384. C61#33619semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
#214589202
#214555412
#473×D612
#214479522
#214479512
#214479502
#214479492
#214479432
#214479422
#214479412
#214476122
#214475672
#362×D612
C61×#22222
#3361961