Finite Group Databasebeta

← #29836533

Subgroups — #29836533

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic, complemented
2C2×#16840193C211yesmaximal, characteristic, complemented
3#21457578C211yesmaximal, characteristic, complemented
4#21457505C211yesmaximal, characteristic, complemented
5#21448085C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447906C211yesmaximal, characteristic, complemented
8#21447881C211yesmaximal, characteristic, complemented
9#21447854C211yesmaximal, characteristic, complemented
10#21447762C211yesmaximal, characteristic, complemented
11#21447745C211yesmaximal, characteristic, complemented
12#21447743C211yesmaximal, characteristic, complemented
13#21447732C211yesmaximal, characteristic, complemented
14#21447646C211yesmaximal, characteristic, complemented
15C61×#2175C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519800C2211yescharacteristic, complemented
18C2×#14519796C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840434—21no—
21#16840407—21no—
22#16840199—24no—
23#16840193—24no—
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic, complemented
26C2×#14519722C2211yescharacteristic, complemented
27C2×#14519720C2211yescharacteristic
28C2×D976C2211yescharacteristic
29C2×D976C2211yescharacteristic
30C2×#14519718C2211yescharacteristic
31C2×#14519718C2211yescharacteristic
32C2×#14519716C2211yescharacteristic
33C2×C16×D61C2211yescharacteristic, complemented
34#16840146C2211yescharacteristic
35#16840139C2211yescharacteristic, complemented
36#16840004C2211yescharacteristic
37#16839999C2211yescharacteristic
38#16839999C2211yescharacteristic
39#16839999C2211yescharacteristic, complemented
40#16839977—21no—
41#16839962—21no—
42#16839914—21no—
43#16839910—21no—
44#16839910—21no—
45#16839908—21no—
46C8×D244C2211yescharacteristic
47C4⋊#11758185C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic
50#16839562C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic
53#16839541C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic, complemented
56C4×D8×C61C2211yescharacteristic
57C61×#366C2211yescharacteristic, complemented
58C61×#358C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic
60C61×#346C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×#11758226—21no—
63C2×#11758224—21no—
64C2×D4×D61—21no—
65#14519800—22no—
66#14519800—22no—
67#14519800—41no—
68#14519800—41no—
69#14519796—22no—
70#14519796—41no—
71C2×Dic244C2311yescharacteristic
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×D488—21no—
77C2×D488C2311yescharacteristic
78C2×D488C2311yescharacteristic
79C2×D488C2311yescharacteristic
80C2×#11758199—21no—
81D8×D61—22no—
82D8×D61—41no—
83C2×C8×D61C2311yescharacteristic
84C2×C8×D61C2311yescharacteristic
85#14519774—21no—
86#14519774—41no—
87#14519765—21no—
88#14519761—41no—
89#14519757—21no—
90#14519752—21no—
91#14519752—21no—
92#14519752—21no—
93#14519752—41no—
94#14519752—41no—
95#14519751—41no—
96#14519747—41no—
97C4×D244—41no—
98C4×D244D411yescharacteristic
99C2×#11758198—41no—
100#14519723—22no—
101#14519722—22no—
102#14519722—41no—
103#14519720—22no—
104D976—22no—
105D976—41no—
106#14519718—22no—
107#14519718—22no—
108#14519716—22no—
109C2×#11758185C2311yescharacteristic
110C2×#11758185C2311yescharacteristic
111C16×D61—22no—
112C4⋊#92418D411yescharacteristic
113#14519700—21no—
114#14519697—21no—
115#14519697—21no—
116#14519697—21no—
117#14519689D411yescharacteristic
118#14519689D411yescharacteristic
119C2×Q16×C61C2311yescharacteristic
120C2×D8×C61C2311yescharacteristic
121C4×Q8×C61—21no—
122C4×D4×C61—21no—
123SD32×C61D414yescomplemented
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#107×C61C2311yescharacteristic
128#103×C61—21no—
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×D4×C61—21no—
134C22×C4×C61—41no—
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C22×Dic61—41no—
138#11758226—42no—
139#11758226—42no—
140#11758226—81no—
141#11758226—81no—
142#11758224—42no—
143#11758224—81no—
144D4×D61—42no—
145D4×D61—81no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—41no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244D811yescharacteristic
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—21no—
162Q16×C61—21no—
163Q16×C61C2×D412yes—
164D8×C61—21no—
165D8×C61C2×D412yes—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C42×C61—41no—
173C42×C61C2×D411yescharacteristic
174#11758201—42no—
175#11758201—42no—
176#11758200—42no—
177#11758199—42no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198D811yescharacteristic
185#11758198D811yescharacteristic
186C4⋊Dic61—41no—
187C4⋊Dic61D811yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192D488—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—81no—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—161no—
215C22×D61C2×D811yescharacteristic
216C22×D61C2×D811yescharacteristic
217Q8×C61—22no—
218Q8×C61—41no—
219D4×C61—22no—
220D4×C61—41no—
221C2×C4×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—
230C2×Dic61—41no—
231C2×Dic61—82no—
232C2×Dic61C2×D811yescharacteristic
233C2×Dic61C2×D811yescharacteristic
234D244—41no—
235D244—41no—
236D244—41no—
237D244—41no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—82no—
242D244—82no—
243D244—82no—
244D244—161no—
245D244—161no—
246D244—161no—
247D244—161no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—82no—
253C4×D61—82no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61C22×D411yescharacteristic
258C8×C61H(5,𝔽2)11yescharacteristic
259#92418—21no—
260#92418—21no—
261#33834—611nomaximal, Sylow
262C22×C61—42no—
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—21no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—164no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—42no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61D4211yescharacteristic
288C4×C61#57611yescharacteristic
289Dic61—21no—
290Dic61—21no—
291Dic61—81no—
292C2×#509—611no—
293C2×#508—611no—
294#3287—611no—
295#3280—611no—
296#2250—611no—
297#2220—611no—
298#2215—611no—
299#2211—611no—
300#2206—611no—
301#2189—611no—
302#2186—611no—
303#2186—611no—
304#2184—611no—
305#2175D6111yescharacteristic, complemented
306#2168—611no—
307C2×C61—82no—
308C2×C61D4×D811yescharacteristic
309C2×C61#341411yescharacteristic
310C2×C61#341511yescharacteristic
311D61—41no—
312D61—41no—
313D61—161no—
314D61—161no—
315D61—161no—
316C2×#135—611no—
317C2×#135—611no—
318C2×#135—611no—
319C22×D8—611no—
320#550—1221no—
321#546—1221no—
322#509—1224no—
323#508—1224no—
324C2×Q32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32C2×D6111yescharacteristic
329C2×D16—611no—
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—611no—
333C2×M32—611no—
334C22×C16—611no—
335#495—611no—
336#493—611no—
337#467—611no—
338#466—611no—
339#466—611no—
340#466—611no—
341#463—1221no—
342#460—1221no—
343#450—1221no—
344#449—1221no—
345#449—1221no—
346#449—1221no—
347C4×Q16C2×D6111yescharacteristic, complemented
348C4×D8C2×D6111yescharacteristic
349C8×D4—611no—
350#366C2×D6111yescharacteristic, complemented
351C4⋊C16—611no—
352#358—611no—
353#358C2×D6111yescharacteristic, complemented
354#357—611no—
355#357—611no—
356#357—611no—
357#357C2×D6111yescharacteristic
358C22⋊C16—611no—
359C22⋊C16—611no—
360#346C2×D6111yescharacteristic, complemented
361C61#3383411yescharacteristic, complemented, Sylow
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C22×D4—1221no—
366#135—1222no—
367#135—1222no—
368#135—1222no—
369#135—2441no—
370#135—2441no—
371#135—2441no—
372C2×Q16—611no—
373C2×Q16C22×D6111yescharacteristic
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—611no—
379C2×D8—611no—
380C2×D8—611no—
381C2×D8—1221no—
382C2×D8—1222no—
383C2×D8—1221no—
384C2×D8—2441no—
385C2×D8C22×D6111yescharacteristic
386C22×C8—611no—
387C22×C8—611no—
388#127—1221no—
389#127—2441no—
390#124—1221no—
391#123—2441no—
392C22⋊Q8—1221no—
393#121—1221no—
394#121—1221no—
395#121—1221no—
396#121—2441no—
397#121—2441no—
398#121—2441no—
399#120—2441no—
400C4×Q8—21no—
401C4×D4—21no—
402C4×D4—611no—
403C4×D4—2441no—
404C2×C22⋊C4—2441no—
405Q32—1222no—
406SD32—1222no—
407SD32—1222no—
408SD32—2441no—
409SD32D24414yescomplemented
410D16—1222no—
411D16—1222no—
412D16—1222no—
413D16—2441no—
414M32—1222no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—1222no—
418C2×C16C22×D6111yescharacteristic
419C2×C16C22×D6111yescharacteristic
420#107C22×D6111yescharacteristic
421#107C22×D6111yescharacteristic
422C4⋊C8—611no—
423#103—21no—
424#103—1221no—
425#102—21no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429C22⋊C8—611no—
430C22⋊C8—611no—
431C4×C8C22×D6111yescharacteristic
432C24—2441no—
433#41—4881no—
434#41—4881no—
435#41—4881no—
436#41—2442no—
437#41—2442no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—1221no—
441C2×D4—21no—
442C2×D4—611no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—4881no—
447C2×D4—2441no—
448C2×D4—2441no—
449C2×D4—2441no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4—2442no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2441no—
457C22×C4—41no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4—2441no—
463Q16—21no—
464Q16—1221no—
465Q16—1221no—
466Q16C2×D24412yes—
467SD16—4881no—
468SD16—2442no—
469SD16—2442no—
470SD16—2442no—
471D8—21no—
472D8—1221no—
473D8—1221no—
474D8—1221no—
475D8—1221no—
476D8—1221no—
477D8—1221no—
478D8—4881no—
479D8—2442no—
480D8C2×D24412yes—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8D4×D6111yescharacteristic
488C2×C8C23×D6111yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—611no—
493C4⋊C4—2441no—
494C22⋊C4—41no—
495C22⋊C4—611no—
496C22⋊C4—611no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C42—41no—
504C42D4×D6111yescharacteristic
505C16—21no—
506C16—1221no—
507C16—1221no—
508C16C2×D24412yes—
509C23—41no—
510C23—611no—
511C23—611no—
512C23—9761no—
513C23—4882no—
514C23—4882no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518Q8—22no—
519Q8—41no—
520Q8—2441no—
521Q8—2441no—
522D4—22no—
523D4—41no—
524D4—9761no—
525D4—9761no—
526D4—9761no—
527D4—9761no—
528D4—4882no—
529D4—4882no—
530D4—4882no—
531D4—4882no—
532D4—4882no—
533D4—2441no—
534D4—2441no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539D4—2441no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—161no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—4882no—
551C2×C4—4882no—
552C2×C4—4882no—
553C2×C4—2441no—
554C2×C4D8×D6111yescharacteristic
555C2×C4D8×D6111yescharacteristic
556C2×C4C2×D4×D6111yescharacteristic
557C8—21no—
558C8—1221no—
559C8—1221no—
560C8C22×D24411yescharacteristic
561C8#1451982311yescharacteristic
562C22—42no—
563C22—81no—
564C22—81no—
565C22—1221no—
566C22—1221no—
567C22—1221no—
568C22—1221no—
569C22—1221no—
570C22—1221no—
571C22—9764no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22C2×D8×D6111yescharacteristic
582C4—21no—
583C4—21no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—1221no—
588C4—1221no—
589C4—4881no—
590C4D4×D24411yescharacteristic
591C4#1684056411yescharacteristic
592C2—82no—
593C2—9761no—
594C2—9761no—
595C2—9761no—
596C2—2441no—
597C2—2441no—
598C2D8×D24411yescharacteristic
599C2#2145895411yescharacteristic
600C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2175C2
socleC22×C61C22×D8
hypercenter#2175D61
hypocenterC61#33834
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×#16840193C2semidirect
3. #21457578C2semidirect
4. #21457505C2semidirect
5. #21448085C2semidirect
6. #21447933C2semidirect
7. #21447906C2semidirect
8. #21447881C2semidirect
9. #21447854C2semidirect
10. #21447762C2semidirect
11. #21447745C2semidirect
12. #21447743C2semidirect
13. #21447732C2semidirect
14. #21447646C2semidirect
15. C61×#2175C2semidirect
17. C2×#14519800C22semidirect
24. C2×#14519723C22semidirect
25. C2×#14519722C22semidirect
26. C2×#14519722C22semidirect
33. C2×C16×D61C22semidirect
35. #16840139C22semidirect
39. #16839999C22semidirect
47. C4⋊#11758185C22semidirect
48. #16839565C22semidirect
50. #16839562C22semidirect
51. #16839562C22semidirect
53. #16839541C22semidirect
55. C4×Q16×C61C22semidirect
57. C61×#366C22semidirect
58. C61×#358C22semidirect
60. C61×#346C22semidirect
123. SD32×C61D4semidirect
305. #2175D61semidirect
347. C4×Q16C2×D61semidirect
350. #366C2×D61semidirect
353. #358C2×D61semidirect
360. #346C2×D61semidirect
361. C61#33834semidirect
409. SD32D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×#168401932
#214575782
#214575052
#214480852
#214479332
#214479062
#214478812
#214478542
#214477622
#214477452
#214477432
#214477322
#214476462
C61×#21752
#3383461