Finite Group Databasebeta

← #29836536

Subgroups — #29836536

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×#16840200C211yesmaximal, characteristic, complemented
2C2×#16840194C211yesmaximal, characteristic, complemented
3#21457579C211yesmaximal, characteristic, complemented
4#21457506C211yesmaximal, characteristic, complemented
5#21448085C211yesmaximal, characteristic, complemented
6#21447932C211yesmaximal, characteristic, complemented
7#21447905C211yesmaximal, characteristic, complemented
8#21447882C211yesmaximal, characteristic, complemented
9#21447855C211yesmaximal, characteristic, complemented
10#21447761C211yesmaximal, characteristic, complemented
11#21447744C211yesmaximal, characteristic, complemented
12#21447744C211yesmaximal, characteristic, complemented
13#21447733C211yesmaximal, characteristic, complemented
14#21447685C211yesmaximal, characteristic, complemented
15C61×#2168C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519799C2211yescharacteristic
18C2×#14519799C2211yescharacteristic, complemented
19C22×D488C2211yescharacteristic
20#16840433—21no—
21#16840408—21no—
22#16840200—24no—
23#16840194—24no—
24C2×Dic488C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic
26C2×#14519722C2211yescharacteristic
27C2×#14519722C2211yescharacteristic, complemented
28C2×#14519722C2211yescharacteristic, complemented
29C2×D976C2211yescharacteristic
30C2×D976C2211yescharacteristic
31C2×D976C2211yescharacteristic
32C2×D976C2211yescharacteristic
33#16840146C2211yescharacteristic
34#16840139C2211yescharacteristic, complemented
35#16840003C2211yescharacteristic
36#16839998C2211yescharacteristic
37#16839998C2211yescharacteristic
38#16839998C2211yescharacteristic, complemented
39#16839978—21no—
40#16839963—21no—
41#16839915—21no—
42#16839909—21no—
43#16839909—21no—
44#16839909—21no—
45C4×Dic244C2211yescharacteristic, complemented
46C4×D488C2211yescharacteristic
47#16839588C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839567C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic
51#16839564C2211yescharacteristic
52#16839564C2211yescharacteristic, complemented
53#16839564C2211yescharacteristic, complemented
54C2×M32×C61C2211yescharacteristic
55C22×C16×C61C2211yescharacteristic, complemented
56C8×D4×C61C2211yescharacteristic
57C61×C4⋊C16C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic
59C61×C22⋊C16C2211yescharacteristic, complemented
60C61×#346C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C2×#11758222—21no—
63C2×#11758222—21no—
64C22×D244—21no—
65#14519799—22no—
66#14519799—22no—
67#14519799—22no—
68#14519799—41no—
69#14519799—41no—
70#14519799—41no—
71C2×Dic244C2311yescharacteristic
72C2×Dic244C2311yescharacteristic
73C2×#11758190—21no—
74C2×#11758190—21no—
75C2×#11758190—21no—
76C2×#11758190—21no—
77C2×D488—22no—
78C2×D488—21no—
79C2×D488—21no—
80C2×D488—41no—
81C2×D488C2311yescharacteristic
82C2×D488C2311yescharacteristic
83C2×D488C2311yescharacteristic
84C2×D488C2311yescharacteristic
85#14519774—21no—
86#14519773—41no—
87#14519765—21no—
88#14519762—41no—
89#14519756—21no—
90#14519751—21no—
91#14519751—21no—
92#14519751—21no—
93#14519750—41no—
94#14519750—41no—
95#14519750—41no—
96#14519745—41no—
97C4×Dic122—21no—
98C4×D244—21no—
99#14519739—41no—
100C22⋊C4×D61—41no—
101Dic488—22no—
102#14519722—22no—
103#14519722—22no—
104#14519722—41no—
105#14519722D414yescomplemented
106D976—22no—
107D976—22no—
108D976—22no—
109D976—41no—
110#14519711C2311yescharacteristic
111#14519711C2311yescharacteristic
112#14519702—21no—
113#14519702—21no—
114#14519699—21no—
115#14519699—21no—
116#14519699—21no—
117#14519699—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61D411yescharacteristic
121M32×C61—22no—
122C2×C16×C61—22no—
123C2×C16×C61C2311yescharacteristic
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126C2×C16×C61C2311yescharacteristic
127C4⋊C8×C61D411yescharacteristic
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C2×D4×C61D811yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C2×C22⋊D61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141#11758222—42no—
142#11758222—42no—
143#11758222—42no—
144#11758222—81no—
145#11758222—81no—
146#11758222—81no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—42no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—81no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2411yescharacteristic
170C4⋊C4×C61D811yescharacteristic
171C22⋊C4×C61D811yescharacteristic
172C22⋊C4×C61D811yescharacteristic
173C42×C61C2×D411yescharacteristic
174C22⋊Dic61—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183#11758196—41no—
184#11758196—41no—
185C4×Dic61—41no—
186Dic244—21no—
187Dic244—21no—
188Dic244—21no—
189Dic244C2×D412yes—
190D488—21no—
191D488—21no—
192D488—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—42no—
198D488—81no—
199D488C2×D412yes—
200#11758190—42no—
201#11758190—42no—
202#11758190—42no—
203#11758190—81no—
204C16×C61—21no—
205C16×C61—21no—
206C16×C61—21no—
207C16×C61C2×D412yes—
208C23×C61C2×D811yescharacteristic
209C23×C61C2×D811yescharacteristic
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—161no—
217D4×C61—41no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61C2×D811yescharacteristic
223C2×C4×C61C2×D811yescharacteristic
224C2×C4×C61C2×D811yescharacteristic
225C2×C4×C61C2×D811yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C22⋊D61—82no—
228C22⋊D61—82no—
229C22⋊D61—82no—
230C22⋊D61—161no—
231C22⋊D61—161no—
232C22⋊D61—161no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237D244—22no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—82no—
246D244—82no—
247D244—161no—
248C4×D61—82no—
249C4×D61—82no—
250C4×D61—82no—
251C4×D61—161no—
252Dic122—22no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61C22×D411yescharacteristic
260C8×C61H(5,𝔽2)11yescharacteristic
261#33834—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61C22×D811yescharacteristic
269C2×D61—42no—
270C2×D61—81no—
271C2×D61—81no—
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—21no—
285C4×C61—21no—
286C4×C61D4211yescharacteristic
287C4×C61#57611yescharacteristic
288Dic61—42no—
289Dic61—81no—
290Dic61—81no—
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#2175—611no—
306#2168D6111yescharacteristic, complemented
307C2×C61—41no—
308C2×C61—41no—
309C2×C61D4×D811yescharacteristic
310C2×C61#341411yescharacteristic
311C2×C61#341511yescharacteristic
312D61—82no—
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×SD32—611no—
329C2×D16—611no—
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—611no—
333C2×M32C2×D6111yescharacteristic
334C22×C16C2×D6111yescharacteristic, complemented
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×Q16—611no—
348C4×D8—611no—
349C8×D4C2×D6111yescharacteristic
350#366—611no—
351C4⋊C16C2×D6111yescharacteristic, complemented
352#358—611no—
353#358—611no—
354#357—611no—
355#357—611no—
356#357—611no—
357#357—611no—
358C22⋊C16C2×D6111yescharacteristic
359C22⋊C16C2×D6111yescharacteristic, complemented
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×Q16—611no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—611no—
379C2×D8—611no—
380C2×D8—611no—
381C2×D8—611no—
382C2×D8—1222no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×D8—2441no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
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—1221no—
401C4×D4—1221no—
402C4×D4—2441no—
403C4×D4D24411yescharacteristic
404C2×C22⋊C4—2441no—
405Q32—1222no—
406SD32—614no—
407SD32—1222no—
408SD32—1222no—
409SD32—2441no—
410D16—1222no—
411D16—1222no—
412D16—1222no—
413D16—2441no—
414M32—22no—
415C2×C16—22no—
416C2×C16C22×D6111yescharacteristic
417C2×C16C22×D6111yescharacteristic
418C2×C16C22×D6111yescharacteristic
419C2×C16C22×D6111yescharacteristic
420#107—611no—
421#107—611no—
422C4⋊C8D24411yescharacteristic
423#103—1221no—
424#103—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429C22⋊C8D24411yescharacteristic
430C22⋊C8D24411yescharacteristic
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—1221no—
440C2×Q8—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—4881no—
446C2×D4—2442no—
447C2×D4—2441no—
448C2×D4—2441no—
449C2×D4—2441no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4D48811yescharacteristic
457C22×C4—2441no—
458C22×C4—2441no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463Q16—612no—
464Q16—1221no—
465Q16—1221no—
466Q16—1221no—
467SD16—4881no—
468SD16—2442no—
469SD16—2442no—
470SD16—2442no—
471D8—612no—
472D8—1221no—
473D8—1221no—
474D8—1221no—
475D8—1221no—
476D8—1221no—
477D8—1221no—
478D8—1221no—
479D8—4881no—
480D8—2442no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8C2×D24411yescharacteristic
486C2×C8C2×D24411yescharacteristic
487C2×C8C2×D24411yescharacteristic
488C2×C8C23×D6111yescharacteristic
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C4⋊C4D48811yescharacteristic
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4D48811yescharacteristic
502C22⋊C4D48811yescharacteristic
503C42—2441no—
504C42C2×D24411yescharacteristic
505C16—21no—
506C16—21no—
507C16—21no—
508C16D4×D6112yes—
509C23—9761no—
510C23—4882no—
511C23—4882no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23C2×D48811yescharacteristic
517C23C2×D48811yescharacteristic
518Q8—1222no—
519Q8—2441no—
520Q8—2441no—
521Q8—2441no—
522D4—41no—
523D4—1222no—
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—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—9761no—
545C2×C4—4882no—
546C2×C4—4882no—
547C2×C4—4882no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C2×D48811yescharacteristic
553C2×C4C2×D48811yescharacteristic
554C2×C4C2×D48811yescharacteristic
555C2×C4C2×D48811yescharacteristic
556C2×C4C22×D24411yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8C2×D4×D6111yescharacteristic
561C8#1451982311yescharacteristic
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—9764no—
569C22—4881no—
570C22—4881no—
571C22—4881no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—2442no—
581C22C22×D48811yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—4881no—
587C4—4881no—
588C4—4881no—
589C4—2442no—
590C4D4×D24411yescharacteristic
591C4#1684056711yescharacteristic
592C2—41no—
593C2—41no—
594C2—9761no—
595C2—9761no—
596C2—9761no—
597C2—4882no—
598C2D4×D48811yescharacteristic
599C2#2145895511yescharacteristic
600C2#2145896011yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2168C2
socleC22×C61C22×D8
hypercenter#2168D61
hypocenterC61#33834
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. C2×#16840194C2semidirect
3. #21457579C2semidirect
4. #21457506C2semidirect
5. #21448085C2semidirect
6. #21447932C2semidirect
7. #21447905C2semidirect
8. #21447882C2semidirect
9. #21447855C2semidirect
10. #21447761C2semidirect
11. #21447744C2semidirect
12. #21447744C2semidirect
13. #21447733C2semidirect
14. #21447685C2semidirect
15. C61×#2168C2semidirect
18. C2×#14519799C22semidirect
24. C2×Dic488C22semidirect
27. C2×#14519722C22semidirect
28. C2×#14519722C22semidirect
34. #16840139C22semidirect
38. #16839998C22semidirect
45. C4×Dic244C22semidirect
47. #16839588C22semidirect
48. #16839567C22semidirect
49. #16839567C22semidirect
52. #16839564C22semidirect
53. #16839564C22semidirect
55. C22×C16×C61C22semidirect
57. C61×C4⋊C16C22semidirect
59. C61×C22⋊C16C22semidirect
60. C61×#346C22semidirect
105. #14519722D4semidirect
306. #2168D61semidirect
334. C22×C16C2×D61semidirect
351. C4⋊C16C2×D61semidirect
359. C22⋊C16C2×D61semidirect
360. #346C2×D61semidirect
361. C61#33834semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C2×#168401942
#214575792
#214575062
#214480852
#214479322
#214479052
#214478822
#214478552
#214477612
#214477442
#214477332
#214476852
C61×#21682
#3383461