Finite Group Databasebeta

← #29836488

Subgroups — #29836488

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457561C211yesmaximal, characteristic, complemented
4D8×D244C211yesmaximal, characteristic, complemented
5#21448085C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447906C211yesmaximal, characteristic, complemented
8#21447874C211yesmaximal, characteristic, complemented
9#21447859C211yesmaximal, characteristic, complemented
10#21447749C211yesmaximal, characteristic, complemented
11#21447745C211yesmaximal, characteristic, complemented
12#21447737C211yesmaximal, characteristic, complemented
13#21447732C211yesmaximal, characteristic, complemented
14#21447646C211yesmaximal, characteristic, complemented
15C61×#2175C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic, complemented
17C2×Q16×D61C2211yescharacteristic, complemented
18C2×D8×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840424—21no—
21D4×D244—21no—
22#16840199—24no—
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic
26C2×#14519720C2211yescharacteristic, complemented
27C2×#14519720C2211yescharacteristic, complemented
28SD32×D61—24no—
29C2×D976C2211yescharacteristic, complemented
30C2×D976C2211yescharacteristic, complemented
31C2×#14519718C2211yescharacteristic
32C2×#14519716C2211yescharacteristic, complemented
33C2×C16×D61C2211yescharacteristic, complemented
34#16840146C2211yescharacteristic, complemented
35#16840139C2211yescharacteristic
36#16840004C2211yescharacteristic, complemented
37#16839999C2211yescharacteristic
38#16839999C2211yescharacteristic
39#16839999C2211yescharacteristic, complemented
40#16839984—21no—
41#16839957—21no—
42#16839927—21no—
43#16839902—21no—
44#16839897—21no—
45#16839897—21no—
46C8×D244C2211yescharacteristic
47C4⋊#11758185C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
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×Q8×D61—21no—
63C22×D244—41no—
64C2×D4×D61—21no—
65C2×D4×D61—21no—
66#14519800—22no—
67#14519800—41no—
68C2×Dic244C2311yescharacteristic
69C2×#11758202—21no—
70Q16×D61—22no—
71Q16×D61—41no—
72C2×#11758190—21no—
73C2×#11758201—21no—
74C2×D488—21no—
75C2×D488C2311yescharacteristic
76C2×D488C2311yescharacteristic
77C2×D488C2311yescharacteristic
78C2×#11758199—21no—
79C2×#11758199—21no—
80D8×D61—22no—
81D8×D61—22no—
82D8×D61—41no—
83D8×D61—41no—
84C2×C8×D61C2311yescharacteristic
85C2×C8×D61C2311yescharacteristic
86#14519774—21no—
87#14519765—21no—
88#14519765—41no—
89#14519757—21no—
90#14519757—41no—
91#14519752—21no—
92#14519752—21no—
93#14519752—21no—
94#14519752—41no—
95#14519751—41no—
96#14519747—41no—
97#14519747—41no—
98C4×D244—41no—
99C4×D244D411yescharacteristic
100#14519722—22no—
101#14519722—41no—
102#14519721—22no—
103#14519720—22no—
104#14519720—22no—
105D976—22no—
106D976—41no—
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—
132C23×D61—41no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C22×C4×C61—41no—
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138#11758226—42no—
139#11758226—81no—
140Q8×D61—42no—
141Q8×D61—81no—
142D4×D61—42no—
143D4×D61—42no—
144D4×D61—81no—
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—84no—
156C2×D244D811yescharacteristic
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—41no—
163Q16×C61—21no—
164Q16×C61C2×D412yescomplemented
165D8×C61—21no—
166D8×C61C2×D412yes—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C42×C61—41no—
174C42×C61C2×D411yescharacteristic
175#11758202—42no—
176#11758201—42no—
177#11758199—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—41no—
188C4⋊Dic61D811yescharacteristic
189C2×#92418C2×D411yescharacteristic
190C2×#92418C2×D411yescharacteristic
191Dic244—21no—
192Dic244—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488—81no—
200#11758190—81no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61C2×D412yescomplemented
207#11758185—21no—
208#11758185—21no—
209C23×C61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—161no—
218C22×D61—161no—
219C22×D61C2×D811yescharacteristic
220C22×D61C2×D811yescharacteristic
221Q8×C61—22no—
222Q8×C61—41no—
223D4×C61—22no—
224D4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61C2×D811yescharacteristic
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C22⋊D61—82no—
233C22⋊D61—82no—
234C2×Dic61—41no—
235C2×Dic61C2×D811yescharacteristic
236C2×Dic61C2×D811yescharacteristic
237D244—22no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—82no—
245D244—82no—
246D244—82no—
247D244—82no—
248D244—82no—
249D244—161no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—82no—
255C4×D61—82no—
256Dic122—41no—
257Dic122—41no—
258Dic122—82no—
259C8×C61—21no—
260C8×C61C22×D411yescharacteristic
261C8×C61H(5,𝔽2)11yescharacteristic
262#92418—21no—
263#92418—21no—
264#33831—611nomaximal, Sylow
265C22×C61—42no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61C22×D811yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—164no—
284C2×D61—164no—
285C2×D61#50912yescomplemented
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61D4211yescharacteristic
292C4×C61#57611yescharacteristic
293Dic61—21no—
294Dic61—21no—
295Dic61—81no—
296C2×#509—611no—
297C22×SD32—611no—
298#3285—611no—
299D4×D8—611no—
300#2250—611no—
301#2220—611no—
302#2215—611no—
303#2210—611no—
304#2207—611no—
305#2187—611no—
306#2186—611no—
307#2185—611no—
308#2184—611no—
309#2175D6111yescharacteristic, complemented
310#2168—611no—
311C2×C61—82no—
312C2×C61D4×D811yescharacteristic
313C2×C61C2×#50911yescharacteristic
314C2×C61#341611yescharacteristic
315D61—22no—
316D61—41no—
317D61—161no—
318D61—161no—
319D61—161no—
320C2×#135—611no—
321C22×Q16—611no—
322C22×D8—611no—
323C22×D8—611no—
324#548—1221no—
325D42—1221no—
326#509—1224no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×SD32—611no—
330C2×SD32—611no—
331C2×SD32—611no—
332C2×SD32—1224no—
333C2×SD32C2×D6111yescharacteristic
334C2×D16—611no—
335C2×D16—611no—
336C2×D16—611no—
337C2×M32—611no—
338C22×C16—611no—
339#495—611no—
340#493—611no—
341#467—611no—
342#466—611no—
343#466—611no—
344#466—611no—
345#464—1221no—
346#459—1221no—
347#452—1221no—
348#448—1221no—
349#447—1221no—
350#447—1221no—
351C4×Q16C2×D6111yescharacteristic, complemented
352C4×D8C2×D6111yescharacteristic
353C8×D4—611no—
354#366C2×D6111yescharacteristic, complemented
355C4⋊C16—611no—
356#358—611no—
357#358C2×D6111yescharacteristic, complemented
358#357—611no—
359#357—611no—
360#357—611no—
361#357C2×D6111yescharacteristic
362C22⋊C16—611no—
363C22⋊C16—611no—
364#346C2×D6111yescharacteristic, complemented
365C61#3383111yescharacteristic, complemented, Sylow
366C2×#41—1221no—
367C22×Q8—1221no—
368C22×D4—1221no—
369C22×D4—1221no—
370C22×D4—2441no—
371#135—1222no—
372#135—2441no—
373C2×Q16—611no—
374C2×Q16—1221no—
375C2×Q16—1222no—
376C2×Q16—2441no—
377C2×Q16C22×D6111yescharacteristic
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×D8—611no—
381C2×D8—611no—
382C2×D8—611no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×D8—1222no—
386C2×D8—1222no—
387C2×D8—1221no—
388C2×D8—2441no—
389C2×D8—2441no—
390C2×D8C22×D6111yescharacteristic
391C22×C8—611no—
392C22×C8—611no—
393#127—1221no—
394#124—1221no—
395#124—2441no—
396C22⋊Q8—1221no—
397C22⋊Q8—2441no—
398#121—1221no—
399#121—1221no—
400#121—1221no—
401#121—2441no—
402#121—2441no—
403#120—2441no—
404#120—2441no—
405C4×Q8—21no—
406C4×D4—21no—
407C4×D4—611no—
408C4×D4—2441no—
409SD32—1222no—
410SD32—1222no—
411SD32—1222no—
412SD32—1222no—
413SD32—2441no—
414SD32D24414yescomplemented
415D16—1222no—
416D16—1222no—
417D16—2441no—
418M32—1222no—
419C2×C16—611no—
420C2×C16—611no—
421C2×C16—1222no—
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424#107C22×D6111yescharacteristic
425#107C22×D6111yescharacteristic
426C4⋊C8—611no—
427#103—21no—
428#103—1221no—
429#102—21no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433C22⋊C8—611no—
434C22⋊C8—611no—
435C4×C8C22×D6111yescharacteristic
436C24—2441no—
437C24—2441no—
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—2441no—
444C2×Q8—2442no—
445C2×D4—21no—
446C2×D4—611no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—4884no—
451C2×D4—4881no—
452C2×D4—4881no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2441no—
462C2×D4—2441no—
463C22×C4—41no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—2441no—
467C22×C4—2441no—
468Q16—21no—
469Q16—1221no—
470Q16—1221no—
471Q16—2442no—
472Q16C2×D24412yescomplemented
473SD16—4881no—
474SD16—2442no—
475D8—21no—
476D8—1221no—
477D8—1221no—
478D8—1221no—
479D8—1221no—
480D8—1221no—
481D8—1221no—
482D8—4881no—
483D8—2442no—
484D8—2442no—
485D8C2×D24412yes—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8D4×D6111yescharacteristic
493C2×C8C23×D6111yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—611no—
498C4⋊C4—2441no—
499C22⋊C4—41no—
500C22⋊C4—611no—
501C22⋊C4—611no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C42—41no—
509C42D4×D6111yescharacteristic
510C16—21no—
511C16—1221no—
512C16—1221no—
513C16C2×D24412yescomplemented
514C23—41no—
515C23—611no—
516C23—611no—
517C23—9761no—
518C23—9761no—
519C23—4882no—
520C23—4882no—
521C23—4882no—
522C23—4882no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526Q8—22no—
527Q8—41no—
528Q8—4882no—
529Q8—2441no—
530Q8—2441no—
531D4—22no—
532D4—41no—
533D4—1222no—
534D4—9761no—
535D4—4882no—
536D4—4882no—
537D4—4882no—
538D4—4882no—
539D4—4882no—
540D4—4882no—
541D4—4882no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—4882no—
559C2×C4—4882no—
560C2×C4—2441no—
561C2×C4D8×D6111yescharacteristic
562C2×C4D8×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C8—21no—
565C8—1221no—
566C8—1221no—
567C8C22×D24411yescharacteristic
568C8#1451982311yescharacteristic
569C22—42no—
570C22—81no—
571C22—81no—
572C22—612no—
573C22—1221no—
574C22—1221no—
575C22—1221no—
576C22—1221no—
577C22—1221no—
578C22—9764no—
579C22—9764no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22C2×D8×D6111yescharacteristic
590C4—21no—
591C4—21no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—1221no—
596C4—1221no—
597C4—4881no—
598C4D4×D24411yescharacteristic
599C4#1684056411yescharacteristic
600C2—82no—
601C2—1222no—
602C2—9761no—
603C2—9761no—
604C2—9761no—
605C2—2441no—
606C2D8×D24411yescharacteristic
607C2#509×D6111yescharacteristic
608C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2175C2
socleC22×C61C22×D8
hypercenter#2175D61
hypocenterC61#33831
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×SD32×D61C2semidirect
3. #21457561C2semidirect
4. D8×D244C2semidirect
5. #21448085C2semidirect
6. #21447933C2semidirect
7. #21447906C2semidirect
8. #21447874C2semidirect
9. #21447859C2semidirect
10. #21447749C2semidirect
11. #21447745C2semidirect
12. #21447737C2semidirect
13. #21447732C2semidirect
14. #21447646C2semidirect
15. C61×#2175C2semidirect
16. C2×#14519800C22semidirect
17. C2×Q16×D61C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519722C22semidirect
26. C2×#14519720C22semidirect
27. C2×#14519720C22semidirect
29. C2×D976C22semidirect
30. C2×D976C22semidirect
32. C2×#14519716C22semidirect
33. C2×C16×D61C22semidirect
34. #16840146C22semidirect
36. #16840004C22semidirect
39. #16839999C22semidirect
47. C4⋊#11758185C22semidirect
48. #16839565C22semidirect
49. #16839562C22semidirect
50. #16839562C22semidirect
51. #16839562C22semidirect
52. #16839541C22semidirect
53. #16839541C22semidirect
55. C4×Q16×C61C22semidirect
57. C61×#366C22semidirect
58. C61×#358C22semidirect
60. C61×#346C22semidirect
123. SD32×C61D4semidirect
164. Q16×C61C2×D4semidirect
206. C16×C61C2×D4semidirect
285. C2×D61#509semidirect
309. #2175D61semidirect
351. C4×Q16C2×D61semidirect
354. #366C2×D61semidirect
357. #358C2×D61semidirect
364. #346C2×D61semidirect
365. C61#33831semidirect
414. SD32D244semidirect
472. Q16C2×D244semidirect
513. C16C2×D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×SD32×D612
#214575612
D8×D2442
#214480852
#214479332
#214479062
#214478742
#214478592
#214477492
#214477452
#214477372
#214477322
#214476462
C61×#21752
#3383161