Finite Group Databasebeta

← #29836491

Subgroups — #29836491

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×#16840200C211yesmaximal, characteristic, complemented
2C22×#14519722C211yesmaximal, characteristic, complemented
3#21457562C211yesmaximal, characteristic, complemented
4D4×D488C211yesmaximal, characteristic, complemented
5#21448085C211yesmaximal, characteristic, complemented
6#21447932C211yesmaximal, characteristic, complemented
7#21447905C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447860C211yesmaximal, characteristic, complemented
10#21447750C211yesmaximal, characteristic, complemented
11#21447744C211yesmaximal, characteristic, complemented
12#21447738C211yesmaximal, characteristic, complemented
13#21447733C211yesmaximal, characteristic, complemented
14#21447685C211yesmaximal, characteristic, complemented
15C61×#2168C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C22×Dic244C2211yescharacteristic, complemented
18C22×D488C2211yescharacteristic
19C22×D488C2211yescharacteristic
20#16840421—21no—
21D4×D244—21no—
22#16840200—24no—
23C2×#14519722—24no—
24C2×#14519722C2211yescharacteristic
25C2×#14519722C2211yescharacteristic
26C2×#14519722C2211yescharacteristic, complemented
27C2×#14519722C2211yescharacteristic, complemented
28C2×#14519722C2211yescharacteristic, complemented
29C2×#14519722C2211yescharacteristic, complemented
30C2×D976C2211yescharacteristic
31C2×D976C2211yescharacteristic, complemented
32C2×D976C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic, complemented
34#16840139C2211yescharacteristic
35#16840003C2211yescharacteristic, complemented
36#16839998C2211yescharacteristic
37#16839998C2211yescharacteristic
38#16839998C2211yescharacteristic, complemented
39#16839985—21no—
40#16839958—21no—
41#16839926—21no—
42#16839903—21no—
43#16839898—21no—
44#16839898—21no—
45C4×Dic244C2211yescharacteristic, complemented
46C4×D488C2211yescharacteristic
47#16839588C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839567C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic
51#16839564C2211yescharacteristic, complemented
52#16839564C2211yescharacteristic, complemented
53#16839564C2211yescharacteristic, complemented
54C2×M32×C61C2211yescharacteristic, complemented
55C22×C16×C61C2211yescharacteristic, complemented
56C8×D4×C61C2211yescharacteristic
57C61×C4⋊C16C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic, complemented
59C61×C22⋊C16C2211yescharacteristic, complemented
60C61×#346C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C22×Dic122—21no—
63C22×D244—21no—
64C22×D244—21no—
65C2×D4×D61—41no—
66#14519799—22no—
67#14519799—41no—
68C2×Dic244—22no—
69C2×Dic244—21no—
70C2×Dic244—41no—
71C2×Dic244C2311yescharacteristic
72C2×Dic244C2311yescharacteristic
73C2×#11758190—21no—
74C2×#11758190—21no—
75C2×D488—22no—
76C2×D488—22no—
77C2×D488—21no—
78C2×D488—21no—
79C2×D488—21no—
80C2×D488—41no—
81C2×D488—41no—
82C2×D488C2311yescharacteristic
83C2×D488C2311yescharacteristic
84C2×D488C2311yescharacteristic
85C2×D488C2311yescharacteristic
86#14519774—21no—
87#14519765—21no—
88#14519763—41no—
89#14519756—21no—
90#14519755—41no—
91#14519752—41no—
92#14519751—21no—
93#14519751—21no—
94#14519751—21no—
95#14519749—41no—
96#14519747—41no—
97#14519747—41no—
98C4×Dic122—21no—
99C4×D244—21no—
100#14519737—41no—
101#14519722—22no—
102#14519722—22no—
103#14519722—22no—
104#14519722—22no—
105#14519722—41no—
106#14519722D414yescomplemented
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—
132C23×D61—41no—
133C2×D4×C61D811yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C2×C22⋊D61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C22×Dic61—41no—
141D4×D61—84no—
142#11758222—42no—
143#11758222—81no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244—81no—
155C2×D244—81no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—42no—
161C2×Dic122—41no—
162C2×Dic122—81no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2411yescharacteristic
171C4⋊C4×C61D811yescharacteristic
172C22⋊C4×C61D811yescharacteristic
173C22⋊C4×C61D811yescharacteristic
174C42×C61C2×D411yescharacteristic
175C22⋊Dic61—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184#11758196—41no—
185#11758196—41no—
186C4×Dic61—41no—
187Dic244—21no—
188Dic244—21no—
189Dic244—21no—
190Dic244—42no—
191Dic244C2×D412yescomplemented
192D488—21no—
193D488—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488—42no—
200D488—42no—
201D488—81no—
202D488C2×D412yes—
203#11758190—42no—
204#11758190—81no—
205C16×C61—21no—
206C16×C61—21no—
207C16×C61—21no—
208C16×C61C2×D412yescomplemented
209C23×C61C2×D811yescharacteristic
210C23×C61C2×D811yescharacteristic
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—161no—
220C22×D61—161no—
221D4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C2×D811yescharacteristic
228C2×C4×C61C2×D811yescharacteristic
229C2×C4×C61C2×D811yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C22⋊D61—82no—
232C22⋊D61—82no—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—82no—
240D244—22no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—82no—
249D244—82no—
250D244—82no—
251D244—161no—
252C4×D61—82no—
253C4×D61—82no—
254Dic122—22no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—82no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61C22×D411yescharacteristic
263C8×C61H(5,𝔽2)11yescharacteristic
264#33831—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61#50912yescomplemented
271C22×C61C22×D811yescharacteristic
272C2×D61—42no—
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—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—164no—
285C2×D61—164no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61D4211yescharacteristic
291C4×C61#57611yescharacteristic
292Dic61—42no—
293Dic61—81no—
294Dic61—81no—
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#2175—611no—
310#2168D6111yescharacteristic, complemented
311C2×C61—22no—
312C2×C61—41no—
313C2×C61D4×D811yescharacteristic
314C2×C61C2×#50911yescharacteristic
315C2×C61#341611yescharacteristic
316D61—82no—
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—611no—
333C2×SD32—1224no—
334C2×D16—611no—
335C2×D16—611no—
336C2×D16—611no—
337C2×M32C2×D6111yescharacteristic, complemented
338C22×C16C2×D6111yescharacteristic, complemented
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×Q16—611no—
352C4×D8—611no—
353C8×D4C2×D6111yescharacteristic
354#366—611no—
355C4⋊C16C2×D6111yescharacteristic, complemented
356#358—611no—
357#358—611no—
358#357—611no—
359#357—611no—
360#357—611no—
361#357—611no—
362C22⋊C16C2×D6111yescharacteristic, complemented
363C22⋊C16C2×D6111yescharacteristic, complemented
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—611no—
375C2×Q16—1222no—
376C2×Q16—1221no—
377C2×Q16—2441no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×D8—611no—
381C2×D8—611no—
382C2×D8—611no—
383C2×D8—611no—
384C2×D8—1222no—
385C2×D8—1222no—
386C2×D8—1221no—
387C2×D8—1221no—
388C2×D8—1221no—
389C2×D8—2441no—
390C2×D8—2441no—
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
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—1221no—
406C4×D4—1221no—
407C4×D4—2441no—
408C4×D4D24411yescharacteristic
409SD32—614no—
410SD32—1222no—
411SD32—1222no—
412SD32—1222no—
413SD32—1222no—
414SD32—2441no—
415D16—1222no—
416D16—1222no—
417D16—2441no—
418M32—22no—
419C2×C16—22no—
420C2×C16C22×D6111yescharacteristic
421C2×C16C22×D6111yescharacteristic
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424#107—611no—
425#107—611no—
426C4⋊C8D24411yescharacteristic
427#103—1221no—
428#103—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433C22⋊C8D24411yescharacteristic
434C22⋊C8D24411yescharacteristic
435C4×C8C22×D6111yescharacteristic
436C24—2441no—
437C24—2441no—
438#41—4881no—
439#41—2442no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—2442no—
444C2×Q8—2441no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—4881no—
450C2×D4—4881no—
451C2×D4—4884no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4D48811yescharacteristic
463C22×C4—2441no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4C2×D24411yescharacteristic
467C22×C4C2×D24411yescharacteristic
468Q16—612no—
469Q16—1221no—
470Q16—1221no—
471Q16—1221no—
472Q16—2442no—
473SD16—4881no—
474SD16—2442no—
475D8—612no—
476D8—1221no—
477D8—1221no—
478D8—1221no—
479D8—1221no—
480D8—1221no—
481D8—1221no—
482D8—1221no—
483D8—4881no—
484D8—2442no—
485D8—2442no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8C2×D24411yescharacteristic
491C2×C8C2×D24411yescharacteristic
492C2×C8C2×D24411yescharacteristic
493C2×C8C23×D6111yescharacteristic
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4D48811yescharacteristic
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C22⋊C4D48811yescharacteristic
507C22⋊C4D48811yescharacteristic
508C42—2441no—
509C42C2×D24411yescharacteristic
510C16—21no—
511C16—21no—
512C16—21no—
513C16D4×D6112yescomplemented
514C23—9761no—
515C23—9761no—
516C23—4882no—
517C23—4882no—
518C23—4882no—
519C23—4882no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23—2441no—
524C23C2×D48811yescharacteristic
525C23C2×D48811yescharacteristic
526Q8—1222no—
527Q8—4882no—
528Q8—2441no—
529Q8—2441no—
530Q8—2441no—
531D4—22no—
532D4—1222no—
533D4—9761no—
534D4—4882no—
535D4—4882no—
536D4—4882no—
537D4—4882no—
538D4—4882no—
539D4—4882no—
540D4—4882no—
541D4—2441no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—4882no—
553C2×C4—4882no—
554C2×C4—4882no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4C2×D48811yescharacteristic
560C2×C4C2×D48811yescharacteristic
561C2×C4C2×D48811yescharacteristic
562C2×C4C2×D48811yescharacteristic
563C2×C4C22×D24411yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8C2×D4×D6111yescharacteristic
568C8#1451982311yescharacteristic
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—9764no—
575C22—9764no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—2442no—
588C22#1684020012yescomplemented
589C22C22×D48811yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—4881no—
595C4—4881no—
596C4—4881no—
597C4—2442no—
598C4D4×D24411yescharacteristic
599C4#1684056711yescharacteristic
600C2—22no—
601C2—41no—
602C2—9761no—
603C2—9761no—
604C2—9761no—
605C2—4882no—
606C2D4×D48811yescharacteristic
607C2C2×#1684020011yescharacteristic
608C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2168C2
socleC22×C61C22×D8
hypercenter#2168D61
hypocenterC61#33831
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. C22×#14519722C2semidirect
3. #21457562C2semidirect
4. D4×D488C2semidirect
5. #21448085C2semidirect
6. #21447932C2semidirect
7. #21447905C2semidirect
8. #21447875C2semidirect
9. #21447860C2semidirect
10. #21447750C2semidirect
11. #21447744C2semidirect
12. #21447738C2semidirect
13. #21447733C2semidirect
14. #21447685C2semidirect
15. C61×#2168C2semidirect
16. C2×#14519799C22semidirect
17. C22×Dic244C22semidirect
26. C2×#14519722C22semidirect
27. C2×#14519722C22semidirect
28. C2×#14519722C22semidirect
29. C2×#14519722C22semidirect
31. C2×D976C22semidirect
32. C2×D976C22semidirect
33. #16840146C22semidirect
35. #16840003C22semidirect
38. #16839998C22semidirect
45. C4×Dic244C22semidirect
47. #16839588C22semidirect
48. #16839567C22semidirect
49. #16839567C22semidirect
51. #16839564C22semidirect
52. #16839564C22semidirect
53. #16839564C22semidirect
54. C2×M32×C61C22semidirect
55. C22×C16×C61C22semidirect
57. C61×C4⋊C16C22semidirect
58. C61×C22⋊C16C22semidirect
59. C61×C22⋊C16C22semidirect
60. C61×#346C22semidirect
106. #14519722D4semidirect
191. Dic244C2×D4semidirect
208. C16×C61C2×D4semidirect
270. C22×C61#509semidirect
310. #2168D61semidirect
337. C2×M32C2×D61semidirect
338. C22×C16C2×D61semidirect
355. C4⋊C16C2×D61semidirect
362. C22⋊C16C2×D61semidirect
363. C22⋊C16C2×D61semidirect
364. #346C2×D61semidirect
365. C61#33831semidirect
513. C16D4×D61semidirect
588. C22#16840200semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C22×#145197222
#214575622
D4×D4882
#214480852
#214479322
#214479052
#214478752
#214478602
#214477502
#214477442
#214477382
#214477332
#214476852
C61×#21682
#3383161