Finite Group Databasebeta

← #29836483

Subgroups — #29836483

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×#16840196C211yesmaximal, characteristic, complemented
2C22×#14519722C211yesmaximal, characteristic, complemented
3#21457557C211yesmaximal, characteristic, complemented
4#21457493C211yesmaximal, characteristic, complemented
5#21448082C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447873C211yesmaximal, characteristic, complemented
8#21447858C211yesmaximal, characteristic, complemented
9#21447749C211yesmaximal, characteristic, complemented
10#21447743C211yesmaximal, characteristic, complemented
11#21447735C211yesmaximal, characteristic, complemented
12#21447730C211yesmaximal, characteristic, complemented
13#21447682C211yesmaximal, characteristic, complemented
14#21447644C211yesmaximal, characteristic, complemented
15C61×#2215C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic, complemented
17C22×Dic244C2211yescharacteristic, complemented
18C22×D488C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840418—21no—
21#16840396—21no—
22#16840196—24no—
23C2×#14519722—24no—
24C2×#14519722C2211yescharacteristic
25C2×#14519722C2211yescharacteristic
26C2×#14519722C2211yescharacteristic, complemented
27C2×#14519722C2211yescharacteristic, complemented
28C2×#14519721C2211yescharacteristic, complemented
29C2×#14519721C2211yescharacteristic, complemented
30C2×#14519718C2211yescharacteristic
31C2×#14519718C2211yescharacteristic, complemented
32C2×#14519716C2211yescharacteristic, complemented
33#16840145C2211yescharacteristic, complemented
34#16840137C2211yescharacteristic
35#16840004C2211yescharacteristic, complemented
36#16839999C2211yescharacteristic
37#16839983—21no—
38#16839956—21no—
39#16839925—21no—
40#16839900—21no—
41#16839897—21no—
42#16839895—21no—
43#16839856C2211yescharacteristic, complemented
44#16839844C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839580C2211yescharacteristic, complemented
47#16839565C2211yescharacteristic, complemented
48#16839565C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic
50#16839562C2211yescharacteristic, complemented
51C22⋊#11758185C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839537C2211yescharacteristic, complemented
54C2×D16×C61C2211yescharacteristic, complemented
55C22×C16×C61C2211yescharacteristic, complemented
56C61×#466C2211yescharacteristic
57C61×#466C2211yescharacteristic, complemented
58C61×#366C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C22×Dic122—21no—
63C22×D244—21no—
64C2×D4×D61—21no—
65C2×D4×D61—41no—
66#14519796—22no—
67#14519796—41no—
68C2×Dic244—22no—
69C2×Dic244—21no—
70C2×Dic244—41no—
71C2×Dic244C2311yescharacteristic
72C2×Dic244C2311yescharacteristic
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×D488—22no—
76C2×D488—21no—
77C2×D488—41no—
78C2×D488C2311yescharacteristic
79C2×D488C2311yescharacteristic
80C2×#11758199—21no—
81C2×#11758199—21no—
82D8×D61—22no—
83D8×D61—41no—
84C2×C8×D61C2311yescharacteristic
85#14519773—21no—
86#14519764—21no—
87#14519763—41no—
88#14519757—21no—
89#14519754—41no—
90#14519752—21no—
91#14519750—41no—
92#14519749—41no—
93#14519747—41no—
94#14519745—41no—
95#14519742—21no—
96D4×Dic61—41no—
97#14519739—21no—
98C4×C22⋊D61D411yescharacteristic
99#14519722—22no—
100#14519722—22no—
101#14519722—22no—
102#14519722D414yescomplemented
103#14519721—22no—
104#14519721—41no—
105#14519718—22no—
106#14519718—41no—
107#14519716—22no—
108C2×#11758185C2311yescharacteristic
109C2×#11758185C2311yescharacteristic
110#14519706D411yescharacteristic
111#14519700—21no—
112#14519700—21no—
113#14519697—21no—
114#14519697—21no—
115C22⋊#92418D411yescharacteristic
116#14519689D411yescharacteristic
117C8×Dic61C2311yescharacteristic
118C2×D8×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121#121×C61—21no—
122#121×C61—21no—
123D16×C61—22no—
124C2×C16×C61—22no—
125C2×C16×C61C2311yescharacteristic
126C2×C16×C61C2311yescharacteristic
127#107×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130#102×C61—21no—
131C23×D61—41no—
132C23×D61—41no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—41no—
136C2×D4×C61—41no—
137C22×C4×C61C2×D411yescharacteristic
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61D811yescharacteristic
143C22×Dic61—41no—
144C22×Dic61—41no—
145#11758224—42no—
146#11758224—81no—
147D4×D61—42no—
148D4×D61—81no—
149D4×D61—84no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—42no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—81no—
156C2×C4×D61—41no—
157C2×C4×D61C2×D411yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—42no—
161C2×Dic122—41no—
162C2×Dic122—81no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61D811yescharacteristic
178#11758200—42no—
179#11758200—81no—
180#11758199—42no—
181#11758199—81no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198D811yescharacteristic
186C4⋊Dic61—41no—
187#11758196—41no—
188#11758196D811yescharacteristic
189C4×Dic61—41no—
190C4×Dic61C2×D411yescharacteristic
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195Dic244—21no—
196Dic244—42no—
197Dic244C2×D412yescomplemented
198D488—21no—
199D488—21no—
200D488—21no—
201D488—42no—
202D488C2×D412yes—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61C2×D412yescomplemented
207#11758185—21no—
208#11758185—21no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C2×D811yescharacteristic
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—161no—
219C22×D61—161no—
220C22×D61C2×D811yescharacteristic
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—82no—
226D4×C61—82no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C22⋊D61—22no—
234C22⋊D61—82no—
235C22⋊D61—82no—
236C22⋊D61—82no—
237C22⋊D61—161no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241C2×Dic61—82no—
242C2×Dic61C2×D811yescharacteristic
243C2×Dic61C2×D811yescharacteristic
244C2×Dic61C2×D811yescharacteristic
245D244—22no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—82no—
250D244—82no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—82no—
254Dic122—22no—
255Dic122—41no—
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—21no—
266C22×C61—21no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61#50912yescomplemented
274C22×C61C22×D811yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—42no—
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—81no—
288C4×C61—81no—
289C4×C61D4211yescharacteristic
290C4×C61#57611yescharacteristic
291Dic61—21no—
292Dic61—21no—
293Dic61—21no—
294Dic61—42no—
295Dic61—81no—
296C2×#509—611no—
297C22×SD32—611no—
298#3285—611no—
299D4×D8—611no—
300#2250—611no—
301#2220—611no—
302#2215D6111yescharacteristic, complemented
303#2210—611no—
304#2207—611no—
305#2187—611no—
306#2186—611no—
307#2185—611no—
308#2184—611no—
309#2175—611no—
310#2168—611no—
311C2×C61—22no—
312C2×C61—161no—
313C2×C61—161no—
314C2×C61D4×D811yescharacteristic
315C2×C61C2×#50911yescharacteristic
316C2×C61#341611yescharacteristic
317D61—41no—
318D61—82no—
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×D16C2×D6111yescharacteristic, complemented
337C2×M32—611no—
338C22×C16C2×D6111yescharacteristic, complemented
339#495—611no—
340#493—611no—
341#467—611no—
342#466—611no—
343#466C2×D6111yescharacteristic
344#466C2×D6111yescharacteristic, complemented
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×D4—611no—
354#366C2×D6111yescharacteristic, complemented
355C4⋊C16—611no—
356#358—611no—
357#358—611no—
358#357—611no—
359#357—611no—
360#357C2×D6111yescharacteristic, complemented
361#357C2×D6111yescharacteristic, complemented
362C22⋊C16—611no—
363C22⋊C16—611no—
364#346—611no—
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—1222no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×D8—1222no—
386C2×D8—1221no—
387C2×D8—2441no—
388C2×D8—2441no—
389C2×D8C22×D6111yescharacteristic
390C2×D8C22×D6111yescharacteristic
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393#127—1221no—
394#124—1221no—
395#124—2441no—
396C22⋊Q8—1221no—
397C22⋊Q8—2441no—
398#121—21no—
399#121—21no—
400#121—1221no—
401#121—2441no—
402#121—2441no—
403#120—2441no—
404#120—2441no—
405C4×Q8—1221no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4—2441no—
409SD32—614no—
410SD32—1222no—
411SD32—1222no—
412SD32—1222no—
413SD32—1222no—
414SD32—2441no—
415D16—22no—
416D16—1222no—
417D16—2441no—
418M32—1222no—
419C2×C16—22no—
420C2×C16—611no—
421C2×C16—611no—
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424#107C22×D6111yescharacteristic
425#107C22×D6111yescharacteristic
426C4⋊C8—611no—
427#103—1221no—
428#103—1221no—
429#102—21no—
430#102—21no—
431#102—1221no—
432#102—1221no—
433C22⋊C8—611no—
434C22⋊C8—611no—
435C4×C8—611no—
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—21no—
446C2×D4—21no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—611no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—4881no—
453C2×D4—4881no—
454C2×D4—4884no—
455C2×D4—2442no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4—2441no—
460C2×D4—2442no—
461C2×D4—2441no—
462C2×D4—2441no—
463C22×C4—611no—
464C22×C4—2441no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4D4×D6111yescharacteristic
468Q16—612no—
469Q16—1221no—
470Q16—1221no—
471Q16—1221no—
472Q16—2442no—
473SD16—4881no—
474SD16—2442no—
475D8—21no—
476D8—21no—
477D8—21no—
478D8—21no—
479D8—612no—
480D8—1221no—
481D8—1221no—
482D8—1221no—
483D8—4881no—
484D8—2442no—
485D8—2442no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8D4×D6111yescharacteristic
493C2×C8C23×D6111yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—611no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—611no—
502C22⋊C4—611no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C42—611no—
509C42—2441no—
510C16—21no—
511C16—1221no—
512C16—1221no—
513C16D4×D6112yescomplemented
514C23—41no—
515C23—41no—
516C23—611no—
517C23—9761no—
518C23—9761no—
519C23—4882no—
520C23—4882no—
521C23—4882no—
522C23—4882no—
523C23—2441no—
524C23—2441no—
525C23D8×D6111yescharacteristic
526Q8—1222no—
527Q8—4882no—
528Q8—2441no—
529Q8—2441no—
530Q8—2441no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—82no—
536D4—82no—
537D4—1222no—
538D4—1222no—
539D4—9761no—
540D4—4882no—
541D4—4882no—
542D4—4882no—
543D4—4882no—
544D4—4882no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—4882no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4D8×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C8—21no—
565C8—1221no—
566C8—1221no—
567C8C2×D4×D6111yescharacteristic
568C8#1451982311yescharacteristic
569C22—21no—
570C22—21no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—1221no—
578C22—1221no—
579C22—1221no—
580C22—9764no—
581C22—9764no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—2442no—
588C22#1684019612yescomplemented
589C22C2×D8×D6111yescharacteristic
590C4—21no—
591C4—81no—
592C4—81no—
593C4—1221no—
594C4—1221no—
595C4—1221no—
596C4—4881no—
597C4—2442no—
598C4#1684039611yescharacteristic
599C4#1684056411yescharacteristic
600C2—22no—
601C2—161no—
602C2—161no—
603C2—9761no—
604C2—4882no—
605C2—2441no—
606C2#2145749311yescharacteristic
607C2C2×#1684019611yescharacteristic
608C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2215C2
socleC22×C61C22×D8
hypercenter#2215D61
hypocenterC61#33831
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. C2×#16840196C2semidirect
2. C22×#14519722C2semidirect
3. #21457557C2semidirect
4. #21457493C2semidirect
5. #21448082C2semidirect
6. #21447933C2semidirect
7. #21447873C2semidirect
8. #21447858C2semidirect
9. #21447749C2semidirect
10. #21447743C2semidirect
11. #21447735C2semidirect
12. #21447730C2semidirect
13. #21447682C2semidirect
14. #21447644C2semidirect
15. C61×#2215C2semidirect
16. C2×#14519796C22semidirect
17. C22×Dic244C22semidirect
26. C2×#14519722C22semidirect
27. C2×#14519722C22semidirect
28. C2×#14519721C22semidirect
29. C2×#14519721C22semidirect
31. C2×#14519718C22semidirect
32. C2×#14519716C22semidirect
33. #16840145C22semidirect
35. #16840004C22semidirect
43. #16839856C22semidirect
46. #16839580C22semidirect
47. #16839565C22semidirect
48. #16839565C22semidirect
50. #16839562C22semidirect
51. C22⋊#11758185C22semidirect
52. #16839541C22semidirect
53. #16839537C22semidirect
54. C2×D16×C61C22semidirect
55. C22×C16×C61C22semidirect
57. C61×#466C22semidirect
58. C61×#366C22semidirect
59. C61×#357C22semidirect
60. C61×#357C22semidirect
102. #14519722D4semidirect
197. Dic244C2×D4semidirect
206. C16×C61C2×D4semidirect
273. C22×C61#509semidirect
302. #2215D61semidirect
336. C2×D16C2×D61semidirect
338. C22×C16C2×D61semidirect
344. #466C2×D61semidirect
354. #366C2×D61semidirect
360. #357C2×D61semidirect
361. #357C2×D61semidirect
365. C61#33831semidirect
513. C16D4×D61semidirect
588. C22#16840196semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
C22×#145197222
#214575572
#214574932
#214480822
#214479332
#214478732
#214478582
#214477492
#214477432
#214477352
#214477302
#214476822
#214476442
C61×#22152
#3383161