Finite Group Databasebeta

← #29836728

Subgroups — #29836728

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840192C211yesmaximal, characteristic, complemented
2C22×#14519723C211yesmaximal, characteristic, complemented
3#21457562C211yesmaximal, characteristic, complemented
4#21448078C211yesmaximal, characteristic, complemented
5#21447908C211yesmaximal, characteristic, complemented
6#21447903C211yesmaximal, characteristic, complemented
7#21447881C211yesmaximal, characteristic, complemented
8#21447878C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447757C211yesmaximal, characteristic, complemented
11#21447736C211yesmaximal, characteristic, complemented
12#21447735C211yesmaximal, characteristic, complemented
13C4×#14519723C211yesmaximal, characteristic, complemented
14D4×#11758185C211yesmaximal, characteristic, complemented
15C61×#3285C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18#16840421—21no—
19#16840192—24no—
20C2×#14519723—24no—
21C2×#14519723C2211yescharacteristic
22C2×#14519723C2211yescharacteristic
23C2×#14519723C2211yescharacteristic
24C2×#14519723C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519720C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C22×#11758185C2211yescharacteristic
31C22×#11758185C2211yescharacteristic
32#16840146C2211yescharacteristic
33#16840003C2211yescharacteristic
34#16839998C2211yescharacteristic
35#16839985—21no—
36#16839926—21no—
37#16839903—21no—
38C4×Dic244C2211yescharacteristic
39#16839587C2211yescharacteristic
40C4⋊#11758185C2211yescharacteristic
41#16839566C2211yescharacteristic
42#16839566C2211yescharacteristic
43#16839565C2211yescharacteristic
44#16839565C2211yescharacteristic
45#16839563C2211yescharacteristic
46#16839562C2211yescharacteristic
47C22⋊#11758185C2211yescharacteristic
48C22⋊#11758185C2211yescharacteristic
49C4×#11758185C2211yescharacteristic
50C2×#135×C61C2211yescharacteristic
51C22×Q16×C61C2211yescharacteristic
52C61×#548—21no—
53C61×#495C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#464—21no—
57C61×#452—21no—
58C61×#448—21no—
59C4×Q16×C61C2211yescharacteristic
60C8×D4×C61C2211yescharacteristic
61C2×#11758222—21no—
62C22×Dic122—21no—
63#14519799—22no—
64#14519799—41no—
65C2×Dic244—22no—
66C2×Dic244—21no—
67C2×Dic244—41no—
68C2×Dic244C2311yescharacteristic
69C2×Dic244C2311yescharacteristic
70C2×#11758190—21no—
71C2×#11758190—21no—
72C2×D488C2311yescharacteristic
73#14519765—21no—
74#14519763—41no—
75#14519756—21no—
76#14519755—41no—
77#14519751—21no—
78#14519749—41no—
79C4×Dic122—21no—
80#14519737—41no—
81#14519723—22no—
82#14519723—22no—
83#14519723—22no—
84#14519723D414yescomplemented
85#14519721—22no—
86#14519721—41no—
87#14519720—22no—
88#14519720—41no—
89#14519718—22no—
90C2×#11758185—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519699—21no—
100C2×#41×C61—21no—
101C22×Q8×C61—21no—
102#135×C61—22no—
103#135×C61—41no—
104C2×Q16×C61—22no—
105C2×Q16×C61—21no—
106C2×Q16×C61—41no—
107C2×Q16×C61C2311yescharacteristic
108C2×Q16×C61C2311yescharacteristic
109C2×SD16×C61—21no—
110C2×SD16×C61—21no—
111C2×D8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114#124×C61—21no—
115#124×C61—41no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—41no—
118#121×C61—21no—
119#121×C61—41no—
120C4×Q8×C61—21no—
121C4×D4×C61—41no—
122C4×D4×C61D411yescharacteristic
123#107×C61C2311yescharacteristic
124C4⋊C8×C61D411yescharacteristic
125#103×C61—21no—
126#103×C61—21no—
127#102×C61—21no—
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131#41×C61—42no—
132#41×C61—81no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—42no—
136C2×Q8×C61—41no—
137C2×Q8×C61—81no—
138C2×D4×C61—21no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61D811yescharacteristic
142C22×C4×C61—41no—
143C22×C4×C61—41no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C22×Dic61—41no—
149#11758222—42no—
150#11758222—81no—
151C2×D244—21no—
152C2×C4×D61—41no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—42no—
156C2×Dic122—41no—
157C2×Dic122—81no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—42no—
162Q16×C61C2×D412yes—
163SD16×C61—42no—
164SD16×C61—81no—
165D8×C61—21no—
166D8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61D811yescharacteristic
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61D811yescharacteristic
184C22⋊C4×C61D811yescharacteristic
185C42×C61—41no—
186C42×C61C2×D411yescharacteristic
187C22⋊Dic61—41no—
188#11758198—41no—
189#11758198—41no—
190#11758198—41no—
191C4⋊Dic61—21no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—41no—
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244—42no—
199Dic244C2×D412yes—
200D488—21no—
201D488—21no—
202#11758190—42no—
203#11758190—81no—
204#11758185—21no—
205#11758185—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61C2×D811yescharacteristic
210C23×C61C2×D811yescharacteristic
211C22×D61—41no—
212Q8×C61—22no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—82no—
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—82no—
221D4×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—82no—
230C2×C4×C61—82no—
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C2×D811yescharacteristic
233C2×C4×C61C2×D811yescharacteristic
234C2×C4×C61C2×D811yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C22⋊D61—82no—
237C22⋊D61—82no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—82no—
242D244—41no—
243D244—41no—
244C4×D61—82no—
245Dic122—22no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—82no—
250C8×C61—21no—
251C8×C61—21no—
252C8×C61—21no—
253C8×C61C22×D411yescharacteristic
254C8×C61H(5,𝔽2)11yescharacteristic
255#33851—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61#50812yescomplemented
265C22×C61C22×D811yescharacteristic
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—42no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61D4211yescharacteristic
277C4×C61#57611yescharacteristic
278Dic61—42no—
279Dic61—81no—
280Dic61—81no—
281C2×#508—611no—
282C22×Q32—611no—
283#3285—611no—
284#3285D6111yescharacteristic, complemented
285#2249—611no—
286#2216—611no—
287#2215—611no—
288#2211—611no—
289#2211—611no—
290#2189—611no—
291#2189—611no—
292#2185—611no—
293#2185—611no—
294C4×Q32—611no—
295D4×C16—611no—
296C2×C61—22no—
297C2×C61—41no—
298C2×C61—161no—
299C2×C61D4×D811yescharacteristic
300C2×C61C2×#50811yescharacteristic
301C2×C61#341711yescharacteristic
302D61—161no—
303C2×#135—611no—
304C2×#135C2×D6111yescharacteristic
305C22×Q16—611no—
306C22×Q16C2×D6111yescharacteristic
307#548—21no—
308#548—1221no—
309#508—1224no—
310C2×Q32—611no—
311C2×Q32—611no—
312C2×Q32—611no—
313C2×Q32—611no—
314C2×Q32—1224no—
315C2×SD32—611no—
316C2×SD32—611no—
317C2×SD32—611no—
318C2×SD32—611no—
319C2×D16—611no—
320C22×C16—611no—
321C22×C16—611no—
322#495—611no—
323#495C2×D6111yescharacteristic
324#467—611no—
325#467C2×D6111yescharacteristic
326#466—611no—
327#466C2×D6111yescharacteristic
328#464—21no—
329#464—1221no—
330#452—21no—
331#452—1221no—
332#448—21no—
333#448—1221no—
334C4×Q16—611no—
335C4×Q16C2×D6111yescharacteristic
336C8×D4C2×D6111yescharacteristic
337#366—611no—
338C4⋊C16—611no—
339#358—611no—
340#358—611no—
341#358—611no—
342#358—611no—
343#357—611no—
344#357—611no—
345C22⋊C16—611no—
346C22⋊C16—611no—
347C4×C16—611no—
348C61#3385111yescharacteristic, complemented, Sylow
349C2×#41—21no—
350C2×#41—1221no—
351C22×Q8—21no—
352C22×Q8—1221no—
353#135—22no—
354#135—41no—
355#135—1222no—
356#135—2441no—
357C2×Q16—22no—
358C2×Q16—21no—
359C2×Q16—41no—
360C2×Q16—611no—
361C2×Q16—611no—
362C2×Q16—1222no—
363C2×Q16—1221no—
364C2×Q16—2441no—
365C2×Q16C22×D6111yescharacteristic
366C2×Q16C22×D6111yescharacteristic
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—611no—
372C2×D8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375#124—21no—
376#124—41no—
377#124—1221no—
378#124—2441no—
379C22⋊Q8—21no—
380C22⋊Q8—41no—
381C22⋊Q8—1221no—
382C22⋊Q8—2441no—
383#121—21no—
384#121—41no—
385#121—1221no—
386#121—2441no—
387C4×Q8—21no—
388C4×Q8—1221no—
389C4×D4—41no—
390C4×D4—2441no—
391C4×D4C22⋊D6111yescharacteristic
392Q32—614no—
393Q32—1222no—
394Q32—1222no—
395Q32—1222no—
396SD32—1222no—
397SD32—1222no—
398SD32—2441no—
399SD32—2441no—
400D16—1222no—
401C2×C16—611no—
402C2×C16—611no—
403C2×C16—611no—
404C2×C16—611no—
405C2×C16—1222no—
406C2×C16—1222no—
407#107—611no—
408#107C22×D6111yescharacteristic
409C4⋊C8C22⋊D6111yescharacteristic
410#103—21no—
411#103—21no—
412#103—1221no—
413#103—1221no—
414#102—21no—
415#102—1221no—
416C22⋊C8C22⋊D6111yescharacteristic
417C22⋊C8C22⋊D6111yescharacteristic
418C4×C8C22×D6111yescharacteristic
419#41—42no—
420#41—81no—
421#41—4881no—
422#41—2442no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—42no—
426C2×Q8—41no—
427C2×Q8—81no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—4881no—
431C2×Q8—2442no—
432C2×Q8—2441no—
433C2×D4—21no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—1221no—
437C2×D4—2441no—
438C2×D4—2441no—
439C2×D4#1175819911yescharacteristic
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446Q16—21no—
447Q16—21no—
448Q16—21no—
449Q16—42no—
450Q16—612no—
451Q16—1221no—
452Q16—1221no—
453Q16—1221no—
454Q16—2442no—
455Q16D4×D6112yes—
456SD16—42no—
457SD16—81no—
458SD16—4881no—
459SD16—2442no—
460D8—21no—
461D8—21no—
462D8—1221no—
463D8—1221no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8C2×C22⋊D6111yescharacteristic
469C2×C8C2×C22⋊D6111yescharacteristic
470C2×C8C2×C22⋊D6111yescharacteristic
471C2×C8C23×D6111yescharacteristic
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—1221no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4#1175819911yescharacteristic
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—2441no—
484C22⋊C4—2441no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C22⋊C4#1175819911yescharacteristic
488C22⋊C4#1175819911yescharacteristic
489C42—41no—
490C42—2441no—
491C42C2×C22⋊D6111yescharacteristic
492C16—612no—
493C16—1221no—
494C16—1221no—
495C16—1221no—
496C23—41no—
497C23—2441no—
498C23C2×#1175819911yescharacteristic
499C23C2×#1175819911yescharacteristic
500Q8—22no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—82no—
505Q8—1222no—
506Q8—4882no—
507Q8—2441no—
508Q8—2441no—
509Q8—2441no—
510D4—22no—
511D4—41no—
512D4—41no—
513D4—82no—
514D4—82no—
515D4—4882no—
516D4—4882no—
517D4—2441no—
518D4—2441no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—4882no—
529C2×C4—4882no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4C2×#1175819911yescharacteristic
534C2×C4C2×#1175819911yescharacteristic
535C2×C4C2×#1175819911yescharacteristic
536C2×C4C2×#1175819911yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C8—21no—
539C8—21no—
540C8—21no—
541C8C2×D4×D6111yescharacteristic
542C8#1451982211yescharacteristic
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22#1684019212yescomplemented
555C22C22×#1175819911yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—4881no—
564C4—4881no—
565C4—2442no—
566C4D4×C22⋊D6111yescharacteristic
567C4#1684056511yescharacteristic
568C2—22no—
569C2—41no—
570C2—161no—
571C2—9761no—
572C2D4×#1175819911yescharacteristic
573C2C2×#1684019211yescharacteristic
574C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3285C2
socleC22×C61C22×D8
hypercenter#3285D61
hypocenterC61#33851
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. C2×#16840192C2semidirect
2. C22×#14519723C2semidirect
3. #21457562C2semidirect
4. #21448078C2semidirect
5. #21447908C2semidirect
6. #21447903C2semidirect
7. #21447881C2semidirect
8. #21447878C2semidirect
9. #21447759C2semidirect
10. #21447757C2semidirect
11. #21447736C2semidirect
12. #21447735C2semidirect
13. C4×#14519723C2semidirect
14. D4×#11758185C2semidirect
15. C61×#3285C2semidirect
84. #14519723D4semidirect
264. C22×C61#508semidirect
284. #3285D61semidirect
348. C61#33851semidirect
554. C22#16840192semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
C22×#145197232
#214575622
#214480782
#214479082
#214479032
#214478812
#214478782
#214477592
#214477572
#214477362
#214477352
C4×#145197232
D4×#117581852
C61×#32852
#3385161