Finite Group Databasebeta

← #29836562

Subgroups — #29836562

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840203C211yesmaximal, characteristic, complemented
2C22×#14519723C211yesmaximal, characteristic, complemented
3#21457562C211yesmaximal, characteristic
4#21448088C211yesmaximal, characteristic, complemented
5#21447925C211yesmaximal, characteristic, complemented
6#21447917C211yesmaximal, characteristic, complemented
7#21447881C211yesmaximal, characteristic, complemented
8#21447866C211yesmaximal, characteristic, complemented
9#21447757C211yesmaximal, characteristic, complemented
10#21447753C211yesmaximal, characteristic, complemented
11#21447752C211yesmaximal, characteristic, complemented
12#21447736C211yesmaximal, characteristic, complemented
13#21447688C211yesmaximal, characteristic, complemented
14#21447645C211yesmaximal, characteristic, complemented
15D4×Q16×C61C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18#16840421—21no—
19#16840203—24no—
20C2×#14519723—24no—
21C2×#14519723C2211yescharacteristic
22C2×#14519723C2211yescharacteristic
23C2×#14519723C2211yescharacteristic
24C2×#14519723C2211yescharacteristic
25C2×#14519723C2211yescharacteristic
26C2×#14519723C2211yescharacteristic
27C2×#14519720C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×#14519720C2211yescharacteristic
30C2×#14519717C2211yescharacteristic
31C22×#11758185C2211yescharacteristic
32#16840146C2211yescharacteristic
33#16840003C2211yescharacteristic
34#16839998C2211yescharacteristic
35#16839985—21no—
36#16839926—21no—
37#16839903—21no—
38C4×Dic244C2211yescharacteristic
39#16839589C2211yescharacteristic
40C4⋊#11758185C2211yescharacteristic
41#16839566C2211yescharacteristic
42#16839566C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839565C2211yescharacteristic
45#16839565C2211yescharacteristic
46#16839562C2211yescharacteristic
47C22⋊#11758185C2211yescharacteristic
48C22⋊#11758185C2211yescharacteristic
49#16839538C2211yescharacteristic
50C22×Q16×C61C2211yescharacteristic
51C22×Q16×C61C2211yescharacteristic
52D4×Q8×C61—21no—
53C61×#494C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#462—21no—
57C61×#451—21no—
58C61×#451—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#14519723—22no—
85#14519723—41no—
86#14519723D414yescomplemented
87#14519720—22no—
88#14519720—22no—
89#14519720—41no—
90#14519717—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519699—21no—
100C22×Q8×C61—21no—
101C22×Q8×C61—21no—
102C2×Q16×C61—22no—
103C2×Q16×C61—22no—
104C2×Q16×C61—21no—
105C2×Q16×C61—21no—
106C2×Q16×C61—21no—
107C2×Q16×C61—41no—
108C2×Q16×C61—41no—
109C2×Q16×C61C2311yescharacteristic
110C2×Q16×C61C2311yescharacteristic
111C2×Q16×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4⋊Q8×C61—21no—
115C4⋊Q8×C61—41no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—21no—
118C22⋊Q8×C61—41no—
119C22⋊Q8×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#103×C61—21no—
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—42no—
135C2×Q8×C61—42no—
136C2×Q8×C61—41no—
137C2×Q8×C61—41no—
138C2×Q8×C61—81no—
139C2×Q8×C61—81no—
140C2×D4×C61D811yescharacteristic
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C22×Dic61—41no—
148#11758222—42no—
149#11758222—81no—
150C2×D244—21no—
151C2×C4×D61—41no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—42no—
155C2×Dic122—41no—
156C2×Dic122—81no—
157Q16×C61—21no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—42no—
163Q16×C61—42no—
164Q16×C61—81no—
165Q16×C61C2×D412yes—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61D811yescharacteristic
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×C61C2×D811yescharacteristic
209C23×C61C2×D811yescharacteristic
210C22×D61—41no—
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—82no—
218Q8×C61—82no—
219D4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61—82no—
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C2×D811yescharacteristic
233C2×C4×C61C2×D811yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C22⋊D61—82no—
236C22⋊D61—82no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241D244—41no—
242D244—41no—
243C4×D61—82no—
244Dic122—22no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—82no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—21no—
252C8×C61C22×D411yescharacteristic
253C8×C61H(5,𝔽2)11yescharacteristic
254#33836—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61#51012yescomplemented
261C22×C61C22×D811yescharacteristic
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—42no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61D4211yescharacteristic
274C4×C61#57611yescharacteristic
275Dic61—42no—
276Dic61—81no—
277Dic61—81no—
278C2×#510—611no—
279C22×Q32—611no—
280D4×Q16D6111yescharacteristic, complemented
281#3285—611no—
282#2251—611no—
283#2219—611no—
284#2217—611no—
285#2211—611no—
286#2209—611no—
287#2189—611no—
288#2188—611no—
289#2188—611no—
290#2185—611no—
291#2176—611no—
292#2168—611no—
293C2×C61—22no—
294C2×C61—41no—
295C2×C61D4×D811yescharacteristic
296C2×C61C2×#51011yescharacteristic
297C2×C61#341711yescharacteristic
298D61—161no—
299C2×#135—611no—
300C22×Q16—611no—
301C22×Q16C2×D6111yescharacteristic
302C22×Q16C2×D6111yescharacteristic
303D4×Q8—21no—
304#548—1221no—
305#510—1224no—
306C2×Q32—611no—
307C2×Q32—611no—
308C2×Q32—611no—
309C2×Q32—611no—
310C2×Q32—611no—
311C2×Q32—611no—
312C2×Q32—1224no—
313C2×SD32—611no—
314C2×SD32—611no—
315C2×SD32—611no—
316C2×M32—611no—
317C22×C16—611no—
318#495—611no—
319#494C2×D6111yescharacteristic
320#467—611no—
321#467C2×D6111yescharacteristic
322#467C2×D6111yescharacteristic
323#466—611no—
324#464—1221no—
325#462—21no—
326#452—1221no—
327#451—21no—
328#451—21no—
329#448—1221no—
330C4×Q16—611no—
331C4×Q16C2×D6111yescharacteristic
332C8×D4C2×D6111yescharacteristic
333#367—611no—
334C4⋊C16—611no—
335#358—611no—
336#358—611no—
337#358—611no—
338#358—611no—
339#358—611no—
340#357—611no—
341C22⋊C16—611no—
342C22⋊C16—611no—
343#346—611no—
344C61#3383611yescharacteristic, complemented, Sylow
345C2×#41—1221no—
346C22×Q8—21no—
347C22×Q8—21no—
348C22×Q8—1221no—
349#135—1222no—
350#135—2441no—
351C2×Q16—22no—
352C2×Q16—22no—
353C2×Q16—21no—
354C2×Q16—21no—
355C2×Q16—21no—
356C2×Q16—41no—
357C2×Q16—41no—
358C2×Q16—611no—
359C2×Q16—611no—
360C2×Q16—1222no—
361C2×Q16—1221no—
362C2×Q16—2441no—
363C2×Q16C22×D6111yescharacteristic
364C2×Q16C22×D6111yescharacteristic
365C2×Q16C22×D6111yescharacteristic
366C2×SD16—1221no—
367C2×SD16—1221no—
368C2×D8—611no—
369C22×C8C22×D6111yescharacteristic
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—21no—
372C4⋊Q8—41no—
373#124—1221no—
374#124—2441no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—41no—
378C22⋊Q8—41no—
379C22⋊Q8—1221no—
380C22⋊Q8—2441no—
381#121—1221no—
382#121—2441no—
383C4×Q8—21no—
384C4×Q8—1221no—
385C4×D4—41no—
386C4×D4—2441no—
387C4×D4C22⋊D6111yescharacteristic
388Q32—614no—
389Q32—1222no—
390Q32—1222no—
391Q32—1222no—
392Q32—1222no—
393Q32—2441no—
394SD32—1222no—
395SD32—1222no—
396SD32—2441no—
397M32—1222no—
398C2×C16—611no—
399C2×C16—611no—
400C2×C16—611no—
401C2×C16—611no—
402C2×C16—1222no—
403#107—611no—
404#107C22×D6111yescharacteristic
405C4⋊C8C22⋊D6111yescharacteristic
406#103—21no—
407#103—21no—
408#103—21no—
409#103—1221no—
410#103—1221no—
411#102—1221no—
412C22⋊C8C22⋊D6111yescharacteristic
413C22⋊C8C22⋊D6111yescharacteristic
414C4×C8C22×D6111yescharacteristic
415#41—4881no—
416#41—2442no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—41no—
423C2×Q8—41no—
424C2×Q8—81no—
425C2×Q8—81no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—4881no—
429C2×Q8—2442no—
430C2×Q8—2441no—
431C2×D4—1221no—
432C2×D4—2441no—
433C2×D4—2441no—
434C2×D4#1175819911yescharacteristic
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—2441no—
438C22×C4—2441no—
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441Q16—21no—
442Q16—21no—
443Q16—21no—
444Q16—21no—
445Q16—21no—
446Q16—42no—
447Q16—42no—
448Q16—81no—
449Q16—612no—
450Q16—1221no—
451Q16—1221no—
452Q16—1221no—
453Q16—2442no—
454Q16D4×D6112yes—
455SD16—4881no—
456SD16—2442no—
457D8—1221no—
458D8—1221no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8C2×C22⋊D6111yescharacteristic
464C2×C8C2×C22⋊D6111yescharacteristic
465C2×C8C2×C22⋊D6111yescharacteristic
466C2×C8C23×D6111yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—1221no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4#1175819911yescharacteristic
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—2441no—
480C22⋊C4—2441no—
481C22⋊C4—2441no—
482C22⋊C4—2441no—
483C22⋊C4#1175819911yescharacteristic
484C22⋊C4#1175819911yescharacteristic
485C42—41no—
486C42—2441no—
487C42C2×C22⋊D6111yescharacteristic
488C16—612no—
489C16—1221no—
490C16—1221no—
491C16—1221no—
492C23—2441no—
493C23C2×#1175819911yescharacteristic
494C23C2×#1175819911yescharacteristic
495Q8—22no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—82no—
503Q8—1222no—
504Q8—4882no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508D4—22no—
509D4—4882no—
510D4—4882no—
511D4—2441no—
512D4—2441no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—4882no—
524C2×C4—4882no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4C2×#1175819911yescharacteristic
529C2×C4C2×#1175819911yescharacteristic
530C2×C4C2×#1175819911yescharacteristic
531C2×C4C2×#1175819911yescharacteristic
532C2×C4C22×C22⋊D6111yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8C2×D4×D6111yescharacteristic
537C8#1451982211yescharacteristic
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22#1684020312yescomplemented
547C22C22×#1175819911yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—4881no—
557C4—4881no—
558C4—2442no—
559C4D4×C22⋊D6111yescharacteristic
560C4#1684056511yescharacteristic
561C2—22no—
562C2—41no—
563C2—9761no—
564C2D4×#1175819911yescharacteristic
565C2C2×#1684020311yescharacteristic
566C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingD4×Q16×C61C2
socleC22×C61C22×D8
prime radical#21457562C2
hypercenterD4×Q16D61
hypocenterC61#33836
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C22×#14519723C2semidirect
4. #21448088C2semidirect
5. #21447925C2semidirect
6. #21447917C2semidirect
7. #21447881C2semidirect
8. #21447866C2semidirect
9. #21447757C2semidirect
10. #21447753C2semidirect
11. #21447752C2semidirect
12. #21447736C2semidirect
13. #21447688C2semidirect
14. #21447645C2semidirect
15. D4×Q16×C61C2semidirect
86. #14519723D4semidirect
260. C22×C61#510semidirect
280. D4×Q16D61semidirect
344. C61#33836semidirect
546. C22#16840203semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C22×#145197232
#214575622
#214480882
#214479252
#214479172
#214478812
#214478662
#214477572
#214477532
#214477522
#214477362
#214476882
#214476452
D4×Q16×C612
#3383661