Finite Group Databasebeta

← #29836655

Subgroups — #29836655

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840203C211yesmaximal, characteristic, complemented
2C2×#16840201C211yesmaximal, characteristic, complemented
3#21457576C211yesmaximal, characteristic
4Q16×C22⋊D61C211yesmaximal, characteristic, complemented
5#21448073C211yesmaximal, characteristic, complemented
6#21447935C211yesmaximal, characteristic, complemented
7#21447930C211yesmaximal, characteristic, complemented
8#21447868C211yesmaximal, characteristic, complemented
9#21447857C211yesmaximal, characteristic, complemented
10#21447761C211yesmaximal, characteristic, complemented
11#21447756C211yesmaximal, characteristic, complemented
12#21447737C211yesmaximal, characteristic, complemented
13#21447675C211yesmaximal, characteristic, complemented
14#21447648C211yesmaximal, characteristic, complemented
15C61×#2185C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519796C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic
19#16840430—21no—
20Q8×C22⋊D61—21no—
21#16840203—24no—
22#16840201—24no—
23C2×Dic488C2211yescharacteristic
24C2×#14519723C2211yescharacteristic
25C2×#14519723C2211yescharacteristic
26C2×#14519722C2211yescharacteristic
27C2×#14519721C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×#14519720C2211yescharacteristic
30C2×#14519717C2211yescharacteristic
31C2×#14519716C2211yescharacteristic
32#16840140C2211yescharacteristic
33#16840137C2211yescharacteristic
34#16840003C2211yescharacteristic
35#16840000C2211yescharacteristic
36#16839999C2211yescharacteristic
37#16839970—21no—
38#16839961—21no—
39#16839918—21no—
40#16839916—21no—
41#16839908—21no—
42#16839905—21no—
43Q16×Dic61C2211yescharacteristic
44#16839856C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839565C2211yescharacteristic
50#16839564C2211yescharacteristic
51#16839563C2211yescharacteristic
52#16839541C2211yescharacteristic
53C16×Dic61C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic
56C2×SD32×C61C2211yescharacteristic
57C61×#466C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C61×#358C2211yescharacteristic
60C61×C22⋊C16C2211yescharacteristic
61C2×#11758222—21no—
62C2×#11758224—21no—
63C2×Q8×D61—21no—
64#14519799—22no—
65#14519799—41no—
66#14519796—22no—
67#14519796—41no—
68C2×Dic244C2311yescharacteristic
69C2×Dic244C2311yescharacteristic
70C2×#11758202—21no—
71C2×#11758202—21no—
72Q16×D61—22no—
73Q16×D61—41no—
74C2×#11758190—21no—
75C2×#11758200—21no—
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78C2×C8×D61C2311yescharacteristic
79#14519775—21no—
80#14519775—41no—
81#14519773—21no—
82#14519773—41no—
83#14519756—21no—
84#14519756—41no—
85#14519753—21no—
86#14519753—41no—
87#14519752—21no—
88#14519750—41no—
89#14519749—41no—
90Q8×Dic61—21no—
91#14519742—21no—
92C4×C22⋊D61—41no—
93C4×C22⋊D61D411yescharacteristic
94#14519737—41no—
95Dic488—22no—
96#14519723—22no—
97#14519723D414yescomplemented
98#14519722—41no—
99#14519721—22no—
100#14519720—22no—
101#14519720—22no—
102#14519717—22no—
103#14519716—22no—
104C2×#11758185C2311yescharacteristic
105C2×#11758185C2311yescharacteristic
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519702—21no—
109#14519701—21no—
110#14519701—21no—
111#14519700—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×Q16×C61—22no—
118C2×Q16×C61—21no—
119C2×Q16×C61—41no—
120C2×Q16×C61C2311yescharacteristic
121C2×Q16×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124#121×C61—21no—
125SD32×C61—22no—
126SD32×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#102×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—42no—
134C2×Q8×C61—41no—
135C2×Q8×C61—81no—
136C2×D4×C61—21no—
137C2×D4×C61—41no—
138C22×C4×C61—41no—
139C22×C4×C61C2×D411yescharacteristic
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61D811yescharacteristic
145C22×Dic61—41no—
146Q8×D61—42no—
147Q8×D61—81no—
148#11758224—42no—
149#11758224—81no—
150#11758222—42no—
151#11758222—81no—
152C2×D244—21no—
153C2×D244—41no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—41no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—21no—
163Q16×C61—42no—
164Q16×C61C2×D412yes—
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—
172C22⋊C4×C61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61D811yescharacteristic
175#11758202—42no—
176#11758202—81no—
177#11758200—42no—
178#11758199—81no—
179#11758198—41no—
180#11758198—41no—
181#11758198D811yescharacteristic
182C4⋊Dic61—21no—
183C4⋊Dic61—41no—
184#11758196—41no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189#11758196D811yescharacteristic
190C4×Dic61—41no—
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244C2×D412yes—
199D488—21no—
200D488—21no—
201#11758190—42no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61C2×D811yescharacteristic
210C22×D61—41no—
211C22×D61C2×D811yescharacteristic
212Q8×C61—22no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—82no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—161no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—82no—
226C2×C4×C61C2×D811yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C22⋊D61—22no—
229C22⋊D61—82no—
230C22⋊D61—82no—
231C22⋊D61—82no—
232C22⋊D61—161no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—82no—
237C2×Dic61C2×D811yescharacteristic
238C2×Dic61C2×D811yescharacteristic
239C2×Dic61C2×D811yescharacteristic
240D244—41no—
241D244—41no—
242D244—161no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—82no—
246C4×D61—82no—
247Dic122—22no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
251Dic122—82no—
252C8×C61—21no—
253C8×C61C22×D411yescharacteristic
254C8×C61H(5,𝔽2)11yescharacteristic
255#92418—21no—
256#92418—21no—
257#33844—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61C22×D811yescharacteristic
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—21no—
272C4×C61—42no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61D4211yescharacteristic
276C4×C61#57611yescharacteristic
277Dic61—21no—
278Dic61—21no—
279Dic61—42no—
280Dic61—81no—
281Dic61—81no—
282Dic61#51012yescomplemented
283C2×#510—611no—
284C2×#510—611no—
285#3287—611no—
286D4×Q16—611no—
287#2248—611no—
288#2220—611no—
289#2220—611no—
290#2209—611no—
291#2207—611no—
292#2189—611no—
293#2189—611no—
294#2185—611no—
295#2185D6111yescharacteristic, complemented
296C4×Q32—611no—
297#2169—611no—
298C2×C61—41no—
299C2×C61—161no—
300C2×C61D4×D811yescharacteristic
301C2×C61C2×#51011yescharacteristic
302C2×C61#341611yescharacteristic
303D61—41no—
304D61—161no—
305C2×#135—611no—
306C2×#135—611no—
307C22×Q16—611no—
308C22×Q16C2×D6111yescharacteristic
309#550—1221no—
310D4×Q8—1221no—
311#510—1224no—
312#510—1224no—
313C2×Q32—611no—
314C2×Q32—611no—
315C2×Q32—611no—
316C2×SD32—611no—
317C2×SD32—611no—
318C2×SD32—611no—
319C2×SD32—611no—
320C2×SD32C2×D6111yescharacteristic
321C2×SD32C2×D6111yescharacteristic
322C2×M32—611no—
323C2×M32—611no—
324#494—611no—
325#493—611no—
326#467—611no—
327#467—611no—
328#466—611no—
329#466C2×D6111yescharacteristic
330#462—1221no—
331#460—1221no—
332#451—1221no—
333#451—1221no—
334#449—1221no—
335#449—1221no—
336C4×Q16—611no—
337C4×Q16—611no—
338C8×D4—611no—
339#366—611no—
340C4⋊C16—611no—
341#358—611no—
342#358—611no—
343#358C2×D6111yescharacteristic
344#358C2×D6111yescharacteristic
345#357—611no—
346#357—611no—
347C22⋊C16—611no—
348C22⋊C16C2×D6111yescharacteristic
349C4×C16—611no—
350C61#3384411yescharacteristic, complemented, Sylow
351C2×#41—1221no—
352C2×#41—1221no—
353C22×Q8—21no—
354C22×Q8—1221no—
355#135—1222no—
356#135—1222no—
357#135—2441no—
358#135—2441no—
359C2×Q16—22no—
360C2×Q16—21no—
361C2×Q16—41no—
362C2×Q16—611no—
363C2×Q16—611no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×Q16—1222no—
367C2×Q16—2441no—
368C2×Q16C22×D6111yescharacteristic
369C2×Q16C22×D6111yescharacteristic
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×D8—611no—
373C2×D8—1221no—
374C2×D8C22×D6111yescharacteristic
375C22×C8—611no—
376C22×C8C22×D6111yescharacteristic
377C4⋊Q8—1221no—
378C4⋊Q8—2441no—
379#127—1221no—
380#127—2441no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—2441no—
384C22⋊Q8—2441no—
385#121—21no—
386#121—1221no—
387#121—2441no—
388#121—2441no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—611no—
392C4×D4—2441no—
393C4×D4—2441no—
394Q32—614no—
395Q32—1222no—
396Q32—1222no—
397SD32—22no—
398SD32—41no—
399SD32—1222no—
400SD32—1222no—
401SD32—1222no—
402SD32—2441no—
403M32—1222no—
404M32—1222no—
405C2×C16—611no—
406C2×C16—611no—
407C2×C16C22×D6111yescharacteristic
408C2×C16C22×D6111yescharacteristic
409#107—611no—
410#107C22×D6111yescharacteristic
411C4⋊C8—611no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#102—21no—
417#102—1221no—
418C22⋊C8—611no—
419C22⋊C8—611no—
420C4×C8—611no—
421#41—4881no—
422#41—4881no—
423#41—2442no—
424#41—2442no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—42no—
428C2×Q8—41no—
429C2×Q8—81no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—4881no—
433C2×Q8—2441no—
434C2×Q8—2442no—
435C2×D4—21no—
436C2×D4—41no—
437C2×D4—611no—
438C2×D4—1221no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
442C2×D4—2441no—
443C2×D4—2441no—
444C22×C4—41no—
445C22×C4—611no—
446C22×C4—2441no—
447C22×C4—2441no—
448C22×C4—2441no—
449C22×C4D4×D6111yescharacteristic
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—42no—
454Q16—612no—
455Q16—1221no—
456Q16—1221no—
457Q16—1221no—
458Q16—4881no—
459Q16—2442no—
460Q16D4×D6112yes—
461SD16—2442no—
462SD16—2442no—
463D8—21no—
464D8—21no—
465D8—1221no—
466D8—1221no—
467D8—4881no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8D4×D6111yescharacteristic
475C2×C8C23×D6111yescharacteristic
476C4⋊C4—21no—
477C4⋊C4—611no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C4⋊C4—2441no—
485C22⋊C4—41no—
486C22⋊C4—611no—
487C22⋊C4—611no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C42—611no—
492C42—2441no—
493C42—2441no—
494C16—21no—
495C16—21no—
496C16—612no—
497C16—1221no—
498C23—41no—
499C23—611no—
500C23—2441no—
501C23D8×D6111yescharacteristic
502Q8—22no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—82no—
507Q8—1222no—
508Q8—4882no—
509Q8—2441no—
510Q8—2441no—
511Q8—2441no—
512D4—41no—
513D4—41no—
514D4—161no—
515D4—1222no—
516D4—9761no—
517D4—9761no—
518D4—4882no—
519D4—4882no—
520D4—4882no—
521D4—2441no—
522D4—2441no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4D8×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C8—21no—
543C8—1221no—
544C8—1221no—
545C8C2×D4×D6111yescharacteristic
546C8#1451982211yescharacteristic
547C22—21no—
548C22—21no—
549C22—21no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—1221no—
554C22—1221no—
555C22—1221no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22C2×D8×D6111yescharacteristic
560C4—21no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4—612no—
565C4—1221no—
566C4—1221no—
567C4—4881no—
568C4—4881no—
569C4—2442no—
570C4#1684039411yescharacteristic
571C4#1684056611yescharacteristic
572C2—41no—
573C2—161no—
574C2—9761no—
575C2—2441no—
576C2#2145748911yescharacteristic
577C2#510×D6111yescharacteristic
578C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2185C2
socleC22×C61C22×D8
prime radical#21457576C2
hypercenter#2185D61
hypocenterC61#33844
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2185

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C2×#16840201C2semidirect
4. Q16×C22⋊D61C2semidirect
5. #21448073C2semidirect
6. #21447935C2semidirect
7. #21447930C2semidirect
8. #21447868C2semidirect
9. #21447857C2semidirect
10. #21447761C2semidirect
11. #21447756C2semidirect
12. #21447737C2semidirect
13. #21447675C2semidirect
14. #21447648C2semidirect
15. C61×#2185C2semidirect
97. #14519723D4semidirect
282. Dic61#510semidirect
295. #2185D61semidirect
350. C61#33844semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C2×#168402012
#214575762
Q16×C22⋊D612
#214480732
#214479352
#214479302
#214478682
#214478572
#214477612
#214477562
#214477372
#214476752
#214476482
C61×#21852
#3384461