Finite Group Databasebeta

← #29836497

Subgroups — #29836497

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840202C211yesmaximal, characteristic, complemented
2C22×#14519721C211yesmaximal, characteristic, complemented
3D4×Dic244C211yesmaximal, characteristic, complemented
4#21448091C211yesmaximal, characteristic, complemented
5#21447930C211yesmaximal, characteristic, complemented
6#21447908C211yesmaximal, characteristic, complemented
7#21447878C211yesmaximal, characteristic, complemented
8#21447864C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447752C211yesmaximal, characteristic, complemented
11#21447748C211yesmaximal, characteristic, complemented
12#21447735C211yesmaximal, characteristic, complemented
13#21447683C211yesmaximal, characteristic, complemented
14#21447645C211yesmaximal, characteristic, complemented
15C61×#3280C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×Dic244C2211yescharacteristic, complemented
18D4×Dic122—21no—
19#16840202—24no—
20C2×#14519723C2211yescharacteristic, complemented
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519723C2211yescharacteristic, complemented
23C2×#14519721—24no—
24C2×#14519721C2211yescharacteristic
25C2×#14519721C2211yescharacteristic
26C2×#14519721C2211yescharacteristic
27C2×#14519721C2211yescharacteristic
28C2×#14519721C2211yescharacteristic
29C2×#14519721C2211yescharacteristic
30C2×#14519717C2211yescharacteristic, complemented
31C22×#11758185C2211yescharacteristic
32#16840142C2211yescharacteristic, complemented
33#16840003C2211yescharacteristic
34#16840003C2211yescharacteristic, complemented
35#16839973—21no—
36#16839920—21no—
37#16839920—21no—
38C4×Dic244C2211yescharacteristic, complemented
39#16839587C2211yescharacteristic, complemented
40C4⋊#11758185C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839565C2211yescharacteristic, complemented
43#16839565C2211yescharacteristic, complemented
44#16839565C2211yescharacteristic, complemented
45#16839563C2211yescharacteristic
46#16839563C2211yescharacteristic
47C22⋊#11758185C2211yescharacteristic
48C22⋊#11758185C2211yescharacteristic, complemented
49#16839538C2211yescharacteristic, complemented
50C2×#135×C61C2211yescharacteristic
51C22×D8×C61C2211yescharacteristic
52C61×#546—21no—
53C61×#495C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#463—21no—
57C61×#450—21no—
58C61×#449—21no—
59C4×D8×C61C2211yescharacteristic
60C8×D4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—21no—
63C2×Dic244—22no—
64C2×Dic244—22no—
65C2×Dic244—21no—
66C2×Dic244—21no—
67C2×Dic244—21no—
68C2×Dic244—41no—
69C2×Dic244—41no—
70C2×Dic244C2311yescharacteristic
71C2×Dic244C2311yescharacteristic
72C2×Dic244C2311yescharacteristic
73#14519777—21no—
74#14519776—41no—
75#14519756—21no—
76#14519756—21no—
77#14519754—41no—
78#14519754—41no—
79C4×Dic122—21no—
80D4×Dic61—41no—
81#14519723—22no—
82#14519723—22no—
83#14519723—41no—
84#14519721—22no—
85#14519721—22no—
86#14519721—22no—
87#14519721—22no—
88#14519721—41no—
89#14519721D414yescomplemented
90#14519717—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519702—21no—
100C2×#41×C61—21no—
101C22×D4×C61—21no—
102#135×C61—22no—
103#135×C61—41no—
104C2×Q16×C61C2311yescharacteristic
105C2×SD16×C61—21no—
106C2×SD16×C61—21no—
107C2×D8×C61—22no—
108C2×D8×C61—21no—
109C2×D8×C61—41no—
110C2×D8×C61C2311yescharacteristic
111C2×D8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114#124×C61—21no—
115#123×C61—41no—
116C22⋊Q8×C61—21no—
117#121×C61—21no—
118#121×C61—41no—
119#120×C61—41no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122C2×C22⋊C4×C61—41no—
123#107×C61C2311yescharacteristic
124C4⋊C8×C61D411yescharacteristic
125#103×C61—21no—
126#102×C61—21no—
127#102×C61—21no—
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C24×C61—41no—
132#41×C61—42no—
133#41×C61—81no—
134C2×Q8×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—42no—
138C2×D4×C61—41no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61—81no—
142C2×D4×C61D811yescharacteristic
143C22×C4×C61—41no—
144C22×C4×C61—41no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×Dic61—41no—
148C22×Dic61—41no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—41no—
155C2×Dic122—41no—
156C2×Dic122—81no—
157C2×Dic122—81no—
158Q16×C61—21no—
159Q16×C61—21no—
160SD16×C61—42no—
161SD16×C61—81no—
162D8×C61—21no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61—42no—
166D8×C61C2×D412yes—
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×C61D811yescharacteristic
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61D811yescharacteristic
184C22⋊C4×C61D811yescharacteristic
185C42×C61C2×D411yescharacteristic
186C22⋊Dic61—41no—
187C22⋊Dic61—41no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61—41no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—41no—
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244—21no—
199Dic244—21no—
200Dic244—42no—
201Dic244—42no—
202Dic244—81no—
203Dic244C2×D412yes—
204#11758185—21no—
205#11758185—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—82no—
211C23×C61—82no—
212C23×C61—161no—
213C23×C61C2×D811yescharacteristic
214C23×C61C2×D811yescharacteristic
215Q8×C61—41no—
216Q8×C61—41no—
217D4×C61—22no—
218D4×C61—22no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—82no—
223D4×C61—82no—
224D4×C61—161no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—82no—
232C2×C4×C61—161no—
233C2×C4×C61C2×D811yescharacteristic
234C2×C4×C61C2×D811yescharacteristic
235C2×C4×C61C2×D811yescharacteristic
236C2×C4×C61C2×D811yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—82no—
243C2×Dic61—82no—
244Dic122—22no—
245Dic122—41no—
246Dic122—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—82no—
251Dic122—82no—
252C8×C61—21no—
253C8×C61—21no—
254C8×C61—21no—
255C8×C61C22×D411yescharacteristic
256C8×C61H(5,𝔽2)11yescharacteristic
257#33832—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—42no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—164no—
270C22×C61#51012yescomplemented
271C22×C61C22×D811yescharacteristic
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61D4211yescharacteristic
279C4×C61#57611yescharacteristic
280Dic61—42no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284C2×#510—611no—
285C22×SD32—611no—
286D4×Q16—611no—
287#3280D6111yescharacteristic
288#2251—611no—
289#2220—611no—
290#2216—611no—
291#2211—611no—
292#2208—611no—
293#2189—611no—
294#2188—611no—
295#2187—611no—
296#2185—611no—
297#2175—611no—
298#2168—611no—
299C2×C61—22no—
300C2×C61—41no—
301C2×C61—82no—
302C2×C61—161no—
303C2×C61D4×D811yescharacteristic
304C2×C61C2×#51011yescharacteristic
305C2×C61#341611yescharacteristic
306C2×#135C2×D6111yescharacteristic
307C22×Q16—611no—
308C22×Q16—611no—
309C22×D8C2×D6111yescharacteristic
310D4×Q8—1221no—
311#546—21no—
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×SD32—611no—
321C2×SD32—611no—
322C2×SD32—1224no—
323C2×M32—611no—
324C22×C16—611no—
325#495C2×D6111yescharacteristic
326#494—611no—
327#467—611no—
328#467—611no—
329#467C2×D6111yescharacteristic
330#466C2×D6111yescharacteristic
331#463—21no—
332#462—1221no—
333#451—1221no—
334#451—1221no—
335#450—21no—
336#449—21no—
337C4×Q16—611no—
338C4×D8C2×D6111yescharacteristic
339C8×D4C2×D6111yescharacteristic
340#366—611no—
341C4⋊C16—611no—
342#358—611no—
343#358—611no—
344#358—611no—
345#358—611no—
346#357—611no—
347#357—611no—
348C22⋊C16—611no—
349C22⋊C16—611no—
350#346—611no—
351C61#3383211yescharacteristic, complemented, Sylow
352C2×#41—21no—
353C22×Q8—1221no—
354C22×Q8—1221no—
355C22×D4—21no—
356#135—22no—
357#135—41no—
358C2×Q16—611no—
359C2×Q16—611no—
360C2×Q16—611no—
361C2×Q16—1222no—
362C2×Q16—1222no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×Q16—2441no—
367C2×Q16—2441no—
368C2×Q16C22×D6111yescharacteristic
369C2×SD16—21no—
370C2×SD16—21no—
371C2×D8—22no—
372C2×D8—21no—
373C2×D8—41no—
374C2×D8C22×D6111yescharacteristic
375C2×D8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C4⋊Q8—1221no—
379C4⋊Q8—2441no—
380#124—21no—
381#123—41no—
382C22⋊Q8—21no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—2441no—
386C22⋊Q8—2441no—
387#121—21no—
388#121—41no—
389#120—41no—
390C4×Q8—1221no—
391C4×D4—21no—
392C4×D4—2441no—
393C4×D4C22⋊D6111yescharacteristic
394C2×C22⋊C4—41no—
395Q32—1222no—
396Q32—1222no—
397Q32—2441no—
398SD32—614no—
399SD32—1222no—
400SD32—1222no—
401SD32—1222no—
402SD32—1222no—
403SD32—2441no—
404M32—1222no—
405C2×C16—611no—
406C2×C16—611no—
407C2×C16—611no—
408C2×C16—611no—
409C2×C16—1222no—
410#107—611no—
411#107C22×D6111yescharacteristic
412C4⋊C8C22⋊D6111yescharacteristic
413#103—21no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—21no—
418#102—21no—
419C22⋊C8C22⋊D6111yescharacteristic
420C22⋊C8C22⋊D6111yescharacteristic
421C4×C8C22×D6111yescharacteristic
422C24—41no—
423#41—42no—
424#41—81no—
425C2×Q8—21no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—4881no—
430C2×Q8—4881no—
431C2×Q8—2442no—
432C2×Q8—2442no—
433C2×Q8—2441no—
434C2×Q8—2441no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—81no—
442C2×D4#1175819911yescharacteristic
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—2441no—
446C22×C4—2441no—
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449Q16—21no—
450Q16—21no—
451Q16—612no—
452Q16—1221no—
453Q16—1221no—
454Q16—1221no—
455Q16—1221no—
456Q16—1221no—
457Q16—4881no—
458Q16—2442no—
459Q16—2442no—
460SD16—42no—
461SD16—81no—
462D8—21no—
463D8—21no—
464D8—21no—
465D8—42no—
466D8D4×D6112yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8C2×C22⋊D6111yescharacteristic
472C2×C8C2×C22⋊D6111yescharacteristic
473C2×C8C2×C22⋊D6111yescharacteristic
474C2×C8C23×D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—1221no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4#1175819911yescharacteristic
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4#1175819911yescharacteristic
492C22⋊C4#1175819911yescharacteristic
493C42—2441no—
494C42C2×C22⋊D6111yescharacteristic
495C16—612no—
496C16—1221no—
497C16—1221no—
498C16—1221no—
499C23—41no—
500C23—41no—
501C23—82no—
502C23—82no—
503C23—161no—
504C23C2×#1175819911yescharacteristic
505C23C2×#1175819911yescharacteristic
506Q8—41no—
507Q8—41no—
508Q8—1222no—
509Q8—4882no—
510Q8—4882no—
511Q8—2441no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515Q8—2441no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—82no—
522D4—82no—
523D4—161no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—82no—
531C2×C4—161no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4C2×#1175819911yescharacteristic
539C2×C4C2×#1175819911yescharacteristic
540C2×C4C2×#1175819911yescharacteristic
541C2×C4C2×#1175819911yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D6111yescharacteristic
547C8#1451982211yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—164no—
560C22#1684020212yescomplemented
561C22C22×#1175819911yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—81no—
567C4—81no—
568C4—4881no—
569C4—4881no—
570C4—4881no—
571C4—2442no—
572C4D4×C22⋊D6111yescharacteristic
573C4#1684056511yescharacteristic
574C2—22no—
575C2—41no—
576C2—82no—
577C2—161no—
578C2D4×#1175819911yescharacteristic
579C2C2×#1684020211yescharacteristic
580C2#2145896611yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#3280C2
socleC22×C61C22×D8
hypercenter#3280D61
hypocenterC61#33832
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
2. C22×#14519721C2semidirect
3. D4×Dic244C2semidirect
4. #21448091C2semidirect
5. #21447930C2semidirect
6. #21447908C2semidirect
7. #21447878C2semidirect
8. #21447864C2semidirect
9. #21447759C2semidirect
10. #21447752C2semidirect
11. #21447748C2semidirect
12. #21447735C2semidirect
13. #21447683C2semidirect
14. #21447645C2semidirect
17. C22×Dic244C22semidirect
20. C2×#14519723C22semidirect
21. C2×#14519723C22semidirect
22. C2×#14519723C22semidirect
30. C2×#14519717C22semidirect
32. #16840142C22semidirect
34. #16840003C22semidirect
38. C4×Dic244C22semidirect
39. #16839587C22semidirect
40. C4⋊#11758185C22semidirect
41. #16839566C22semidirect
42. #16839565C22semidirect
43. #16839565C22semidirect
44. #16839565C22semidirect
48. C22⋊#11758185C22semidirect
49. #16839538C22semidirect
89. #14519721D4semidirect
270. C22×C61#510semidirect
351. C61#33832semidirect
560. C22#16840202semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C22×#145197212
D4×Dic2442
#214480912
#214479302
#214479082
#214478782
#214478642
#214477592
#214477522
#214477482
#214477352
#214476832
#214476452
C61×#32802
#3383261