Finite Group Databasebeta

← #29836502

Subgroups — #29836502

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×#16840203C211yesmaximal, characteristic, complemented
2C22×#14519720C211yesmaximal, characteristic, complemented
3#21457506C211yesmaximal, characteristic
4#21448088C211yesmaximal, characteristic, complemented
5#21447934C211yesmaximal, characteristic, complemented
6#21447908C211yesmaximal, characteristic, complemented
7#21447881C211yesmaximal, characteristic, complemented
8#21447861C211yesmaximal, characteristic, complemented
9#21447757C211yesmaximal, characteristic, complemented
10#21447753C211yesmaximal, characteristic, complemented
11#21447747C211yesmaximal, characteristic, complemented
12#21447736C211yesmaximal, characteristic, complemented
13#21447681C211yesmaximal, characteristic, complemented
14#21447645C211yesmaximal, characteristic, complemented
15D4×Q16×C61C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C22×D488C2211yescharacteristic
18#16840408—21no—
19#16840203—24no—
20C2×#14519723C2211yescharacteristic, complemented
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519723C2211yescharacteristic, complemented
23C2×#14519720—24no—
24C2×#14519720C2211yescharacteristic
25C2×#14519720C2211yescharacteristic
26C2×#14519720C2211yescharacteristic
27C2×#14519720C2211yescharacteristic
28C2×#14519720C2211yescharacteristic
29C2×#14519720C2211yescharacteristic
30C2×#14519717C2211yescharacteristic, complemented
31C22×#11758185C2211yescharacteristic
32#16840146C2211yescharacteristic
33#16840003C2211yescharacteristic
34#16839998C2211yescharacteristic
35#16839978—21no—
36#16839915—21no—
37#16839909—21no—
38C4×D488C2211yescharacteristic
39#16839587C2211yescharacteristic, complemented
40C4⋊#11758185C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839566C2211yescharacteristic, complemented
44#16839565C2211yescharacteristic, complemented
45#16839562C2211yescharacteristic
46#16839562C2211yescharacteristic
47C22⋊#11758185C2211yescharacteristic
48C22⋊#11758185C2211yescharacteristic, complemented
49#16839538C2211yescharacteristic, complemented
50C22×Q16×C61C2211yescharacteristic
51C22×Q16×C61C2211yescharacteristic, complemented
52D4×Q8×C61—21no—
53C61×#494C2211yescharacteristic, complemented
54C61×#467C2211yescharacteristic
55C61×#467C2211yescharacteristic, complemented
56C61×#462—21no—
57C61×#451—21no—
58C61×#451—21no—
59C4×Q16×C61C2211yescharacteristic, complemented
60C8×D4×C61C2211yescharacteristic
61C2×#11758222—21no—
62C22×D244—21no—
63#14519799—22no—
64#14519799—41no—
65C2×Dic244C2311yescharacteristic
66C2×#11758190—21no—
67C2×#11758190—21no—
68C2×D488—22no—
69C2×D488—21no—
70C2×D488—41no—
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73#14519765—21no—
74#14519762—41no—
75#14519756—21no—
76#14519751—21no—
77#14519750—41no—
78#14519745—41no—
79C4×D244—21no—
80C22⋊C4×D61—41no—
81#14519723—22no—
82#14519723—22no—
83#14519723—41no—
84#14519720—22no—
85#14519720—22no—
86#14519720—22no—
87#14519720—22no—
88#14519720—41no—
89#14519720D414yescomplemented
90#14519717—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519702—21no—
98#14519699—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
131C23×D61—41no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—42no—
136C2×Q8×C61—42no—
137C2×Q8×C61—41no—
138C2×Q8×C61—41no—
139C2×Q8×C61—81no—
140C2×Q8×C61—81no—
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—
148#11758222—42no—
149#11758222—81no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—42no—
153C2×D244—41no—
154C2×D244—81no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×Dic122—21no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162Q16×C61—21no—
163Q16×C61—42no—
164Q16×C61—42no—
165Q16×C61—81no—
166Q16×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×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61D811yescharacteristic
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61D811yescharacteristic
185C22⋊C4×C61D811yescharacteristic
186C42×C61—41no—
187C42×C61C2×D411yescharacteristic
188C22⋊Dic61—41no—
189#11758198—41no—
190#11758198—41no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—21no—
194#11758196—41no—
195Dic244—21no—
196Dic244—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—42no—
201D488C2×D412yes—
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—
211C22×D61—41no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—161no—
215Q8×C61—22no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—82no—
222Q8×C61—82no—
223D4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—82no—
233C2×C4×C61—82no—
234C2×C4×C61C2×D811yescharacteristic
235C2×C4×C61C2×D811yescharacteristic
236C2×C4×C61C2×D811yescharacteristic
237C2×C4×C61C2×D811yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—82no—
240C22⋊D61—161no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243D244—22no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—82no—
248C4×D61—82no—
249C4×D61—161no—
250Dic122—41no—
251Dic122—41no—
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#51012yescomplemented
264C22×C61C22×D811yescharacteristic
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—164no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—42no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61D4211yescharacteristic
281C4×C61#57611yescharacteristic
282Dic61—81no—
283Dic61—81no—
284C2×#510—611no—
285C22×SD32—611no—
286D4×Q16D6111yescharacteristic, complemented
287#3280—611no—
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×C61D4×D811yescharacteristic
302C2×C61C2×#51011yescharacteristic
303C2×C61#341611yescharacteristic
304D61—82no—
305D61—161no—
306C2×#135—611no—
307C22×Q16C2×D6111yescharacteristic
308C22×Q16C2×D6111yescharacteristic, complemented
309C22×D8—611no—
310D4×Q8—21no—
311#546—1221no—
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#495—611no—
326#494C2×D6111yescharacteristic, complemented
327#467—611no—
328#467C2×D6111yescharacteristic
329#467C2×D6111yescharacteristic, complemented
330#466—611no—
331#463—1221no—
332#462—21no—
333#451—21no—
334#451—21no—
335#450—1221no—
336#449—1221no—
337C4×Q16C2×D6111yescharacteristic, complemented
338C4×D8—611no—
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—1221no—
353C22×Q8—21no—
354C22×Q8—21no—
355C22×D4—1221no—
356#135—1222no—
357#135—2441no—
358C2×Q16—22no—
359C2×Q16—22no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—41no—
364C2×Q16—41no—
365C2×Q16—611no—
366C2×Q16C22×D6111yescharacteristic
367C2×Q16C22×D6111yescharacteristic
368C2×Q16C22×D6111yescharacteristic
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—611no—
372C2×D8—611no—
373C2×D8—1222no—
374C2×D8—1221no—
375C2×D8—2441no—
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C4⋊Q8—21no—
379C4⋊Q8—41no—
380#124—1221no—
381#123—2441no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—41no—
385C22⋊Q8—41no—
386C22⋊Q8—1221no—
387#121—1221no—
388#121—2441no—
389#120—2441no—
390C4×Q8—21no—
391C4×D4—41no—
392C4×D4—1221no—
393C4×D4C22⋊D6111yescharacteristic
394C2×C22⋊C4—2441no—
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—21no—
415#103—21no—
416#103—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8C22⋊D6111yescharacteristic
420C22⋊C8C22⋊D6111yescharacteristic
421C4×C8C22×D6111yescharacteristic
422C24—2441no—
423#41—4881no—
424#41—2442no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—4881no—
438C2×D4—2442no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
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—21no—
452Q16—21no—
453Q16—21no—
454Q16—42no—
455Q16—42no—
456Q16—81no—
457Q16—1221no—
458Q16—1221no—
459Q16D4×D6112yes—
460SD16—4881no—
461SD16—2442no—
462D8—612no—
463D8—1221no—
464D8—1221no—
465D8—1221no—
466D8—2442no—
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—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—1221no—
482C4⋊C4—2441no—
483C4⋊C4#1175819911yescharacteristic
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—2441no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C22⋊C4#1175819911yescharacteristic
492C22⋊C4#1175819911yescharacteristic
493C42—41no—
494C42C2×C22⋊D6111yescharacteristic
495C16—612no—
496C16—1221no—
497C16—1221no—
498C16—1221no—
499C23—9761no—
500C23—4882no—
501C23—4882no—
502C23—2441no—
503C23—2441no—
504C23C2×#1175819911yescharacteristic
505C23C2×#1175819911yescharacteristic
506Q8—22no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—82no—
514Q8—2441no—
515Q8—2441no—
516D4—22no—
517D4—1222no—
518D4—9761no—
519D4—4882no—
520D4—4882no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—9761no—
535C2×C4—4882no—
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—9764no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—2442no—
560C22#1684020312yescomplemented
561C22C22×#1175819911yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—4881no—
571C4—4881no—
572C4D4×C22⋊D6111yescharacteristic
573C4#1684056511yescharacteristic
574C2—22no—
575C2—41no—
576C2—9761no—
577C2—4882no—
578C2D4×#1175819911yescharacteristic
579C2C2×#1684020311yescharacteristic
580C2#2145896311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingD4×Q16×C61C2
socleC22×C61C22×D8
prime radical#21457506C2
hypercenterD4×Q16D61
hypocenterC61#33832
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×#14519720C2semidirect
4. #21448088C2semidirect
5. #21447934C2semidirect
6. #21447908C2semidirect
7. #21447881C2semidirect
8. #21447861C2semidirect
9. #21447757C2semidirect
10. #21447753C2semidirect
11. #21447747C2semidirect
12. #21447736C2semidirect
13. #21447681C2semidirect
14. #21447645C2semidirect
15. D4×Q16×C61C2semidirect
20. C2×#14519723C22semidirect
21. C2×#14519723C22semidirect
22. C2×#14519723C22semidirect
30. C2×#14519717C22semidirect
39. #16839587C22semidirect
40. C4⋊#11758185C22semidirect
41. #16839566C22semidirect
42. #16839566C22semidirect
43. #16839566C22semidirect
44. #16839565C22semidirect
48. C22⋊#11758185C22semidirect
49. #16839538C22semidirect
51. C22×Q16×C61C22semidirect
53. C61×#494C22semidirect
55. C61×#467C22semidirect
59. C4×Q16×C61C22semidirect
89. #14519720D4semidirect
263. C22×C61#510semidirect
286. D4×Q16D61semidirect
308. C22×Q16C2×D61semidirect
326. #494C2×D61semidirect
329. #467C2×D61semidirect
337. C4×Q16C2×D61semidirect
351. C61#33832semidirect
560. 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×#145197202
#214575062
#214480882
#214479342
#214479082
#214478812
#214478612
#214477572
#214477532
#214477472
#214477362
#214476812
#214476452
D4×Q16×C612
#3383261