Finite Group Databasebeta

← #29836599

Subgroups — #29836599

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840200C211yesmaximal, characteristic, complemented
2C2×#16840199C211yesmaximal, characteristic, complemented
3#21457507C211yesmaximal, characteristic, complemented
4#21457501C211yesmaximal, characteristic, complemented
5#21448079C211yesmaximal, characteristic, complemented
6#21447933C211yesmaximal, characteristic, complemented
7#21447877C211yesmaximal, characteristic, complemented
8#21447873C211yesmaximal, characteristic, complemented
9#21447762C211yesmaximal, characteristic, complemented
10#21447757C211yesmaximal, characteristic, complemented
11#21447732C211yesmaximal, characteristic, complemented
12#21447730C211yesmaximal, characteristic, complemented
13#21447665C211yesmaximal, characteristic, complemented
14#21447649C211yesmaximal, characteristic, complemented
15C61×#2220C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×#14519799C2211yescharacteristic
18C22×D488C2211yescharacteristic, complemented
19C2×D8×D61C2211yescharacteristic, complemented
20#16840409—21no—
21#16840403—21no—
22#16840200—24no—
23#16840199—24no—
24C2×#14519723C2211yescharacteristic, complemented
25C2×#14519722C2211yescharacteristic, complemented
26C2×#14519722C2211yescharacteristic, complemented
27C2×#14519720C2211yescharacteristic, complemented
28C2×D976C2211yescharacteristic
29C2×D976C2211yescharacteristic
30C2×#14519718C2211yescharacteristic
31C2×#14519718C2211yescharacteristic
32C2×#14519716C2211yescharacteristic, complemented
33#16840147C2211yescharacteristic, complemented
34#16840145C2211yescharacteristic, complemented
35#16840004C2211yescharacteristic, complemented
36#16839999C2211yescharacteristic
37#16839980—21no—
38#16839976—21no—
39#16839914—21no—
40#16839912—21no—
41#16839910—21no—
42#16839905—21no—
43#16839844C2211yescharacteristic, complemented
44#16839844C2211yescharacteristic, complemented
45C8×C22⋊D61C2211yescharacteristic
46#16839580C2211yescharacteristic, complemented
47#16839565C2211yescharacteristic, complemented
48#16839562C2211yescharacteristic, complemented
49#16839562C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic, complemented
51C22⋊#11758185C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53C16×Dic61C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C2×M32×C61C2211yescharacteristic, complemented
56C61×#467C2211yescharacteristic, complemented
57C61×#466C2211yescharacteristic
58C61×#366C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×#11758222—21no—
63C22×D244—21no—
64C2×D4×D61—21no—
65#14519800—22no—
66#14519800—41no—
67#14519799—22no—
68#14519799—41no—
69C2×Dic244C2311yescharacteristic
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×D488—22no—
75C2×D488—21no—
76C2×D488—41no—
77C2×D488C2311yescharacteristic
78C2×D488C2311yescharacteristic
79C2×D488C2311yescharacteristic
80C2×#11758199—21no—
81D8×D61—22no—
82D8×D61—41no—
83C2×C8×D61C2311yescharacteristic
84#14519766—21no—
85#14519764—21no—
86#14519762—41no—
87#14519759—41no—
88#14519757—21no—
89#14519752—21no—
90#14519752—41no—
91#14519748—41no—
92#14519747—41no—
93#14519745—41no—
94#14519739—21no—
95#14519739—21no—
96C4×C22⋊D61D411yescharacteristic
97C22⋊C4×D61—41no—
98C22⋊C4×D61—41no—
99#14519723—41no—
100#14519722—22no—
101#14519722—22no—
102#14519720—22no—
103D976—22no—
104D976D414yescomplemented
105#14519718—22no—
106#14519718—41no—
107#14519716—22no—
108C2×#11758185C2311yescharacteristic
109C2×#11758185C2311yescharacteristic
110#14519706D411yescharacteristic
111#14519700—21no—
112#14519697—21no—
113#14519697—21no—
114#14519697—21no—
115C22⋊#92418D411yescharacteristic
116#14519689D411yescharacteristic
117C8×Dic61C2311yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22⋊Q8×C61—21no—
122#121×C61—21no—
123SD32×C61—22no—
124M32×C61—22no—
125C2×C16×C61C2311yescharacteristic
126C2×C16×C61C2311yescharacteristic
127#107×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#103×C61—21no—
130#102×C61—21no—
131C23×D61—41no—
132C23×D61—41no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—41no—
136C22×C4×C61C2×D411yescharacteristic
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61D811yescharacteristic
140#11758226—42no—
141#11758226—81no—
142D4×D61—42no—
143D4×D61—81no—
144#11758222—42no—
145#11758222—81no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—42no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244—41no—
153C2×D244—81no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162D8×C61—21no—
163D8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C22⋊Dic61D811yescharacteristic
176#11758201—42no—
177#11758201—81no—
178#11758200—81no—
179#11758199—42no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198D811yescharacteristic
187C4⋊Dic61—41no—
188#11758196D811yescharacteristic
189C4×Dic61C2×D411yescharacteristic
190C2×#92418C2×D411yescharacteristic
191C2×#92418C2×D411yescharacteristic
192Dic244—21no—
193Dic244—21no—
194D488—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488—42no—
199D488C2×D412yes—
200D488C2×D412yes—
201#11758190—42no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61C2×D811yescharacteristic
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—161no—
218C22×D61—161no—
219C22×D61C2×D811yescharacteristic
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—161no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C22⋊D61—41no—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C2×Dic61—41no—
236C2×Dic61C2×D811yescharacteristic
237C2×Dic61C2×D811yescharacteristic
238C2×Dic61C2×D811yescharacteristic
239D244—22no—
240D244—22no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—82no—
246D244—82no—
247D244—161no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—82no—
251C4×D61—82no—
252C4×D61—161no—
253C4×D61—161no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61C22×D411yescharacteristic
258C8×C61H(5,𝔽2)11yescharacteristic
259#92418—21no—
260#92418—21no—
261#33839—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61C22×D811yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—164no—
282C2×D61—164no—
283C4×C61—21no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61D4211yescharacteristic
288C4×C61#57611yescharacteristic
289Dic61—21no—
290Dic61—21no—
291Dic61—21no—
292Dic61—81no—
293C2×#509—611no—
294C2×#509—611no—
295#3280—611no—
296#3280—611no—
297#2249—611no—
298#2220—611no—
299#2220D6111yescharacteristic, complemented
300#2210—611no—
301#2210—611no—
302#2189—611no—
303#2189—611no—
304#2184—611no—
305#2184—611no—
306C4×D16—611no—
307#2169—611no—
308C2×C61—41no—
309C2×C61—161no—
310C2×C61D4×D811yescharacteristic
311C2×C61#341411yescharacteristic
312C2×C61#341511yescharacteristic
313D61—41no—
314D61—82no—
315D61—82no—
316D61—161no—
317C2×#135—611no—
318C2×#135—611no—
319C22×D8—611no—
320C22×D8—611no—
321#546—1221no—
322#546—1221no—
323#509—1224no—
324#509—1224no—
325C2×Q32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×SD32C2×D6111yescharacteristic, complemented
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—611no—
333C2×D16—611no—
334C2×M32—611no—
335C2×M32C2×D6111yescharacteristic, complemented
336#495—611no—
337#495—611no—
338#467—611no—
339#467C2×D6111yescharacteristic, complemented
340#466—611no—
341#466C2×D6111yescharacteristic
342#463—1221no—
343#463—1221no—
344#450—1221no—
345#450—1221no—
346#449—1221no—
347#449—1221no—
348C4×D8—611no—
349C4×D8—611no—
350C8×D4—611no—
351#366C2×D6111yescharacteristic, complemented
352C4⋊C16—611no—
353#358—611no—
354#358C2×D6111yescharacteristic, complemented
355#357—611no—
356#357—611no—
357#357—611no—
358#357C2×D6111yescharacteristic, complemented
359C22⋊C16—611no—
360C22⋊C16—611no—
361C4×C16—611no—
362C61#3383911yescharacteristic, complemented, Sylow
363C2×#41—1221no—
364C2×#41—1221no—
365C22×D4—1221no—
366C22×D4—1221no—
367#135—1222no—
368#135—1222no—
369#135—2441no—
370#135—2441no—
371C2×Q16—611no—
372C2×Q16C22×D6111yescharacteristic
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×D8—611no—
378C2×D8—611no—
379C2×D8—611no—
380C2×D8—1222no—
381C2×D8—1221no—
382C2×D8—1222no—
383C2×D8—1221no—
384C2×D8—2441no—
385C2×D8—2441no—
386C2×D8C22×D6111yescharacteristic
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389#124—1221no—
390#124—1221no—
391#123—2441no—
392#123—2441no—
393C22⋊Q8—21no—
394C22⋊Q8—1221no—
395#121—21no—
396#121—1221no—
397#121—2441no—
398#121—2441no—
399#120—2441no—
400#120—2441no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—1221no—
404C2×C22⋊C4—2441no—
405C2×C22⋊C4—2441no—
406Q32—2441no—
407SD32—22no—
408SD32—1222no—
409SD32—1222no—
410SD32—1222no—
411D16—614no—
412D16—1222no—
413D16—1222no—
414D16—2441no—
415M32—22no—
416M32—1222no—
417C2×C16—611no—
418C2×C16—611no—
419C2×C16C22×D6111yescharacteristic
420C2×C16C22×D6111yescharacteristic
421#107C22×D6111yescharacteristic
422#107C22×D6111yescharacteristic
423C4⋊C8—611no—
424#103—21no—
425#103—1221no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430C22⋊C8—611no—
431C22⋊C8—611no—
432C4×C8—611no—
433C24—2441no—
434C24—2441no—
435#41—4881no—
436#41—4881no—
437#41—2442no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—1221no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—611no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—4881no—
448C2×D4—4881no—
449C2×D4—2442no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4—2442no—
454C2×D4—2441no—
455C2×D4—2441no—
456C22×C4—611no—
457C22×C4—2441no—
458C22×C4—2441no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4D4×D6111yescharacteristic
462Q16—21no—
463Q16—21no—
464Q16—1221no—
465Q16—1221no—
466SD16—4881no—
467SD16—4881no—
468SD16—2442no—
469SD16—2442no—
470D8—21no—
471D8—21no—
472D8—612no—
473D8—612no—
474D8—1221no—
475D8—1221no—
476D8—1221no—
477D8—1221no—
478D8—2442no—
479D8—2442no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8D4×D6111yescharacteristic
487C2×C8C23×D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—611no—
492C4⋊C4—2441no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—611no—
496C22⋊C4—611no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C42—611no—
506C16—21no—
507C16—1221no—
508C16—1221no—
509C16D4×D6112yes—
510C23—41no—
511C23—611no—
512C23—9761no—
513C23—9761no—
514C23—4882no—
515C23—4882no—
516C23—4882no—
517C23—4882no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23D8×D6111yescharacteristic
522Q8—41no—
523Q8—41no—
524Q8—2441no—
525Q8—2441no—
526D4—41no—
527D4—41no—
528D4—161no—
529D4—1222no—
530D4—1222no—
531D4—9761no—
532D4—4882no—
533D4—4882no—
534D4—4882no—
535D4—4882no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—9761no—
552C2×C4—9761no—
553C2×C4—4882no—
554C2×C4—4882no—
555C2×C4—2441no—
556C2×C4D8×D6111yescharacteristic
557C2×C4C2×D4×D6111yescharacteristic
558C8—21no—
559C8—1221no—
560C8—1221no—
561C8C2×D4×D6111yescharacteristic
562C8#1451982311yescharacteristic
563C22—21no—
564C22—21no—
565C22—21no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—1221no—
570C22—1221no—
571C22—1221no—
572C22—9764no—
573C22—9764no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—2442no—
582C22—2442no—
583C22C2×D8×D6111yescharacteristic
584C4—21no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—1221no—
589C4—1221no—
590C4—1221no—
591C4—4881no—
592C4#1684039611yescharacteristic
593C4#1684056411yescharacteristic
594C2—41no—
595C2—161no—
596C2—9761no—
597C2—4882no—
598C2—4882no—
599C2—2441no—
600C2#2145749311yescharacteristic
601C2#2145895411yescharacteristic
602C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2220C2
socleC22×C61C22×D8
hypercenter#2220D61
hypocenterC61#33839
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2220

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. C2×#16840199C2semidirect
3. #21457507C2semidirect
4. #21457501C2semidirect
5. #21448079C2semidirect
6. #21447933C2semidirect
7. #21447877C2semidirect
8. #21447873C2semidirect
9. #21447762C2semidirect
10. #21447757C2semidirect
11. #21447732C2semidirect
12. #21447730C2semidirect
13. #21447665C2semidirect
14. #21447649C2semidirect
15. C61×#2220C2semidirect
18. C22×D488C22semidirect
19. C2×D8×D61C22semidirect
24. C2×#14519723C22semidirect
25. C2×#14519722C22semidirect
26. C2×#14519722C22semidirect
27. C2×#14519720C22semidirect
32. C2×#14519716C22semidirect
33. #16840147C22semidirect
34. #16840145C22semidirect
35. #16840004C22semidirect
43. #16839844C22semidirect
44. #16839844C22semidirect
46. #16839580C22semidirect
47. #16839565C22semidirect
48. #16839562C22semidirect
49. #16839562C22semidirect
50. #16839562C22semidirect
51. C22⋊#11758185C22semidirect
52. #16839541C22semidirect
53. C16×Dic61C22semidirect
54. C2×SD32×C61C22semidirect
55. C2×M32×C61C22semidirect
56. C61×#467C22semidirect
58. C61×#366C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
104. D976D4semidirect
299. #2220D61semidirect
329. C2×SD32C2×D61semidirect
335. C2×M32C2×D61semidirect
339. #467C2×D61semidirect
351. #366C2×D61semidirect
354. #358C2×D61semidirect
358. #357C2×D61semidirect
362. C61#33839semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C2×#168401992
#214575072
#214575012
#214480792
#214479332
#214478772
#214478732
#214477622
#214477572
#214477322
#214477302
#214476652
#214476492
C61×#22202
#3383961