Finite Group Databasebeta

← #29836671

Subgroups — #29836671

Order 15616.

718 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×#16840193C211yesmaximal, characteristic, complemented
2C2×D16×D61C211yesmaximal, characteristic, complemented
3#21457505C211yesmaximal, characteristic, complemented
4#21457503C211yesmaximal, characteristic, complemented
5#21448078C211yesmaximal, characteristic, complemented
6#21447909C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447749C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11#21447745C211yesmaximal, characteristic, complemented
12#21447739C211yesmaximal, characteristic, complemented
13C4×#14519718C211yesmaximal, characteristic, complemented
14#21447638C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic, complemented
17C2×#14519796C2211yescharacteristic, complemented
18C2×D8×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840407—21no—
21#16840405—21no—
22#16840193—24no—
23C2×#14519723C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic, complemented
26C2×#14519720C2211yescharacteristic, complemented
27C2×D976C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30D16×D61—24no—
31C2×C16×D61C2211yescharacteristic
32C2×C16×D61C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic, complemented
34#16840004C2211yescharacteristic
35#16840000C2211yescharacteristic
36#16839999C2211yescharacteristic, complemented
37#16839995C2211yescharacteristic, complemented
38#16839977—21no—
39#16839974—21no—
40#16839914—21no—
41#16839911—21no—
42#16839908—21no—
43#16839906—21no—
44C4×D488C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839564C2211yescharacteristic, complemented
49#16839563C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic, complemented
51#16839541C2211yescharacteristic
52#16839541C2211yescharacteristic, complemented
53C4×#11758185C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×#11758224—21no—
63C2×D4×D61—21no—
64C2×D4×D61—21no—
65#14519800—22no—
66#14519800—41no—
67#14519796—22no—
68#14519796—41no—
69C2×Dic244C2311yescharacteristic
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×D488C2311yescharacteristic
74C2×D488C2311yescharacteristic
75C2×#11758199—21no—
76C2×#11758199—21no—
77D8×D61—22no—
78D8×D61—22no—
79D8×D61—41no—
80D8×D61—41no—
81C2×C8×D61C2311yescharacteristic
82C2×C8×D61C2311yescharacteristic
83#14519765—21no—
84#14519761—41no—
85#14519759—41no—
86#14519757—21no—
87#14519753—21no—
88#14519752—21no—
89#14519751—41no—
90#14519750—41no—
91#14519748—21no—
92#14519747—41no—
93#14519747—41no—
94C4×D244—21no—
95C4×D244D411yescharacteristic
96C2×#11758198—41no—
97C22⋊C4×D61—41no—
98#14519723—22no—
99#14519722—22no—
100#14519721—41no—
101#14519720—41no—
102D976—22no—
103#14519718—22no—
104#14519718D414yescomplemented
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519701—21no—
112#14519700—21no—
113#14519699—21no—
114#14519698—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#124×C61—21no—
123C4×D4×C61—21no—
124SD32×C61—22no—
125D16×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C22×C4×C61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C22×Dic61—41no—
142#11758226—42no—
143#11758226—81no—
144#11758224—42no—
145#11758224—81no—
146D4×D61—42no—
147D4×D61—42no—
148D4×D61—81no—
149D4×D61—81no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244D811yescharacteristic
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162SD16×C61—81no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61C2×D412yes—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2411yescharacteristic
169C4⋊C4×C61—21no—
170C22⋊C4×C61—41no—
171C22⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C42×C61C2×D411yescharacteristic
174C22⋊Dic61—41no—
175#11758201—42no—
176#11758200—42no—
177#11758199—42no—
178#11758199—42no—
179#11758198—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198D811yescharacteristic
186#11758198D811yescharacteristic
187C4⋊Dic61—21no—
188C4⋊Dic61—41no—
189C4⋊Dic61D811yescharacteristic
190#11758196—41no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244—21no—
194Dic244—21no—
195D488—21no—
196D488—21no—
197D488—21no—
198D488C2×D412yes—
199#11758190—81no—
200C8×D61—21no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—161no—
217C22×D61—161no—
218C22×D61C2×D811yescharacteristic
219C22×D61C2×D811yescharacteristic
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—22no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—161no—
229C2×C4×C61C2×D811yescharacteristic
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C22⋊D61—82no—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C22⋊D61—161no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—82no—
239C2×Dic61C2×D811yescharacteristic
240C2×Dic61C2×D811yescharacteristic
241D244—22no—
242D244—22no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—82no—
247D244—161no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—82no—
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#33846—611nomaximal, Sylow
262C22×C61—42no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61C22×D811yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—42no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—164no—
281C2×D61—164no—
282C2×D61#50812yescomplemented
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61D4211yescharacteristic
288C4×C61#57611yescharacteristic
289Dic61—21no—
290Dic61—21no—
291Dic61—81no—
292Dic61—81no—
293C2×#508—611no—
294C22×D16—611no—
295#3280—611no—
296#3280—611no—
297#2249—611no—
298#2216—611no—
299#2215—611no—
300#2210—611no—
301#2210D6111yescharacteristic, complemented
302#2187—611no—
303#2187—611no—
304#2186—611no—
305#2186—611no—
306C4×D16—611no—
307D4×C16—611no—
308C2×C61—82no—
309C2×C61—161no—
310C2×C61D4×D811yescharacteristic
311C2×C61C2×#50811yescharacteristic
312C2×C61#341511yescharacteristic
313D61—22no—
314D61—41no—
315D61—82no—
316D61—161no—
317C2×#135—611no—
318C2×#135—611no—
319C22×D8—611no—
320C22×D8—611no—
321#546—1221no—
322#546—1221no—
323#508—1224no—
324C2×Q32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32C2×D6111yescharacteristic, complemented
329C2×D16—611no—
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—1224no—
333C2×D16C2×D6111yescharacteristic
334C22×C16—611no—
335C22×C16—611no—
336#495—611no—
337#495C2×D6111yescharacteristic, complemented
338#467—611no—
339#467—611no—
340#466—611no—
341#466—611no—
342#463—1221no—
343#463—1221no—
344#450—1221no—
345#450—1221no—
346#449—1221no—
347#449—1221no—
348C4×D8—611no—
349C4×D8C2×D6111yescharacteristic, complemented
350C8×D4—611no—
351#366—611no—
352C4⋊C16C2×D6111yescharacteristic, complemented
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#3384611yescharacteristic, 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—21no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×D8—611no—
378C2×D8—611no—
379C2×D8—1221no—
380C2×D8—1221no—
381C2×D8—1222no—
382C2×D8—1222no—
383C2×D8—2441no—
384C2×D8—2441no—
385C2×D8C22×D6111yescharacteristic
386C2×D8C22×D6111yescharacteristic
387C22×C8—611no—
388C22×C8—611no—
389#124—21no—
390#124—1221no—
391#123—2441no—
392#123—2441no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395#121—1221no—
396#121—1221no—
397#121—2441no—
398#121—2441no—
399#120—2441no—
400#120—2441no—
401C4×D4—21no—
402C4×D4—611no—
403C4×D4—1221no—
404C2×C22⋊C4—2441no—
405C2×C22⋊C4—2441no—
406Q32—1222no—
407SD32—22no—
408SD32—1222no—
409SD32—2441no—
410SD32—2441no—
411D16—22no—
412D16—614no—
413D16—1222no—
414D16—1222no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—1222no—
418C2×C16—1222no—
419C2×C16C22×D6111yescharacteristic
420C2×C16C22×D6111yescharacteristic
421#107—611no—
422#107C22×D6111yescharacteristic
423C4⋊C8—611no—
424#103—1221no—
425#103—1221no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430C22⋊C8—611no—
431C22⋊C8—611no—
432C4×C8C22×D6111yescharacteristic
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—21no—
443C2×D4—611no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—4881no—
447C2×D4—4881no—
448C2×D4—2441no—
449C2×D4—2441no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4—2441no—
455C2×D4—2441no—
456C22×C4—41no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2441no—
462Q16—21no—
463Q16—21no—
464Q16—1221no—
465Q16—1221no—
466SD16—81no—
467SD16—4881no—
468SD16—2442no—
469SD16—2442no—
470D8—21no—
471D8—21no—
472D8—21no—
473D8—612no—
474D8—1221no—
475D8—1221no—
476D8—1221no—
477D8—2442no—
478D8—2442no—
479D8D4×D6112yes—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8D4×D6111yescharacteristic
487C2×C8C23×D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—611no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—611no—
497C22⋊C4—611no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C42D4×D6111yescharacteristic
506C16—21no—
507C16—21no—
508C16—612no—
509C16—1221no—
510C23—41no—
511C23—41no—
512C23—611no—
513C23—611no—
514C23—9761no—
515C23—9761no—
516C23—4882no—
517C23—4882no—
518C23—4882no—
519C23—4882no—
520C23—2441no—
521C23—2441no—
522Q8—41no—
523Q8—41no—
524Q8—2441no—
525Q8—2441no—
526D4—22no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—1222no—
531D4—1222no—
532D4—9761no—
533D4—9761no—
534D4—4882no—
535D4—4882no—
536D4—4882no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—161no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—9761no—
551C2×C4—4882no—
552C2×C4—4882no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4D8×D6111yescharacteristic
556C2×C4D8×D6111yescharacteristic
557C2×C4C2×D4×D6111yescharacteristic
558C8—21no—
559C8—1221no—
560C8—1221no—
561C8C2×D4×D6111yescharacteristic
562C8#1451982311yescharacteristic
563C22—42no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—612no—
570C22—1221no—
571C22—1221no—
572C22—1221no—
573C22—1221no—
574C22—1221no—
575C22—9764no—
576C22—9764no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—2442no—
583C22C2×D8×D6111yescharacteristic
584C4—21no—
585C4—21no—
586C4—81no—
587C4—81no—
588C4—1221no—
589C4—1221no—
590C4—4881no—
591C4—4881no—
592C4#1684039611yescharacteristic
593C4#1684056611yescharacteristic
594C2—82no—
595C2—161no—
596C2—1222no—
597C2—9761no—
598C2—4882no—
599C2—2441no—
600C2#2145749111yescharacteristic
601C2#508×D6111yescharacteristic
602C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
hypercenter#2210D61
hypocenterC61#33846
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#16840193C2semidirect
2. C2×D16×D61C2semidirect
3. #21457505C2semidirect
4. #21457503C2semidirect
5. #21448078C2semidirect
6. #21447909C2semidirect
7. #21447904C2semidirect
8. #21447875C2semidirect
9. #21447749C2semidirect
10. #21447746C2semidirect
11. #21447745C2semidirect
12. #21447739C2semidirect
13. C4×#14519718C2semidirect
14. #21447638C2semidirect
15. C61×#2210C2semidirect
16. C2×#14519800C22semidirect
17. C2×#14519796C22semidirect
23. C2×#14519723C22semidirect
24. C2×#14519722C22semidirect
25. C2×#14519721C22semidirect
26. C2×#14519720C22semidirect
32. C2×C16×D61C22semidirect
33. #16840146C22semidirect
36. #16839999C22semidirect
37. #16839995C22semidirect
44. C4×D488C22semidirect
46. #16839587C22semidirect
47. #16839567C22semidirect
48. #16839564C22semidirect
49. #16839563C22semidirect
50. #16839562C22semidirect
52. #16839541C22semidirect
53. C4×#11758185C22semidirect
54. C2×SD32×C61C22semidirect
56. C61×#495C22semidirect
57. C4×D8×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
104. #14519718D4semidirect
282. C2×D61#508semidirect
301. #2210D61semidirect
328. C2×SD32C2×D61semidirect
337. #495C2×D61semidirect
349. C4×D8C2×D61semidirect
352. C4⋊C16C2×D61semidirect
354. #358C2×D61semidirect
358. #357C2×D61semidirect
362. C61#33846semidirect

Maximal subgroups

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

subgroupindex
C2×#168401932
C2×D16×D612
#214575052
#214575032
#214480782
#214479092
#214479042
#214478752
#214477492
#214477462
#214477452
#214477392
C4×#145197182
#214476382
C61×#22102
#3384661