Finite Group Databasebeta

← #29836481

Subgroups — #29836481

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457561C211yesmaximal, characteristic, complemented
4#21457494C211yesmaximal, characteristic, complemented
5#21448083C211yesmaximal, characteristic, complemented
6#21447931C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447875C211yesmaximal, characteristic, complemented
9#21447749C211yesmaximal, characteristic, complemented
10#21447741C211yesmaximal, characteristic, complemented
11#21447734C211yesmaximal, characteristic, complemented
12#21447732C211yesmaximal, characteristic, complemented
13#21447681C211yesmaximal, characteristic, complemented
14#21447643C211yesmaximal, characteristic, complemented
15C61×#2207C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic, complemented
17C2×Q16×D61C2211yescharacteristic, complemented
18C2×D8×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840424—21no—
21#16840397—21no—
22#16840199—24no—
23C2×#14519722C2211yescharacteristic
24C2×#14519721C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic, complemented
26C2×#14519720C2211yescharacteristic
27SD32×D61—24no—
28C2×D976C2211yescharacteristic
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519718C2211yescharacteristic, complemented
31C2×#14519716C2211yescharacteristic, complemented
32C2×C16×D61C2211yescharacteristic, complemented
33#16840146C2211yescharacteristic, complemented
34#16840004C2211yescharacteristic, complemented
35#16839999C2211yescharacteristic, complemented
36#16839995C2211yescharacteristic
37#16839995C2211yescharacteristic
38#16839984—21no—
39#16839954—21no—
40#16839927—21no—
41#16839902—21no—
42#16839894—21no—
43#16839894—21no—
44C4×D488C2211yescharacteristic
45C8×D244C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839567C2211yescharacteristic, complemented
48#16839566C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839562C2211yescharacteristic
51#16839541C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839538C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic, complemented
55C2×SD32×C61C2211yescharacteristic, complemented
56C61×#493C2211yescharacteristic
57C4×Q16×C61C2211yescharacteristic, complemented
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×Q8×D61—21no—
63C2×D4×D61—21no—
64C2×D4×D61—21no—
65C2×D4×D61—41no—
66#14519800—22no—
67#14519800—41no—
68C2×Dic244C2311yescharacteristic
69C2×#11758202—21no—
70Q16×D61—22no—
71Q16×D61—41no—
72C2×#11758190—21no—
73C2×#11758201—21no—
74C2×D488C2311yescharacteristic
75C2×D488C2311yescharacteristic
76C2×#11758199—21no—
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61—22no—
80D8×D61—41no—
81D8×D61—41no—
82C2×C8×D61C2311yescharacteristic
83C2×C8×D61C2311yescharacteristic
84#14519765—21no—
85#14519765—41no—
86#14519757—21no—
87#14519757—41no—
88#14519752—21no—
89#14519752—41no—
90#14519748—21no—
91#14519748—21no—
92#14519748—41no—
93#14519745—41no—
94#14519745—41no—
95C4×D244—21no—
96C4×D244—41no—
97C4×D244D411yescharacteristic
98#14519722—22no—
99#14519721—22no—
100#14519721—41no—
101#14519720D414yescomplemented
102D976—22no—
103#14519718—22no—
104#14519718—41no—
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519700—21no—
112#14519699—21no—
113#14519698—21no—
114#14519698—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×D8×C61—21no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#127×C61—21no—
123C4×Q8×C61—21no—
124SD32×C61—22no—
125SD32×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#103×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—41no—
137C2×D4×C61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142#11758226—42no—
143#11758226—81no—
144Q8×D61—42no—
145Q8×D61—81no—
146D4×D61—42no—
147D4×D61—42no—
148D4×D61—81no—
149D4×D61—81no—
150D4×D61—84no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244D811yescharacteristic
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—41no—
163Q16×C61—21no—
164Q16×C61C2×D412yescomplemented
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—81no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C42×C61—41no—
175C42×C61C2×D411yescharacteristic
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178#11758202—42no—
179#11758201—42no—
180#11758199—42no—
181#11758199—42no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198D811yescharacteristic
188#11758198D811yescharacteristic
189C4⋊Dic61—21no—
190C4⋊Dic61—41no—
191C4⋊Dic61D811yescharacteristic
192C2×#92418C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194Dic244—21no—
195Dic244—21no—
196D488—21no—
197D488—21no—
198D488—21no—
199D488C2×D412yes—
200#11758190—81no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61—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
220C22×D61C2×D811yescharacteristic
221Q8×C61—22no—
222Q8×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—82no—
228D4×C61—161no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C2×D811yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C22⋊D61—82no—
235C22⋊D61—82no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61C2×D811yescharacteristic
241C2×Dic61C2×D811yescharacteristic
242D244—22no—
243D244—22no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—82no—
248D244—82no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—82no—
254C4×D61—82no—
255C4×D61—82no—
256Dic122—41no—
257Dic122—41no—
258Dic122—82no—
259C8×C61—21no—
260C8×C61C22×D411yescharacteristic
261C8×C61H(5,𝔽2)11yescharacteristic
262#92418—21no—
263#92418—21no—
264#33831—611nomaximal, Sylow
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61C22×D811yescharacteristic
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—42no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—164no—
284C2×D61—164no—
285C2×D61#50912yescomplemented
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61D4211yescharacteristic
291C4×C61#57611yescharacteristic
292Dic61—21no—
293Dic61—21no—
294Dic61—81no—
295Dic61—81no—
296C2×#509—611no—
297C22×SD32—611no—
298#3285—611no—
299D4×D8—611no—
300#2250—611no—
301#2220—611no—
302#2215—611no—
303#2210—611no—
304#2207D6111yescharacteristic, complemented
305#2187—611no—
306#2186—611no—
307#2185—611no—
308#2184—611no—
309#2175—611no—
310#2168—611no—
311C2×C61—161no—
312C2×C61—161no—
313C2×C61D4×D811yescharacteristic
314C2×C61C2×#50911yescharacteristic
315C2×C61#341611yescharacteristic
316D61—22no—
317D61—41no—
318D61—82no—
319D61—161no—
320C2×#135—611no—
321C22×Q16—611no—
322C22×D8—611no—
323C22×D8—611no—
324#548—1221no—
325D42—1221no—
326#509—1224no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×SD32—611no—
330C2×SD32—611no—
331C2×SD32—1224no—
332C2×SD32C2×D6111yescharacteristic, complemented
333C2×SD32C2×D6111yescharacteristic, complemented
334C2×D16—611no—
335C2×D16—611no—
336C2×D16—611no—
337C2×M32—611no—
338C22×C16—611no—
339#495—611no—
340#493C2×D6111yescharacteristic
341#467—611no—
342#466—611no—
343#466—611no—
344#466—611no—
345#464—1221no—
346#459—1221no—
347#452—1221no—
348#448—1221no—
349#447—1221no—
350#447—1221no—
351C4×Q16C2×D6111yescharacteristic, complemented
352C4×D8—611no—
353C8×D4—611no—
354#366—611no—
355C4⋊C16C2×D6111yescharacteristic, complemented
356#358—611no—
357#358—611no—
358#357—611no—
359#357—611no—
360#357C2×D6111yescharacteristic, complemented
361#357C2×D6111yescharacteristic, complemented
362C22⋊C16—611no—
363C22⋊C16—611no—
364#346—611no—
365C61#3383111yescharacteristic, complemented, Sylow
366C2×#41—1221no—
367C22×Q8—1221no—
368C22×D4—1221no—
369C22×D4—1221no—
370C22×D4—2441no—
371#135—1222no—
372#135—2441no—
373C2×Q16—611no—
374C2×Q16—1221no—
375C2×Q16—1222no—
376C2×Q16—2441no—
377C2×Q16C22×D6111yescharacteristic
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×D8—21no—
381C2×D8—611no—
382C2×D8—611no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×D8—1222no—
386C2×D8—1222no—
387C2×D8—2441no—
388C2×D8—2441no—
389C2×D8C22×D6111yescharacteristic
390C2×D8C22×D6111yescharacteristic
391C22×C8—611no—
392C22×C8—611no—
393#127—21no—
394#124—1221no—
395#124—2441no—
396C22⋊Q8—1221no—
397C22⋊Q8—2441no—
398#121—1221no—
399#121—1221no—
400#121—1221no—
401#121—2441no—
402#121—2441no—
403#120—2441no—
404#120—2441no—
405C4×Q8—21no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4—2441no—
409SD32—22no—
410SD32—22no—
411SD32—614no—
412SD32—1222no—
413SD32—1222no—
414SD32—2441no—
415D16—1222no—
416D16—1222no—
417D16—2441no—
418M32—1222no—
419C2×C16—611no—
420C2×C16—611no—
421C2×C16—1222no—
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424#107—611no—
425#107C22×D6111yescharacteristic
426C4⋊C8—611no—
427#103—21no—
428#103—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433C22⋊C8—611no—
434C22⋊C8—611no—
435C4×C8C22×D6111yescharacteristic
436C24—2441no—
437C24—2441no—
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—2441no—
444C2×Q8—2442no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—611no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—4881no—
453C2×D4—4881no—
454C2×D4—4884no—
455C2×D4—2441no—
456C2×D4—2441no—
457C2×D4—2441no—
458C2×D4—2441no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2441no—
462C2×D4—2441no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—2441no—
466C22×C4—2441no—
467C22×C4—2441no—
468Q16—21no—
469Q16—1221no—
470Q16—1221no—
471Q16—2442no—
472Q16D4×D6112yescomplemented
473SD16—4881no—
474SD16—2442no—
475D8—21no—
476D8—21no—
477D8—21no—
478D8—21no—
479D8—81no—
480D8—612no—
481D8—1221no—
482D8—1221no—
483D8—1221no—
484D8—2442no—
485D8—2442no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8D4×D6111yescharacteristic
493C2×C8C23×D6111yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—611no—
497C4⋊C4—1221no—
498C4⋊C4—2441no—
499C22⋊C4—611no—
500C22⋊C4—611no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C42—41no—
509C42D4×D6111yescharacteristic
510C16—21no—
511C16—21no—
512C16—612no—
513C16—1221no—
514C23—41no—
515C23—41no—
516C23—611no—
517C23—611no—
518C23—9761no—
519C23—9761no—
520C23—4882no—
521C23—4882no—
522C23—4882no—
523C23—4882no—
524C23—2441no—
525C23—2441no—
526Q8—22no—
527Q8—41no—
528Q8—4882no—
529Q8—2441no—
530Q8—2441no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—82no—
536D4—161no—
537D4—1222no—
538D4—1222no—
539D4—4882no—
540D4—4882no—
541D4—4882no—
542D4—4882no—
543D4—4882no—
544D4—4882no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—4882no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4D8×D6111yescharacteristic
562C2×C4D8×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C8—21no—
565C8—1221no—
566C8—1221no—
567C8C2×D4×D6111yescharacteristic
568C8#1451982311yescharacteristic
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—612no—
576C22—1221no—
577C22—1221no—
578C22—1221no—
579C22—1221no—
580C22—1221no—
581C22—9764no—
582C22—9764no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—2442no—
589C22C2×D8×D6111yescharacteristic
590C4—21no—
591C4—21no—
592C4—42no—
593C4—81no—
594C4—1221no—
595C4—1221no—
596C4—4881no—
597C4—4881no—
598C4#1684039611yescharacteristic
599C4#1684056611yescharacteristic
600C2—161no—
601C2—161no—
602C2—1222no—
603C2—9761no—
604C2—4882no—
605C2—2441no—
606C2#2145749111yescharacteristic
607C2#509×D6111yescharacteristic
608C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2207C2
socleC22×C61C22×D8
hypercenter#2207D61
hypocenterC61#33831
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. C2×SD32×D61C2semidirect
3. #21457561C2semidirect
4. #21457494C2semidirect
5. #21448083C2semidirect
6. #21447931C2semidirect
7. #21447904C2semidirect
8. #21447875C2semidirect
9. #21447749C2semidirect
10. #21447741C2semidirect
11. #21447734C2semidirect
12. #21447732C2semidirect
13. #21447681C2semidirect
14. #21447643C2semidirect
15. C61×#2207C2semidirect
16. C2×#14519800C22semidirect
17. C2×Q16×D61C22semidirect
24. C2×#14519721C22semidirect
25. C2×#14519721C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519718C22semidirect
31. C2×#14519716C22semidirect
32. C2×C16×D61C22semidirect
33. #16840146C22semidirect
34. #16840004C22semidirect
35. #16839999C22semidirect
46. #16839587C22semidirect
47. #16839567C22semidirect
48. #16839566C22semidirect
49. #16839564C22semidirect
51. #16839541C22semidirect
52. #16839541C22semidirect
53. #16839538C22semidirect
54. C2×SD32×C61C22semidirect
55. C2×SD32×C61C22semidirect
57. C4×Q16×C61C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#357C22semidirect
60. C61×#357C22semidirect
101. #14519720D4semidirect
164. Q16×C61C2×D4semidirect
208. #11758185C2×D4semidirect
285. C2×D61#509semidirect
304. #2207D61semidirect
332. C2×SD32C2×D61semidirect
333. C2×SD32C2×D61semidirect
351. C4×Q16C2×D61semidirect
355. C4⋊C16C2×D61semidirect
360. #357C2×D61semidirect
361. #357C2×D61semidirect
365. C61#33831semidirect
472. Q16D4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×SD32×D612
#214575612
#214574942
#214480832
#214479312
#214479042
#214478752
#214477492
#214477412
#214477342
#214477322
#214476812
#214476432
C61×#22072
#3383161