Finite Group Databasebeta

← #29671403

Subgroups — #29671403

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458725C211yesmaximal, characteristic, complemented
2#21457645C211yesmaximal, characteristic, complemented
3#21457439C211yesmaximal, characteristic
4#21445410C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445301C211yesmaximal, characteristic, complemented
7#21445235C211yesmaximal, characteristic, complemented
8#21445225C211yesmaximal, characteristic, complemented
9#21444946C211yesmaximal, characteristic, complemented
10#21444934C211yesmaximal, characteristic, complemented
11#21444897C211yesmaximal, characteristic, complemented
12#21444890C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14C8×#11758201C211yesmaximal, characteristic, complemented
15C61×#3362C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840472C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840168—21no—
20#16840165C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840136C2211yescharacteristic
23#16840134C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic
25#16840082—21no—
26#16840069—21no—
27#16840057C2211yescharacteristic, complemented
28#16840052C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16840039—21no—
31#16840013—21no—
32#16839994—21no—
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839911—21no—
37#16839907—21no—
38#16839895—21no—
39#16839893C2211yescharacteristic, complemented
40#16839860—21no—
41C4×#11758201C2211yescharacteristic
42#16839840C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#102×D61—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56Q82×C61C2211yescharacteristic, complemented
57C61×#550C2211yescharacteristic
58C61×#486—21no—
59C61×#486C2211yescharacteristic, complemented
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#460—21no—
63C61×#460C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C22×D244—21no—
68C2×D4×D61—21no—
69#14519802—44no—
70C2×#11758200—41no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—41no—
74C2×#11758201D411yescharacteristic
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758194—21no—
79C2×C8×D61—21no—
80#14519777C2311yescharacteristic
81#14519774C2311yescharacteristic
82#14519769—21no—
83#14519755—41no—
84#14519753—21no—
85#14519752—21no—
86#14519751—21no—
87#14519747—41no—
88C4×D244—21no—
89C4×D244C2311yescharacteristic
90C4⋊C4×D61—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701—21no—
102#14519701D411yescharacteristic
103#14519699—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123#127×C61—21no—
124#127×C61C2311yescharacteristic
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#106×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C2×C22⋊D61—41no—
161D4×D61—42no—
162D4×D61—81no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—42no—
167C2×D244—41no—
168C2×D244—81no—
169C2×D244C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—41no—
172C2×Dic122—21no—
173SD16×C61—41no—
174SD16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—41no—
192C42×C61C2411yescharacteristic
193#11758201—42no—
194#11758201—82no—
195#11758201—81no—
196#11758201SD1614yescomplemented
197#11758200—82no—
198#11758199—42no—
199#11758199—42no—
200#11758199—81no—
201#11758198—21no—
202#11758198—41no—
203#11758198—41no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C2×D411yescharacteristic
207#11758196—41no—
208#11758196—41no—
209#11758194—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216C8×D61—81no—
217C23×C61—21no—
218C23×C61—41no—
219C22×D61—21no—
220C22×D61—41no—
221C22×D61—82no—
222C22×D61—82no—
223C22×D61—161no—
224Q8×C61—21no—
225Q8×C61—21no—
226Q8×C61—21no—
227Q8×C61—81no—
228Q8×C61—161no—
229Q8×C61C2×SD1612yes—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—82no—
237D4×C61—81no—
238D4×C61—161no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—82no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C22⋊D61—82no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254D244—21no—
255D244—41no—
256D244—41no—
257D244—41no—
258D244—41no—
259D244—81no—
260D244—82no—
261D244C2×SD1612yes—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—161no—
265Dic122—81no—
266C8×C61—21no—
267C8×C61—21no—
268C8×C61—41no—
269#92418—22no—
270#92418—21no—
271#92418—41no—
272#92418—41no—
273#92418—41no—
274#92418C2×SD1612yes—
275#19893—611nomaximal, Sylow
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61D4211yescharacteristic
283C2×D61—22no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—164no—
290C4×C61—22no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61C22×SD1611yescharacteristic
298C4×C61#57511yescharacteristic
299C4×C61#57711yescharacteristic
300C4×C61#57711yescharacteristic
301Dic61—41no—
302Dic61—81no—
303#3385—611no—
304#3362D6111yescharacteristic, complemented
305#3295—611no—
306#3274—611no—
307#1706—611no—
308#1695—611no—
309#1688—611no—
310#1678—611no—
311#1677—611no—
312#1631—611no—
313#1629—611no—
314#1623—611no—
315#1622—611no—
316#1590—611no—
317C8×SD16—611no—
318C2×C61—81no—
319C2×C61—161no—
320C2×C61#327511yescharacteristic
321C2×C61D4×SD1611yescharacteristic
322C2×C61#329211yescharacteristic
323D61—42no—
324D61—161no—
325C2×AGL(1,ℤ/8)—1221no—
326Q82C2×D6111yescharacteristic, complemented
327#554—611no—
328#550C2×D6111yescharacteristic
329D42—611no—
330#501—1221no—
331#500—611no—
332#493—611no—
333#492—611no—
334#492—611no—
335#486—21no—
336#486—611no—
337#486C2×D6111yescharacteristic, complemented
338#481—1221no—
339#479—1221no—
340#477—611no—
341#476—21no—
342#476—611no—
343#476—611no—
344#476C2×D6111yescharacteristic
345#474—1221no—
346#469—1221no—
347#465—1221no—
348#460—21no—
349#460—611no—
350#460—611no—
351#460C2×D6111yescharacteristic, complemented
352#459—611no—
353#450—1221no—
354#449—1221no—
355#447—1221no—
356#446—611no—
357C8×Q8C2×D6111yescharacteristic, complemented
358#440—1221no—
359C4×SD16—21no—
360C4×SD16—611no—
361C4×SD16C2×D6111yescharacteristic
362#436—611no—
363C8×D4—611no—
364C2×#102—1221no—
365#334—611no—
366#334—611no—
367#333—611no—
368#333—611no—
369#331—611no—
370#331—611no—
371Q8⋊C8—611no—
372Q8⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375C82—611no—
376C61#1989311yescharacteristic, complemented, Sylow
377C2×#41—21no—
378C22×D4—1221no—
379C22×D4—1221no—
380AGL(1,ℤ/8)—2444no—
381C2×SD16—21no—
382C2×SD16—611no—
383C2×SD16—1221no—
384C2×SD16—1221no—
385C2×SD16—2441no—
386C2×SD16—2441no—
387C2×SD16C22⋊D6111yescharacteristic
388C2×D8—1221no—
389C2×D8—1221no—
390C2×D8—1221no—
391C2×M16—1221no—
392C22×C8—1221no—
393C4⋊Q8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—41no—
396C4⋊Q8—611no—
397C4⋊Q8C22×D6111yescharacteristic
398#127—21no—
399#127—611no—
400#127C22×D6111yescharacteristic
401#125—1221no—
402C22⋊Q8—1221no—
403C22⋊Q8—2441no—
404#121—21no—
405#121—41no—
406#121—1221no—
407#121—1221no—
408#120—2441no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8C22×D6111yescharacteristic
412C4×Q8C22×D6111yescharacteristic
413C4×D4—21no—
414C4×D4—611no—
415C4×D4—1221no—
416C4×D4C22×D6111yescharacteristic
417C2×C4⋊C4—1221no—
418#107—1221no—
419#107—1221no—
420#107—1221no—
421#106—21no—
422#106—611no—
423#106—1221no—
424#106—1221no—
425#106C22⋊D6111yescharacteristic
426C4⋊C8—21no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432#103—611no—
433#103—1221no—
434#103C22⋊D6111yescharacteristic
435#102—21no—
436#102—21no—
437#102—21no—
438#102—21no—
439#102—611no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102C22⋊D6111yescharacteristic
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450#97—1221no—
451C4×C8—21no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8C22×D6111yescharacteristic
455C24—2441no—
456#41—42no—
457#41—81no—
458C2×Q8—21no—
459C2×Q8—41no—
460C2×Q8—1221no—
461C2×Q8D4×D6111yescharacteristic
462C2×Q8C2×C22⋊D6111yescharacteristic
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—611no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—4881no—
473C2×D4—4881no—
474C2×D4—2441no—
475C2×D4—2442no—
476C2×D4—2441no—
477C2×D4—2442no—
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—41no—
481C22×C4—1221no—
482C22×C4—2441no—
483SD16—41no—
484SD16—42no—
485SD16—614no—
486SD16—4882no—
487SD16—4882no—
488SD16—4881no—
489SD16—2442no—
490D8—4881no—
491D8—2442no—
492D8—2442no—
493M16—2442no—
494C2×C8—21no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—4881no—
502C2×C8C2×C22⋊D6111yescharacteristic
503C2×C8C2×C22⋊D6111yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—2441no—
514C4⋊C4—2441no—
515C4⋊C4D4×D6111yescharacteristic
516C4⋊C4C2×C22⋊D6111yescharacteristic
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C22⋊C4—21no—
519C22⋊C4—41no—
520C22⋊C4—1221no—
521C22⋊C4—2441no—
522C22⋊C4—2441no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42C23×D6111yescharacteristic
527C23—21no—
528C23—41no—
529C23—1221no—
530C23—9761no—
531C23—4882no—
532C23—4882no—
533C23—2441no—
534Q8—21no—
535Q8—21no—
536Q8—21no—
537Q8—81no—
538Q8—161no—
539Q8—4881no—
540Q8SD16×D6112yes—
541D4—21no—
542D4—21no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—82no—
548D4—81no—
549D4—161no—
550D4—612no—
551D4—1221no—
552D4—4881no—
553D4—4882no—
554D4—4882no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—82no—
568C2×C4—1221no—
569C2×C4—9761no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—21no—
578C8—41no—
579C8—612no—
580C8—1222no—
581C8—1221no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—1222no—
592C22—9764no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—2441no—
597C22—2441no—
598C22D4×C22⋊D6111yescharacteristic
599C4—22no—
600C4—21no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—4881no—
607C4—2441no—
608C4C2×SD16×D6111yescharacteristic
609C4#1684056011yescharacteristic
610C4#1684056811yescharacteristic
611C4#1684057011yescharacteristic
612C2—81no—
613C2—161no—
614C2—9761no—
615C2—2442no—
616C2#2145745111yescharacteristic
617C2SD16×C22⋊D6111yescharacteristic
618C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3362C2
socleC22×C61D42
prime radical#21457439C2
hypercenter#3362D61
hypocenterC61#19893
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3362

Complements

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

NH ≅ G/Ntype
1. #21458725C2semidirect
2. #21457645C2semidirect
4. #21445410C2semidirect
5. #21445340C2semidirect
6. #21445301C2semidirect
7. #21445235C2semidirect
8. #21445225C2semidirect
9. #21444946C2semidirect
10. #21444934C2semidirect
11. #21444897C2semidirect
12. #21444890C2semidirect
13. #21444710C2semidirect
14. C8×#11758201C2semidirect
15. C61×#3362C2semidirect
20. #16840165C22semidirect
23. #16840134C22semidirect
27. #16840057C22semidirect
39. #16839893C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839510C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. C8×#92418C22semidirect
56. Q82×C61C22semidirect
59. C61×#486C22semidirect
63. C61×#460C22semidirect
64. C8×Q8×C61C22semidirect
196. #11758201SD16semidirect
304. #3362D61semidirect
326. Q82C2×D61semidirect
337. #486C2×D61semidirect
351. #460C2×D61semidirect
357. C8×Q8C2×D61semidirect
376. C61#19893semidirect

Maximal subgroups

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

subgroupindex
#214587252
#214576452
#214574392
#214454102
#214453402
#214453012
#214452352
#214452252
#214449462
#214449342
#214448972
#214448902
#214447102
C8×#117582012
C61×#33622
#1989361