Finite Group Databasebeta

← #29676833

Subgroups — #29676833

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458445C211yesmaximal, characteristic, complemented
2#21458063C211yesmaximal, characteristic, complemented
3#21457439C211yesmaximal, characteristic, complemented
4#21445410C211yesmaximal, characteristic, complemented
5#21445359C211yesmaximal, characteristic, complemented
6#21445349C211yesmaximal, characteristic, complemented
7#21445340C211yesmaximal, characteristic, complemented
8#21445296C211yesmaximal, characteristic, complemented
9#21444946C211yesmaximal, characteristic, complemented
10#21444934C211yesmaximal, characteristic, complemented
11#21444901C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444709C211yesmaximal, characteristic, complemented
14#21444704C211yesmaximal, characteristic, complemented
15C61×#3366C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840477C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840138C2211yescharacteristic
21#16840136C2211yescharacteristic
22#16840134C2211yescharacteristic, complemented
23#16840109C2211yescharacteristic
24#16840100—21no—
25#16840089—21no—
26#16840075—21no—
27#16840057C2211yescharacteristic, complemented
28#16840052C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16840014—21no—
31#16840013—21no—
32#16840010—21no—
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839907—21no—
37#16839895—21no—
38#16839894—21no—
39C4×#11758201C2211yescharacteristic
40#16839841C2211yescharacteristic
41#16839840C2211yescharacteristic
42#16839840C2211yescharacteristic, complemented
43#16839802—21no—
44#16839765—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#554C2211yescharacteristic, complemented
57C61×#550C2211yescharacteristic
58C61×#490—21no—
59C61×#486C2211yescharacteristic, complemented
60C61×#478—21no—
61C61×#476C2211yescharacteristic
62C61×#459—21no—
63C61×#459C2211yescharacteristic, complemented
64C61×#446C2211yescharacteristic, complemented
65C61×#442—21no—
66C4×D8×C61C2211yescharacteristic
67C22×D244—21no—
68C2×D4×D61—21no—
69#14519802—44no—
70C2×#11758200—41no—
71C2×#11758201—21no—
72C2×#11758201—41no—
73C2×#11758201D411yescharacteristic
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758194—21no—
79C2×#11758194—21no—
80C2×#11758189—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519763—41no—
84#14519752—21no—
85#14519751—21no—
86#14519751—21no—
87#14519748—21no—
88#14519747—41no—
89C4×D244C2311yescharacteristic
90#14519736—21no—
91#14519735—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708D411yescharacteristic
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—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×D8×C61—21no—
118C2×D8×C61D411yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#127×C61—21no—
121#127×C61C2311yescharacteristic
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61—41no—
155C2×D4×C61C2×D411yescharacteristic
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C22×C4×C61—41no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163D4×D61—42no—
164D4×D61—81no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—42no—
169C2×D244—41no—
170C2×D244—81no—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×Dic122—21no—
174D8×C61—41no—
175D8×C61—42no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
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—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194#11758201—22no—
195#11758201—42no—
196#11758201—82no—
197#11758200—82no—
198#11758199—42no—
199#11758199—42no—
200#11758199—81no—
201#11758199—81no—
202#11758198—21no—
203#11758198—41no—
204#11758198—41no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D411yescharacteristic
208C4×Dic61—41no—
209#11758194—42no—
210#11758194—42no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217#11758189—81no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—41no—
221C22×D61—21no—
222C22×D61—41no—
223C22×D61—82no—
224C22×D61—82no—
225C22×D61—161no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—81no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—81no—
238D4×C61—82no—
239D4×C61—161no—
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—41no—
248C2×C4×C61—41no—
249C2×C4×C61—82no—
250C2×C4×C61—161no—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C22⋊D61—82no—
255C22⋊D61—161no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258D244—21no—
259D244—21no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—81no—
265D244—82no—
266C4×D61—41no—
267C4×D61—41no—
268Dic122—81no—
269C8×C61—21no—
270C8×C61—21no—
271C8×C61—41no—
272#92418—22no—
273#92418—21no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#92418AGL(1,ℤ/8)12yes—
278#20286—611nomaximal, Sylow
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61D4211yescharacteristic
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—164no—
296C4×C61—21no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—81no—
302C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C61#57511yescharacteristic
304C4×C61#57711yescharacteristic
305C4×C61#57711yescharacteristic
306Dic61—41no—
307Dic61—81no—
308#3366D6111yescharacteristic, complemented
309#3357—611no—
310#3325—611no—
311#3274—611no—
312#1706—611no—
313#1698—611no—
314#1697—611no—
315#1695—611no—
316#1687—611no—
317#1631—611no—
318#1629—611no—
319#1624—611no—
320#1622—611no—
321#1590—611no—
322#1589—611no—
323C2×C61—81no—
324C2×C61—81no—
325C2×C61—161no—
326C2×C61#327211yescharacteristic
327C2×C61#327511yescharacteristic
328C2×C61#329211yescharacteristic
329D61—81no—
330D61—161no—
331C2×AGL(1,ℤ/8)—1221no—
332#555—611no—
333#554C2×D6111yescharacteristic, complemented
334#550C2×D6111yescharacteristic
335D42—611no—
336#500—611no—
337#493—611no—
338#492—611no—
339#492—611no—
340#490—21no—
341#486—611no—
342#486C2×D6111yescharacteristic, complemented
343#485—1221no—
344#482—1221no—
345#480—1221no—
346#478—21no—
347#477—611no—
348#476—611no—
349#476—611no—
350#476C2×D6111yescharacteristic
351#469—1221no—
352#469—1221no—
353#468—1221no—
354#460—611no—
355#459—21no—
356#459—611no—
357#459—611no—
358#459C2×D6111yescharacteristic, complemented
359#449—1221no—
360#447—1221no—
361#447—1221no—
362#446C2×D6111yescharacteristic, complemented
363#442—21no—
364C4×SD16—611no—
365C4×D8C2×D6111yescharacteristic
366#436—611no—
367#436—611no—
368#436—611no—
369#428—1221no—
370#418—1221no—
371#335—611no—
372#334—611no—
373#333—611no—
374#333—611no—
375#331—611no—
376#331—611no—
377Q8⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380D4⋊C8—611no—
381C82—611no—
382C61#2028611yescharacteristic, complemented, Sylow
383C2×#41—21no—
384C22×D4—1221no—
385C22×D4—1221no—
386AGL(1,ℤ/8)—2444no—
387C2×SD16—611no—
388C2×SD16—1221no—
389C2×SD16—2441no—
390C2×SD16—2441no—
391C2×D8—21no—
392C2×D8—1221no—
393C2×D8—1221no—
394C2×D8—1221no—
395C2×D8—1221no—
396C2×D8C22⋊D6111yescharacteristic
397C2×M16—1221no—
398C2×M16—1221no—
399C2×M16—1221no—
400C4⋊Q8—611no—
401C4⋊Q8C22×D6111yescharacteristic
402#127—21no—
403#127—611no—
404#127C22×D6111yescharacteristic
405#125—21no—
406#124—2441no—
407C22⋊Q8—21no—
408C22⋊Q8—41no—
409#121—21no—
410#121—41no—
411#121—1221no—
412#121—1221no—
413#121—1221no—
414#121—1221no—
415#120—2441no—
416C4×Q8C22×D6111yescharacteristic
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—611no—
420C4×D4C22×D6111yescharacteristic
421C4×D4C22×D6111yescharacteristic
422#117—1221no—
423#117—1221no—
424C2×C4⋊C4—21no—
425#107—1221no—
426#107—1221no—
427#107—1221no—
428#107C22⋊D6111yescharacteristic
429#106—21no—
430#106—611no—
431#106—1221no—
432#106—1221no—
433C4⋊C8—21no—
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8C22×D6111yescharacteristic
438#103—611no—
439#103—1221no—
440#102—21no—
441#102—21no—
442#102—21no—
443#102—21no—
444#102—611no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102C22⋊D6111yescharacteristic
453#102C22⋊D6111yescharacteristic
454C22⋊C8—1221no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457#97—21no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8C22×D6111yescharacteristic
461C24—2441no—
462#41—42no—
463#41—81no—
464C2×Q8—21no—
465C2×Q8—1221no—
466C2×Q8D4×D6111yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—611no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—4881no—
477C2×D4—4881no—
478C2×D4—2441no—
479C2×D4—2442no—
480C2×D4—2441no—
481C2×D4—2442no—
482C2×D4—2441no—
483C2×D4C2×C22⋊D6111yescharacteristic
484C2×D4C2×C22⋊D6111yescharacteristic
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—1221no—
490SD16—1222no—
491SD16—4882no—
492SD16—4882no—
493SD16—2442no—
494D8—41no—
495D8—42no—
496D8—4881no—
497D8—4881no—
498D8—2442no—
499D8—2442no—
500M16—4881no—
501M16—2442no—
502M16—2442no—
503C2×C8—21no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8C2×C22⋊D6111yescharacteristic
511C2×C8C2×C22⋊D6111yescharacteristic
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—611no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4D4×D6111yescharacteristic
521C4⋊C4C2×C22⋊D6111yescharacteristic
522C4⋊C4C2×C22⋊D6111yescharacteristic
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—1221no—
528C22⋊C4—2441no—
529C22⋊C4—2441no—
530C22⋊C4—2441no—
531C42—21no—
532C42—2441no—
533C42C23×D6111yescharacteristic
534C23—21no—
535C23—21no—
536C23—41no—
537C23—1221no—
538C23—9761no—
539C23—4882no—
540C23—4882no—
541C23—2441no—
542Q8—21no—
543Q8—21no—
544Q8—81no—
545Q8—4881no—
546D4—21no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—81no—
555D4—82no—
556D4—161no—
557D4—1221no—
558D4—1221no—
559D4—9761no—
560D4—4881no—
561D4—4882no—
562D4—4882no—
563D4—2441no—
564D4—2441no—
565D4—2441no—
566D4—2441no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4—161no—
578C2×C4—1221no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4#1451982211yescharacteristic
585C8—21no—
586C8—21no—
587C8—41no—
588C8—612no—
589C8—1222no—
590C8—1221no—
591C8—2441no—
592C8—2441no—
593C8—2441no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—9764no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—2441no—
608C22—2441no—
609C22—2441no—
610C22D4×C22⋊D6111yescharacteristic
611C4—21no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—4881no—
618C4—2441no—
619C4AGL(1,ℤ/8)×D6111yescharacteristic
620C4#1684056011yescharacteristic
621C4#1684057011yescharacteristic
622C4#1684057211yescharacteristic
623C2—81no—
624C2—81no—
625C2—161no—
626C2—9761no—
627C2—4881no—
628C2#2145741311yescharacteristic
629C2#2145745111yescharacteristic
630C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3366C2
socleC22×C61D42
hypercenter#3366D61
hypocenterC61#20286
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3366

Complements

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

NH ≅ G/Ntype
1. #21458445C2semidirect
2. #21458063C2semidirect
3. #21457439C2semidirect
4. #21445410C2semidirect
5. #21445359C2semidirect
6. #21445349C2semidirect
7. #21445340C2semidirect
8. #21445296C2semidirect
9. #21444946C2semidirect
10. #21444934C2semidirect
11. #21444901C2semidirect
12. #21444891C2semidirect
13. #21444709C2semidirect
14. #21444704C2semidirect
15. C61×#3366C2semidirect
19. #16840165C22semidirect
22. #16840134C22semidirect
27. #16840057C22semidirect
42. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839510C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
56. C61×#554C22semidirect
59. C61×#486C22semidirect
63. C61×#459C22semidirect
64. C61×#446C22semidirect
308. #3366D61semidirect
333. #554C2×D61semidirect
342. #486C2×D61semidirect
358. #459C2×D61semidirect
362. #446C2×D61semidirect
382. C61#20286semidirect

Maximal subgroups

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

subgroupindex
#214584452
#214580632
#214574392
#214454102
#214453592
#214453492
#214453402
#214452962
#214449462
#214449342
#214449012
#214448912
#214447092
#214447042
C61×#33662
#2028661