Finite Group Databasebeta

← #29669491

Subgroups — #29669491

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458740C211yesmaximal, characteristic, complemented
2#21458459C211yesmaximal, characteristic, complemented
3D4×#11758200C211yesmaximal, characteristic
4#21445439C211yesmaximal, characteristic, complemented
5#21445289C211yesmaximal, characteristic, complemented
6#21445242C211yesmaximal, characteristic, complemented
7#21445206C211yesmaximal, characteristic, complemented
8#21445164C211yesmaximal, characteristic, complemented
9#21445082C211yesmaximal, characteristic, complemented
10#21445080C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic
12#21444913C211yesmaximal, characteristic, complemented
13#21444656C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic, complemented
15D4×SD16×C61C211yesmaximal, characteristic
16C22×#11758200—21no—
17#16840490C2211yescharacteristic, complemented
18D4×Dic122C2211yescharacteristic
19#16840168—21no—
20#16840165C2211yescharacteristic, complemented
21#16840158C2211yescharacteristic, complemented
22#16840134C2211yescharacteristic
23#16840113C2211yescharacteristic, complemented
24#16840111—21no—
25#16840108C2211yescharacteristic
26#16840062—21no—
27#16840054—21no—
28#16840047C2211yescharacteristic, complemented
29#16840046—21no—
30#16840044C2211yescharacteristic, complemented
31#16840042C2211yescharacteristic
32#16839989—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839965—21no—
36#16839959C2211yescharacteristic
37#16839913—21no—
38#16839900—21no—
39#16839893C2211yescharacteristic, complemented
40C8×Dic122C2211yescharacteristic, complemented
41#16839862—21no—
42#16839853—21no—
43C4×#11758200C2211yescharacteristic
44D4×#92418C2211yescharacteristic
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C22×SD16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#465—21no—
61C61×#461C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C61×#448—21no—
65C4×SD16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—44no—
74C2×#11758200D411yescharacteristic
75C22×#92418—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519775—21no—
80#14519769—21no—
81#14519767—41no—
82#14519756—21no—
83#14519754—41no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86#14519742—21no—
87D4×Dic61—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708—41no—
94#14519708D411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519706—21no—
100#14519702—21no—
101#14519702—21no—
102#14519701—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698—41no—
110#14519698D411yescharacteristic
111C22⋊#92418—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C8×Dic61—21no—
116C22×Q8×C61—21no—
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×SD16×C61—44no—
124C2×SD16×C61D411yescharacteristic
125C22×C8×C61—21no—
126C4⋊Q8×C61—21no—
127C4⋊Q8×C61C2311yescharacteristic
128#127×C61C2311yescharacteristic
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131#121×C61—21no—
132#121×C61—21no—
133#120×C61—41no—
134C4×Q8×C61C2311yescharacteristic
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137C4×D4×C61C2311yescharacteristic
138#106×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61—21no—
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
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×C61—42no—
158C2×D4×C61—42no—
159C2×D4×C61—81no—
160C2×D4×C61—81no—
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—41no—
166C22×Dic61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—42no—
170C2×Dic122—41no—
171C2×Dic122—81no—
172C2×Dic122C2×D411yescharacteristic
173SD16×C61—22no—
174SD16×C61—42no—
175SD16×C61—42no—
176SD16×C61—42no—
177SD16×C61—81no—
178C2×C8×C61—21no—
179C2×C8×C61—42no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61C2411yescharacteristic
195C22⋊Dic61—41no—
196#11758200—42no—
197#11758200—42no—
198#11758200—42no—
199#11758200—42no—
200#11758200—81no—
201#11758200SD1614yescomplemented
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C2×D411yescharacteristic
207#11758196—21no—
208#11758196—41no—
209#11758196—41no—
210#11758196—41no—
211#11758196—41no—
212C4×Dic61—21no—
213C4×Dic61—41no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—42no—
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—41no—
224C23×C61—82no—
225C23×C61—82no—
226C23×C61—161no—
227Q8×C61—21no—
228Q8×C61—21no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—82no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—81no—
239D4×C61—82no—
240D4×C61—82no—
241D4×C61C2×SD1612yes—
242D4×C61AGL(1,ℤ/8)12yes—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—82no—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—82no—
261Dic122—21no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265Dic122—41no—
266Dic122—82no—
267Dic122C2×SD1612yescomplemented
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418C2×SD1612yescomplemented
277#19764—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—164no—
288C22×C61D4211yescharacteristic
289C4×C61—21no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61C22×SD1611yescharacteristic
296C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C4×C61#57711yescharacteristic
298C4×C61#57711yescharacteristic
299Dic61—22no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303#3386—611no—
304#3359—611no—
305D4×SD16—611no—
306D4×SD16D6111yescharacteristic
307#1714—611no—
308#1686—611no—
309#1679—611no—
310#1673—611no—
311#1666—611no—
312#1651—611no—
313#1651—611no—
314#1639—611no—
315#1626—611no—
316#1582—611no—
317SD16⋊C8—611no—
318C2×C61—42no—
319C2×C61—42no—
320C2×C61—161no—
321C2×C61#326511yescharacteristic
322C2×C61D4×SD1611yescharacteristic
323C2×C61#328911yescharacteristic
324C22×SD16—21no—
325C22×SD16—1221no—
326#557—611no—
327D4×Q8—611no—
328D4×Q8C2×D6111yescharacteristic
329D42C2×D6111yescharacteristic
330#501—1221no—
331#500—611no—
332#498—611no—
333#492—611no—
334#492C2×D6111yescharacteristic
335#487—611no—
336#487—1221no—
337#486—611no—
338#478—1221no—
339#477—1221no—
340#475—611no—
341#475—611no—
342#475—1221no—
343#474—611no—
344#465—21no—
345#465—1221no—
346#461—611no—
347#461—611no—
348#461—1221no—
349#461C2×D6111yescharacteristic
350#460—611no—
351#460C2×D6111yescharacteristic
352#450—21no—
353#450—1221no—
354#448—21no—
355#448—1221no—
356#446—611no—
357C8×Q8—611no—
358#440—1221no—
359C4×SD16—611no—
360C4×SD16—1221no—
361C4×SD16C2×D6111yescharacteristic
362C8×D4—611no—
363C8×D4C2×D6111yescharacteristic
364#335—611no—
365#334—611no—
366#332—611no—
367#332—611no—
368#332—611no—
369#331—611no—
370Q8⋊C8—611no—
371Q8⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8—611no—
374#322—611no—
375C61#1976411yescharacteristic, complemented, Sylow
376C22×Q8—21no—
377C22×Q8—1221no—
378C22×D4—21no—
379C22×D4—21no—
380C2×SD16—21no—
381C2×SD16—21no—
382C2×SD16—21no—
383C2×SD16—21no—
384C2×SD16—44no—
385C2×SD16—611no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16—2444no—
392C2×SD16C22⋊D6111yescharacteristic
393C22×C8—21no—
394C22×C8—1221no—
395C4⋊Q8—21no—
396C4⋊Q8—611no—
397C4⋊Q8—611no—
398C4⋊Q8—1221no—
399C4⋊Q8—1221no—
400C4⋊Q8C22×D6111yescharacteristic
401#127C22×D6111yescharacteristic
402#125—1221no—
403#125—2441no—
404C22⋊Q8—21no—
405C22⋊Q8—41no—
406C22⋊Q8—1221no—
407C22⋊Q8—2441no—
408#121—21no—
409#121—21no—
410#120—41no—
411C4×Q8—611no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×Q8C22×D6111yescharacteristic
415C4×D4—21no—
416C4×D4—1221no—
417C4×D4C22×D6111yescharacteristic
418C4×D4C22×D6111yescharacteristic
419#107—1221no—
420#107—1221no—
421#107—1221no—
422#106—611no—
423#106—1221no—
424#106—1221no—
425#106—2441no—
426#106C22⋊D6111yescharacteristic
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—1221no—
431C4⋊C8—1221no—
432C4⋊C8C22×D6111yescharacteristic
433#103—21no—
434#103—611no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103C22⋊D6111yescharacteristic
442#102—21no—
443#102—611no—
444#102—1221no—
445#102—1221no—
446#102—2441no—
447#102C22⋊D6111yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—1221no—
450#97—1221no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—1221no—
454C4×C8C22×D6111yescharacteristic
455C24—41no—
456C2×Q8—21no—
457C2×Q8—41no—
458C2×Q8—42no—
459C2×Q8—81no—
460C2×Q8—611no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—4881no—
464C2×Q8—2441no—
465C2×Q8—2442no—
466C2×Q8C2×C22⋊D6111yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—81no—
475C2×D4—81no—
476C2×D4D4×D6111yescharacteristic
477C2×D4C2×C22⋊D6111yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4—2441no—
482SD16—22no—
483SD16—42no—
484SD16—42no—
485SD16—42no—
486SD16—81no—
487SD16—614no—
488SD16—4881no—
489SD16—2442no—
490SD16—2442no—
491SD16—2442no—
492SD16—2442no—
493C2×C8—21no—
494C2×C8—42no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—2442no—
502C2×C8C2×C22⋊D6111yescharacteristic
503C2×C8C2×C22⋊D6111yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—41no—
508C4⋊C4—611no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4—2441no—
518C4⋊C4D4×D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C4⋊C4C2×C22⋊D6111yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—2441no—
526C42—21no—
527C42—1221no—
528C42—2441no—
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23—41no—
533C23—82no—
534C23—82no—
535C23—161no—
536Q8—21no—
537Q8—21no—
538Q8—41no—
539Q8—41no—
540Q8—82no—
541Q8—612no—
542Q8—1221no—
543Q8—4882no—
544Q8—2441no—
545Q8—2441no—
546Q8—2441no—
547Q8—2441no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—81no—
555D4—82no—
556D4—82no—
557D4SD16×D6112yes—
558D4#1451980312yes—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—82no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—4882no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C22×C22⋊D6111yescharacteristic
576C2×C4#1451982211yescharacteristic
577C8—21no—
578C8—21no—
579C8—41no—
580C8—612no—
581C8—1221no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C8—2441no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—164no—
596C22D4×C22⋊D6111yescharacteristic
597C4—21no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—81no—
603C4—1222no—
604C4—4881no—
605C4—4881no—
606C4—2441no—
607C4C2×SD16×D6111yescharacteristic
608C4C2×#1451980311yescharacteristic
609C4#1684056811yescharacteristic
610C4#1684057311yescharacteristic
611C2—42no—
612C2—42no—
613C2—161no—
614C2#2145731011yescharacteristic
615C2SD16×C22⋊D6111yescharacteristic
616C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×SD16×C61C2
socleC22×C61D42
prime radicalD42×C61C22
hypercenterD4×SD16D61
hypocenterC61#19764
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×SD16

Complements

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

NH ≅ G/Ntype
1. #21458740C2semidirect
2. #21458459C2semidirect
4. #21445439C2semidirect
5. #21445289C2semidirect
6. #21445242C2semidirect
7. #21445206C2semidirect
8. #21445164C2semidirect
9. #21445082C2semidirect
10. #21445080C2semidirect
12. #21444913C2semidirect
13. #21444656C2semidirect
14. #21444631C2semidirect
17. #16840490C22semidirect
20. #16840165C22semidirect
21. #16840158C22semidirect
23. #16840113C22semidirect
28. #16840047C22semidirect
30. #16840044C22semidirect
39. #16839893C22semidirect
40. C8×Dic122C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839484C22semidirect
201. #11758200SD16semidirect
267. Dic122C2×SD16semidirect
276. #92418C2×SD16semidirect
375. C61#19764semidirect

Maximal subgroups

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

subgroupindex
#214587402
#214584592
D4×#117582002
#214454392
#214452892
#214452422
#214452062
#214451642
#214450822
#214450802
#214449982
#214449132
#214446562
#214446312
D4×SD16×C612
#1976461