Finite Group Databasebeta

← #29671417

Subgroups — #29671417

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458738C211yesmaximal, characteristic, complemented
2#21458461C211yesmaximal, characteristic, complemented
3#21457678C211yesmaximal, characteristic
4#21445413C211yesmaximal, characteristic, complemented
5#21445332C211yesmaximal, characteristic, complemented
6#21445310C211yesmaximal, characteristic, complemented
7#21445251C211yesmaximal, characteristic, complemented
8#21445244C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic
10#21445094C211yesmaximal, characteristic, complemented
11#21445082C211yesmaximal, characteristic, complemented
12#21445080C211yesmaximal, characteristic, complemented
13#21444707C211yesmaximal, characteristic, complemented
14SD16×#92418C211yesmaximal, characteristic, complemented
15C61×#3276C211yesmaximal, characteristic
16#16840491C2211yescharacteristic
17D4×Dic122C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840158C2211yescharacteristic
22#16840114—21no—
23#16840114C2211yescharacteristic
24#16840114C2211yescharacteristic
25#16840092—21no—
26#16840060—21no—
27#16840055—21no—
28#16840047C2211yescharacteristic
29#16840044—21no—
30#16840044C2211yescharacteristic
31#16840044C2211yescharacteristic
32#16840042C2211yescharacteristic
33#16839989—21no—
34#16839967—21no—
35#16839967C2211yescharacteristic
36#16839967C2211yescharacteristic
37#16839900—21no—
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839863—21no—
41C4×#11758200—21no—
42C4×#11758200C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519700—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C2×#137×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58C61×#547C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#470—21no—
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C61×#451—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C22×Dic122—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200D411yescharacteristic
71C22×#92418—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519776—21no—
76#14519768—21no—
77#14519768—41no—
78#14519756—21no—
79#14519754—41no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2311yescharacteristic
83D4×Dic61—21no—
84#14519711—21no—
85#14519711—21no—
86#14519710—21no—
87#14519708—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—41no—
91#14519708—41no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519701—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700D411yescharacteristic
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117#137×C61—44no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61D411yescharacteristic
123C2×M16×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126C4⋊Q8×C61C2311yescharacteristic
127#123×C61—41no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131#121×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C22×Dic61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—42no—
164C2×Dic122—41no—
165C2×Dic122—81no—
166C2×Dic122C2×D411yescharacteristic
167Q16×C61—42no—
168Q16×C61—42no—
169Q16×C61—81no—
170SD16×C61—42no—
171SD16×C61SD1614yescomplemented
172M16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758200—41no—
194#11758200—42no—
195#11758200—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61—41no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—81no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C23×C61—21no—
216C23×C61—21no—
217Q8×C61—21no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—82no—
223Q8×C61C2×SD1612yes—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—82no—
228D4×C61C2×SD1612yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—82no—
240C2×C4×C61—82no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—82no—
249Dic122—21no—
250Dic122—21no—
251Dic122—41no—
252Dic122—41no—
253Dic122—41no—
254Dic122—41no—
255Dic122—82no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61C2×SD1612yes—
259#92418—22no—
260#92418—21no—
261#92418—21no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#19894—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61D4211yescharacteristic
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C22×SD1611yescharacteristic
281C4×C61#57511yescharacteristic
282C4×C61#57711yescharacteristic
283C4×C61#57711yescharacteristic
284Dic61—41no—
285Dic61—41no—
286Dic61—81no—
287Dic61—81no—
288#3386—611no—
289#3359—611no—
290#3298—611no—
291#3276D6111yescharacteristic
292#1707—611no—
293#1693—611no—
294#1690—611no—
295#1680—611no—
296#1679—611no—
297#1654—611no—
298#1653—611no—
299#1651—611no—
300#1651—611no—
301#1590—611no—
302C8×SD16—611no—
303C2×C61—42no—
304C2×C61—81no—
305C2×C61#327511yescharacteristic
306C2×C61D4×SD1611yescharacteristic
307C2×C61#329311yescharacteristic
308C2×#137—21no—
309#557—611no—
310D4×Q8—611no—
311D4×Q8C2×D6111yescharacteristic
312#547C2×D6111yescharacteristic
313#501—1221no—
314#500—611no—
315#498—611no—
316#498—611no—
317#494C2×D6111yescharacteristic
318#487—611no—
319#487—611no—
320#487—1221no—
321#483—1221no—
322#478—1221no—
323#477—1221no—
324#475—611no—
325#475—611no—
326#475—611no—
327#475—1221no—
328#474—611no—
329#470—21no—
330#465—1221no—
331#462C2×D6111yescharacteristic
332#461—611no—
333#461—611no—
334#461—1221no—
335#461C2×D6111yescharacteristic
336#452—21no—
337#451—21no—
338#448—1221no—
339#446—611no—
340C8×Q8—611no—
341#440—1221no—
342C4×SD16—611no—
343C4×SD16—1221no—
344C4×SD16C2×D6111yescharacteristic
345#436C2×D6111yescharacteristic
346C8×D4—611no—
347C2×#103—1221no—
348#334—611no—
349#334—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353#332—611no—
354Q8⋊C8—611no—
355Q8⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358C82—611no—
359C61#1989411yescharacteristic, complemented, Sylow
360C2×#41—21no—
361C22×Q8—21no—
362C22×Q8—1221no—
363#137—44no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×Q16—21no—
367C2×SD16—21no—
368C2×SD16—611no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×SD16C22⋊D6111yescharacteristic
372C2×M16—21no—
373C22×C8—1221no—
374C4⋊Q8—21no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#125—1221no—
382#125—2441no—
383#123—41no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—41no—
387C22⋊Q8—1221no—
388C22⋊Q8—2441no—
389#121—21no—
390C4×Q8—611no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8C22×D6111yescharacteristic
394C4×D4—21no—
395C4×D4—1221no—
396C4×D4C22×D6111yescharacteristic
397C4×D4C22×D6111yescharacteristic
398C2×C4⋊C4—21no—
399#107—1221no—
400#107—1221no—
401#107—1221no—
402#106—611no—
403#106—1221no—
404#106—1221no—
405#106—1221no—
406#106—2441no—
407#106—2441no—
408#106C22⋊D6111yescharacteristic
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415#103—21no—
416#103—21no—
417#103—611no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103C22⋊D6111yescharacteristic
428#102—611no—
429#102—1221no—
430#102C22⋊D6111yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—1221no—
433#97—1221no—
434C4×C8—611no—
435C4×C8—611no—
436C4×C8—1221no—
437C4×C8C22×D6111yescharacteristic
438#41—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8—611no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—4881no—
449C2×Q8—2441no—
450C2×Q8—2442no—
451C2×Q8C2×C22⋊D6111yescharacteristic
452C2×D4—41no—
453C2×D4D4×D6111yescharacteristic
454C2×D4C2×C22⋊D6111yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—2441no—
460Q16—42no—
461Q16—42no—
462Q16—81no—
463SD16—42no—
464SD16—4881no—
465SD16—2441no—
466SD16—2442no—
467SD16#1175820014yescomplemented
468M16—42no—
469C2×C8—21no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—4881no—
477C2×C8C2×C22⋊D6111yescharacteristic
478C2×C8C2×C22⋊D6111yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—611no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C4⋊C4—2441no—
495C4⋊C4—2441no—
496C4⋊C4D4×D6111yescharacteristic
497C4⋊C4C2×C22⋊D6111yescharacteristic
498C4⋊C4C2×C22⋊D6111yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—2441no—
504C42—21no—
505C42—1221no—
506C42—2441no—
507C42C23×D6111yescharacteristic
508C23—21no—
509C23—21no—
510Q8—21no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—82no—
516Q8—1221no—
517Q8—1221no—
518Q8—4882no—
519Q8—2441no—
520Q8—2441no—
521Q8—2441no—
522Q8—2441no—
523Q8C2×#1175820012yes—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—82no—
528D4C2×#1175820012yes—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—4882no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4#1451982211yescharacteristic
549C8—21no—
550C8—41no—
551C8—1222no—
552C8—1221no—
553C8—1221no—
554C8—2441no—
555C8—2441no—
556C8—2441no—
557C8C2×#1175820012yes—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22D4×C22⋊D6111yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—4881no—
573C4—4881no—
574C4—2441no—
575C4—2441no—
576C4C22×#1175820011yescharacteristic
577C4#1684056111yescharacteristic
578C4#1684057111yescharacteristic
579C4#1684057311yescharacteristic
580C2—42no—
581C2—81no—
582C2#2145744411yescharacteristic
583C2D4×#1175820011yescharacteristic
584C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3276C2
socleC22×C61D42
prime radicalC61×#547C22
hypercenter#3276D61
hypocenterC61#19894
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3276

Complements

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

NH ≅ G/Ntype
1. #21458738C2semidirect
2. #21458461C2semidirect
4. #21445413C2semidirect
5. #21445332C2semidirect
6. #21445310C2semidirect
7. #21445251C2semidirect
8. #21445244C2semidirect
10. #21445094C2semidirect
11. #21445082C2semidirect
12. #21445080C2semidirect
13. #21444707C2semidirect
14. SD16×#92418C2semidirect
171. SD16×C61SD16semidirect
359. C61#19894semidirect
467. SD16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214587382
#214584612
#214576782
#214454132
#214453322
#214453102
#214452512
#214452442
#214451032
#214450942
#214450822
#214450802
#214447072
SD16×#924182
C61×#32762
#1989461