Finite Group Databasebeta

← #29677234

Subgroups — #29677234

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458537C211yesmaximal, characteristic, complemented
2#21458522C211yesmaximal, characteristic
3#21458436C211yesmaximal, characteristic, complemented
4#21445418C211yesmaximal, characteristic, complemented
5#21445375C211yesmaximal, characteristic, complemented
6#21445357C211yesmaximal, characteristic, complemented
7#21445318C211yesmaximal, characteristic, complemented
8#21445281C211yesmaximal, characteristic, complemented
9#21445031C211yesmaximal, characteristic, complemented
10#21445019C211yesmaximal, characteristic, complemented
11#21444984C211yesmaximal, characteristic, complemented
12#21444928C211yesmaximal, characteristic, complemented
13#21444716C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3278C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840472C2211yescharacteristic
18#16840160C2211yescharacteristic
19#16840126—21no—
20#16840124—21no—
21#16840114C2211yescharacteristic
22#16840114C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840101—21no—
26#16840091—21no—
27#16840062—21no—
28#16840057C2211yescharacteristic
29#16840054—21no—
30#16840052C2211yescharacteristic
31#16840047C2211yescharacteristic
32#16840039C2211yescharacteristic
33#16840017—21no—
34#16839977—21no—
35#16839967C2211yescharacteristic
36#16839954—21no—
37#16839954C2211yescharacteristic
38#16839893C2211yescharacteristic
39#16839893C2211yescharacteristic
40#16839872—21no—
41#16839862—21no—
42C4×#11758199C2211yescharacteristic
43#16839841C2211yescharacteristic
44#16839803—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C2×#137×C61—21no—
57C61×#550C2211yescharacteristic
58D4×Q8×C61C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#468—21no—
61C61×#462C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#452—21no—
64C61×#448—21no—
65C4×Q16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758199—21no—
69C2×#11758199D411yescharacteristic
70C2×#11758189—21no—
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519772—41no—
74#14519769—21no—
75#14519765—21no—
76#14519755—41no—
77#14519753—21no—
78#14519753—21no—
79C4×Dic122—21no—
80C4×D244—21no—
81C4×D244C2311yescharacteristic
82#14519735—21no—
83C4⋊C4×D61—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710D411yescharacteristic
87#14519709—21no—
88#14519709—21no—
89#14519708—21no—
90#14519708—21no—
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519701—21no—
97#14519701—21no—
98#14519700—21no—
99#14519700—21no—
100#14519700—21no—
101#14519700—41no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111#14519688—21no—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×Q8×C61—21no—
117#137×C61—44no—
118C2×Q16×C61—21no—
119C2×Q16×C61D411yescharacteristic
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×M16×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61C2311yescharacteristic
126#127×C61—21no—
127#127×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—41no—
130#121×C61—21no—
131#121×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—81no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167Q16×C61—22no—
168Q16×C61—42no—
169SD16×C61—42no—
170SD16×C61—42no—
171SD16×C61—81no—
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—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61C2411yescharacteristic
190#11758200—42no—
191#11758199—41no—
192#11758199—42no—
193#11758198—21no—
194#11758198—41no—
195#11758198—41no—
196#11758198—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—41no—
203#11758196—41no—
204#11758196—41no—
205C4×Dic61—41no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212#11758189—42no—
213C23×C61—21no—
214C23×C61—41no—
215C22×D61—21no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—82no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230D4×C61—82no—
231D4×C61—161no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—82no—
241C2×C4×C61—82no—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248D244—21no—
249D244—21no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61—161no—
253Dic122—41no—
254Dic122—41no—
255Dic122—81no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61AGL(1,ℤ/8)12yes—
259#92418—21no—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#20316—611nomaximal, Sylow
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61D4211yescharacteristic
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—41no—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C61#57511yescharacteristic
285C4×C61#57711yescharacteristic
286C4×C61#57811yescharacteristic
287Dic61—41no—
288Dic61—81no—
289Dic61—81no—
290#3367—611no—
291#3366—611no—
292#3356—611no—
293#3278D6111yescharacteristic, complemented
294#1708—611no—
295#1700—611no—
296#1698—611no—
297#1691—611no—
298#1685—611no—
299#1644—611no—
300#1642—611no—
301#1637—611no—
302#1628—611no—
303#1591—611no—
304#1589—611no—
305C2×C61—81no—
306C2×C61—161no—
307C2×C61#327211yescharacteristic
308C2×C61#327711yescharacteristic
309C2×C61#329011yescharacteristic
310D61—81no—
311C2×#137—21no—
312#556—611no—
313#554—611no—
314#550C2×D6111yescharacteristic
315D4×Q8C2×D6111yescharacteristic
316#498—611no—
317#492C2×D6111yescharacteristic
318#490—1221no—
319#489—1221no—
320#487—611no—
321#487—611no—
322#486—611no—
323#486—611no—
324#485—1221no—
325#483—1221no—
326#478—1221no—
327#477—611no—
328#477—1221no—
329#476—611no—
330#475—611no—
331#474—611no—
332#470—1221no—
333#468—21no—
334#463—1221no—
335#462C2×D6111yescharacteristic
336#461—611no—
337#460C2×D6111yescharacteristic
338#459—611no—
339#459—1221no—
340#452—21no—
341#448—21no—
342#446—611no—
343#446—611no—
344#442—1221no—
345#440—1221no—
346C4×Q16C2×D6111yescharacteristic
347C4×D8—611no—
348#436—611no—
349#436C2×D6111yescharacteristic
350#428—1221no—
351#335—611no—
352#335—611no—
353#333—611no—
354#332—611no—
355#332—611no—
356#331—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361C82—611no—
362C61#2031611yescharacteristic, complemented, Sylow
363C2×#41—21no—
364C22×Q8—21no—
365#137—44no—
366C2×Q16—21no—
367C2×Q16C22⋊D6111yescharacteristic
368C2×SD16—21no—
369C2×SD16—21no—
370C2×SD16—21no—
371C2×SD16—1221no—
372C2×D8—611no—
373C2×D8—1221no—
374C2×M16—21no—
375C2×M16—1221no—
376C4⋊Q8—21no—
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379C4⋊Q8C22×D6111yescharacteristic
380#127—21no—
381#127C22×D6111yescharacteristic
382#126—2441no—
383#125—1221no—
384#124—1221no—
385C22⋊Q8—21no—
386C22⋊Q8—41no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—2441no—
390#121—21no—
391#121—41no—
392C4×Q8—1221no—
393C4×Q8C22×D6111yescharacteristic
394C4×Q8C22×D6111yescharacteristic
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—611no—
398C4×D4—1221no—
399C4×D4C22×D6111yescharacteristic
400#117—1221no—
401C2×C4⋊C4—1221no—
402#107—611no—
403#107—1221no—
404#107—1221no—
405#107C22⋊D6111yescharacteristic
406#106—1221no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8C22×D6111yescharacteristic
416#103—21no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—2441no—
423#103C22⋊D6111yescharacteristic
424#103C22⋊D6111yescharacteristic
425#102—21no—
426#102—611no—
427#102—611no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—2441no—
434C22⋊C8—21no—
435C22⋊C8—1221no—
436#97—1221no—
437#97—1221no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8C22×D6111yescharacteristic
441#41—42no—
442#41—81no—
443C2×Q8—21no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—81no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8C2×C22⋊D6111yescharacteristic
450C2×Q8C2×C22⋊D6111yescharacteristic
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—611no—
457C2×D4D4×D6111yescharacteristic
458C22×C4—21no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—1221no—
462C22×C4—2441no—
463Q16—22no—
464Q16—42no—
465SD16—42no—
466SD16—42no—
467SD16—81no—
468SD16—2442no—
469D8—2441no—
470D8—2442no—
471M16—42no—
472M16—2442no—
473C2×C8—21no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8C2×C22⋊D6111yescharacteristic
481C2×C8C2×C22⋊D6111yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—611no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C4⋊C4—2441no—
495C4⋊C4D4×D6111yescharacteristic
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C4⋊C4C2×C22⋊D6111yescharacteristic
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—1221no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C42—21no—
506C42—21no—
507C42—2441no—
508C42C23×D6111yescharacteristic
509C23—21no—
510C23—41no—
511C23—1221no—
512Q8—21no—
513Q8—21no—
514Q8—21no—
515Q8—21no—
516Q8—41no—
517Q8—41no—
518Q8—82no—
519Q8—4881no—
520Q8—2441no—
521Q8—2441no—
522D4—21no—
523D4—21no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—81no—
529D4—82no—
530D4—161no—
531D4—1221no—
532D4—1221no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—1221no—
544C2×C4—9761no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4#1451982211yescharacteristic
552C8—21no—
553C8—41no—
554C8—1221no—
555C8—1221no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C8#1451980312yes—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—2441no—
568C22—2441no—
569C22—2441no—
570C22D4×C22⋊D6111yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—4881no—
579C4—4881no—
580C4—2441no—
581C4C2×#1451980311yescharacteristic
582C4#1684056111yescharacteristic
583C4#1684057311yescharacteristic
584C4#1684057811yescharacteristic
585C2—81no—
586C2—161no—
587C2—4881no—
588C2#2145741511yescharacteristic
589C2#2145747011yescharacteristic
590C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3278C2
socleC22×C61D42
prime radical#21458522C2
hypercenter#3278D61
hypocenterC61#20316
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3278

Complements

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

NH ≅ G/Ntype
1. #21458537C2semidirect
3. #21458436C2semidirect
4. #21445418C2semidirect
5. #21445375C2semidirect
6. #21445357C2semidirect
7. #21445318C2semidirect
8. #21445281C2semidirect
9. #21445031C2semidirect
10. #21445019C2semidirect
11. #21444984C2semidirect
12. #21444928C2semidirect
13. #21444716C2semidirect
14. #21444702C2semidirect
15. C61×#3278C2semidirect
293. #3278D61semidirect
362. C61#20316semidirect

Maximal subgroups

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

subgroupindex
#214585372
#214585222
#214584362
#214454182
#214453752
#214453572
#214453182
#214452812
#214450312
#214450192
#214449842
#214449282
#214447162
#214447022
C61×#32782
#2031661