Finite Group Databasebeta

← #29676675

Subgroups — #29676675

Order 15616.

634 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458835C211yesmaximal, characteristic
2#21458537C211yesmaximal, characteristic, complemented
3#21458251C211yesmaximal, characteristic, complemented
4#21445438C211yesmaximal, characteristic, complemented
5#21445375C211yesmaximal, characteristic, complemented
6#21445332C211yesmaximal, characteristic, complemented
7#21445289C211yesmaximal, characteristic, complemented
8#21445157C211yesmaximal, characteristic, complemented
9#21445087C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445019C211yesmaximal, characteristic, complemented
12#21444928C211yesmaximal, characteristic, complemented
13#21444728C211yesmaximal, characteristic, complemented
14#21444646C211yesmaximal, characteristic, complemented
15C61×#3278C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17#16840485C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840149—21no—
21#16840134C2211yescharacteristic
22#16840124—21no—
23#16840114C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840102—21no—
26#16840102—21no—
27#16840054—21no—
28#16840047C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840039C2211yescharacteristic
31#16840000—21no—
32#16839977—21no—
33#16839977—21no—
34#16839967C2211yescharacteristic
35#16839967C2211yescharacteristic
36#16839959—21no—
37#16839954C2211yescharacteristic
38#16839893C2211yescharacteristic
39#16839893C2211yescharacteristic
40#16839862—21no—
41#16839862—21no—
42C4×#11758199C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#16839799—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839507C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
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×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758199—21no—
72C2×#11758199D411yescharacteristic
73C2×C8×D61—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519772—41no—
77#14519772—41no—
78#14519765—21no—
79#14519765—21no—
80#14519753—21no—
81#14519753—21no—
82C4×Dic122—21no—
83C4×Dic122—21no—
84C4×D244C2311yescharacteristic
85#14519735—21no—
86#14519735—21no—
87#14519711—21no—
88#14519711—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93#14519708—41no—
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—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×Dic122—21no—
165C2×Dic122—21no—
166Q16×C61—22no—
167Q16×C61—42no—
168SD16×C61—42no—
169SD16×C61—42no—
170SD16×C61—81no—
171M16×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C2411yescharacteristic
189#11758200—42no—
190#11758200—42no—
191#11758200—81no—
192#11758200—81no—
193#11758199—41no—
194#11758199—81no—
195#11758198—21no—
196#11758198—41no—
197#11758198—41no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206#11758196—41no—
207C4×Dic61—41no—
208C4×Dic61—41no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C8×D61—42no—
216C23×C61—21no—
217C23×C61—41no—
218C22×D61—21no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—82no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—161no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—82no—
244C2×C4×C61—82no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251D244—21no—
252D244—21no—
253C4×D61—41no—
254C4×D61—41no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61AGL(1,ℤ/8)12yes—
262#92418—21no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#20275—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61D4211yescharacteristic
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C61#57511yescharacteristic
288C4×C61#57711yescharacteristic
289C4×C61#57711yescharacteristic
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293#3397—611no—
294#3367—611no—
295#3342—611no—
296#3278D6111yescharacteristic, complemented
297#1714—611no—
298#1700—611no—
299#1693—611no—
300#1686—611no—
301#1665—611no—
302#1652—611no—
303#1651—611no—
304#1642—611no—
305#1628—611no—
306#1593—611no—
307#1581—611no—
308C2×C61—81no—
309C2×C61—161no—
310C2×C61#327111yescharacteristic
311C2×C61#327211yescharacteristic
312C2×C61#328911yescharacteristic
313D61—81no—
314C2×#137—21no—
315#556—611no—
316#556—611no—
317#550C2×D6111yescharacteristic
318D4×Q8C2×D6111yescharacteristic
319#500—611no—
320#498—611no—
321#496—1221no—
322#492—611no—
323#492C2×D6111yescharacteristic
324#489—1221no—
325#487—611no—
326#486—611no—
327#485—1221no—
328#485—1221no—
329#477—1221no—
330#475—611no—
331#475—611no—
332#474—611no—
333#468—21no—
334#467—1221no—
335#463—1221no—
336#463—1221no—
337#462C2×D6111yescharacteristic
338#461—611no—
339#461—611no—
340#460—1221no—
341#460C2×D6111yescharacteristic
342#459—611no—
343#452—21no—
344#448—21no—
345#446—611no—
346#446—611no—
347#440—1221no—
348#440—1221no—
349C4×Q16C2×D6111yescharacteristic
350C4×D8—611no—
351#436C2×D6111yescharacteristic
352C8×D4—611no—
353#427—1221no—
354#335—611no—
355#334—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#331—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364#322—611no—
365C61#2027511yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C22×Q8—21no—
368#137—44no—
369C2×Q16—21no—
370C2×Q16C22⋊D6111yescharacteristic
371C2×SD16—21no—
372C2×SD16—21no—
373C2×SD16—21no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—611no—
379C2×D8—1221no—
380C2×M16—21no—
381C22×C8—1221no—
382C4⋊Q8—21no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8C22×D6111yescharacteristic
386#127—21no—
387#127C22×D6111yescharacteristic
388#126—2441no—
389#126—2441no—
390#124—1221no—
391#124—1221no—
392C22⋊Q8—21no—
393C22⋊Q8—41no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396#121—21no—
397#121—41no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8C22×D6111yescharacteristic
401C4×Q8C22×D6111yescharacteristic
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—611no—
405C4×D4C22×D6111yescharacteristic
406#117—1221no—
407#117—1221no—
408#107—611no—
409#107—1221no—
410#107—1221no—
411#107—1221no—
412#107C22⋊D6111yescharacteristic
413#106—1221no—
414#106—1221no—
415#106—2441no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422#103—21no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103C22⋊D6111yescharacteristic
430#103C22⋊D6111yescharacteristic
431#102—21no—
432#102—611no—
433#102—611no—
434#102—1221no—
435#102—1221no—
436#102—2441no—
437C22⋊C8—21no—
438C22⋊C8—1221no—
439#97—1221no—
440#97—1221no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8C22×D6111yescharacteristic
444#41—42no—
445#41—81no—
446C2×Q8—21no—
447C2×Q8—41no—
448C2×Q8—42no—
449C2×Q8—81no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8C2×C22⋊D6111yescharacteristic
453C2×Q8C2×C22⋊D6111yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—611no—
460C2×D4D4×D6111yescharacteristic
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—1221no—
465Q16—22no—
466Q16—42no—
467SD16—42no—
468SD16—42no—
469SD16—81no—
470SD16—4881no—
471SD16—4881no—
472SD16—2442no—
473SD16—2442no—
474D8—4881no—
475D8—2441no—
476M16—42no—
477C2×C8—21no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—2442no—
485C2×C8C2×C22⋊D6111yescharacteristic
486C2×C8C2×C22⋊D6111yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4D4×D6111yescharacteristic
500C4⋊C4C2×C22⋊D6111yescharacteristic
501C4⋊C4C2×C22⋊D6111yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—1221no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C22⋊C4—2441no—
509C22⋊C4—2441no—
510C42—21no—
511C42—21no—
512C42—2441no—
513C42—2441no—
514C42C23×D6111yescharacteristic
515C23—21no—
516C23—41no—
517C23—1221no—
518Q8—21no—
519Q8—21no—
520Q8—21no—
521Q8—21no—
522Q8—41no—
523Q8—41no—
524Q8—82no—
525Q8—2441no—
526Q8—2441no—
527Q8—2441no—
528Q8—2441no—
529D4—21no—
530D4—21no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—81no—
536D4—82no—
537D4—161no—
538D4—1221no—
539D4—1221no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—1221no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—21no—
559C8—41no—
560C8—1221no—
561C8—1221no—
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8#1451980312yes—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22D4×C22⋊D6111yescharacteristic
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—4881no—
585C4—4881no—
586C4—2441no—
587C4C2×#1451980311yescharacteristic
588C4#1684056111yescharacteristic
589C4#1684057111yescharacteristic
590C4#1684057311yescharacteristic
591C2—81no—
592C2—161no—
593C2—4881no—
594C2#2145739811yescharacteristic
595C2#2145741511yescharacteristic
596C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3278C2
socleC22×C61D42
prime radical#21458835C2
hypercenter#3278D61
hypocenterC61#20275
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3278

Complements

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

NH ≅ G/Ntype
2. #21458537C2semidirect
3. #21458251C2semidirect
4. #21445438C2semidirect
5. #21445375C2semidirect
6. #21445332C2semidirect
7. #21445289C2semidirect
8. #21445157C2semidirect
9. #21445087C2semidirect
10. #21445082C2semidirect
11. #21445019C2semidirect
12. #21444928C2semidirect
13. #21444728C2semidirect
14. #21444646C2semidirect
15. C61×#3278C2semidirect
296. #3278D61semidirect
365. C61#20275semidirect

Maximal subgroups

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

subgroupindex
#214588352
#214585372
#214582512
#214454382
#214453752
#214453322
#214452892
#214451572
#214450872
#214450822
#214450192
#214449282
#214447282
#214446462
C61×#32782
#2027561