Finite Group Databasebeta

← #29670680

Subgroups — #29670680

Order 15616.

644 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#21458546C211yesmaximal, characteristic, complemented
2#21458497C211yesmaximal, characteristic, complemented
3#21457575C211yesmaximal, characteristic
4#21445459C211yesmaximal, characteristic, complemented
5#21445356C211yesmaximal, characteristic, complemented
6#21445272C211yesmaximal, characteristic, complemented
7#21445263C211yesmaximal, characteristic, complemented
8#21445177C211yesmaximal, characteristic, complemented
9#21445012C211yesmaximal, characteristic, complemented
10#21444945C211yesmaximal, characteristic, complemented
11#21444914C211yesmaximal, characteristic, complemented
12#21444897C211yesmaximal, characteristic, complemented
13#21444669C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3362C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840491C2211yescharacteristic
18#16840433C2211yescharacteristic
19#16840138C2211yescharacteristic
20#16840136C2211yescharacteristic
21#16840125—21no—
22#16840114—21no—
23#16840114C2211yescharacteristic
24#16840109C2211yescharacteristic
25#16840108C2211yescharacteristic
26#16840061—21no—
27#16840054—21no—
28#16840054C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16839996—21no—
31#16839972—21no—
32#16839972C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839960—21no—
36#16839959C2211yescharacteristic
37#16839907—21no—
38#16839905—21no—
39#16839892C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41#16839865—21no—
42C4×#11758202—21no—
43C4×#11758202C2211yescharacteristic
44D4×#92418C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839507C2211yescharacteristic
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57C61×#550C2211yescharacteristic
58C61×#486—21no—
59C61×#486C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#460—21no—
63C61×#460C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758202D411yescharacteristic
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201—41no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C22×#92418—21no—
78#14519777—21no—
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519773—21no—
82#14519768—21no—
83#14519768—41no—
84#14519751—21no—
85#14519750—41no—
86Q8×Dic61—21no—
87C4×Dic122—21no—
88C4×Dic122C2311yescharacteristic
89#14519739—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519701D411yescharacteristic
99#14519700—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700—21no—
103#14519700D411yescharacteristic
104#14519699—21no—
105#14519699—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519697—21no—
110#14519697—21no—
111C22⋊#92418—21no—
112#14519687—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123#127×C61—21no—
124#127×C61C2311yescharacteristic
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#106×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—41no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C2×C22⋊D61—41no—
160C2×C22⋊D61—41no—
161#11758222—42no—
162#11758222—81no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C2×Dic122C2×D411yescharacteristic
169SD16×C61—22no—
170SD16×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—41no—
188C42×C61C2411yescharacteristic
189#11758202—22no—
190#11758202—42no—
191#11758201—42no—
192#11758201—42no—
193#11758201—82no—
194#11758200—42no—
195#11758199—42no—
196#11758199—81no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—42no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C23×C61—21no—
215C23×C61—41no—
216C22×D61—41no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—81no—
220Q8×C61—161no—
221Q8×C61#13512yescomplemented
222Q8×C61AGL(1,ℤ/8)12yes—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—82no—
230D4×C61—81no—
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×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C22⋊D61—82no—
245C22⋊D61—161no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—81no—
254C4×D61—82no—
255Dic122—21no—
256Dic122—21no—
257Dic122—41no—
258Dic122—41no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—41no—
262#92418—21no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#19843—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61D4211yescharacteristic
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61C2×#13511yescharacteristic
287C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C4×C61#57711yescharacteristic
289C4×C61#57811yescharacteristic
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293#3368—611no—
294#3363—611no—
295#3362D6111yescharacteristic, complemented
296#3287—611no—
297#1717—611no—
298#1698—611no—
299#1683—611no—
300#1682—611no—
301#1668—611no—
302#1641—611no—
303#1631—611no—
304#1626—611no—
305#1623—611no—
306#1584—611no—
307SD16⋊C8—611no—
308C2×C61—81no—
309C2×C61—161no—
310C2×C61#326511yescharacteristic
311C2×C61#328511yescharacteristic
312C2×C61#328811yescharacteristic
313D61—161no—
314C2×#135—1221no—
315Q82C2×D6111yescharacteristic
316#557—611no—
317#550—611no—
318#550C2×D6111yescharacteristic
319#493—611no—
320#492—611no—
321#490—1221no—
322#487—611no—
323#487—1221no—
324#486—21no—
325#486—611no—
326#486—611no—
327#486C2×D6111yescharacteristic
328#478—1221no—
329#477—611no—
330#477—1221no—
331#476—21no—
332#476C2×D6111yescharacteristic
333#474—611no—
334#466—1221no—
335#462—611no—
336#462—1221no—
337#460—21no—
338#460—611no—
339#460—611no—
340#460—611no—
341#460—1221no—
342#460C2×D6111yescharacteristic
343#449—1221no—
344#449—1221no—
345#446—611no—
346C8×Q8—611no—
347C8×Q8C2×D6111yescharacteristic
348#440—1221no—
349C4×Q16—611no—
350C4×Q16—1221no—
351C4×SD16—21no—
352C4×SD16C2×D6111yescharacteristic
353C8×D4—611no—
354#335—611no—
355#335—611no—
356#333—611no—
357#332—611no—
358#331—611no—
359#331—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362Q8⋊C8—611no—
363D4⋊C8—611no—
364#322—611no—
365C61#1984311yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C2×#41—1221no—
368#135—2444no—
369C2×Q16—611no—
370C2×Q16—1221no—
371C2×SD16—21no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—2441no—
376C2×SD16C22⋊D6111yescharacteristic
377C2×D8—1221no—
378C2×D8—1221no—
379C22×C8—1221no—
380C4⋊Q8—21no—
381C4⋊Q8—21no—
382C4⋊Q8—41no—
383C4⋊Q8—611no—
384C4⋊Q8—1221no—
385C4⋊Q8C22×D6111yescharacteristic
386#127—21no—
387#127—611no—
388#127—1221no—
389#127C22×D6111yescharacteristic
390#125—1221no—
391#125—2441no—
392#121—21no—
393#121—41no—
394#121—1221no—
395#121—2441no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—611no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8C22×D6111yescharacteristic
402C4×Q8C22×D6111yescharacteristic
403C4×D4—21no—
404C4×D4—1221no—
405C4×D4C22×D6111yescharacteristic
406#107—611no—
407#107—1221no—
408#107—1221no—
409#106—21no—
410#106C22⋊D6111yescharacteristic
411C4⋊C8—21no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418#103—611no—
419#103—611no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103C22⋊D6111yescharacteristic
425#102—21no—
426#102—21no—
427#102—21no—
428#102—21no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—2441no—
436#102C22⋊D6111yescharacteristic
437C22⋊C8—1221no—
438#97—1221no—
439C4×C8—21no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8C22×D6111yescharacteristic
444#41—42no—
445#41—81no—
446#41—4881no—
447#41—2442no—
448C2×Q8—21no—
449C2×Q8—41no—
450C2×Q8—611no—
451C2×Q8—1221no—
452C2×Q8D4×D6111yescharacteristic
453C2×Q8C2×C22⋊D6111yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4C2×C22⋊D6111yescharacteristic
465C22×C4—21no—
466C22×C4—41no—
467C22×C4—2441no—
468Q16—1222no—
469Q16—2442no—
470SD16—22no—
471SD16—42no—
472SD16—4882no—
473SD16—2442no—
474SD16—2442no—
475SD16—2442no—
476D8—4881no—
477D8—2442no—
478C2×C8—21no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—2442no—
486C2×C8C2×C22⋊D6111yescharacteristic
487C2×C8C2×C22⋊D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—41no—
506C22⋊C4—2441no—
507C42—21no—
508C42—21no—
509C42—41no—
510C42—1221no—
511C42—2441no—
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—41no—
515C23—2441no—
516Q8—21no—
517Q8—21no—
518Q8—81no—
519Q8—161no—
520Q8—1221no—
521Q8—1221no—
522Q8—2441no—
523Q8—2441no—
524Q8#1451979712yescomplemented
525Q8#1451980412yes—
526D4—21no—
527D4—21no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—82no—
533D4—81no—
534D4—161no—
535D4—9761no—
536D4—4881no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—82no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—21no—
559C8—21no—
560C8—41no—
561C8—1221no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22D4×C22⋊D6111yescharacteristic
577C4—22no—
578C4—22no—
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—4881no—
585C4—2441no—
586C4—2441no—
587C4C2×#1451979711yescharacteristic
588C4C2×#1451980411yescharacteristic
589C4#1684056811yescharacteristic
590C4#1684057711yescharacteristic
591C2—81no—
592C2—161no—
593C2—9761no—
594C2#2145731411yescharacteristic
595C2#2145755611yescharacteristic
596C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3362C2
socleC22×C61D42
prime radical#21457575C2
hypercenter#3362D61
hypocenterC61#19843
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3362

Complements

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

NH ≅ G/Ntype
1. #21458546C2semidirect
2. #21458497C2semidirect
4. #21445459C2semidirect
5. #21445356C2semidirect
6. #21445272C2semidirect
7. #21445263C2semidirect
8. #21445177C2semidirect
9. #21445012C2semidirect
10. #21444945C2semidirect
11. #21444914C2semidirect
12. #21444897C2semidirect
13. #21444669C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3362C2semidirect
221. Q8×C61#135semidirect
295. #3362D61semidirect
365. C61#19843semidirect
524. Q8#14519797semidirect

Maximal subgroups

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

subgroupindex
#214585462
#214584972
#214575752
#214454592
#214453562
#214452722
#214452632
#214451772
#214450122
#214449452
#214449142
#214448972
#214446692
SD16⋊#924182
C61×#33622
#1984361