Finite Group Databasebeta

← #29663017

Subgroups — #29663017

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456537C211yesmaximal, characteristic, complemented
2#21455890C211yesmaximal, characteristic, complemented
3#21455274C211yesmaximal, characteristic
4#21445380C211yesmaximal, characteristic, complemented
5#21445366C211yesmaximal, characteristic, complemented
6#21445332C211yesmaximal, characteristic, complemented
7#21445284C211yesmaximal, characteristic, complemented
8#21445263C211yesmaximal, characteristic, complemented
9#21445249C211yesmaximal, characteristic, complemented
10#21445213C211yesmaximal, characteristic, complemented
11#21445167C211yesmaximal, characteristic, complemented
12#21444571C211yesmaximal, characteristic, complemented
13#21444525C211yesmaximal, characteristic, complemented
14#21444520C211yesmaximal, characteristic, complemented
15C61×#2971C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840319C2211yescharacteristic
18#16840312C2211yescharacteristic
19#16840167C2211yescharacteristic
20#16840135C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840103—21no—
24#16840047C2211yescharacteristic
25#16840042C2211yescharacteristic
26#16840008—21no—
27#16840008—21no—
28#16839984—21no—
29#16839977—21no—
30#16839972C2211yescharacteristic
31#16839972C2211yescharacteristic
32#16839967C2211yescharacteristic
33#16839967C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839924—21no—
37#16839864—21no—
38#16839864—21no—
39C4×#11758202—21no—
40C4×#11758200—21no—
41#16839815C2211yescharacteristic
42C2×C4⋊#92418C2211yescharacteristic
43#16839756—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839507C2211yescharacteristic
51#16839507C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C4×#41×C61C2211yescharacteristic
57C8×Q8×C61C2211yescharacteristic
58C8×Q8×C61C2211yescharacteristic
59C61×#436—21no—
60C61×#436—21no—
61C61×#436C2211yescharacteristic
62C61×#436C2211yescharacteristic
63C61×#432—21no—
64C61×#432C2211yescharacteristic
65C4×M16×C61—21no—
66C4×M16×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758201—21no—
76C2×#11758199—21no—
77C22×#92418—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519774C2311yescharacteristic
82#14519769—21no—
83#14519765—21no—
84#14519756—21no—
85#14519756—21no—
86#14519751—21no—
87#14519751—21no—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×D244—21no—
91C4×D244—21no—
92#14519736—21no—
93#14519711—21no—
94#14519711—21no—
95#14519710—21no—
96#14519708—21no—
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519702—21no—
103#14519702—21no—
104#14519701—21no—
105#14519700—21no—
106#14519700—21no—
107#14519700—21no—
108#14519700—21no—
109#14519699—21no—
110#14519699—21no—
111#14519698—21no—
112#14519697—21no—
113#14519697—21no—
114C22⋊#92418—21no—
115C4×#92418—21no—
116C2×#41×C61D411yescharacteristic
117C2×M16×C61—21no—
118C2×M16×C61—21no—
119C2×M16×C61D411yescharacteristic
120C2×M16×C61D411yescharacteristic
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61C2311yescharacteristic
123C4×D4×C61—21no—
124C4×D4×C61—21no—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61C2311yescharacteristic
127#117×C61—21no—
128#117×C61D411yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—21no—
132C4⋊C8×C61—21no—
133C4⋊C8×C61C2311yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61—21no—
139#97×C61D411yescharacteristic
140#97×C61D411yescharacteristic
141C4×C8×C61—21no—
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145#41×C61—22no—
146#41×C61—41no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C2×C22⋊D61—41no—
157#11758222—42no—
158#11758222—42no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166M16×C61—22no—
167M16×C61—41no—
168M16×C61—42no—
169M16×C61—42no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61#4111yescharacteristic
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191#11758202—42no—
192#11758202—42no—
193#11758201—42no—
194#11758200—42no—
195#11758200—42no—
196#11758199—42no—
197#11758198—41no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205#11758196—41no—
206#11758196—41no—
207C4×Dic61—41no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—42no—
213#11758190—42no—
214#11758190—42no—
215C23×C61—21no—
216C23×C61#12111yescharacteristic
217C22×D61—41no—
218Q8×C61—21no—
219Q8×C61AGL(1,ℤ/8)12yes—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—42no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—81no—
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C22⋊D61—82no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245D244—41no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—81no—
250C4×D61—82no—
251Dic122—41no—
252Dic122—41no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256Dic122—41no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61—41no—
262C8×C61#13512yescomplemented
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#19182—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61C2×#12111yescharacteristic
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—22no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61C2×#13511yescharacteristic
286C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C61#57511yescharacteristic
288C4×C61#57511yescharacteristic
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292#3182—611no—
293#3107—611no—
294#3035—611no—
295#2971D6111yescharacteristic, complemented
296#1701—611no—
297#1699—611no—
298#1693—611no—
299#1685—611no—
300#1682—611no—
301#1680—611no—
302#1674—611no—
303#1666—611no—
304#1565—611no—
305#1553—611no—
306#1552—611no—
307C2×C61—41no—
308C2×C61—81no—
309C2×C61#303511yescharacteristic
310C2×C61#305511yescharacteristic
311C2×C61#306911yescharacteristic
312D61—161no—
313C2×#135—1221no—
314#533—611no—
315#532—611no—
316C4×#41C2×D6111yescharacteristic
317#500—611no—
318#492—611no—
319#487—611no—
320#486—611no—
321#485—1221no—
322#475—611no—
323#474—611no—
324#468—1221no—
325#468—1221no—
326#464—1221no—
327#463—1221no—
328#462—611no—
329#462—611no—
330#461—611no—
331#461—611no—
332#460—611no—
333#459—611no—
334#452—1221no—
335C8×Q8C2×D6111yescharacteristic
336C8×Q8C2×D6111yescharacteristic
337#440—1221no—
338#440—1221no—
339C4×Q16—1221no—
340C4×SD16—1221no—
341#436—21no—
342#436—21no—
343#436C2×D6111yescharacteristic
344#436C2×D6111yescharacteristic
345#432—21no—
346#432C2×D6111yescharacteristic
347#431—611no—
348C2×C4⋊C8—611no—
349#416—1221no—
350C4×M16—21no—
351C4×M16C2×D6111yescharacteristic
352#335—611no—
353#335—611no—
354#334—611no—
355#334—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#332—611no—
360#332—611no—
361#332—611no—
362#331—611no—
363#331—611no—
364C61#1918211yescharacteristic, complemented, Sylow
365C2×#41—1221no—
366C2×#41D24411yescharacteristic
367#135—2444no—
368C2×Q16—1221no—
369C2×Q16—1221no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×D8—1221no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16C22⋊D6111yescharacteristic
379C2×M16C22⋊D6111yescharacteristic
380C22×C8—1221no—
381C4⋊Q8—611no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384#127—611no—
385#125—1221no—
386#124—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389#121—1221no—
390#121—1221no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8D24411yescharacteristic
394C4×Q8C22×D6111yescharacteristic
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—1221no—
398C4×D4—1221no—
399C4×D4D24411yescharacteristic
400C4×D4C22×D6111yescharacteristic
401#117—21no—
402#117—1221no—
403#117D24411yescharacteristic
404C2×C42—21no—
405C2×C42C22×D6111yescharacteristic
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#106—1221no—
410C4⋊C8—21no—
411C4⋊C8—21no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—21no—
434C22⋊C8—1221no—
435#97—21no—
436#97C22⋊D6111yescharacteristic
437#97C22⋊D6111yescharacteristic
438C4×C8—21no—
439C4×C8—21no—
440C4×C8—1221no—
441C4×C8C22×D6111yescharacteristic
442C4×C8C22×D6111yescharacteristic
443#41—22no—
444#41—41no—
445#41—2442no—
446#41—2442no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8C2×D24411yescharacteristic
451C2×D4—21no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—2441no—
456C2×D4C2×D24411yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—2441no—
461C22×C4C2×D24411yescharacteristic
462C22×C4#1175822211yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464Q16—2442no—
465Q16—2442no—
466SD16—2442no—
467SD16—2442no—
468SD16—2442no—
469SD16—2442no—
470SD16—2442no—
471D8—2442no—
472M16—22no—
473M16—41no—
474M16—42no—
475M16—42no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—2442no—
483C2×C8C2×C22⋊D6111yescharacteristic
484C2×C8C2×C22⋊D6111yescharacteristic
485C2×C8C2×C22⋊D6111yescharacteristic
486C2×C8C2×C22⋊D6111yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—2441no—
496C4⋊C4—2441no—
497C4⋊C4C2×D24411yescharacteristic
498C4⋊C4C2×D24411yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—2441no—
509C42C2×D24411yescharacteristic
510C42#1175822211yescharacteristic
511C42C2×C22⋊D6111yescharacteristic
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—2441no—
515C23#1451975111yescharacteristic
516Q8—21no—
517Q8—2441no—
518Q8—2441no—
519Q8—2441no—
520Q8—2441no—
521Q8—2441no—
522Q8—2441no—
523Q8#1451980512yes—
524D4—21no—
525D4—21no—
526D4—42no—
527D4—4881no—
528D4—4882no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532D4—2441no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—81no—
545C2×C4—4882no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4#1451975111yescharacteristic
550C2×C4#1451975111yescharacteristic
551C2×C4#1451975111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4C22×D24411yescharacteristic
554C2×C4C2×#1175822211yescharacteristic
555C8—21no—
556C8—21no—
557C8—21no—
558C8—41no—
559C8—41no—
560C8—2441no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C8#1451979712yescomplemented
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—4881no—
572C22—4881no—
573C22—4881no—
574C22C2×#1451975111yescharacteristic
575C4—21no—
576C4—22no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—41no—
581C4—41no—
582C4—4881no—
583C4—4881no—
584C4—4881no—
585C4C2×#1451979711yescharacteristic
586C4C2×#1451980511yescharacteristic
587C4#1684056011yescharacteristic
588C4#1684056211yescharacteristic
589C2—41no—
590C2—81no—
591C2—9761no—
592C2#2145527411yescharacteristic
593C2#2145543811yescharacteristic
594C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2971C2
socleC22×C61C2×#121
prime radical#21455274C2
hypercenter#2971D61
hypocenterC61#19182
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2971

Complements

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

NH ≅ G/Ntype
1. #21456537C2semidirect
2. #21455890C2semidirect
4. #21445380C2semidirect
5. #21445366C2semidirect
6. #21445332C2semidirect
7. #21445284C2semidirect
8. #21445263C2semidirect
9. #21445249C2semidirect
10. #21445213C2semidirect
11. #21445167C2semidirect
12. #21444571C2semidirect
13. #21444525C2semidirect
14. #21444520C2semidirect
15. C61×#2971C2semidirect
262. C8×C61#135semidirect
295. #2971D61semidirect
364. C61#19182semidirect
564. C8#14519797semidirect

Maximal subgroups

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

subgroupindex
#214565372
#214558902
#214552742
#214453802
#214453662
#214453322
#214452842
#214452632
#214452492
#214452132
#214451672
#214445712
#214445252
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C61×#29712
#1918261