Finite Group Databasebeta

← #29662429

Subgroups — #29662429

Order 15616.

672 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#21455601C211yesmaximal, characteristic, complemented
2#21455268C211yesmaximal, characteristic
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21445246C211yesmaximal, characteristic, complemented
5#21445239C211yesmaximal, characteristic, complemented
6#21445232C211yesmaximal, characteristic, complemented
7#21445225C211yesmaximal, characteristic, complemented
8#21445198C211yesmaximal, characteristic, complemented
9#21445191C211yesmaximal, characteristic, complemented
10#21445164C211yesmaximal, characteristic, complemented
11#21445157C211yesmaximal, characteristic, complemented
12C2×#16839513C211yesmaximal, characteristic, complemented
13#21444476C211yesmaximal, characteristic, complemented
14#21444468C211yesmaximal, characteristic, complemented
15C61×#3167C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17C4×#11758222C2211yescharacteristic
18#16840066—21no—
19#16840059—21no—
20#16840054C2211yescharacteristic
21#16840049C2211yescharacteristic
22#16840044C2211yescharacteristic
23#16840039C2211yescharacteristic
24#16839981—21no—
25#16839974—21no—
26#16839969C2211yescharacteristic
27#16839964C2211yescharacteristic
28#16839959C2211yescharacteristic
29#16839954C2211yescharacteristic
30C8×Dic122C2211yescharacteristic
31C8×Dic122C2211yescharacteristic
32C8×D244C2211yescharacteristic
33C8×D244C2211yescharacteristic
34C8×C22⋊D61—21no—
35C8×C22⋊D61—21no—
36#16839815C2211yescharacteristic
37#16839813—21no—
38#16839798—21no—
39C2×C4⋊#92418C2211yescharacteristic
40C2×C4⋊#92418C2211yescharacteristic
41#16839758—21no—
42#16839756—21no—
43C4×C8×D61—21no—
44#16839513C2211yescharacteristic
45#16839513C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#532C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#465—21no—
63C61×#465—21no—
64C4×SD16×C61—21no—
65C4×SD16×C61—21no—
66C2×C4×C8×C61C2211yescharacteristic
67C2×#11758222D411yescharacteristic
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758201—21no—
72C2×#11758199—21no—
73C22×#92418—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C4×Dic122D411yescharacteristic
78C4×Dic122C2311yescharacteristic
79C4×D244D411yescharacteristic
80C4×D244C2311yescharacteristic
81C4×C22⋊D61—21no—
82C4×C22⋊D61—21no—
83#14519736D411yescharacteristic
84#14519734—21no—
85C42×D61—21no—
86#14519710—21no—
87#14519710—21no—
88#14519708—21no—
89#14519708—21no—
90C4⋊#92418—21no—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519706—21no—
100#14519701—21no—
101#14519701—21no—
102#14519700—21no—
103#14519700—21no—
104#14519698—21no—
105#14519698—21no—
106#14519697—21no—
107#14519697—21no—
108C22⋊#92418—21no—
109#14519689—21no—
110#14519689—21no—
111C4×#92418—21no—
112C8×Dic61—21no—
113C8×Dic61—21no—
114C2×#41×C61—21no—
115C2×SD16×C61—21no—
116C2×SD16×C61—21no—
117C22×C8×C61D411yescharacteristic
118C22×C8×C61D411yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#125×C61—21no—
124#124×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—21no—
127#121×C61—21no—
128#121×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#106×C61—21no—
136#106×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141C4×C8×C61D411yescharacteristic
142C4×C8×C61D411yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C22×C4×C61—41no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4111yescharacteristic
158C2×C22⋊D61—21no—
159#11758222—22no—
160#11758222SD1614yescomplemented
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×Dic122C2×D411yescharacteristic
166SD16×C61—42no—
167SD16×C61—42no—
168C2×C8×C61—22no—
169C2×C8×C61—41no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
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—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—41no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61#4111yescharacteristic
189C42×C61C2411yescharacteristic
190C22⋊Dic61—21no—
191#11758202—42no—
192#11758201—42no—
193#11758200—42no—
194#11758199—42no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61C2×D411yescharacteristic
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—42no—
211C2×#92418—42no—
212C8×D61—42no—
213C8×D61—42no—
214C23×C61—41no—
215C23×C61#12111yescharacteristic
216C22×D61—21no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—81no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—81no—
227D4×C61—82no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—82no—
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61C2×SD1612yes—
239C2×C4×C61#13512yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C22⋊D61—24no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247D244—21no—
248D244C2×SD1612yes—
249C4×D61—24no—
250C4×D61—42no—
251C4×D61—41no—
252C4×D61—41no—
253Dic122—21no—
254Dic122C2×SD1612yes—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61#13512yescomplemented
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#19124—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61#46512yes—
272C22×C61C2×#12111yescharacteristic
273C2×D61—22no—
274C2×D61—41no—
275C2×D61—41no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61#46512yes—
283C4×C61C22×SD1611yescharacteristic
284C4×C61C2×#13511yescharacteristic
285C4×C61C2×#13511yescharacteristic
286C4×C61#57511yescharacteristic
287Dic61—22no—
288Dic61—41no—
289Dic61—41no—
290#3167D6111yescharacteristic, complemented
291#3076—611no—
292#3035—611no—
293C8×#41—611no—
294#1680—611no—
295#1679—611no—
296#1678—611no—
297#1677—611no—
298#1672—611no—
299#1671—611no—
300#1666—611no—
301#1665—611no—
302C2×#334—611no—
303#1542—611no—
304#1540—611no—
305C2×C61—22no—
306C2×C61—161no—
307C2×C61#303511yescharacteristic
308C2×C61C2×#46511yescharacteristic
309C2×C61#306711yescharacteristic
310D61—42no—
311C2×#135—1221no—
312#533C2×D6111yescharacteristic
313#532C2×D6111yescharacteristic
314C4×#41—611no—
315#498C2×D6111yescharacteristic
316#492C2×D6111yescharacteristic
317#487C2×D6111yescharacteristic
318#486C2×D6111yescharacteristic
319#479—1221no—
320#478—1221no—
321#477—611no—
322#476—611no—
323#475—611no—
324#474—611no—
325#465—21no—
326#465—21no—
327#464—1221no—
328#463—1221no—
329#462—611no—
330#461—611no—
331#460—611no—
332#459—611no—
333C8×Q8—611no—
334C8×Q8—611no—
335C4×SD16—21no—
336C4×SD16—21no—
337C8×D4—611no—
338C8×D4—611no—
339C8×D4—1221no—
340C8×D4—1221no—
341#431—611no—
342#431—1221no—
343#427—1221no—
344C2×C4⋊C8—611no—
345C2×C4⋊C8—611no—
346#416—1221no—
347#416—1221no—
348C2×C4×C8—1221no—
349C2×C4×C8C2×D6111yescharacteristic
350#334—611no—
351#334—611no—
352#334—611no—
353#334—611no—
354#333—611no—
355#333—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#332—611no—
360#331—611no—
361#331—611no—
362C61#1912411yescharacteristic, complemented, Sylow
363C2×#41—21no—
364C2×#41—611no—
365#135—2444no—
366C2×Q16—1221no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—1221no—
372C22×C8—1221no—
373C22×C8—1221no—
374C22×C8—1221no—
375C22×C8—1221no—
376C22×C8C22⋊D6111yescharacteristic
377C22×C8C22⋊D6111yescharacteristic
378C4⋊Q8C22×D6111yescharacteristic
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#127C22×D6111yescharacteristic
382#125—21no—
383#124—21no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386#121—21no—
387#121—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—611no—
391C4×Q8—611no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—611no—
395C4×D4—611no—
396C4×D4—1221no—
397C4×D4—1221no—
398#117—21no—
399#117—611no—
400#117—1221no—
401C2×C42—1221no—
402C2×C42C22×D6111yescharacteristic
403#107—1221no—
404#107—1221no—
405#106—21no—
406#106—21no—
407#106—1221no—
408#106—1221no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419#103—21no—
420#103—21no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#102—21no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8—1221no—
437C4×C8C22⋊D6111yescharacteristic
438C4×C8C22⋊D6111yescharacteristic
439C4×C8C22×D6111yescharacteristic
440C4×C8C22×D6111yescharacteristic
441#41—42no—
442#41—42no—
443#41—614no—
444#41—1222no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—611no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4—611no—
454C2×D4—1221no—
455C22×C4—41no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4D4×D6111yescharacteristic
460C22×C4#1175822411yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462Q16—2442no—
463SD16—42no—
464SD16—42no—
465SD16—2442no—
466SD16—2442no—
467D8—2442no—
468C2×C8—22no—
469C2×C8—41no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—2442no—
477C2×C8—2442no—
478C2×C8—2442no—
479C2×C8—2442no—
480C2×C8C2×C22⋊D6111yescharacteristic
481C2×C8C2×C22⋊D6111yescharacteristic
482C2×C8C2×C22⋊D6111yescharacteristic
483C2×C8C2×C22⋊D6111yescharacteristic
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—1221no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C42—41no—
503C42—1221no—
504C42—1221no—
505C42—1221no—
506C42D4×D6111yescharacteristic
507C42#1175822411yescharacteristic
508C42C2×C22⋊D6111yescharacteristic
509C42C23×D6111yescharacteristic
510C23—41no—
511C23—1221no—
512C23#1451974811yescharacteristic
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—81no—
518Q8—612no—
519Q8—1221no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—81no—
525D4—82no—
526D4—612no—
527D4—1224no—
528D4—1221no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—1221no—
537C2×C4—1224no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—2442no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4#1451974811yescharacteristic
544C2×C4#1451974811yescharacteristic
545C2×C4#1451974811yescharacteristic
546C2×C4SD16×D6112yes—
547C2×C4#1451979812yes—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C2×#1175822411yescharacteristic
551C8—21no—
552C8—21no—
553C8—21no—
554C8—2441no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C8#1451979712yescomplemented
561C22—21no—
562C22—21no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—1222no—
567C22—2441no—
568C22—2441no—
569C22#1683998812yes—
570C22C2×#1451974811yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—1222no—
578C4—2441no—
579C4—2441no—
580C4#1683998812yes—
581C4C2×SD16×D6111yescharacteristic
582C4C2×#1451979711yescharacteristic
583C4C2×#1451979811yescharacteristic
584C4#1684056011yescharacteristic
585C2—22no—
586C2—161no—
587C2—2442no—
588C2#2145526811yescharacteristic
589C2C2×#1683998811yescharacteristic
590C2#2145553511yescharacteristic
subgroupG/H
centerC2×C4#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3167C2
socleC22×C61C2×#121
prime radical#21455268C2
hypercenter#3167D61
hypocenterC61#19124
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21455601C2semidirect
3. C8×#11758222C2semidirect
4. #21445246C2semidirect
5. #21445239C2semidirect
6. #21445232C2semidirect
7. #21445225C2semidirect
8. #21445198C2semidirect
9. #21445191C2semidirect
10. #21445164C2semidirect
11. #21445157C2semidirect
12. C2×#16839513C2semidirect
13. #21444476C2semidirect
14. #21444468C2semidirect
15. C61×#3167C2semidirect
160. #11758222SD16semidirect
258. C8×C61#135semidirect
290. #3167D61semidirect
362. C61#19124semidirect
560. C8#14519797semidirect

Maximal subgroups

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

subgroupindex
#214556012
#214552682
C8×#117582222
#214452462
#214452392
#214452322
#214452252
#214451982
#214451912
#214451642
#214451572
C2×#168395132
#214444762
#214444682
C61×#31672
#1912461