Finite Group Databasebeta

← #29662439

Subgroups — #29662439

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#21456379C211yesmaximal, characteristic, complemented
2#21455274C211yesmaximal, characteristic
3#41×#92418C211yesmaximal, characteristic, complemented
4#21445247C211yesmaximal, characteristic, complemented
5#21445240C211yesmaximal, characteristic, complemented
6#21445237C211yesmaximal, characteristic, complemented
7#21445230C211yesmaximal, characteristic, complemented
8#21445203C211yesmaximal, characteristic, complemented
9#21445196C211yesmaximal, characteristic, complemented
10#21445169C211yesmaximal, characteristic, complemented
11#21445162C211yesmaximal, characteristic, complemented
12C2×#16839512C211yesmaximal, characteristic, complemented
13#21444476C211yesmaximal, characteristic, complemented
14#21444469C211yesmaximal, characteristic, complemented
15C61×#3076C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840312C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840136C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16839993—21no—
23#16839993—21no—
24#16839984—21no—
25#16839977—21no—
26#16839972C2211yescharacteristic
27#16839967C2211yescharacteristic
28#16839962C2211yescharacteristic
29#16839957C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31Q8×#92418C2211yescharacteristic
32C4×#11758201—21no—
33C4×#11758201—21no—
34D4×#92418—21no—
35D4×#92418—21no—
36D4×#92418C2211yescharacteristic
37D4×#92418C2211yescharacteristic
38#16839815—21no—
39#16839815C2211yescharacteristic
40C2×C4⋊#92418C2211yescharacteristic
41#16839756—21no—
42C2×C4×#92418—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839512C2211yescharacteristic
45#16839512C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#479—21no—
59C61×#478—21no—
60C61×#477C2211yescharacteristic
61C61×#476C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#427—21no—
65C2×C4⋊C8×C61C2211yescharacteristic
66C61×#416—21no—
67C2×#11758222—21no—
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758199—21no—
75C22×#92418—21no—
76C22×#92418—21no—
77C22×#92418—21no—
78C22×#92418D411yescharacteristic
79C22×#92418D411yescharacteristic
80#14519777C2311yescharacteristic
81#14519774C2311yescharacteristic
82#14519765—21no—
83#14519751—21no—
84#14519751—21no—
85C4×D244—21no—
86C4×D244—21no—
87#14519708—21no—
88#14519708—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519701—21no—
97#14519701—21no—
98#14519700—21no—
99#14519700—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—21no—
103#14519697—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107C4×#92418—21no—
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418D411yescharacteristic
111C4×#92418D411yescharacteristic
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C2×#41×C61D411yescharacteristic
115C22×C8×C61—21no—
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#125×C61—21no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61D411yescharacteristic
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61D411yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131#117×C61D411yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—21no—
135#107×C61—21no—
136#106×C61—21no—
137#106×C61—21no—
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—21no—
142#103×C61—21no—
143#102×C61—21no—
144#102×C61—21no—
145#41×C61—22no—
146#41×C61SD1614yescomplemented
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4111yescharacteristic
158C2×C22⋊D61—41no—
159#11758222—42no—
160#11758222—42no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—42no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—41no—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61#4111yescharacteristic
190C42×C61C2411yescharacteristic
191#11758202—42no—
192#11758201—42no—
193#11758201—42no—
194#11758201—42no—
195#11758200—42no—
196#11758199—42no—
197#11758198—41no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C2×#92418—22no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—41no—
207C2×#92418—42no—
208C2×#92418—42no—
209C2×#92418—42no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214C23×C61—21no—
215C23×C61#12111yescharacteristic
216C22×D61—41no—
217Q8×C61—21no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—81no—
221Q8×C61C2×SD1612yes—
222D4×C61—24no—
223D4×C61—21no—
224D4×C61C2×SD1612yes—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—24no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61C2×SD1612yes—
241C2×C4×C61#13512yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C22⋊D61—82no—
246C2×Dic61—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—81no—
252C4×D61—82no—
253Dic122—41no—
254Dic122—41no—
255C8×C61—41no—
256C8×C61—41no—
257#92418—21no—
258#92418—21no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418#13512yescomplemented
265#19124—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—22no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61#46512yes—
272C22×C61C2×#12111yescharacteristic
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C4×C61—21no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61#46512yes—
285C4×C61C22×SD1611yescharacteristic
286C4×C61C2×#13511yescharacteristic
287C4×C61C2×#13511yescharacteristic
288C4×C61#57511yescharacteristic
289Dic61—81no—
290#3167—611no—
291#3076D6111yescharacteristic, complemented
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—42no—
307C2×C61#303511yescharacteristic
308C2×C61C2×#46511yescharacteristic
309C2×C61#306711yescharacteristic
310D61—161no—
311C2×#135—1221no—
312#533C2×D6111yescharacteristic
313#532—611no—
314C4×#41C2×D6111yescharacteristic
315#498—611no—
316#492—611no—
317#487—611no—
318#486—611no—
319#479—21no—
320#478—21no—
321#477C2×D6111yescharacteristic
322#476C2×D6111yescharacteristic
323#475C2×D6111yescharacteristic
324#474C2×D6111yescharacteristic
325#465—1221no—
326#465—1221no—
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—1221no—
336C4×SD16—1221no—
337C8×D4—611no—
338C8×D4—611no—
339C8×D4—1221no—
340C8×D4—1221no—
341#431—611no—
342#431—1221no—
343#427—21no—
344C2×C4⋊C8—611no—
345C2×C4⋊C8C2×D6111yescharacteristic
346#416—21no—
347#416—1221no—
348C2×C4×C8—611no—
349C2×C4×C8—1221no—
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—1221no—
364C2×#41C22⋊D6111yescharacteristic
365#135—2444no—
366C2×Q16—1221no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×D8—1221no—
372C22×C8—21no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—1221no—
376C22×C8—1221no—
377C22×C8—1221no—
378C4⋊Q8—611no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#127—611no—
382#125—21no—
383#124—1221no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386#121—1221no—
387#121—1221no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8C22⋊D6111yescharacteristic
391C4×Q8C22×D6111yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4C22⋊D6111yescharacteristic
397C4×D4C22×D6111yescharacteristic
398#117—21no—
399#117—21no—
400#117C22⋊D6111yescharacteristic
401C2×C42—21no—
402C2×C42C22×D6111yescharacteristic
403#107—21no—
404#107—21no—
405#106—21no—
406#106—21no—
407#106—1221no—
408#106—1221no—
409C4⋊C8—21no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
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—611no—
435C4×C8—611no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8—1221no—
439C4×C8—1221no—
440C4×C8—1221no—
441#41—22no—
442#41—2442no—
443#41—2442no—
444#41#1175820114yescomplemented
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—1221no—
448C2×Q8C2×C22⋊D6111yescharacteristic
449C2×D4—21no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—2441no—
454C2×D4C2×C22⋊D6111yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—2441no—
459C22×C4D4×D6111yescharacteristic
460C22×C4#1175822411yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462Q16—2442no—
463SD16—2442no—
464SD16—2442no—
465SD16—2442no—
466SD16—2442no—
467D8—2442no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—42no—
471C2×C8—611no—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—611no—
475C2×C8—1222no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—2442no—
481C2×C8—2441no—
482C2×C8—2442no—
483C2×C8—2442no—
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—1221no—
493C4⋊C4—1221no—
494C4⋊C4C2×C22⋊D6111yescharacteristic
495C4⋊C4C2×C22⋊D6111yescharacteristic
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—41no—
506C42D4×D6111yescharacteristic
507C42#1175822411yescharacteristic
508C42C2×C22⋊D6111yescharacteristic
509C42C23×D6111yescharacteristic
510C23—21no—
511C23—2441no—
512C23#1451974911yescharacteristic
513Q8—21no—
514Q8—41no—
515Q8—41no—
516Q8—81no—
517Q8—2441no—
518Q8—2441no—
519Q8C2×#1175820112yes—
520D4—24no—
521D4—21no—
522D4—4881no—
523D4—4882no—
524D4—2441no—
525D4—2441no—
526D4—2441no—
527D4—2441no—
528D4C2×#1175820112yes—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—4882no—
542C2×C4—2441no—
543C2×C4#1451974911yescharacteristic
544C2×C4#1451974911yescharacteristic
545C2×C4#1451974911yescharacteristic
546C2×C4C2×#1175820112yes—
547C2×C4#1451979712yes—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C2×#1175822411yescharacteristic
551C8—41no—
552C8—41no—
553C8—612no—
554C8—1221no—
555C8—1221no—
556C8—1221no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C8—2441no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22#1683999312yes—
570C22C2×#1451974911yescharacteristic
571C4—22no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—4881no—
580C4#1683999312yes—
581C4C22×#1175820111yescharacteristic
582C4#135×D6111yescharacteristic
583C4C2×#1451979711yescharacteristic
584C4#1684055911yescharacteristic
585C2—22no—
586C2—42no—
587C2—9761no—
588C2#2145527011yescharacteristic
589C2C2×#1683999311yescharacteristic
590C2#2145553611yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3076C2
socleC22×C61C2×#121
prime radical#21455274C2
hypercenter#3076D61
hypocenterC61#19124
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #21456379C2semidirect
3. #41×#92418C2semidirect
4. #21445247C2semidirect
5. #21445240C2semidirect
6. #21445237C2semidirect
7. #21445230C2semidirect
8. #21445203C2semidirect
9. #21445196C2semidirect
10. #21445169C2semidirect
11. #21445162C2semidirect
12. C2×#16839512C2semidirect
13. #21444476C2semidirect
14. #21444469C2semidirect
15. C61×#3076C2semidirect
146. #41×C61SD16semidirect
264. #92418#135semidirect
291. #3076D61semidirect
362. C61#19124semidirect
444. #41#11758201semidirect

Maximal subgroups

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

subgroupindex
#214563792
#214552742
#41×#924182
#214452472
#214452402
#214452372
#214452302
#214452032
#214451962
#214451692
#214451622
C2×#168395122
#214444762
#214444692
C61×#30762
#1912461