Finite Group Databasebeta

← #29662718

Subgroups — #29662718

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456394C211yesmaximal, characteristic, complemented
2#21455274C211yesmaximal, characteristic, complemented
3#21454786C211yesmaximal, characteristic, complemented
4#21445389C211yesmaximal, characteristic, complemented
5#21445385C211yesmaximal, characteristic, complemented
6#21445382C211yesmaximal, characteristic, complemented
7#21445378C211yesmaximal, characteristic, complemented
8#21445335C211yesmaximal, characteristic, complemented
9#21445330C211yesmaximal, characteristic, complemented
10#21445301C211yesmaximal, characteristic, complemented
11#21445296C211yesmaximal, characteristic, complemented
12C2×#16839512C211yesmaximal, characteristic, complemented
13#21444469C211yesmaximal, characteristic, complemented
14#21444468C211yesmaximal, characteristic, complemented
15C61×#3035C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17#16840312C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840138C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840108C2211yescharacteristic
22#16839996—21no—
23#16839996—21no—
24#16839984—21no—
25#16839977—21no—
26#16839972C2211yescharacteristic
27#16839967C2211yescharacteristic
28#16839962C2211yescharacteristic
29#16839957C2211yescharacteristic
30#16839893C2211yescharacteristic
31#16839893C2211yescharacteristic
32C4×#11758199—21no—
33C4×#11758199—21no—
34#16839840C2211yescharacteristic
35#16839840C2211yescharacteristic
36#16839834—21no—
37#16839834—21no—
38#16839815—21no—
39#16839815C2211yescharacteristic
40C2×C4⋊#92418C2211yescharacteristic
41#16839756—21no—
42C4×#11758194—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839512C2211yescharacteristic
45#16839512C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839506C2211yescharacteristic
53#16839506C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#532C2211yescharacteristic
58C4×#41×C61C2211yescharacteristic
59C61×#464—21no—
60C61×#463—21no—
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#460C2211yescharacteristic
64C61×#459C2211yescharacteristic
65C2×C4⋊C8×C61C2211yescharacteristic
66C61×#416—21no—
67C2×#11758222—21no—
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758201—21no—
72C2×#11758199—21no—
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758194—21no—
76C2×#11758194—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#14519710—21no—
88#14519710—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519700—21no—
97#14519700—21no—
98#14519698—21no—
99#14519698—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—21no—
103#14519697—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107#14519688—21no—
108#14519688—21no—
109C4×#92418—21no—
110C4×#92418D411yescharacteristic
111C4×#92418D411yescharacteristic
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C2×#41×C61—21no—
115C2×#41×C61D411yescharacteristic
116#135×C61—44no—
117C2×Q16×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×D8×C61—21no—
121C22×C8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123#127×C61C2311yescharacteristic
124#124×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127C4×Q8×C61D411yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61D411yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#117×C61—21no—
136#117×C61D411yescharacteristic
137C2×C42×C61—21no—
138C2×C42×C61C2311yescharacteristic
139C4⋊C8×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#103×C61—21no—
143#103×C61—21no—
144#102×C61—21no—
145#102×C61—21no—
146#41×C61—22no—
147#41×C61—22no—
148#41×C61—42no—
149#41×C61—42no—
150C2×Q8×C61—21no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—41no—
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—41no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61#4111yescharacteristic
165C2×C22⋊D61—41no—
166#11758222—42no—
167#11758222—42no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×C4×D61—41no—
172C2×Dic122—21no—
173Q16×C61—42no—
174SD16×C61—42no—
175SD16×C61—42no—
176D8×C61—42no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—42no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61#4111yescharacteristic
197C42×C61C2411yescharacteristic
198#11758202—42no—
199#11758201—42no—
200#11758200—42no—
201#11758199—42no—
202#11758199—42no—
203#11758199—42no—
204#11758198—41no—
205#11758198—41no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208#11758194—42no—
209#11758194—42no—
210C2×#92418—22no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—41no—
216C2×#92418—42no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C2×#92418C2×D411yescharacteristic
220C2×#92418C2×D411yescharacteristic
221C23×C61—21no—
222C23×C61—41no—
223C23×C61#12111yescharacteristic
224C22×D61—41no—
225Q8×C61—21no—
226Q8×C61—21no—
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—42no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—81no—
237D4×C61—82no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—41no—
249C2×C4×C61—82no—
250C2×C4×C61#12111yescharacteristic
251C2×C4×C61#12111yescharacteristic
252C2×C4×C61#12111yescharacteristic
253C2×C4×C61AGL(1,ℤ/8)12yes—
254C2×C4×C61AGL(1,ℤ/8)12yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—82no—
259C2×Dic61—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—81no—
265C4×D61—82no—
266Dic122—41no—
267Dic122—41no—
268C8×C61—41no—
269C8×C61—41no—
270#92418—21no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#92418#13512yescomplemented
278#19157—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61#46912yes—
288C22×C61C2×#12111yescharacteristic
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61#46912yes—
300C4×C61C2×#13511yescharacteristic
301C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C61#57511yescharacteristic
304Dic61—81no—
305#3169—611no—
306#3035—611no—
307#3035D6111yescharacteristic, complemented
308#2965—611no—
309#1702—611no—
310#1702—611no—
311#1701—611no—
312#1701—611no—
313#1694—611no—
314#1693—611no—
315#1688—611no—
316#1687—611no—
317C2×#334—611no—
318#1540—611no—
319#1540—611no—
320C2×C61—22no—
321C2×C61—81no—
322C2×C61—161no—
323C2×C61#303511yescharacteristic
324C2×C61C2×#46911yescharacteristic
325C2×C61#306711yescharacteristic
326D61—161no—
327C2×#135—21no—
328C2×#135—1221no—
329#532—611no—
330#532C2×D6111yescharacteristic
331C4×#41C2×D6111yescharacteristic
332#500—611no—
333#493—611no—
334#486—611no—
335#486—611no—
336#466—1221no—
337#466—1221no—
338#464—21no—
339#464—1221no—
340#463—21no—
341#463—1221no—
342#462—611no—
343#462C2×D6111yescharacteristic
344#461—611no—
345#461C2×D6111yescharacteristic
346#460—611no—
347#460C2×D6111yescharacteristic
348#459—611no—
349#459C2×D6111yescharacteristic
350#446—611no—
351#446—611no—
352C4×D8—1221no—
353C4×D8—1221no—
354#436—611no—
355#436—611no—
356#435—1221no—
357#435—1221no—
358#431—611no—
359#431—1221no—
360C2×C4⋊C8—611no—
361C2×C4⋊C8C2×D6111yescharacteristic
362#416—21no—
363#416—1221no—
364C4×M16—1221no—
365C2×C4×C8—611no—
366#334—611no—
367#334—611no—
368#334—611no—
369#334—611no—
370#332—611no—
371#332—611no—
372#332—611no—
373#332—611no—
374#331—611no—
375#331—611no—
376#331—611no—
377#331—611no—
378C61#1915711yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C2×#41—1221no—
381C2×#41C22⋊D6111yescharacteristic
382#135—44no—
383#135—2444no—
384C2×Q16—21no—
385C2×Q16—1221no—
386C2×SD16—21no—
387C2×SD16—21no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×D8—21no—
391C2×D8—1221no—
392C2×D8—1221no—
393C2×D8—1221no—
394C2×M16—1221no—
395C2×M16—1221no—
396C22×C8—21no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—1221no—
400C4⋊Q8—611no—
401C4⋊Q8C22×D6111yescharacteristic
402#127—611no—
403#127C22×D6111yescharacteristic
404#124—21no—
405#124—1221no—
406#121—21no—
407#121—21no—
408#121—1221no—
409#121—1221no—
410C4×Q8C22⋊D6111yescharacteristic
411C4×Q8C22×D6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4C22⋊D6111yescharacteristic
419C4×D4C22×D6111yescharacteristic
420#117—21no—
421#117C22⋊D6111yescharacteristic
422C2×C42—21no—
423C2×C42C22×D6111yescharacteristic
424#107—1221no—
425#107—1221no—
426C4⋊C8—21no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8—1221no—
434C4⋊C8C22×D6111yescharacteristic
435C4⋊C8C22×D6111yescharacteristic
436#103—21no—
437#103—21no—
438#103—1221no—
439#103—1221no—
440#102—21no—
441#102—21no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451#97—1221no—
452#97—1221no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8—611no—
457C4×C8—1221no—
458#41—22no—
459#41—22no—
460#41—42no—
461#41—42no—
462#41—2442no—
463#41—2442no—
464C2×Q8—21no—
465C2×Q8—1221no—
466C2×Q8C2×C22⋊D6111yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—41no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—2441no—
476C2×D4C2×C22⋊D6111yescharacteristic
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4—2441no—
482C22×C4D4×D6111yescharacteristic
483C22×C4#1175822411yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485Q16—42no—
486Q16—2442no—
487SD16—42no—
488SD16—42no—
489SD16—2442no—
490SD16—2442no—
491D8—42no—
492D8—2442no—
493D8—2442no—
494D8—2442no—
495M16—2442no—
496M16—2442no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—42no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—611no—
504C2×C8—1222no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—2441no—
510C2×C8—2442no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42D4×D6111yescharacteristic
530C42#1175822411yescharacteristic
531C42C2×C22⋊D6111yescharacteristic
532C42C23×D6111yescharacteristic
533C23—21no—
534C23—41no—
535C23—2441no—
536C23#1451974911yescharacteristic
537Q8—21no—
538Q8—21no—
539Q8—41no—
540Q8—41no—
541Q8—2441no—
542Q8—2441no—
543D4—21no—
544D4—21no—
545D4—42no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—81no—
551D4—82no—
552D4—4881no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—82no—
570C2×C4—4882no—
571C2×C4—2441no—
572C2×C4#1451974911yescharacteristic
573C2×C4#1451974911yescharacteristic
574C2×C4#1451974911yescharacteristic
575C2×C4#1451980212yes—
576C2×C4#1451980312yes—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4C2×#1175822411yescharacteristic
580C8—41no—
581C8—41no—
582C8—612no—
583C8—1221no—
584C8—1221no—
585C8—1221no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C8—2441no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22#1684001312yes—
602C22C2×#1451974911yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—4881no—
611C4#1684001312yes—
612C4#135×D6111yescharacteristic
613C4C2×#1451980211yescharacteristic
614C4C2×#1451980311yescharacteristic
615C4#1684055911yescharacteristic
616C2—22no—
617C2—81no—
618C2—161no—
619C2—9761no—
620C2#2145527011yescharacteristic
621C2C2×#1684001311yescharacteristic
622C2#2145553611yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3035C2
socleC22×C61C2×#121
hypercenter#3035D61
hypocenterC61#19157
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #21456394C2semidirect
2. #21455274C2semidirect
3. #21454786C2semidirect
4. #21445389C2semidirect
5. #21445385C2semidirect
6. #21445382C2semidirect
7. #21445378C2semidirect
8. #21445335C2semidirect
9. #21445330C2semidirect
10. #21445301C2semidirect
11. #21445296C2semidirect
12. C2×#16839512C2semidirect
13. #21444469C2semidirect
14. #21444468C2semidirect
15. C61×#3035C2semidirect
277. #92418#135semidirect
307. #3035D61semidirect
378. C61#19157semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214552742
#214547862
#214453892
#214453852
#214453822
#214453782
#214453352
#214453302
#214453012
#214452962
C2×#168395122
#214444692
#214444682
C61×#30352
#1915761