Finite Group Databasebeta

← #29663041

Subgroups — #29663041

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455873C211yesmaximal, characteristic, complemented
2#21455268C211yesmaximal, characteristic, complemented
3#21454837C211yesmaximal, characteristic, complemented
4#21445377C211yesmaximal, characteristic, complemented
5#21445349C211yesmaximal, characteristic, complemented
6#21445315C211yesmaximal, characteristic, complemented
7#21445295C211yesmaximal, characteristic, complemented
8#21445260C211yesmaximal, characteristic, complemented
9#21445232C211yesmaximal, characteristic, complemented
10#21445198C211yesmaximal, characteristic, complemented
11#21445176C211yesmaximal, characteristic, complemented
12#21444571C211yesmaximal, characteristic, complemented
13#21444524C211yesmaximal, characteristic, complemented
14#21444514C211yesmaximal, characteristic, complemented
15C61×#3184C211yesmaximal, characteristic, complemented
16C2×#14519797—21no—
17C4×#11758222C2211yescharacteristic
18#16840083—21no—
19#16840054C2211yescharacteristic
20#16840049C2211yescharacteristic
21#16839981—21no—
22#16839974—21no—
23#16839969C2211yescharacteristic
24#16839964C2211yescharacteristic
25#16839959C2211yescharacteristic
26#16839959C2211yescharacteristic
27#16839954C2211yescharacteristic
28#16839954C2211yescharacteristic
29#16839905—21no—
30C8×Dic122C2211yescharacteristic
31C8×Dic122C2211yescharacteristic
32C4×#11758200—21no—
33C4×#11758199—21no—
34#16839841C2211yescharacteristic
35#16839841C2211yescharacteristic
36#16839838—21no—
37#16839838—21no—
38#16839819C2211yescharacteristic
39#16839817—21no—
40#16839815C2211yescharacteristic
41C2×C4⋊#92418C2211yescharacteristic
42#16839758—21no—
43#16839714—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839506C2211yescharacteristic
53#16839506C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C61×#532C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#469—21no—
63C61×#469—21no—
64C61×#442—21no—
65C61×#442—21no—
66C4×M16×C61C2211yescharacteristic
67C2×#11758222D411yescharacteristic
68#14519797—44no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C22×#92418—21no—
78C4×Dic122D411yescharacteristic
79C4×Dic122C2311yescharacteristic
80C4×D244D411yescharacteristic
81C4×D244C2311yescharacteristic
82C4×C22⋊D61—21no—
83C4×C22⋊D61—21no—
84#14519736D411yescharacteristic
85#14519734—21no—
86C42×D61—21no—
87#14519710—21no—
88#14519708—21no—
89C4⋊#92418—21no—
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519706—21no—
98#14519701—21no—
99#14519701—21no—
100#14519700—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519697—21no—
106C22⋊#92418—21no—
107C22⋊#92418—21no—
108#14519689—21no—
109#14519689—21no—
110#14519688—21no—
111C4×#92418—21no—
112C8×Dic61—21no—
113C8×Dic61—21no—
114C2×#41×C61—21no—
115C2×#41×C61—21no—
116C2×D8×C61—21no—
117C2×D8×C61—21no—
118C2×M16×C61D411yescharacteristic
119C2×M16×C61D411yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#127×C61C2311yescharacteristic
124#124×C61—21no—
125#124×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128#121×C61—21no—
129#121×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#106×C61—21no—
136#106×C61—21no—
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#97×C61D411yescharacteristic
142#97×C61D411yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—82no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C2×C22⋊D61—21no—
164#11758222—22no—
165#11758222—41no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×Dic122C2×D411yescharacteristic
171D8×C61—42no—
172D8×C61—42no—
173M16×C61—22no—
174M16×C61—41no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61#4111yescharacteristic
190C42×C61C2411yescharacteristic
191C22⋊Dic61—21no—
192#11758202—42no—
193#11758201—42no—
194#11758200—42no—
195#11758200—42no—
196#11758199—42no—
197#11758199—42no—
198#11758198—21no—
199#11758198—21no—
200C4⋊Dic61C2×D411yescharacteristic
201C4⋊Dic61C2×D411yescharacteristic
202#11758196—21no—
203#11758196—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—42no—
214#11758189—42no—
215#11758189—42no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61#12111yescharacteristic
219C22×D61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—81no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—81no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—82no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—82no—
243C2×C4×C61—82no—
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C22⋊D61—42no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254D244—21no—
255D244—21no—
256C4×D61—42no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—81no—
260Dic122—21no—
261Dic122AGL(1,ℤ/8)12yes—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61#13512yescomplemented
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19184—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61C2×#12111yescharacteristic
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61C2×#13511yescharacteristic
293C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C61#57511yescharacteristic
295C4×C61#57511yescharacteristic
296Dic61—22no—
297Dic61—41no—
298Dic61—41no—
299#3184D6111yescharacteristic, complemented
300#3106—611no—
301#3035—611no—
302#2971—611no—
303#1701—611no—
304#1697—611no—
305#1691—611no—
306#1687—611no—
307#1682—611no—
308#1678—611no—
309#1672—611no—
310#1668—611no—
311#1565—611no—
312#1553—611no—
313#1551—611no—
314C2×C61—41no—
315C2×C61—161no—
316C2×C61—161no—
317C2×C61#303511yescharacteristic
318C2×C61#305511yescharacteristic
319C2×C61#306811yescharacteristic
320D61—81no—
321C2×#135—1221no—
322#532C2×D6111yescharacteristic
323#532C2×D6111yescharacteristic
324C4×#41—611no—
325#498C2×D6111yescharacteristic
326#493C2×D6111yescharacteristic
327#486C2×D6111yescharacteristic
328#486C2×D6111yescharacteristic
329#482—1221no—
330#477—611no—
331#476—611no—
332#469—21no—
333#469—21no—
334#464—1221no—
335#463—1221no—
336#462—611no—
337#461—611no—
338#460—611no—
339#460—611no—
340#459—611no—
341#459—611no—
342#449—1221no—
343C8×Q8—611no—
344C8×Q8—611no—
345#442—21no—
346#442—21no—
347C4×SD16—1221no—
348C4×D8—1221no—
349#436—611no—
350#436—611no—
351#436—1221no—
352#436—1221no—
353#432—611no—
354#432—1221no—
355#431—611no—
356C2×C4⋊C8—611no—
357#416—1221no—
358C4×M16—1221no—
359C4×M16C2×D6111yescharacteristic
360#335—611no—
361#335—611no—
362#334—611no—
363#334—611no—
364#333—611no—
365#333—611no—
366#332—611no—
367#332—611no—
368#331—611no—
369#331—611no—
370#331—611no—
371#331—611no—
372C61#1918411yescharacteristic, complemented, Sylow
373C2×#41—21no—
374C2×#41—21no—
375C2×#41—611no—
376#135—2444no—
377C2×Q16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×D8—21no—
382C2×D8—21no—
383C2×D8—1221no—
384C2×D8—1221no—
385C2×M16—1221no—
386C2×M16—1221no—
387C2×M16C22⋊D6111yescharacteristic
388C2×M16C22⋊D6111yescharacteristic
389C22×C8—1221no—
390C4⋊Q8C22×D6111yescharacteristic
391C4⋊Q8C22×D6111yescharacteristic
392#127C22×D6111yescharacteristic
393#127C22×D6111yescharacteristic
394#124—21no—
395#124—21no—
396#121—21no—
397#121—21no—
398#121—21no—
399#121—21no—
400C4×Q8—611no—
401C4×Q8—611no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—611no—
407C4×D4—611no—
408C4×D4—1221no—
409C4×D4—1221no—
410#117—611no—
411#117—1221no—
412C2×C42—1221no—
413C2×C42C22×D6111yescharacteristic
414#107—1221no—
415#106—21no—
416#106—21no—
417#106—1221no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426C4⋊C8—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439C22⋊C8—1221no—
440C22⋊C8—1221no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443#97—1221no—
444#97C22⋊D6111yescharacteristic
445#97C22⋊D6111yescharacteristic
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8—1221no—
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451#41—42no—
452#41—42no—
453#41—82no—
454#41—1222no—
455#41—2441no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—611no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—611no—
468C2×D4—1221no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4D4×D6111yescharacteristic
475C22×C4#1175822411yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477Q16—2442no—
478SD16—2442no—
479SD16—2442no—
480SD16—2442no—
481D8—42no—
482D8—42no—
483D8—2442no—
484D8—2442no—
485M16—22no—
486M16—41no—
487M16—2442no—
488M16—2442no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—2442no—
496C2×C8C2×C22⋊D6111yescharacteristic
497C2×C8C2×C22⋊D6111yescharacteristic
498C2×C8C2×C22⋊D6111yescharacteristic
499C2×C8C2×C22⋊D6111yescharacteristic
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—611no—
505C4⋊C4—611no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42D4×D6111yescharacteristic
519C42#1175822411yescharacteristic
520C42C2×C22⋊D6111yescharacteristic
521C42C23×D6111yescharacteristic
522C23—41no—
523C23—41no—
524C23—1221no—
525C23#1451974811yescharacteristic
526Q8—41no—
527Q8—41no—
528Q8—81no—
529Q8—612no—
530Q8—1221no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—81no—
540D4—81no—
541D4—82no—
542D4—82no—
543D4—1221no—
544D4—1221no—
545D4—2442no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—4881no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4#1451974811yescharacteristic
563C2×C4#1451974811yescharacteristic
564C2×C4#1451974811yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C8—21no—
569C8—21no—
570C8—21no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C8—2441no—
575C8—2441no—
576C8—2441no—
577C8#1451979712yescomplemented
578C22—21no—
579C22—21no—
580C22—21no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22C2×#1451974811yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—81no—
596C4—81no—
597C4—1222no—
598C4—2441no—
599C4—2441no—
600C4C2×#1451979711yescharacteristic
601C4AGL(1,ℤ/8)×D6111yescharacteristic
602C4#1684055811yescharacteristic
603C4#1684056011yescharacteristic
604C2—41no—
605C2—161no—
606C2—161no—
607C2—4881no—
608C2#2145526811yescharacteristic
609C2#2145543211yescharacteristic
610C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3184C2
socleC22×C61C2×#121
hypercenter#3184D61
hypocenterC61#19184
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3184

Complements

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

NH ≅ G/Ntype
1. #21455873C2semidirect
2. #21455268C2semidirect
3. #21454837C2semidirect
4. #21445377C2semidirect
5. #21445349C2semidirect
6. #21445315C2semidirect
7. #21445295C2semidirect
8. #21445260C2semidirect
9. #21445232C2semidirect
10. #21445198C2semidirect
11. #21445176C2semidirect
12. #21444571C2semidirect
13. #21444524C2semidirect
14. #21444514C2semidirect
15. C61×#3184C2semidirect
265. C8×C61#135semidirect
299. #3184D61semidirect
372. C61#19184semidirect
577. C8#14519797semidirect

Maximal subgroups

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

subgroupindex
#214558732
#214552682
#214548372
#214453772
#214453492
#214453152
#214452952
#214452602
#214452322
#214451982
#214451762
#214445712
#214445242
#214445142
C61×#31842
#1918461