Finite Group Databasebeta

← #29706594

Subgroups — #29706594

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456285C211yesmaximal, characteristic
2#21446314C211yesmaximal, characteristic, complemented
3#21446312C211yesmaximal, characteristic, complemented
4#21446307C211yesmaximal, characteristic, complemented
5#21446305C211yesmaximal, characteristic, complemented
6#21446300C211yesmaximal, characteristic, complemented
7#21446298C211yesmaximal, characteristic, complemented
8#21446293C211yesmaximal, characteristic, complemented
9#21446291C211yesmaximal, characteristic, complemented
10#21446221C211yesmaximal, characteristic, complemented
11#21446206C211yesmaximal, characteristic, complemented
12#21446201C211yesmaximal, characteristic, complemented
13#21446201C211yesmaximal, characteristic, complemented
14#21445968C211yesmaximal, characteristic, complemented
15C61×#1841C211yesmaximal, characteristic, complemented
16C2×#14519810C2211yescharacteristic
17C2×#14519810C2211yescharacteristic
18C2×#14519805C2211yescharacteristic
19C2×#14519805C2211yescharacteristic
20#16840528C2211yescharacteristic
21#16840528C2211yescharacteristic
22#16840312C2211yescharacteristic
23C2×#14519765C2211yescharacteristic
24#16840239C2211yescharacteristic
25#16840236C2211yescharacteristic
26#16840156C412yescomplemented
27#16840151C412yescomplemented
28#16840146C412yescomplemented
29#16840146C412yescomplemented
30#16839780C2211yescharacteristic
31#16839780C2211yescharacteristic
32#16839778C2211yescharacteristic
33#16839778C2211yescharacteristic
34#16839772C2211yescharacteristic
35#16839770C2211yescharacteristic
36#16839768C2211yescharacteristic
37#16839766C2211yescharacteristic
38C2×#14519695C2211yescharacteristic
39C2×#14519693C2211yescharacteristic
40#16839735C2211yescharacteristic
41#16839735C2211yescharacteristic
42#16839533C2211yescharacteristic
43#16839533C2211yescharacteristic
44#16839522C2211yescharacteristic
45#16839522C2211yescharacteristic
46#16839520C2211yescharacteristic
47#16839519C2211yescharacteristic
48C4×M16×C61C2211yescharacteristic
49C61×#344C2211yescharacteristic
50C61×#344C2211yescharacteristic
51C61×#339C2211yescharacteristic
52C61×#339C2211yescharacteristic
53C61×#338C2211yescharacteristic
54C61×H(3,ℤ/4)C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56C2×#11758222C2311yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×D244C2311yescharacteristic
59#14519810—41no—
60#14519810—41no—
61#14519810—41no—
62#14519810—41no—
63#14519805—41no—
64#14519805—41no—
65#14519805—41no—
66#14519805—41no—
67C2×Dic244C2×C412yescomplemented
68C2×Dic244C2×C412yescomplemented
69C2×#11758190C2×C412yes—
70C2×#11758190C2×C412yes—
71C2×#11758190C2×C412yescomplemented
72C2×#11758190C2×C412yescomplemented
73C2×D488C2×C412yes—
74C2×D488C2×C412yes—
75#14519785—41no—
76#14519785—41no—
77#14519783—41no—
78#14519783—41no—
79C2×#11758194C2311yescharacteristic
80C2×#11758194C2311yescharacteristic
81M16×D61—41no—
82M16×D61—41no—
83#14519777C2×C411yescharacteristic, complemented
84#14519774C2×C411yescharacteristic, complemented
85#14519765—41no—
86#14519765C2×C412yes—
87#14519765C2×C412yes—
88#14519765C2×C411yescharacteristic, complemented
89#14519765C2×C411yescharacteristic, complemented
90#14519751—41no—
91C4×Dic122—41no—
92#14519742—41no—
93C4×D244—41no—
94C4×D244—41no—
95#14519739—41no—
96#14519736C2311yescharacteristic
97#14519736C2311yescharacteristic
98#14519733—41no—
99C2×#11758198—41no—
100C22⋊C4×D61—41no—
101#14519705—41no—
102#14519705—41no—
103#14519703—41no—
104#14519703—41no—
105#14519702—41no—
106#14519700—41no—
107#14519699—41no—
108#14519697—41no—
109#14519695—41no—
110#14519693—41no—
111#14519691—41no—
112#14519691—41no—
113C2×M16×C61C2311yescharacteristic
114C2×M16×C61C2311yescharacteristic
115C2×M16×C61C2311yescharacteristic
116C2×M16×C61C2311yescharacteristic
117#117×C61C2311yescharacteristic
118#117×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#97×C61C2×C412yescomplemented
121C4×C8×C61C2×C412yescomplemented
122C23×D61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—21no—
126C2×C22⋊D61—21no—
127C22×Dic61—21no—
128#11758222—21no—
129#11758222—21no—
130#11758222—21no—
131#11758222—21no—
132#11758222—21no—
133#11758222—21no—
134#11758222—21no—
135#11758222—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—41no—
142C2×D244C22×C412yes—
143C2×D244C22×C411yescharacteristic
144C2×D244C22×C411yescharacteristic
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122C22×C412yes—
154C2×Dic122C22×C411yescharacteristic
155C2×Dic122C22×C411yescharacteristic
156M16×C61—21no—
157M16×C61—21no—
158M16×C61—21no—
159M16×C61—21no—
160M16×C61—21no—
161M16×C61—21no—
162M16×C61—21no—
163M16×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C22×C412yes—
167C2×C8×C61C22×C412yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61C22×C411yescharacteristic
176C42×C61C22×C411yescharacteristic
177C22⋊Dic61—21no—
178C22⋊Dic61—21no—
179#11758198—42no—
180#11758198—42no—
181#11758198—41no—
182#11758198—41no—
183#11758198—42no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187#11758196—42no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—42no—
191#11758194—21no—
192#11758194—21no—
193#11758194—21no—
194#11758194—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197Dic244—42no—
198Dic244—42no—
199D488—42no—
200D488—42no—
201#11758190—42no—
202#11758190—42no—
203#11758190—42no—
204#11758190—42no—
205#11758189—42no—
206#11758189—42no—
207C8×D61—42no—
208C8×D61—42no—
209C23×C61H(5,𝔽2)11yescharacteristic
210C22×D61—22no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—82no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—81no—
221C2×C4×C61C23×C411yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C22⋊D61—41no—
228C22⋊D61—41no—
229C22⋊D61—41no—
230C22⋊D61—41no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238D244—21no—
239D244—21no—
240D244—22no—
241D244—22no—
242D244—21no—
243D244—21no—
244D244—41no—
245D244—41no—
246D244—82no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—162no—
252Dic122—21no—
253Dic122—21no—
254Dic122—22no—
255Dic122—22no—
256Dic122—21no—
257Dic122—21no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—22no—
261C8×C61—22no—
262C8×C61—41no—
263C8×C61—41no—
264#92418—41no—
265#92418—41no—
266#23362—611nomaximal, Sylow
267C22×C61—41no—
268C22×C61#51811yescharacteristic
269C22×C61#56011yescharacteristic
270C22×C61#56011yescharacteristic
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—42no—
274C2×D61—82no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—82no—
278C2×D61—161no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61#51811yescharacteristic
284C4×C61#51811yescharacteristic
285C4×C61#56011yescharacteristic
286C4×C61#56011yescharacteristic
287Dic61—42no—
288Dic61—41no—
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292#3153—611no—
293#1905—611no—
294#1905—611no—
295#1904—611no—
296#1904—611no—
297#1903—611no—
298#1903—611no—
299#1902—611no—
300#1902—611no—
301#1888—611no—
302#1885—611no—
303#1884—611no—
304#1884—611no—
305#1841—611no—
306#1841D6111yescharacteristic, complemented
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61#238011yescharacteristic
311D61—81no—
312D61—81no—
313D61—82no—
314C2×#137—611no—
315C2×#137—611no—
316C2×AGL(1,ℤ/8)—611no—
317C2×AGL(1,ℤ/8)—611no—
318#568—611no—
319#568—611no—
320#532—611no—
321C2×#124—611no—
322#519—611no—
323#518—611no—
324#497—612no—
325#496—612no—
326#495—612no—
327#495—612no—
328#421—611no—
329#421—611no—
330#421—611no—
331#421—611no—
332#419—611no—
333#419—611no—
334#418—611no—
335#418—611no—
336C2×#101—611no—
337C2×#100—611no—
338#410—611no—
339#410—611no—
340C4×M16C2×D6111yescharacteristic
341#344—611no—
342#344—611no—
343#344C2×D6111yescharacteristic
344#344C2×D6111yescharacteristic
345#339—611no—
346#339—611no—
347#339C2×D6111yescharacteristic
348#339C2×D6111yescharacteristic
349#338—611no—
350#338C2×D6111yescharacteristic
351H(3,ℤ/4)—611no—
352H(3,ℤ/4)C2×D6111yescharacteristic
353C61#2336211yescharacteristic, complemented, Sylow
354C2×#41—611no—
355C2×#41—611no—
356C22×Q8—611no—
357C22×D4—611no—
358#137—2441no—
359#137—2441no—
360#137—2441no—
361#137—2441no—
362AGL(1,ℤ/8)—2441no—
363AGL(1,ℤ/8)—2441no—
364AGL(1,ℤ/8)—2441no—
365AGL(1,ℤ/8)—2441no—
366C2×Q16—612no—
367C2×Q16—612no—
368C2×SD16—612no—
369C2×SD16—612no—
370C2×SD16—612no—
371C2×SD16—612no—
372C2×D8—612no—
373C2×D8—612no—
374#131—2441no—
375#131—2441no—
376#131—2441no—
377#131—2441no—
378C2×M16—611no—
379C2×M16—611no—
380C2×M16—2441no—
381C2×M16—2441no—
382C2×M16C22×D6111yescharacteristic
383C2×M16C22×D6111yescharacteristic
384C2×M16C22×D6111yescharacteristic
385C2×M16C22×D6111yescharacteristic
386C4⋊Q8—611no—
387#127—611no—
388#124—612no—
389#124—612no—
390#124—611no—
391#124—611no—
392#124—2441no—
393#121—2441no—
394C4×Q8—2441no—
395C4×Q8—2441no—
396C4×D4—2441no—
397C4×D4—2441no—
398C4×D4—2441no—
399#117—611no—
400#117—611no—
401#117—2441no—
402#117C22×D6111yescharacteristic
403#117C22×D6111yescharacteristic
404C2×C22⋊C4—2441no—
405C2×C22⋊C4—2441no—
406C2×C42C22×D6111yescharacteristic
407#104—2441no—
408#104—2441no—
409#104—2441no—
410#104—2441no—
411#103—2441no—
412#103—2441no—
413#102—2441no—
414#102—2441no—
415#101—2441no—
416#100—2441no—
417#99—2441no—
418#99—2441no—
419#97C4×D6112yescomplemented
420C4×C8C4×D6112yescomplemented
421C24—1221no—
422#41—1221no—
423#41—1221no—
424#41—1221no—
425#41—1221no—
426#41—1221no—
427#41—1221no—
428#41—1221no—
429#41—1221no—
430C2×Q8—612no—
431C2×Q8—611no—
432C2×Q8—611no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×D4—612no—
439C2×D4—611no—
440C2×D4—611no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—2441no—
449C22×C4—21no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—2442no—
454C22×C4C23×D6111yescharacteristic
455Q16—2442no—
456Q16—2442no—
457SD16—2442no—
458SD16—2442no—
459SD16—2442no—
460SD16—2442no—
461D8—2442no—
462D8—2442no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—21no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16—2442no—
476M16—2442no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—2442no—
482C2×C8—2442no—
483C2×C8C2×C4×D6112yes—
484C2×C8C2×C4×D6112yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—2441no—
490C4⋊C4—2442no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2442no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—1221no—
504C42—1221no—
505C42—2442no—
506C42C2×C4×D6111yescharacteristic
507C42C2×C4×D6111yescharacteristic
508C23—1222no—
509C23—1221no—
510C23—1221no—
511C23—4882no—
512C23—2441no—
513C23—2441no—
514C23#1451982311yescharacteristic
515Q8—1221no—
516Q8—1221no—
517Q8—1222no—
518Q8—1222no—
519Q8—1221no—
520Q8—1221no—
521Q8—2441no—
522Q8—2441no—
523D4—1221no—
524D4—1221no—
525D4—1222no—
526D4—1222no—
527D4—1221no—
528D4—1221no—
529D4—4882no—
530D4—2441no—
531D4—2441no—
532D4—2441no—
533D4—2441no—
534D4—2441no—
535D4—2441no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—81no—
541C2×C4—1222no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—9762no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C22×C4×D6111yescharacteristic
554C2×C4#1451982311yescharacteristic
555C2×C4#1451982311yescharacteristic
556C2×C4#1451982311yescharacteristic
557C2×C4#1451982311yescharacteristic
558C2×C4#1451982311yescharacteristic
559C8—22no—
560C8—22no—
561C8—41no—
562C8—41no—
563C8—2441no—
564C8—2441no—
565C22—41no—
566C22—9761no—
567C22—4881no—
568C22—4881no—
569C22—4882no—
570C22—4882no—
571C22—2441no—
572C22—2441no—
573C22—2442no—
574C22#1684023611yescharacteristic
575C22#1684050311yescharacteristic
576C22#1684050311yescharacteristic
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—4881no—
582C4—4881no—
583C4—2441no—
584C4—2441no—
585C4—2442no—
586C4#1684023611yescharacteristic
587C4#1684023611yescharacteristic
588C4#1684050311yescharacteristic
589C4#1684050311yescharacteristic
590C2—21no—
591C2—21no—
592C2—21no—
593C2—4882no—
594C2—4881no—
595C2—4881no—
596C2#2144899511yescharacteristic
subgroupG/H
centerC2#21448995
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1841C2
socleC2×C61#2380
prime radical#21456285C2
cosocleC22×C4×C61C24
hypercenter#1841D61
hypocenterC61#23362
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1841

Complements

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

NH ≅ G/Ntype
2. #21446314C2semidirect
3. #21446312C2semidirect
4. #21446307C2semidirect
5. #21446305C2semidirect
6. #21446300C2semidirect
7. #21446298C2semidirect
8. #21446293C2semidirect
9. #21446291C2semidirect
10. #21446221C2semidirect
11. #21446206C2semidirect
12. #21446201C2semidirect
13. #21446201C2semidirect
14. #21445968C2semidirect
15. C61×#1841C2semidirect
26. #16840156C4semidirect
27. #16840151C4semidirect
28. #16840146C4semidirect
29. #16840146C4semidirect
67. C2×Dic244C2×C4semidirect
68. C2×Dic244C2×C4semidirect
71. C2×#11758190C2×C4semidirect
72. C2×#11758190C2×C4semidirect
83. #14519777C2×C4semidirect
84. #14519774C2×C4semidirect
88. #14519765C2×C4semidirect
89. #14519765C2×C4semidirect
120. #97×C61C2×C4semidirect
121. C4×C8×C61C2×C4semidirect
306. #1841D61semidirect
353. C61#23362semidirect
419. #97C4×D61semidirect
420. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214562852
#214463142
#214463122
#214463072
#214463052
#214463002
#214462982
#214462932
#214462912
#214462212
#214462062
#214462012
#214459682
C61×#18412
#2336261