Finite Group Databasebeta

← #29706589

Subgroups — #29706589

Order 15616.

756 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456267C211yesmaximal, characteristic
2#21446308C211yesmaximal, characteristic, complemented
3#21446305C211yesmaximal, characteristic, complemented
4#21446304C211yesmaximal, characteristic, complemented
5#21446303C211yesmaximal, characteristic, complemented
6#21446302C211yesmaximal, characteristic, complemented
7#21446300C211yesmaximal, characteristic, complemented
8#21446299C211yesmaximal, characteristic, complemented
9#21446215C211yesmaximal, characteristic, complemented
10#21446214C211yesmaximal, characteristic, complemented
11#21446202C211yesmaximal, characteristic, complemented
12#21446199C211yesmaximal, characteristic, complemented
13#21445969C211yesmaximal, characteristic, complemented
14#21445967C211yesmaximal, characteristic, complemented
15C61×#1903C211yesmaximal, characteristic, complemented
16C2×#14519805C2211yescharacteristic
17C2×#14519803C2211yescharacteristic
18C2×#14519802C2211yescharacteristic
19#16840528C2211yescharacteristic
20#16840308C2211yescharacteristic
21C2×#14519773C2211yescharacteristic
22#16840237C2211yescharacteristic
23#16840236C2211yescharacteristic
24#16840150C412yescomplemented
25#16840148C412yescomplemented
26#16840137C412yescomplemented
27#16840133C412yescomplemented
28#16839779C2211yescharacteristic
29#16839778C2211yescharacteristic
30#16839778C2211yescharacteristic
31#16839768C2211yescharacteristic
32#16839768C2211yescharacteristic
33#16839767C2211yescharacteristic
34C2×#14519693C2211yescharacteristic
35#16839736C2211yescharacteristic
36#16839735C2211yescharacteristic
37M16×Dic61C2211yescharacteristic
38#16839534C2211yescharacteristic
39#16839534C2211yescharacteristic
40#16839533C2211yescharacteristic
41#16839523C2211yescharacteristic
42#16839523C2211yescharacteristic
43#16839523C2211yescharacteristic
44#16839521C2211yescharacteristic
45#16839520C2211yescharacteristic
46#16839519C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
49C61×#568C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#418C2211yescharacteristic
52C2×#100×C61C2211yescharacteristic
53C61×#344C2211yescharacteristic
54C61×#338C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—41no—
58#14519805—41no—
59#14519805—41no—
60#14519803—41no—
61#14519803—41no—
62#14519802—41no—
63#14519802—41no—
64C2×#11758190C2×C412yescomplemented
65C2×#11758200C2×C412yescomplemented
66C2×#11758201C2×C412yescomplemented
67C2×D488C2×C412yescomplemented
68C2×#11758199C2×C412yescomplemented
69C2×#11758199C2×C412yescomplemented
70#14519785—41no—
71#14519783—41no—
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74M16×D61—41no—
75#14519775C2×C411yescharacteristic, complemented
76#14519773—41no—
77#14519773C2×C412yes—
78#14519773C2×C412yes—
79#14519773C2×C411yescharacteristic, complemented
80#14519766C2×C411yescharacteristic, complemented
81#14519764C2×C411yescharacteristic, complemented
82#14519749—41no—
83C4×D244—41no—
84D4×Dic61—41no—
85D4×Dic61—41no—
86#14519739—41no—
87C4×C22⋊D61—41no—
88#14519736C2311yescharacteristic
89#14519736C2311yescharacteristic
90#14519736C2311yescharacteristic
91C2×C22⋊Dic61—41no—
92C22⋊C4×D61—41no—
93C2×C4×Dic61C2311yescharacteristic
94#14519704—41no—
95#14519703—41no—
96#14519703—41no—
97#14519699—41no—
98#14519699—41no—
99#14519698—41no—
100#14519693—41no—
101#14519692—41no—
102#14519691—41no—
103#14519687C2×C412yescomplemented
104C8×Dic61C2×C412yescomplemented
105C2×#41×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107AGL(1,ℤ/8)×C61—41no—
108AGL(1,ℤ/8)×C61—41no—
109C2×SD16×C61C2×C412yescomplemented
110C2×D8×C61C2×C412yescomplemented
111#131×C61—41no—
112#131×C61—41no—
113C2×M16×C61—41no—
114C2×M16×C61C2311yescharacteristic
115C2×M16×C61C2311yescharacteristic
116C2×M16×C61C2311yescharacteristic
117C2×M16×C61C2311yescharacteristic
118#117×C61C2311yescharacteristic
119#104×C61—41no—
120#102×C61—41no—
121#100×C61—41no—
122C24×C61—21no—
123C23×D61—21no—
124#41×C61—21no—
125#41×C61—21no—
126#41×C61—21no—
127#41×C61—21no—
128C2×Q8×C61C22×C411yescharacteristic
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61C22×C411yescharacteristic
136C2×D4×C61C22×C412yes—
137C22×C4×C61—21no—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—21no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—41no—
143C2×C22⋊D61—84no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146#11758222—21no—
147#11758222—21no—
148#11758222—21no—
149#11758222—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C22×C412yes—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—42no—
159C2×Dic122C22×C411yescharacteristic
160SD16×C61—42no—
161D8×C61—42no—
162M16×C61—21no—
163M16×C61—21no—
164M16×C61—21no—
165M16×C61—21no—
166M16×C61—21no—
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C2×C8×C61C22×C412yes—
176C4⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C42×C61—21no—
179C22⋊Dic61—21no—
180C22⋊Dic61—21no—
181C22⋊Dic61—21no—
182C22⋊Dic61—41no—
183C22⋊Dic61—42no—
184#11758201—42no—
185#11758200—42no—
186#11758199—42no—
187#11758199—42no—
188#11758198—41no—
189#11758198—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193#11758196—41no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61C22×C411yescharacteristic
199C4×Dic61C22×C411yescharacteristic
200#11758194—21no—
201#11758194—21no—
202#11758194—21no—
203#11758194—21no—
204C2×#92418—21no—
205C2×#92418C22×C412yes—
206D488—42no—
207#11758190—42no—
208#11758189—42no—
209C8×D61—42no—
210C23×C61—21no—
211C23×C61—22no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—82no—
215C23×C61H(5,𝔽2)11yescharacteristic
216C22×D61—21no—
217C22×D61—22no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—82no—
221Q8×C61—21no—
222Q8×C61—21no—
223D4×C61—21no—
224D4×C61—22no—
225D4×C61—21no—
226D4×C61—22no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C22⋊D61—41no—
242C22⋊D61—41no—
243C22⋊D61—82no—
244C22⋊D61—84no—
245C22⋊D61—84no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—42no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—162no—
254D244—22no—
255D244—21no—
256D244—21no—
257D244—22no—
258D244—41no—
259D244—41no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—162no—
263Dic122—21no—
264Dic122—21no—
265C8×C61—22no—
266C8×C61—41no—
267C8×C61—41no—
268C8×C61—41no—
269#92418—22no—
270#92418—41no—
271#23361—611nomaximal, Sylow
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—82no—
276C22×C61—81no—
277C22×C61—82no—
278C22×C61—161no—
279C22×C61#51811yescharacteristic
280C22×C61#56011yescharacteristic
281C22×C61#56011yescharacteristic
282C2×D61—41no—
283C2×D61—42no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—81no—
287C2×D61—161no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61#51811yescharacteristic
291C4×C61#51811yescharacteristic
292C4×C61#56011yescharacteristic
293C4×C61#56011yescharacteristic
294Dic61—22no—
295Dic61—41no—
296Dic61—41no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300#3151—611no—
301#1904—611no—
302#1904—611no—
303#1904—611no—
304#1904—611no—
305#1903—611no—
306#1903—611no—
307#1903—611no—
308#1903D6111yescharacteristic, complemented
309#1887—611no—
310#1887—611no—
311#1884—611no—
312#1884—611no—
313#1841—611no—
314#1841—611no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—81no—
319C2×C61—82no—
320C2×C61#238011yescharacteristic
321D61—81no—
322D61—82no—
323C2×AGL(1,ℤ/8)—611no—
324C2×AGL(1,ℤ/8)—611no—
325C2×AGL(1,ℤ/8)—611no—
326C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
327#568—611no—
328#568C2×D6111yescharacteristic
329#532—611no—
330C2×#127—611no—
331#518—611no—
332#518—611no—
333#496—612no—
334#496—612no—
335#493—612no—
336#492—612no—
337#421—611no—
338#421—611no—
339#421—611no—
340#421C2×D6111yescharacteristic
341#418—611no—
342#418—611no—
343#418—611no—
344#418C2×D6111yescharacteristic
345C2×#100—611no—
346C2×#100C2×D6111yescharacteristic
347#410—611no—
348#410—611no—
349C4×M16—611no—
350#344—611no—
351#344—611no—
352#344—611no—
353#344C2×D6111yescharacteristic
354#339—611no—
355#339—611no—
356#339—611no—
357#339—611no—
358#338—611no—
359#338C2×D6111yescharacteristic
360H(3,ℤ/4)—611no—
361H(3,ℤ/4)—611no—
362C61#2336111yescharacteristic, complemented, Sylow
363C2×#41—611no—
364C2×#41C22×D6111yescharacteristic
365C22×D4—611no—
366C22×D4—2441no—
367C22×D4C22×D6111yescharacteristic
368AGL(1,ℤ/8)—41no—
369AGL(1,ℤ/8)—41no—
370AGL(1,ℤ/8)—2441no—
371AGL(1,ℤ/8)—2441no—
372AGL(1,ℤ/8)—2441no—
373AGL(1,ℤ/8)—2441no—
374AGL(1,ℤ/8)—2441no—
375AGL(1,ℤ/8)—2441no—
376C2×SD16—612no—
377C2×SD16—612no—
378C2×SD16—612no—
379C2×SD16C4×D6112yescomplemented
380C2×D8—612no—
381C2×D8—612no—
382C2×D8—612no—
383C2×D8C4×D6112yescomplemented
384#131—41no—
385#131—41no—
386#131—2441no—
387#131—2441no—
388C2×M16—41no—
389C2×M16—611no—
390C2×M16—611no—
391C2×M16—2441no—
392C2×M16C22×D6111yescharacteristic
393C2×M16C22×D6111yescharacteristic
394C2×M16C22×D6111yescharacteristic
395C2×M16C22×D6111yescharacteristic
396C4⋊Q8—611no—
397#127—612no—
398#127—612no—
399#127—611no—
400#127—2441no—
401#124—611no—
402#124—611no—
403#121—2441no—
404C4×D4—2441no—
405C4×D4—2441no—
406C4×D4—2441no—
407C4×D4—2441no—
408C4×D4—2441no—
409#117—611no—
410#117—611no—
411#117—611no—
412#117C22×D6111yescharacteristic
413C2×C22⋊C4—2441no—
414C2×C22⋊C4—2441no—
415C2×C42—611no—
416#104—41no—
417#104—2441no—
418#104—2441no—
419#104—2441no—
420#102—41no—
421#102—2441no—
422#102—2441no—
423#102—2441no—
424#100—41no—
425#100—2441no—
426#99—2441no—
427#99—2441no—
428#97—612no—
429C4×C8—612no—
430C24—21no—
431C24—1221no—
432#41—21no—
433#41—21no—
434#41—21no—
435#41—21no—
436#41—1221no—
437#41—1221no—
438#41—1221no—
439#41—1221no—
440C2×Q8—611no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—611no—
449C2×D4—612no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—4884no—
457C2×D4—2441no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4C2×C4×D6111yescharacteristic
461C2×D4C2×C4×D6112yes—
462C22×C4—21no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4C23×D6111yescharacteristic
468SD16—42no—
469SD16—2442no—
470SD16—2442no—
471SD16—2442no—
472D8—42no—
473D8—2442no—
474D8—2442no—
475D8—2442no—
476M16—21no—
477M16—21no—
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—21no—
483M16—21no—
484M16—42no—
485M16—1221no—
486M16—1221no—
487M16—1221no—
488M16—1221no—
489M16—2442no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—42no—
494C2×C8—612no—
495C2×C8—1221no—
496C2×C8—2442no—
497C2×C8C2×C4×D6112yes—
498C4⋊C4—21no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2441no—
503C22⋊C4—21no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—2441no—
508C22⋊C4—2442no—
509C22⋊C4—2441no—
510C22⋊C4—2442no—
511C42—21no—
512C42—611no—
513C42—611no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—82no—
523C23—1221no—
524C23—1222no—
525C23—4882no—
526C23—2441no—
527C23—2441no—
528C23#1451982311yescharacteristic
529Q8—21no—
530Q8—21no—
531Q8—1221no—
532Q8—1221no—
533D4—21no—
534D4—22no—
535D4—21no—
536D4—22no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—1221no—
542D4—1222no—
543D4—1221no—
544D4—1222no—
545D4—4884no—
546D4—4882no—
547D4—4884no—
548D4—2441no—
549D4—2441no—
550D4—2441no—
551D4—2441no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—9762no—
560C2×C4—9762no—
561C2×C4—2442no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C22×C4×D6111yescharacteristic
568C2×C4#1451982211yescharacteristic
569C2×C4#1451982211yescharacteristic
570C2×C4#1451982211yescharacteristic
571C2×C4#1451982211yescharacteristic
572C2×C4#1451982311yescharacteristic
573C8—22no—
574C8—41no—
575C8—41no—
576C8—41no—
577C8—1222no—
578C8—2441no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—82no—
583C22—81no—
584C22—82no—
585C22—161no—
586C22—9761no—
587C22—4882no—
588C22—4881no—
589C22—4882no—
590C22—2441no—
591C22—2442no—
592C22#1684023411yescharacteristic
593C22#1684050011yescharacteristic
594C22#1684050211yescharacteristic
595C4—41no—
596C4—81no—
597C4—1222no—
598C4—4881no—
599C4—4881no—
600C4—4881no—
601C4—2441no—
602C4—2441no—
603C4#1684023411yescharacteristic
604C4#1684023511yescharacteristic
605C4#1684050011yescharacteristic
606C4#1684050111yescharacteristic
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—81no—
612C2—4882no—
613C2—4881no—
614C2#2144899411yescharacteristic
subgroupG/H
centerC2#21448994
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1903C2
socleC2×C61#2380
prime radical#21456267C2
cosocleC22×C4×C61C24
hypercenter#1903D61
hypocenterC61#23361
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1903

Complements

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

NH ≅ G/Ntype
2. #21446308C2semidirect
3. #21446305C2semidirect
4. #21446304C2semidirect
5. #21446303C2semidirect
6. #21446302C2semidirect
7. #21446300C2semidirect
8. #21446299C2semidirect
9. #21446215C2semidirect
10. #21446214C2semidirect
11. #21446202C2semidirect
12. #21446199C2semidirect
13. #21445969C2semidirect
14. #21445967C2semidirect
15. C61×#1903C2semidirect
24. #16840150C4semidirect
25. #16840148C4semidirect
26. #16840137C4semidirect
27. #16840133C4semidirect
64. C2×#11758190C2×C4semidirect
65. C2×#11758200C2×C4semidirect
66. C2×#11758201C2×C4semidirect
67. C2×D488C2×C4semidirect
68. C2×#11758199C2×C4semidirect
69. C2×#11758199C2×C4semidirect
75. #14519775C2×C4semidirect
79. #14519773C2×C4semidirect
80. #14519766C2×C4semidirect
81. #14519764C2×C4semidirect
103. #14519687C2×C4semidirect
104. C8×Dic61C2×C4semidirect
109. C2×SD16×C61C2×C4semidirect
110. C2×D8×C61C2×C4semidirect
308. #1903D61semidirect
362. C61#23361semidirect
379. C2×SD16C4×D61semidirect
383. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214562672
#214463082
#214463052
#214463042
#214463032
#214463022
#214463002
#214462992
#214462152
#214462142
#214462022
#214461992
#214459692
#214459672
C61×#19032
#2336161