Finite Group Databasebeta

← #29648191

Subgroups — #29648191

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459775C211yesmaximal, characteristic
2#21454837C211yesmaximal, characteristic, complemented
3#21454471C211yesmaximal, characteristic, complemented
4#21444447C212yesmaximal, complemented
5#21444441C212yesmaximal, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444424C211yesmaximal, characteristic, complemented
8#21444423C212yesmaximal, complemented
9#21444415C212yesmaximal, complemented
10#21444408C211yesmaximal, characteristic, complemented
11C61×#1525C211yesmaximal, characteristic, complemented
12#16840498C411yescharacteristic
13#16840486C411yescharacteristic
14#16840476—21no—
15#16840441—22no—
16#16840427C411yescharacteristic
17#16840419C412yes—
18#16840409—21no—
19#16840400C412yes—
20#16840397C411yescharacteristic
21#16840377—21no—
22#16840344—22no—
23#16840296C2211yescharacteristic
24#16840291C2212yes—
25#124×D61C2212yes—
26#16840234—21no—
27C4×#11758222C2211yescharacteristic
28C8×Dic122—21no—
29#16839841—21no—
30#16839838—22no—
31#16839821—22no—
32#16839819C2211yescharacteristic, complemented
33#16839817C2212yescomplemented
34#16839817C2212yescomplemented
35#16839817C2212yescomplemented
36#16839790—21no—
37#16839730—21no—
38#16839714C2212yescomplemented
39C2×#14519688C2211yescharacteristic, complemented
40#16839503C2211yescharacteristic, complemented
41#16839503C2211yescharacteristic, complemented
42#16839503C2212yescomplemented
43#16839502C2212yescomplemented
44#16839502C2212yescomplemented
45#16839501C2211yescharacteristic
46#16839501C2211yescharacteristic
47#16839501C2212yes—
48C61×#532C2211yescharacteristic
49C4×M16×C61C2211yescharacteristic, complemented
50C4×M16×C61C2211yescharacteristic, complemented
51C61×#329C2212yescomplemented
52C61×#329C2212yescomplemented
53C2×#11758222D411yescharacteristic
54C2×Q8×D61C2×C412yes—
55C2×D4×D61—21no—
56C2×D4×D61C2×C412yes—
57C2×#11758189—22no—
58C22×#92418—21no—
59#14519776—21no—
60#14519772—22no—
61#14519771—21no—
62#14519770—22no—
63#14519769C2311yescharacteristic
64#14519766C2×C412yes—
65#14519765C2311yescharacteristic
66#14519764—21no—
67#14519764—22no—
68#14519764C2×C412yes—
69#14519763—22no—
70#14519763—22no—
71#14519760—21no—
72#14519760—22no—
73#14519759—22no—
74#14519758—22no—
75#14519757—21no—
76#14519756—22no—
77#14519756C2×C411yescharacteristic
78#14519755—22no—
79#14519753C2×C412yes—
80#14519751C2×C411yescharacteristic
81#14519749—22no—
82#14519748—22no—
83#14519748—21no—
84#14519748C2×C412yes—
85#14519745—22no—
86#14519745—22no—
87Q8×Dic61C2×C412yes—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122C2×C411yescharacteristic
90C4×Dic122D411yescharacteristic
91C4×D244C2×C411yescharacteristic
92C4×D244C2×C411yescharacteristic
93C4×D244D411yescharacteristic
94D4×Dic61—21no—
95D4×Dic61C2×C412yes—
96#14519739—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61C2×C412yes—
100C4×C22⋊D61D412yes—
101#14519737—22no—
102#14519736D411yescharacteristic
103#14519735—21no—
104#14519734—21no—
105#14519734D412yes—
106#14519733—22no—
107C2×#11758198—22no—
108C22⋊C4×D61—22no—
109C22⋊C4×D61—22no—
110C42×D61C2312yes—
111C4⋊#92418—21no—
112#14519706—22no—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418—21no—
116#14519689—22no—
117#14519689—22no—
118#14519689—22no—
119#14519688—21no—
120#14519688C2311yescharacteristic
121#14519688C2311yescharacteristic
122#14519688C2312yes—
123C8×Dic61—22no—
124C2×#41×C61C2×C412yes—
125C2×M16×C61—21no—
126C2×M16×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128#127×C61C2×C411yescharacteristic
129#124×C61C2312yes—
130#121×C61—21no—
131#121×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#97×C61C2311yescharacteristic
136#97×C61C2312yes—
137#97×C61C2311yescharacteristic
138C4×C8×C61—21no—
139C4×C8×C61—21no—
140C23×D61—22no—
141#41×C61—82no—
142C2×Q8×C61C22×C412yes—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—22no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61C22⋊C412yes—
155C22×Dic61—22no—
156Q8×D61—82no—
157D4×D61—44no—
158D4×D61—82no—
159#11758222—42no—
160C2×D244C22⋊C411yescharacteristic
161C2×D244C22×C411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×C4×D61C22×C412yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D412yes—
167C2×Dic122—22no—
168C2×Dic122C22⋊C411yescharacteristic
169C2×Dic122C22×C411yescharacteristic
170M16×C61—42no—
171M16×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61—22no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C42×C61C22×C411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61C22⋊C412yes—
189#11758198—22no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193#11758198—21no—
194#11758198—22no—
195#11758198—22no—
196#11758198C22⋊C412yes—
197#11758198C22⋊C412yes—
198C4⋊Dic61—22no—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C411yescharacteristic
203#11758196—21no—
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207#11758196—22no—
208#11758196C22⋊C412yes—
209#11758196C22⋊C412yes—
210C4×Dic61—21no—
211C4×Dic61—22no—
212C4×Dic61C22×C412yes—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C2×#92418—22no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—42no—
219#11758189—44no—
220C23×C61—22no—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61—21no—
223C22×D61—44no—
224C22×D61—82no—
225C22×D61—82no—
226C22×D61C2×C22⋊C412yes—
227Q8×C61—42no—
228D4×C61—42no—
229D4×C61—42no—
230D4×C61—81no—
231D4×C61—82no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—82no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C22⋊D61—42no—
244C22⋊D61—44no—
245C22⋊D61—82no—
246C22⋊D61—81no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—82no—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254D244—22no—
255D244—41no—
256C4×D61—42no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61—82no—
261Dic122—22no—
262Dic122—41no—
263Dic122—82no—
264C8×C61—22no—
265C8×C61—42no—
266C8×C61—41no—
267#92418—42no—
268#92418—41no—
269#92418—41no—
270#17681—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—164no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—42no—
291C4×C61#42112yes—
292Dic61—22no—
293Dic61—22no—
294Dic61—22no—
295Dic61—41no—
296Dic61—41no—
297Dic61—41no—
298#3514—611no—
299#2971—611no—
300#2923—611no—
301#1535—612no—
302#1533—612no—
303#1530—611no—
304#1529—611no—
305#1529—612no—
306#1527—612no—
307#1525D6111yescharacteristic, complemented
308#1524—611no—
309C2×C61—41no—
310C2×C61—82no—
311C2×C61#288811yescharacteristic
312C2×C61C2×#42111yescharacteristic
313C2×C61#290111yescharacteristic
314D61—42no—
315D61—81no—
316#559—611no—
317#556—611no—
318#555—1221no—
319#551—1222no—
320D4×Q8—611no—
321#548—612no—
322#546—612no—
323#546—1221no—
324D42—611no—
325#542—1221no—
326#538—1222no—
327#532C2×D6111yescharacteristic
328#529—612no—
329#529—611no—
330C2×#124—612no—
331#518—1221no—
332C4×#41—611no—
333C8×Q8—1221no—
334#436—1221no—
335#436—1222no—
336#433—1222no—
337#432—611no—
338#432—612no—
339#432—612no—
340#432—612no—
341#424—1221no—
342#408—1221no—
343C4×M16—612no—
344C4×M16C2×D6111yescharacteristic, complemented
345C4×M16C2×D6111yescharacteristic, complemented
346C2×#97—611no—
347#330—611no—
348#330—611no—
349#330—612no—
350#329—612no—
351#329—612no—
352#329—611no—
353#329—611no—
354#329—612no—
355#329C2×D6112yescomplemented
356#329C2×D6112yescomplemented
357C61#1768111yescharacteristic, complemented, Sylow
358C2×#41—611no—
359C2×#41C4×D6112yes—
360C22×Q8—612no—
361C22×D4—612no—
362C22×D4—1221no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—1222no—
366C22×C8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8C4×D6111yescharacteristic
369#127C4×D6111yescharacteristic
370#126—1222no—
371#126—1222no—
372#126—1221no—
373#125—611no—
374#124—612no—
375#124—612no—
376#124—611no—
377#124—1222no—
378#124—1222no—
379#124—1222no—
380#124—1221no—
381#124C22×D6112yes—
382#123—1222no—
383#123—1222no—
384#123—1222no—
385#123—1221no—
386C22⋊Q8—612no—
387C22⋊Q8—611no—
388C22⋊Q8—1222no—
389C22⋊Q8—1221no—
390C22⋊Q8—1222no—
391#121—21no—
392#121—21no—
393#121—612no—
394#121—611no—
395#121—1222no—
396#121—1221no—
397#121—1222no—
398#120—1222no—
399#120—1222no—
400C4×Q8—611no—
401C4×Q8—611no—
402C4×Q8—612no—
403C4×Q8—611no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—612no—
407C4×D4—612no—
408C4×D4—612no—
409C4×D4—611no—
410C4×D4—611no—
411C4×D4—611no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4—1221no—
417#117—611no—
418#117—612no—
419#117—1222no—
420#117—1221no—
421#117—1221no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1222no—
425C2×C42—612no—
426C2×C42C22×D6111yescharacteristic
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429C4⋊C8—1222no—
430C4⋊C8—1221no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1222no—
434C22⋊C8—1221no—
435#97—611no—
436#97—611no—
437#97—612no—
438#97—1221no—
439#97C22×D6111yescharacteristic
440#97C22×D6111yescharacteristic
441#97C22×D6112yes—
442C4×C8—21no—
443C4×C8—21no—
444C4×C8—1222no—
445C24—1222no—
446#41—82no—
447#41—2442no—
448C2×Q8—611no—
449C2×Q8—611no—
450C2×Q8—1222no—
451C2×Q8—4882no—
452C2×Q8C2×C4×D6112yes—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—612no—
457C2×D4—611no—
458C2×D4—611no—
459C2×D4—1222no—
460C2×D4—1221no—
461C2×D4—1222no—
462C2×D4—4882no—
463C2×D4—2444no—
464C2×D4C2×C4×D6112yes—
465C22×C4—22no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4C2×C4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475M16—42no—
476M16—42no—
477M16—2444no—
478C2×C8—21no—
479C2×C8—22no—
480C2×C8—21no—
481C2×C8—1222no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—2442no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—612no—
488C4⋊C4—612no—
489C4⋊C4—611no—
490C4⋊C4—611no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—612no—
502C22⋊C4—612no—
503C22⋊C4—612no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C42—612no—
517C42—612no—
518C42—612no—
519C42—1222no—
520C42—1221no—
521C42C2×C4×D6111yescharacteristic
522C42D4×D6111yescharacteristic
523C42D4×D6111yescharacteristic
524C42C23×D6111yescharacteristic
525C23—22no—
526C23—612no—
527C23—1221no—
528C23—4882no—
529C23—4882no—
530C23—2444no—
531C23C22⋊C4×D6111yescharacteristic
532Q8—42no—
533Q8—1222no—
534Q8—4882no—
535Q8—2441no—
536D4—42no—
537D4—42no—
538D4—81no—
539D4—82no—
540D4—1222no—
541D4—4882no—
542D4—4881no—
543D4—2441no—
544D4—2442no—
545D4—2444no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—82no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4C22⋊C4×D6111yescharacteristic
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C8—22no—
570C8—42no—
571C8—41no—
572C8—2441no—
573C8—2441no—
574C8—2442no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—1222no—
582C22—1222no—
583C22—1222no—
584C22—9764no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22C2×C22⋊C4×D6111yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—1222no—
596C4—1222no—
597C4—1222no—
598C4—2441no—
599C4—2441no—
600C4—2441no—
601C4#1683977712yes—
602C2—41no—
603C2—82no—
604C2—4881no—
605C2—2442no—
606C2#2145429511yescharacteristic
607C2C2×#1683977711yescharacteristic
608C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1525C2
socleC22×C61C22×C22⋊C4
prime radical#21459775C2
cosocleC42×C61C24
hypercenter#1525D61
hypocenterC61#17681
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
2. #21454837C2semidirect
3. #21454471C2semidirect
4. #21444447C2semidirect
5. #21444441C2semidirect
6. #21444430C2semidirect
7. #21444424C2semidirect
8. #21444423C2semidirect
9. #21444415C2semidirect
10. #21444408C2semidirect
11. C61×#1525C2semidirect
32. #16839819C22semidirect
33. #16839817C22semidirect
34. #16839817C22semidirect
35. #16839817C22semidirect
38. #16839714C22semidirect
39. C2×#14519688C22semidirect
40. #16839503C22semidirect
41. #16839503C22semidirect
42. #16839503C22semidirect
43. #16839502C22semidirect
44. #16839502C22semidirect
49. C4×M16×C61C22semidirect
50. C4×M16×C61C22semidirect
51. C61×#329C22semidirect
52. C61×#329C22semidirect
307. #1525D61semidirect
344. C4×M16C2×D61semidirect
345. C4×M16C2×D61semidirect
355. #329C2×D61semidirect
356. #329C2×D61semidirect
357. C61#17681semidirect

Maximal subgroups

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

subgroupindex
#214597752
#214548372
#214544712
#214444472
#214444412
#214444302
#214444242
#214444232
#214444152
#214444082
C61×#15252
#1768161