Finite Group Databasebeta

← #29779453

Subgroups — #29779453

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450429C211yesmaximal, characteristic, complemented
2#21449693C211yesmaximal, characteristic
3#21447318C211yesmaximal, characteristic, complemented
4#21447232C211yesmaximal, characteristic
5#21447132C211yesmaximal, characteristic
6#21446970C211yesmaximal, characteristic, complemented
7#21446784C211yesmaximal, characteristic, complemented
8#21446513C211yesmaximal, characteristic
9#21446511C211yesmaximal, characteristic, complemented
10#21446506C211yesmaximal, characteristic, complemented
11#21446399C211yesmaximal, characteristic, complemented
12#21446155C211yesmaximal, characteristic
13#21445693C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic
15C61×#1968C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519706C2211yescharacteristic
19C2×#14519701C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519687C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C61⋊#401—21no—
26#16839703—21no—
27#16839698—21no—
28#16839685—21no—
29#16839669C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839644—21no—
36#16839634—21no—
37#16839629—21no—
38#16839628C2211yescharacteristic
39#16839620—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519769—22no—
63#14519769—41no—
64D4×Dic61—22no—
65D4×Dic61—41no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196D411yescharacteristic
76C2×#11758196D411yescharacteristic
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83#14519706—22no—
84#14519701—22no—
85#14519701—41no—
86#14519698—22no—
87#14519698—22no—
88#14519687—22no—
89#14519687—41no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102C22×Q8×C61C2311yescharacteristic
103C22×D4×C61C2311yescharacteristic
104C2×SD16×C61—22no—
105C2×SD16×C61—22no—
106C2×SD16×C61—41no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#106×C61—22no—
115#103×C61—22no—
116#102×C61—22no—
117C4×C8×C61—41no—
118#95×C61—21no—
119#95×C61—21no—
120C24×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61—81no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150SD16×C61—44no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61#4112yes—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C22⋊C4×C61—81no—
162C22⋊C4×C61—81no—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167C22⋊Dic61—81no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—81no—
176C4⋊Dic61#4112yes—
177#11758196—41no—
178#11758196—41no—
179#11758196—42no—
180#11758196—42no—
181#11758196—42no—
182#11758196—81no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—42no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C23×C61—22no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61C22×D411yescharacteristic
202Q8×C61—41no—
203Q8×C61—41no—
204Q8×C61—41no—
205Q8×C61—41no—
206D4×C61—22no—
207D4×C61—22no—
208D4×C61—41no—
209D4×C61—41no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—84no—
256C8×C61—22no—
257C8×C61—22no—
258#92418—41no—
259#92418—41no—
260#29100—611nomaximal, Sylow
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61#53211yescharacteristic
269C22×C61#54611yescharacteristic
270C22×C61#54811yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57711yescharacteristic
274C22×C61#57711yescharacteristic
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61#52911yescharacteristic
281C4×C61#53911yescharacteristic
282C4×C61#55111yescharacteristic
283C4×C61#55611yescharacteristic
284Dic61—41no—
285Dic61—41no—
286Dic61—41no—
287Dic61—42no—
288Dic61—41no—
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292#2514—611no—
293#2454—611no—
294#2088—611no—
295#2073—611no—
296#2055—611no—
297#2026—611no—
298#1991—611no—
299#1968D6111yescharacteristic
300#1936—611no—
301#1935—611no—
302#1935—611no—
303#1919—611no—
304#1877—611no—
305#1771—611no—
306#1762—611no—
307C2×C61—82no—
308C2×C61#256011yescharacteristic
309C2×C61#317411yescharacteristic
310C2×C61#318311yescharacteristic
311C2×C61#331111yescharacteristic
312C2×C61#331711yescharacteristic
313C2×C61#337611yescharacteristic
314C2×C61#337611yescharacteristic
315C22×SD16C2×D6111yescharacteristic
316C2×#125—611no—
317C2×C4×D4—611no—
318C2×#106C2×D6111yescharacteristic
319C2×C4⋊C8—611no—
320C2×#103—611no—
321C2×#103—611no—
322C2×#103C2×D6111yescharacteristic
323C2×#102—611no—
324C2×#102—611no—
325C2×#102C2×D6111yescharacteristic
326C2×#97—611no—
327C2×#97—611no—
328C2×C4×C8C2×D6111yescharacteristic
329#401—1221no—
330#400—1221no—
331#399—1221no—
332#396—1221no—
333#391—611no—
334#391—611no—
335#391C2×D6111yescharacteristic
336#390—611no—
337#390C2×D6111yescharacteristic
338#389—611no—
339#389—611no—
340#389—611no—
341#387—1221no—
342#385—1221no—
343#383—1221no—
344#382—611no—
345#380—1221no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#336—611no—
357#336—611no—
358C61#2910011yescharacteristic, complemented, Sylow
359C22×Q8C22×D6111yescharacteristic
360C22×D4C22×D6111yescharacteristic
361C23×C4—1221no—
362C2×SD16—22no—
363C2×SD16—22no—
364C2×SD16—41no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369#125—1222no—
370#125—2441no—
371C4×D4—1222no—
372C4×D4—2441no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#106—22no—
396C4⋊C8—1222no—
397#103—22no—
398#103—1222no—
399#103—2441no—
400#102—22no—
401#102—1222no—
402#102—1222no—
403#97—1222no—
404#97—2441no—
405C4×C8—41no—
406#95—21no—
407#95—21no—
408#95—611no—
409#95—611no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4#1175822212yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—4881no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4D4×D6111yescharacteristic
448C22×C4D4×D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450SD16—44no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8#1175822412yes—
462C2×C8#1175822412yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—612no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—4881no—
476C4⋊C4—4881no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2442no—
480C4⋊C4—2442no—
481C4⋊C4—2442no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—4881no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42—2442no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23C2×D4×D6111yescharacteristic
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—4884no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4#41×D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C2×C4#1451982211yescharacteristic
555C2×C4#1451982411yescharacteristic
556C8—22no—
557C8—22no—
558C8—2441no—
559C8—2441no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22#1684031011yescharacteristic
568C22#1684040111yescharacteristic
569C22#1684042111yescharacteristic
570C22#1684055911yescharacteristic
571C22#1684055911yescharacteristic
572C22#1684056811yescharacteristic
573C22#1684056811yescharacteristic
574C4—41no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—4881no—
580C4—4881no—
581C4—4881no—
582C4—2441no—
583C4—2441no—
584C4—2441no—
585C4—2441no—
586C4—2442no—
587C4#1684029211yescharacteristic
588C4#1684035711yescharacteristic
589C4#1684043811yescharacteristic
590C4#1684048111yescharacteristic
591C2—82no—
592C2#2145096811yescharacteristic
593C2#2145644411yescharacteristic
594C2#2145654511yescharacteristic
595C2#2145786111yescharacteristic
596C2#2145795111yescharacteristic
597C2#2145864211yescharacteristic
598C2#2145864211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1968C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1968D61
hypocenterC61#29100
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
1. #21450429C2semidirect
3. #21447318C2semidirect
6. #21446970C2semidirect
7. #21446784C2semidirect
9. #21446511C2semidirect
10. #21446506C2semidirect
11. #21446399C2semidirect
13. #21445693C2semidirect
358. C61#29100semidirect

Maximal subgroups

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

subgroupindex
#214504292
#214496932
#214473182
#214472322
#214471322
#214469702
#214467842
#214465132
#214465112
#214465062
#214463992
#214461552
#214456932
#214456462
C61×#19682
#2910061