Finite Group Databasebeta

← #29779429

Subgroups — #29779429

Order 15616.

726 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#21450394C211yesmaximal, characteristic, complemented
2#21447321C211yesmaximal, characteristic, complemented
3#21447222C211yesmaximal, characteristic
4#21447136C211yesmaximal, characteristic
5#21446957C211yesmaximal, characteristic, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446672C211yesmaximal, characteristic
8#21446519C211yesmaximal, characteristic
9#21446508C211yesmaximal, characteristic, complemented
10#21446508C211yesmaximal, characteristic, complemented
11#21446401C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic
13#21445694C211yesmaximal, characteristic, complemented
14#21445644C211yesmaximal, characteristic
15C61×#2452C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519708C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519700C2211yescharacteristic
20C2×#14519700C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519688C2211yescharacteristic
26C2×#14519688C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28#16839704—21no—
29#16839701—21no—
30#16839668C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839648—21no—
35#16839645—21no—
36#16839632C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×C22⋊Q8×C61—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C61×#396—21no—
52C61×#393—21no—
53C61×#390C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#385—21no—
58C4×C4⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758200—22no—
61C2×#11758200—22no—
62C2×#11758200—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×C4×Dic61—21no—
74#14519708—22no—
75C4⋊#92418—22no—
76#14519700—22no—
77#14519700—22no—
78#14519700—41no—
79#14519698—22no—
80#14519698—22no—
81#14519698—22no—
82#14519688—22no—
83#14519688—41no—
84C4×#92418—41no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93C22×Q8×C61—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—41no—
98C22⋊Q8×C61—44no—
99C4×D4×C61—22no—
100C4×D4×C61—41no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61C2311yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118#95×C61—21no—
119#95×C61D411yescharacteristic
120#95×C61D411yescharacteristic
121C24×C61—21no—
122C2×Q8×C61—42no—
123C2×Q8×C61—42no—
124C2×Q8×C61—81no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61#4112yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—81no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61#4112yes—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—81no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176#11758200—44no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61—81no—
185#11758196—81no—
186C4×Dic61—81no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418#4112yes—
198C2×#92418#4112yes—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207Q8×C61—84no—
208D4×C61—22no—
209D4×C61—22no—
210D4×C61—41no—
211D4×C61—41no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—84no—
238C2×C4×C61AGL(1,ℤ/8)12yes—
239C2×C4×C61#13712yes—
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C612−1+411yescharacteristic
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—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261Dic122—41no—
262#92418—22no—
263#92418—22no—
264#92418—41no—
265#92418—41no—
266#29099—611nomaximal, Sylow
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61#53211yescharacteristic
275C22×C61#54611yescharacteristic
276C22×C61#54811yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57711yescharacteristic
280C22×C61#57711yescharacteristic
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61#52911yescharacteristic
291C4×C61#53911yescharacteristic
292C4×C61#55111yescharacteristic
293C4×C61#55611yescharacteristic
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298#2511—611no—
299#2452D6111yescharacteristic
300#2088—611no—
301#2071—611no—
302#2055—611no—
303#2024—611no—
304#1990—611no—
305#1968—611no—
306#1936—611no—
307#1935—611no—
308#1935—611no—
309#1920—611no—
310#1877—611no—
311#1771—611no—
312#1762—611no—
313C2×C61—82no—
314C2×C61#256011yescharacteristic
315C2×C61#317311yescharacteristic
316C2×C61#318211yescharacteristic
317C2×C61#331011yescharacteristic
318C2×C61#331611yescharacteristic
319C2×C61#337211yescharacteristic
320C2×C61#337311yescharacteristic
321C22×SD16—611no—
322C2×C4⋊Q8C2×D6111yescharacteristic
323C2×C22⋊Q8—21no—
324C2×C4×D4C2×D6111yescharacteristic
325C2×#106—611no—
326C2×C4⋊C8—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×#97—611no—
334C2×#97—611no—
335C2×C4×C8—611no—
336#400—1221no—
337#400—1221no—
338#396—21no—
339#393—21no—
340#391—611no—
341#391—611no—
342#391—611no—
343#390C2×D6111yescharacteristic
344#390C2×D6111yescharacteristic
345#389—611no—
346#389C2×D6111yescharacteristic
347#389C2×D6111yescharacteristic
348#387—1221no—
349#387—1221no—
350#385—21no—
351#384—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4C2×D6111yescharacteristic
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#336—611no—
363#336—611no—
364C61#2909911yescharacteristic, complemented, Sylow
365C22×Q8—21no—
366C22×Q8—611no—
367C22×D4C22×D6111yescharacteristic
368C23×C4—21no—
369C2×SD16—1222no—
370C2×SD16—1222no—
371C2×SD16—2441no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—611no—
376C4⋊Q8—22no—
377C4⋊Q8—41no—
378C22⋊Q8—44no—
379C4×D4—22no—
380C4×D4—41no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4C22⋊D6111yescharacteristic
391C2×C4⋊C4C22⋊D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1221no—
401C2×C42C22×D6111yescharacteristic
402C2×C42C22×D6111yescharacteristic
403C2×C42C22×D6111yescharacteristic
404#106—1222no—
405C4⋊C8—1222no—
406#103—1222no—
407#103—1222no—
408#103—2441no—
409#102—1222no—
410#102—1222no—
411#102—1222no—
412#97—1222no—
413#97—2441no—
414C4×C8—2441no—
415#95—21no—
416#95—21no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95C22⋊D6111yescharacteristic
426#95C22⋊D6111yescharacteristic
427C24—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—81no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4#1175822612yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460SD16—2444no—
461C2×C8—612no—
462C2×C8—612no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—2442no—
492C4⋊C4#1175822412yes—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—81no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—4881no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C22×C22⋊D6111yescharacteristic
513Q8—84no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4#1451980312yes—
561C2×C4#1451980712yes—
562C2×C4#41×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4C2×#1175822611yescharacteristic
566C2×C4#1451982211yescharacteristic
567C2×C4#1451982511yescharacteristic
568C8—1222no—
569C8—1222no—
570C8—2441no—
571C8—2441no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22#1684030811yescharacteristic
580C22#1684040611yescharacteristic
581C22#1684042311yescharacteristic
582C22C2×#1451980311yescharacteristic
583C22C2×#1451980711yescharacteristic
584C22#1684057011yescharacteristic
585C22#1684057311yescharacteristic
586C4—22no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—4881no—
596C4—4881no—
597C4—4881no—
598C4—4881no—
599C4#1684029811yescharacteristic
600C4#1684035811yescharacteristic
601C4#1684043911yescharacteristic
602C4#1684048311yescharacteristic
603C2—82no—
604C2#2145095911yescharacteristic
605C2#2145643011yescharacteristic
606C2#2145653311yescharacteristic
607C2#2145784011yescharacteristic
608C2#2145793011yescharacteristic
609C2#2145860011yescharacteristic
610C2#2145861111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2452C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2452D61
hypocenterC61#29099
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
1. #21450394C2semidirect
2. #21447321C2semidirect
5. #21446957C2semidirect
6. #21446781C2semidirect
9. #21446508C2semidirect
10. #21446508C2semidirect
11. #21446401C2semidirect
13. #21445694C2semidirect
364. C61#29099semidirect

Maximal subgroups

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

subgroupindex
#214503942
#214473212
#214472222
#214471362
#214469572
#214467812
#214466722
#214465192
#214465082
#214464012
#214461532
#214456942
#214456442
C61×#24522
#2909961