Finite Group Databasebeta

← #29753131

Subgroups — #29753131

Order 15616.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448844C211yesmaximal, characteristic
2#21447238C211yesmaximal, characteristic, complemented
3#21447197C211yesmaximal, characteristic, complemented
4#21447022C211yesmaximal, characteristic
5#21447015C211yesmaximal, characteristic
6#21446585C211yesmaximal, characteristic, complemented
7#21446554C211yesmaximal, characteristic, complemented
8#21446434C211yesmaximal, characteristic
9#21446408C211yesmaximal, characteristic, complemented
10#21446404C211yesmaximal, characteristic, complemented
11#21446072C211yesmaximal, characteristic, complemented
12#21446016C211yesmaximal, characteristic
13#21445911C211yesmaximal, characteristic
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2494C211yesmaximal, characteristic
16C2×#14519756C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839656—21no—
34#16839650—21no—
35#16839644—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839613—21no—
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C61×#401—21no—
49C61×#397—21no—
50C61×#389C2211yescharacteristic
51C61×#389C2211yescharacteristic
52C61×#388—21no—
53C61×#387—21no—
54C61×#382C2211yescharacteristic
55C61×#380C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519756—22no—
65#14519756—22no—
66C4×Dic122—22no—
67C4×Dic122—41no—
68#14519736—22no—
69#14519736—41no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519710—22no—
82#14519710—22no—
83#14519708—22no—
84#14519708—41no—
85C4⋊#92418—22no—
86#14519700—22no—
87#14519700—22no—
88#14519700—22no—
89#14519700—41no—
90C22⋊#92418—22no—
91C22⋊#92418—41no—
92#14519688—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C23×C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61D411yescharacteristic
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111#95×C61—21no—
112#95×C61—21no—
113#95×C61—21no—
114#95×C61—21no—
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122#4112yes—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61—81no—
157C22⋊C4×C61—41no—
158C22⋊C4×C61—41no—
159C22⋊C4×C61—42no—
160C22⋊C4×C61—81no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—81no—
166C22⋊Dic61—24no—
167C22⋊Dic61—42no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61#4112yes—
173C4⋊Dic61#4112yes—
174#11758196—24no—
175#11758196—42no—
176#11758196—42no—
177C4×Dic61—42no—
178C4×Dic61—42no—
179C4×Dic61—42no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418#4112yes—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61C22×D411yescharacteristic
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
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×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249Dic122—22no—
250Dic122—22no—
251Dic122—41no—
252Dic122—41no—
253#92418—22no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#27061—611nomaximal, Sylow
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61#52511yescharacteristic
266C22×C61#54611yescharacteristic
267C22×C61#54711yescharacteristic
268C22×C61#57511yescharacteristic
269C22×C61#57511yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57811yescharacteristic
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#52911yescharacteristic
282C4×C61#53911yescharacteristic
283C4×C61#55311yescharacteristic
284C4×C61#55611yescharacteristic
285Dic61—42no—
286Dic61—42no—
287Dic61—42no—
288Dic61—81no—
289#2494D6111yescharacteristic
290#2362—611no—
291#2074—611no—
292#2066—611no—
293#2036—611no—
294#2035—611no—
295#1949—611no—
296#1943—611no—
297#1924—611no—
298#1921—611no—
299#1920—611no—
300#1863—611no—
301#1854—611no—
302#1825—611no—
303#1763—611no—
304C2×C61—81no—
305C2×C61#246611yescharacteristic
306C2×C61#311111yescharacteristic
307C2×C61#312111yescharacteristic
308C2×C61#330711yescharacteristic
309C2×C61#332711yescharacteristic
310C2×C61#334911yescharacteristic
311C2×C61#335011yescharacteristic
312C2×C22⋊Q8—611no—
313C2×C4×Q8—611no—
314C2×#117—611no—
315C2×#107—611no—
316C2×#107—611no—
317C2×#106—611no—
318C2×#106—611no—
319C2×C4⋊C8—611no—
320C2×#103—611no—
321C2×#103—611no—
322C2×#103—611no—
323C2×#103—611no—
324C2×C22⋊C8—611no—
325C2×C22⋊C8—611no—
326C2×#97—611no—
327#401—21no—
328#397—21no—
329#391—611no—
330#389—611no—
331#389C2×D6111yescharacteristic
332#389C2×D6111yescharacteristic
333#388—21no—
334#388—1221no—
335#388—1221no—
336#387—21no—
337#387—1221no—
338#382C2×D6111yescharacteristic
339#380C2×D6111yescharacteristic
340#380C2×D6111yescharacteristic
341C4×C4⋊C4—611no—
342C4×C4⋊C4—1221no—
343C4×C4⋊C4C2×D6111yescharacteristic
344C4×C22⋊C4C2×D6111yescharacteristic
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#336—611no—
354#336—611no—
355C61#2706111yescharacteristic, complemented, Sylow
356C22×Q8—611no—
357C23×C4C22×D6111yescharacteristic
358C22×C8—611no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—611no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C4×Q8—1222no—
365C4×Q8—2441no—
366#117—1222no—
367#117—2441no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4C22⋊D6111yescharacteristic
385C2×C22⋊C4C22⋊D6111yescharacteristic
386C2×C42—21no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42C22×D6111yescharacteristic
391C2×C42C22×D6111yescharacteristic
392#107—1222no—
393#107—1222no—
394#106—1222no—
395#106—2441no—
396C4⋊C8—1222no—
397#103—1222no—
398#103—1222no—
399#103—1222no—
400#103—2441no—
401C22⋊C8—1222no—
402C22⋊C8—2441no—
403#97—2441no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95C22⋊D6111yescharacteristic
416#95C22⋊D6111yescharacteristic
417C24C2×C22⋊D6111yescharacteristic
418C2×Q8—612no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—612no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—1224no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—2442no—
474C4⋊C4—2442no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—1224no—
480C22⋊C4—2442no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—81no—
486C42—2442no—
487C42—2442no—
488C42—2442no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C22×C22⋊D6111yescharacteristic
497Q8—1222no—
498Q8—1222no—
499Q8—2441no—
500Q8—2441no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
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—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4#41×D6111yescharacteristic
545C2×C4#41×D6111yescharacteristic
546C2×C4#41×D6111yescharacteristic
547C2×C4#41×D6111yescharacteristic
548C2×C4#1451982311yescharacteristic
549C2×C4#1451982411yescharacteristic
550C8—1222no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1684026911yescharacteristic
562C22#1684040011yescharacteristic
563C22#1684041111yescharacteristic
564C22#1684056011yescharacteristic
565C22#1684056011yescharacteristic
566C22#1684057311yescharacteristic
567C22#1684057711yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—4881no—
578C4—2442no—
579C4—2442no—
580C4—2442no—
581C4#1684029511yescharacteristic
582C4#1684035311yescharacteristic
583C4#1684046311yescharacteristic
584C4#1684047911yescharacteristic
585C2—81no—
586C2#2144985111yescharacteristic
587C2#2145592211yescharacteristic
588C2#2145604211yescharacteristic
589C2#2145778911yescharacteristic
590C2#2145808111yescharacteristic
591C2#2145834311yescharacteristic
592C2#2145835811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2494C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2494D61
hypocenterC61#27061
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2494

Complements

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

NH ≅ G/Ntype
2. #21447238C2semidirect
3. #21447197C2semidirect
6. #21446585C2semidirect
7. #21446554C2semidirect
9. #21446408C2semidirect
10. #21446404C2semidirect
11. #21446072C2semidirect
14. #21445652C2semidirect
355. C61#27061semidirect

Maximal subgroups

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

subgroupindex
#214488442
#214472382
#214471972
#214470222
#214470152
#214465852
#214465542
#214464342
#214464082
#214464042
#214460722
#214460162
#214459112
#214456522
C61×#24942
#2706161