Finite Group Databasebeta

← #29654422

Subgroups — #29654422

Order 15616.

1276 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#21456616C212yesmaximal, complemented
2#21451703C211yesmaximal, characteristic, complemented
3#21444856C212yesmaximal, complemented
4#21444839C212yesmaximal, complemented
5#21444754C212yesmaximal, complemented
6#21444737C212yesmaximal, complemented
7#21444348C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#3496C211yesmaximal, characteristic, complemented
10C2×#14519751C2211yescharacteristic, complemented
11#16840080—22no—
12#16840078C2214yescomplemented
13#16840013—24no—
14#16839896—22no—
15#16839895C2214yescomplemented
16#16839726C2212yescomplemented
17#16839701C2211yescharacteristic, complemented
18#16839698C2211yescharacteristic, complemented
19#16839679C2211yescharacteristic, complemented
20#16839677—21no—
21#16839671C2211yescharacteristic, complemented
22#16839670—21no—
23#16839653—21no—
24#16839641—21no—
25#16839625C2211yescharacteristic, complemented
26#16839496C2214yescomplemented
27#16839496C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yescomplemented
30C2×H(5,𝔽2)×C61—21no—
31C61×#547—22no—
32C61×#546—24no—
33D42×C61—22no—
34C61×#537C2212yes—
35C61×#534—21no—
36C61×#534C2212yes—
37C61×#532—21no—
38C2×#121×C61—21no—
39C2×#121×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C22×D244C2311yescharacteristic
42C22×C22⋊D61—21no—
43C22×C22⋊D61—21no—
44C2×#11758199—24no—
45C2×#11758194—22no—
46#14519751—22no—
47#14519751—22no—
48C2×C4⋊Dic61C2311yescharacteristic
49C2×#11758196—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56#14519708—24no—
57#14519698—24no—
58#14519697—24no—
59C22⋊#92418—22no—
60C22⋊#92418C2314yes—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66H(5,𝔽2)×C61—44no—
67H(5,𝔽2)×C61—44no—
68C2×#41×C61—24no—
69C2×#41×C61—22no—
70C22×D4×C61—24no—
71C22×D4×C61—22no—
72C22×D4×C61—22no—
73C22×D4×C61—21no—
74C22×D4×C61D411yescharacteristic
75C22×D4×C61D412yescomplemented
76C23×C4×C61D411yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C4⋊Q8×C61—21no—
79#127×C61—21no—
80#124×C61—22no—
81#123×C61—24no—
82#123×C61—24no—
83C22⋊Q8×C61—24no—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61—24no—
87#121×C61—22no—
88#121×C61—22no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61D414yescomplemented
92#121×C61C2314yes—
93#120×C61—24no—
94#120×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D412yescomplemented
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C42×C61D411yescharacteristic
107C24×C61—22no—
108C24×C61—22no—
109C24×C61C2×D411yescharacteristic
110C24×C61C2×D411yescharacteristic
111C23×D61—21no—
112#41×C61—44no—
113#41×C61—416no—
114#41×C61—44no—
115C2×Q8×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—24no—
119C2×D4×C61—22no—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—416no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—44no—
130C2×D4×C61—44no—
131C2×D4×C61—82no—
132C2×D4×C61—82no—
133C2×D4×C61C2×D414yes—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—81no—
140C22×C4×C61—81no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—81no—
151C2×C22⋊D61—81no—
152C2×C22⋊D61—81no—
153C2×C22⋊D61—81no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C2×D244—22no—
158C2×D244—81no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C2×D414yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C42×C61—21no—
171C42×C61—21no—
172C22⋊Dic61—81no—
173C22⋊Dic61—81no—
174#11758199—84no—
175#11758198—41no—
176#11758198—41no—
177#11758198—42no—
178#11758198—81no—
179#11758198—81no—
180C4⋊Dic61—22no—
181#11758196—81no—
182#11758194—82no—
183C2×#92418—24no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—24no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—44no—
194C23×C61—44no—
195C23×C61—82no—
196C23×C61—82no—
197C23×C61—82no—
198C23×C61—82no—
199C23×C61#12012yes—
200C23×C61AGL(1,ℤ/8)12yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209Q8×C61—44no—
210D4×C61—28no—
211D4×C61—44no—
212D4×C61—44no—
213D4×C61—44no—
214D4×C61—48no—
215D4×C61—48no—
216D4×C61—44no—
217D4×C61—44no—
218D4×C61—82no—
219D4×C61—82no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—24no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—48no—
231C2×C4×C61—44no—
232C2×C4×C61—44no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—81no—
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61AGL(1,ℤ/8)12yes—
239C2×C4×C61C22×D412yes—
240C22⋊D61—84no—
241C22⋊D61—84no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251D244—44no—
252#92418—44no—
253#18350—611nomaximal, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—42no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—44no—
264C22×C61—44no—
265C22×C61—44no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—164no—
270C22×C61—164no—
271C22×C61C2×#12011yescharacteristic
272C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—44no—
286C4×C61—41no—
287C4×C61—41no—
288Dic61—81no—
289Dic61—81no—
290Dic61—81no—
291#3496D6111yescharacteristic, complemented
292#3192—612no—
293#2635—611no—
294#1616—612no—
295#1613—612no—
296#1598—612no—
297#1595—612no—
298#1510—612no—
299#1465—611no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—42no—
303C2×C61—42no—
304C2×C61—84no—
305C2×C61#300311yescharacteristic
306C2×C61#302111yescharacteristic
307C2×C61#302611yescharacteristic
308D61—161no—
309C2×H(5,𝔽2)—21no—
310#547—22no—
311#546—24no—
312D42—22no—
313#537C2×D6112yes—
314#534—21no—
315#534C2×D6112yes—
316#532—21no—
317C2×#121—21no—
318C2×#121—611no—
319C2×#121C2×D6112yes—
320C2×C4×D4C2×D6111yescharacteristic
321#481—614no—
322#481—1222no—
323#469—1224no—
324#447—614no—
325#447—1222no—
326#407—612no—
327#400—611no—
328#399—611no—
329#394—611no—
330#394—1221no—
331#392—611no—
332#392—1221no—
333#388—1221no—
334#386—1221no—
335#381—611no—
336#327—614no—
337#327—614no—
338#324—612no—
339#323—612no—
340C61#1835011yescharacteristic, complemented, Sylow
341H(5,𝔽2)—44no—
342H(5,𝔽2)—44no—
343C2×#41—24no—
344C2×#41—22no—
345C22×D4—24no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—21no—
349C22×D4—611no—
350C22×D4—1221no—
351C22×D4—1221no—
352C22×D4C22⋊D6111yescharacteristic
353C22×D4C22⋊D6112yescomplemented
354C23×C4C22⋊D6111yescharacteristic
355C23×C4C22×D6111yescharacteristic
356C2×D8—1224no—
357C2×M16—1222no—
358C4⋊Q8—21no—
359#127—21no—
360#124—22no—
361#123—24no—
362#123—24no—
363C22⋊Q8—24no—
364#121—22no—
365#121—22no—
366#121—24no—
367#121—22no—
368#121—22no—
369#121—22no—
370#121—22no—
371#121—1222no—
372#121—1222no—
373#121C22⋊D6114yescomplemented
374#121C22×D6114yes—
375#120—24no—
376#120—24no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22⋊D6112yescomplemented
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4C22⋊D6111yescharacteristic
395C2×C22⋊C4C22×D6111yescharacteristic
396C2×C42C22⋊D6111yescharacteristic
397#106—1224no—
398#102—1224no—
399#102—1224no—
400C22⋊C8—614no—
401C22⋊C8—1222no—
402#95—611no—
403#95—611no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407C24—22no—
408C24—22no—
409C24—1221no—
410C24C2×C22⋊D6111yescharacteristic
411C24C2×C22⋊D6111yescharacteristic
412#41—44no—
413#41—416no—
414#41—44no—
415C2×Q8—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—416no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—1222no—
434C2×D4—4881no—
435C2×D4—4881no—
436C2×D4—4881no—
437C2×D4—4881no—
438C2×D4—4881no—
439C2×D4—2442no—
440C2×D4—2442no—
441C2×D4C2×C22⋊D6114yes—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6112yes—
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C23×D6111yescharacteristic
459D8—4884no—
460M16—4882no—
461C2×C8—1224no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—1222no—
467C4⋊C4—4881no—
468C4⋊C4C2×C22⋊D6114yes—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—24no—
475C22⋊C4—4881no—
476C22⋊C4—4881no—
477C22⋊C4—4881no—
478C22⋊C4—4881no—
479C22⋊C4—2441no—
480C22⋊C4—2441no—
481C22⋊C4—2442no—
482C42—21no—
483C42—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—44no—
494C23—44no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23#1451974612yes—
507C23#1451980312yes—
508C23C22×C22⋊D6111yescharacteristic
509Q8—44no—
510D4—28no—
511D4—44no—
512D4—44no—
513D4—44no—
514D4—48no—
515D4—48no—
516D4—44no—
517D4—44no—
518D4—82no—
519D4—82no—
520D4—4884no—
521D4—4884no—
522D4—2444no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—81no—
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#1451974611yescharacteristic
549C2×C4#1451974611yescharacteristic
550C2×C4#1451980312yes—
551C2×C4C22×C22⋊D6112yes—
552C8—2444no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—41no—
561C22—41no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—164no—
569C22—164no—
570C22—4881no—
571C22—4881no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22C2×#1451974611yescharacteristic
578C22C2×#1451980311yescharacteristic
579C22C2×#1451980311yescharacteristic
580C4—22no—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—44no—
587C4—4881no—
588C4—4881no—
589C4—4881no—
590C2—21no—
591C2—21no—
592C2—42no—
593C2—42no—
594C2—84no—
595C2—9761no—
596C2#2145504611yescharacteristic
597C2#2145519111yescharacteristic
598C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18350
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21456616C2semidirect
2. #21451703C2semidirect
3. #21444856C2semidirect
4. #21444839C2semidirect
5. #21444754C2semidirect
6. #21444737C2semidirect
7. #21444348C2semidirect
8. #21444157C2semidirect
9. C61×#3496C2semidirect
10. C2×#14519751C22semidirect
12. #16840078C22semidirect
15. #16839895C22semidirect
16. #16839726C22semidirect
17. #16839701C22semidirect
18. #16839698C22semidirect
19. #16839679C22semidirect
21. #16839671C22semidirect
25. #16839625C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
28. #16839488C22semidirect
29. #16839486C22semidirect
75. C22×D4×C61D4semidirect
91. #121×C61D4semidirect
100. C2×C4⋊C4×C61D4semidirect
291. #3496D61semidirect
340. C61#18350semidirect
353. C22×D4C22⋊D61semidirect
373. #121C22⋊D61semidirect
384. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214566162
#214517032
#214448562
#214448392
#214447542
#214447372
#214443482
#214441572
C61×#34962
#1835061