Finite Group Databasebeta

← #29652776

Subgroups — #29652776

Order 15616.

950 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#21455986C212yesmaximal, complemented
2#21449087C211yesmaximal, characteristic, complemented
3#21444850C212yesmaximal, complemented
4#21444844C211yesmaximal, characteristic, complemented
5#21444839C211yesmaximal, characteristic
6#21444782C212yesmaximal, complemented
7#21444776C211yesmaximal, characteristic, complemented
8#21444771C211yesmaximal, characteristic
9#21444344C212yesmaximal, complemented
10#21444153C211yesmaximal, characteristic, complemented
11C61×#3491C211yesmaximal, characteristic
12C2×C4×Dic122C2211yescharacteristic, complemented
13#16840095C2212yescomplemented
14#16840078C2212yes—
15#16840066—22no—
16#16839981—22no—
17#16839917C2212yescomplemented
18#16839900C2212yes—
19#16839867—22no—
20#16839862—22no—
21#16839819C2212yescomplemented
22#16839701C2211yescharacteristic, complemented
23#16839698C2211yescharacteristic
24#16839695—21no—
25#16839685—21no—
26#16839682C2211yescharacteristic, complemented
27#16839674C2211yescharacteristic
28C4×#11758196—21no—
29C4×C4⋊Dic61C2211yescharacteristic, complemented
30C4×C22⋊Dic61—21no—
31#16839499C2212yescomplemented
32#16839499C2212yescomplemented
33#16839496C2212yes—
34#16839496C2212yes—
35#16839488C2212yescomplemented
36#16839486C2211yescharacteristic
37#16839486C2211yescharacteristic
38C61×#552—22no—
39C61×#552—22no—
40C61×#551—22no—
41C61×#551—22no—
42C61×#543C2211yescharacteristic
43C61×#539—21no—
44C61×#539C2212yes—
45C61×#534C2211yescharacteristic
46C61×#529—21no—
47C61×#529C2212yes—
48C2×#123×C61—21no—
49C61×#518—21no—
50C2×#117×C61C2211yescharacteristic
51C22×Dic122C2311yescharacteristic
52C2×#11758202—22no—
53C2×#11758200—22no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65#14519710—22no—
66#14519708—22no—
67C4⋊#92418—22no—
68#14519701—22no—
69#14519700—22no—
70#14519700—22no—
71#14519698—22no—
72C22⋊#92418C2312yes—
73C22⋊#92418C2312yes—
74#14519688—22no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684C2311yescharacteristic
80#14519684C2311yescharacteristic
81C22×D4×C61—21no—
82C23×C4×C61D411yescharacteristic
83#126×C61—22no—
84#126×C61—22no—
85#126×C61—22no—
86#126×C61—22no—
87#125×C61—22no—
88#125×C61—21no—
89#125×C61—22no—
90#125×C61D412yes—
91#124×C61—21no—
92#124×C61D412yes—
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—22no—
96#123×C61—22no—
97#123×C61—22no—
98#123×C61—22no—
99#123×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61C2312yes—
106#121×C61—21no—
107#121×C61C2312yes—
108#120×C61—22no—
109C4×Q8×C61—21no—
110C4×Q8×C61D412yes—
111C4×D4×C61—21no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C4×D4×C61—22no—
116C4×D4×C61D412yes—
117#117×C61—22no—
118#117×C61—22no—
119#117×C61—22no—
120#117×C61—22no—
121#117×C61—22no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—22no—
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61D411yescharacteristic
131C2×C22⋊C4×C61C2311yescharacteristic
132C2×C42×C61D411yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C24×C61—21no—
135C24×C61C2×D411yescharacteristic
136C2×Q8×C61—21no—
137C2×Q8×C61C2×D412yes—
138C2×D4×C61—22no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61C2×D412yes—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—22no—
152C22×C4×C61—41no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C2×Dic122—22no—
164C2×Dic122—42no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C42×C61—22no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—22no—
195C42×C61C2×D412yes—
196C42×C61C2×D412yes—
197C22⋊Dic61—42no—
198C22⋊Dic61—42no—
199#11758202—44no—
200#11758200—44no—
201C4⋊Dic61—22no—
202C4⋊Dic61—42no—
203#11758196—42no—
204#11758196—42no—
205#11758196—42no—
206C4×Dic61—42no—
207C4×Dic61—41no—
208C4×Dic61—41no—
209C2×#92418—22no—
210C2×#92418—22no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—42no—
217C23×C61—81no—
218C23×C61—81no—
219C23×C61#13712yes—
220C23×C61C22×D411yescharacteristic
221Q8×C61—42no—
222Q8×C61—81no—
223D4×C61—42no—
224D4×C61—44no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—81no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—82no—
246C2×C4×C61—81no—
247C2×C4×C61—82no—
248C2×C4×C61—81no—
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12012yes—
252C2×C4×C61C22×D412yes—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265Dic122—24no—
266Dic122—42no—
267#92418—42no—
268#92418—42no—
269#18177—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—22no—
273C22×C61—42no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—81no—
280C22×C61—162no—
281C22×C61C2×#12011yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57511yescharacteristic
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—42no—
290C4×C61—41no—
291C4×C61—42no—
292C4×C61—41no—
293C4×C61—42no—
294Dic61—42no—
295Dic61—42no—
296Dic61—81no—
297Dic61—81no—
298#3491D6111yescharacteristic
299#3117—612no—
300#2396—611no—
301#1615—612no—
302#1614—611no—
303#1613—611no—
304#1603—612no—
305#1602—611no—
306#1601—611no—
307#1509—612no—
308#1464—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—42no—
312C2×C61—82no—
313C2×C61#300711yescharacteristic
314C2×C61#301811yescharacteristic
315C2×C61#302511yescharacteristic
316#552—22no—
317#552—22no—
318#551—22no—
319#551—22no—
320#543C2×D6111yescharacteristic
321#539—21no—
322#539C2×D6112yes—
323#534C2×D6111yescharacteristic
324#529—21no—
325#529C2×D6112yes—
326C2×#123—21no—
327#518—21no—
328C2×C4×Q8—611no—
329C2×#117C2×D6111yescharacteristic
330#484—612no—
331#481—612no—
332#479—1222no—
333#464—1222no—
334#451—612no—
335#448—612no—
336#441—1222no—
337#440—1222no—
338#432—612no—
339#400—611no—
340#399—611no—
341#398—1221no—
342#396—1221no—
343#395—611no—
344#393—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—1221no—
347C4×C22⋊C4—1221no—
348#328—612no—
349#328—612no—
350#327—612no—
351#327—612no—
352#324—612no—
353#323—611no—
354#323—611no—
355C61#1817711yescharacteristic, complemented, Sylow
356C22×Q8—611no—
357C22×D4—21no—
358C23×C4C22⋊D6111yescharacteristic
359C2×Q16—1222no—
360C2×SD16—1222no—
361#126—22no—
362#126—22no—
363#126—22no—
364#126—22no—
365#125—21no—
366#125—22no—
367#125—22no—
368#125C22⋊D6112yes—
369#124—21no—
370#124C22⋊D6112yes—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—22no—
376#123—22no—
377#123—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8C22×D6112yes—
384#121—21no—
385#121C22×D6112yes—
386#120—22no—
387C4×Q8—21no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8C22⋊D6112yes—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4C22⋊D6112yes—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—22no—
401#117—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22⋊D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C22⋊C4—22no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4C22⋊D6111yescharacteristic
418C2×C22⋊C4C22×D6111yescharacteristic
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42C22⋊D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423#107—1222no—
424#106—1222no—
425C4⋊C8—1222no—
426#103—1222no—
427#103—1222no—
428#103—1222no—
429#102—1222no—
430C22⋊C8—612no—
431C22⋊C8—612no—
432#97—1222no—
433#95—611no—
434#95—611no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439C24—21no—
440C24C2×C22⋊D6111yescharacteristic
441C2×Q8—21no—
442C2×Q8—1222no—
443C2×Q8—2442no—
444C2×Q8C2×C22⋊D6112yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4C2×C22⋊D6112yes—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470Q16—2444no—
471SD16—2444no—
472C2×C8—1222no—
473C2×C8—1222no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—1222no—
487C4⋊C4—2442no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4C2×C22⋊D6112yes—
492C4⋊C4C2×C22⋊D6112yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—2442no—
512C42—2441no—
513C42—2441no—
514C42C2×C22⋊D6112yes—
515C42C2×C22⋊D6112yes—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—81no—
523C23—81no—
524C23#1451980712yes—
525C23C22×C22⋊D6111yescharacteristic
526Q8—42no—
527Q8—81no—
528Q8—1224no—
529Q8—2442no—
530D4—42no—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—81no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4#1451974611yescharacteristic
569C2×C4#1451974611yescharacteristic
570C2×C4#1451974612yes—
571C2×C4C22×C22⋊D6112yes—
572C8—2442no—
573C8—2442no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—162no—
585C22C2×#1451974611yescharacteristic
586C22C2×#1451980711yescharacteristic
587C22#1684056011yescharacteristic
588C4—22no—
589C4—22no—
590C4—21no—
591C4—21no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—4881no—
599C4—4881no—
600C4—2442no—
601C4—2442no—
602C2—21no—
603C2—21no—
604C2—42no—
605C2—82no—
606C2#2145507911yescharacteristic
607C2#2145516811yescharacteristic
608C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3491C2
socleC22×C61C2×#120
prime radicalC61×#534C22
hypercenter#3491D61
hypocenterC61#18177
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3491

Complements

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

NH ≅ G/Ntype
1. #21455986C2semidirect
2. #21449087C2semidirect
3. #21444850C2semidirect
4. #21444844C2semidirect
6. #21444782C2semidirect
7. #21444776C2semidirect
9. #21444344C2semidirect
10. #21444153C2semidirect
12. C2×C4×Dic122C22semidirect
13. #16840095C22semidirect
17. #16839917C22semidirect
21. #16839819C22semidirect
22. #16839701C22semidirect
26. #16839682C22semidirect
29. C4×C4⋊Dic61C22semidirect
31. #16839499C22semidirect
32. #16839499C22semidirect
35. #16839488C22semidirect
355. C61#18177semidirect

Maximal subgroups

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

subgroupindex
#214559862
#214490872
#214448502
#214448442
#214448392
#214447822
#214447762
#214447712
#214443442
#214441532
C61×#34912
#1817761