Finite Group Databasebeta

← #29652170

Subgroups — #29652170

Order 15616.

1000 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455728C212yesmaximal, complemented
2#21449078C211yesmaximal, characteristic, complemented
3#21444850C212yesmaximal, complemented
4#21444844C211yesmaximal, characteristic, complemented
5#21444839C211yesmaximal, characteristic, complemented
6#21444816C212yesmaximal, complemented
7#21444810C211yesmaximal, characteristic, complemented
8#21444805C211yesmaximal, characteristic, complemented
9#21444310C212yesmaximal, complemented
10#21444137C211yesmaximal, characteristic, complemented
11C61×#3459C211yesmaximal, characteristic
12C2×#14519736C2211yescharacteristic, complemented
13#16840100—22no—
14#16840095C2212yescomplemented
15#16840090C2212yescomplemented
16#16840083—22no—
17#16840078C2212yes—
18#16840073C2212yes—
19#16839798—22no—
20#16839758—22no—
21C2×C22⋊#92418C2212yescomplemented
22#16839701C2211yescharacteristic, complemented
23#16839698C2211yescharacteristic
24#16839692C2211yescharacteristic, complemented
25#16839690—21no—
26#16839687C2211yescharacteristic
27#16839676—21no—
28C4×#11758196—21no—
29C4×C22⋊Dic61—21no—
30C4×C22⋊Dic61C2211yescharacteristic, complemented
31#16839499C2212yescomplemented
32#16839499C2212yescomplemented
33#16839496C2212yes—
34#16839496C2212yes—
35#16839488C2212yescomplemented
36#16839486C2211yescharacteristic
37#16839486C2211yescharacteristic
38C61×#543C2211yescharacteristic
39C61×#539—22no—
40C61×#539C2212yes—
41C61×#538—22no—
42C61×#538—21no—
43C61×#534—21no—
44C61×#534C2211yescharacteristic
45C61×#529—22no—
46C61×#529—21no—
47C61×#525—22no—
48C61×#525C2212yes—
49C2×C4×D4×C61—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C22×#92418—22no—
52#14519736—22no—
53#14519736—22no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—21no—
64#14519710—22no—
65#14519710—22no—
66#14519708—22no—
67#14519708—22no—
68#14519701—22no—
69#14519701—22no—
70#14519698—22no—
71#14519698—22no—
72C22⋊#92418—22no—
73C22⋊#92418C2312yes—
74C22⋊#92418C2312yes—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684C2311yescharacteristic
80#14519684C2311yescharacteristic
81C2×#41×C61D412yes—
82C22×D4×C61—21no—
83C22×D4×C61D411yescharacteristic
84C23×C4×C61—21no—
85C23×C4×C61D411yescharacteristic
86C23×C4×C61C2311yescharacteristic
87#126×C61—22no—
88#126×C61—22no—
89#126×C61—22no—
90#125×C61—21no—
91#125×C61—22no—
92#124×C61—22no—
93#124×C61—21no—
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—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61D412yes—
105C22⋊Q8×C61C2312yes—
106#121×C61—22no—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61D412yes—
111#121×C61C2312yes—
112#120×C61—22no—
113#120×C61—22no—
114#120×C61—22no—
115C4×Q8×C61—21no—
116C4×D4×C61—22no—
117C4×D4×C61—22no—
118C4×D4×C61—21no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124#117×C61—22no—
125#117×C61D412yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61D411yescharacteristic
131C2×C22⋊C4×C61C2311yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61D411yescharacteristic
134C24×C61—21no—
135C24×C61C2×D411yescharacteristic
136C24×C61C2×D411yescharacteristic
137#41×C61—44no—
138C2×Q8×C61—21no—
139C2×Q8×C61C2×D412yes—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61—41no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61C2×D412yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—22no—
155C22×C4×C61—21no—
156C22×C4×C61—22no—
157C22×C4×C61—21no—
158C22×C4×C61—42no—
159C22×C4×C61—42no—
160C22×C4×C61—82no—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2411yescharacteristic
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C2×D412yes—
182C4⋊C4×C61C2×D412yes—
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—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C42×C61—22no—
196C42×C61—22no—
197C42×C61—21no—
198C42×C61—21no—
199C22⋊Dic61—42no—
200C22⋊Dic61—42no—
201C22⋊Dic61—42no—
202C22⋊Dic61—42no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205#11758196—42no—
206#11758196—42no—
207C4×Dic61—42no—
208C4×Dic61—41no—
209C4×Dic61—41no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—44no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—22no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—42no—
223C23×C61—81no—
224C23×C61—81no—
225C23×C61#12012yes—
226C23×C61#13512yes—
227C23×C61C22×D411yescharacteristic
228Q8×C61—42no—
229Q8×C61—81no—
230D4×C61—42no—
231D4×C61—44no—
232D4×C61—44no—
233D4×C61—42no—
234D4×C61—42no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61—81no—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—42no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—44no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—81no—
256C2×C4×C61—81no—
257C2×C4×C61—82no—
258C2×C4×C61—81no—
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61#12011yescharacteristic
261C2×C4×C61C22×D412yes—
262C2×Dic61—22no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274#92418—42no—
275#92418—42no—
276#18107—611nomaximal, Sylow
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—84no—
293C22×C61C2×#12011yescharacteristic
294C22×C61C2×#13511yescharacteristic
295C22×C61#57511yescharacteristic
296C4×C61—22no—
297C4×C61—21no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61—41no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61#45812yes—
305Dic61—42no—
306Dic61—81no—
307Dic61—81no—
308Dic61—81no—
309#3459D6111yescharacteristic
310#3091—612no—
311#2395—611no—
312#1615—612no—
313#1614—611no—
314#1613—611no—
315#1609—612no—
316#1608—611no—
317#1607—611no—
318#1501—612no—
319#1459—611no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—42no—
323C2×C61—42no—
324C2×C61—82no—
325C2×C61#300211yescharacteristic
326C2×C61#301811yescharacteristic
327C2×C61C2×#45811yescharacteristic
328#543C2×D6111yescharacteristic
329#539—22no—
330#539C2×D6112yes—
331#538—22no—
332#538—21no—
333#534—21no—
334#534C2×D6111yescharacteristic
335#529—22no—
336#529—21no—
337#525—22no—
338#525C2×D6112yes—
339C2×C4×D4—21no—
340C2×C4×D4C2×D6111yescharacteristic
341C2×#117—611no—
342#485—1222no—
343#484—612no—
344#483—612no—
345#482—1222no—
346#481—612no—
347#480—612no—
348#427—1222no—
349#416—1222no—
350C2×C22⋊C8—612no—
351#400—611no—
352#399—611no—
353#398—611no—
354#397—611no—
355#397—1221no—
356#394—1221no—
357C4×C4⋊C4—1221no—
358C4×C22⋊C4—611no—
359C4×C22⋊C4—1221no—
360#328—612no—
361#328—612no—
362#327—612no—
363#327—612no—
364#324—612no—
365#323—611no—
366#323—611no—
367C61#1810711yescharacteristic, complemented, Sylow
368C2×#41C22⋊D6112yes—
369C22×D4—21no—
370C22×D4C22⋊D6111yescharacteristic
371C23×C4—21no—
372C23×C4C22⋊D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C22×C8—1222no—
375#126—22no—
376#126—22no—
377#126—22no—
378#125—21no—
379#125—22no—
380#124—22no—
381#124—21no—
382#123—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—22no—
387#123—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—21no—
391C22⋊Q8—22no—
392C22⋊Q8C22⋊D6112yes—
393C22⋊Q8C22×D6112yes—
394#121—22no—
395#121—22no—
396#121—22no—
397#121—21no—
398#121C22⋊D6112yes—
399#121C22×D6112yes—
400#120—22no—
401#120—22no—
402#120—22no—
403C4×Q8—21no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412#117—22no—
413#117—1222no—
414#117—1222no—
415#117C22⋊D6112yes—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4C22⋊D6111yescharacteristic
429C2×C22⋊C4C22×D6111yescharacteristic
430C2×C42—21no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22⋊D6111yescharacteristic
434#107—1222no—
435#107—1222no—
436#106—1222no—
437#106—1222no—
438#103—1222no—
439#103—1222no—
440#102—1222no—
441#102—1222no—
442C22⋊C8—612no—
443C22⋊C8—612no—
444C22⋊C8—1222no—
445#95—611no—
446#95—611no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451C24—21no—
452C24C2×C22⋊D6111yescharacteristic
453C24C2×C22⋊D6111yescharacteristic
454#41—44no—
455C2×Q8—21no—
456C2×Q8C2×C22⋊D6112yes—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C2×C22⋊D6112yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6112yes—
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6112yes—
488C22×C4C23×D6111yescharacteristic
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—2444no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4C2×C22⋊D6112yes—
506C4⋊C4C2×C22⋊D6112yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C42—22no—
524C42—21no—
525C42—22no—
526C42—21no—
527C42—2442no—
528C42—2441no—
529C42—2441no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—81no—
541C23—81no—
542C23#1451974612yes—
543C23#1451979712yes—
544C23C22×C22⋊D6111yescharacteristic
545Q8—42no—
546Q8—81no—
547D4—42no—
548D4—44no—
549D4—44no—
550D4—42no—
551D4—42no—
552D4—82no—
553D4—82no—
554D4—81no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—1222no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4#1451974611yescharacteristic
589C2×C4#1451974611yescharacteristic
590C2×C4C22×C22⋊D6112yes—
591C8—2442no—
592C8—2442no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—84no—
609C22C2×#1451974611yescharacteristic
610C22C2×#1451979711yescharacteristic
611C22#1684056011yescharacteristic
612C4—22no—
613C4—21no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—4881no—
621C4—4881no—
622C4—4881no—
623C4—2442no—
624C4#1683995212yes—
625C2—21no—
626C2—21no—
627C2—42no—
628C2—42no—
629C2—82no—
630C2#2145503711yescharacteristic
631C2#2145516811yescharacteristic
632C2C2×#1683995211yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3459C2
socleC22×C61C2×#120
hypercenter#3459D61
hypocenterC61#18107
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3459

Complements

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

NH ≅ G/Ntype
1. #21455728C2semidirect
2. #21449078C2semidirect
3. #21444850C2semidirect
4. #21444844C2semidirect
5. #21444839C2semidirect
6. #21444816C2semidirect
7. #21444810C2semidirect
8. #21444805C2semidirect
9. #21444310C2semidirect
10. #21444137C2semidirect
12. C2×#14519736C22semidirect
14. #16840095C22semidirect
15. #16840090C22semidirect
21. C2×C22⋊#92418C22semidirect
22. #16839701C22semidirect
24. #16839692C22semidirect
30. C4×C22⋊Dic61C22semidirect
31. #16839499C22semidirect
32. #16839499C22semidirect
35. #16839488C22semidirect
367. C61#18107semidirect

Maximal subgroups

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

subgroupindex
#214557282
#214490782
#214448502
#214448442
#214448392
#214448162
#214448102
#214448052
#214443102
#214441372
C61×#34592
#1810761