Finite Group Databasebeta

← #29656648

Subgroups — #29656648

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456201C211yesmaximal, characteristic, complemented
3C2×#16840052C211yesmaximal, characteristic, complemented
4#21444991C212yesmaximal, complemented
5#21444984C212yesmaximal, complemented
6#21444970C212yesmaximal, complemented
7#21444970C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3520C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—21no—
14#16840132—22no—
15#16840121—21no—
16#16840114C2212yescomplemented
17#16840114C2212yescomplemented
18#16840090—21no—
19#16840073—21no—
20#16840057C2212yescomplemented
21#16840052—21no—
22#16840052—21no—
23#16840052C2212yes—
24#16840052C2212yes—
25#16840052C2212yescomplemented
26#16839819C2211yescharacteristic
27C2×#14519708—21no—
28C2×C4⋊#92418C2211yescharacteristic
29#16839770—22no—
30C2×#14519698—21no—
31C4×#11758194C2211yescharacteristic
32#16839508C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35D4⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yescomplemented
38C61×#556C2211yescharacteristic
39C61×#554—21no—
40C61×#552—21no—
41C61×#551—22no—
42D4×Q8×C61C2211yescharacteristic
43C61×#548—21no—
44C61×#547C2211yescharacteristic
45C61×#547C2211yescharacteristic
46C61×#546—21no—
47C61×#542—21no—
48C61×#539—22no—
49C61×#533C2211yescharacteristic
50C61×#529C2211yescharacteristic
51C61×#525—21no—
52C2×C22⋊Q8×C61—21no—
53C61×#520—21no—
54C2×C4×D4×C61C2211yescharacteristic
55C22×Dic122—21no—
56C2×#11758194—21no—
57C22×#92418—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777C2312yes—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64#14519710—22no—
65#14519708—22no—
66#14519708—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418C2312yes—
70C4⋊#92418C2312yes—
71#14519701—22no—
72#14519700—24no—
73#14519700—24no—
74#14519698—22no—
75#14519698—22no—
76#14519698—22no—
77C22⋊#92418—21no—
78#14519688—21no—
79#14519688—21no—
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C2×#41×C61—21no—
83C22×Q8×C61—21no—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#126×C61—22no—
90#126×C61—22no—
91#126×C61—21no—
92#125×C61—22no—
93#125×C61D412yes—
94#125×C61D411yescharacteristic
95#124×C61—22no—
96#124×C61—21no—
97#124×C61D411yescharacteristic
98#123×C61—22no—
99#123×C61—22no—
100#123×C61—21no—
101#123×C61—22no—
102#123×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—21no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61D411yescharacteristic
114#121×C61—22no—
115#121×C61—21no—
116#121×C61—21no—
117#121×C61D411yescharacteristic
118#120×C61—22no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61C2311yescharacteristic
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C4×D4×C61—22no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61D411yescharacteristic
131C4×D4×C61D411yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#117×C61—22no—
136#117×C61—21no—
137#117×C61—21no—
138#117×C61D412yes—
139C2×C4⋊C4×C61—22no—
140C2×C4⋊C4×C61—21no—
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61D411yescharacteristic
143C2×C22⋊C4×C61—22no—
144C2×C22⋊C4×C61—21no—
145C2×C42×C61C2311yescharacteristic
146C24×C61—21no—
147#41×C61—42no—
148#41×C61—42no—
149#41×C61—82no—
150C2×Q8×C61—22no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—82no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61—22no—
156C2×D4×C61—22no—
157C2×D4×C61—21no—
158C2×D4×C61—22no—
159C2×D4×C61—21no—
160C2×D4×C61—41no—
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C2×D4×C61C2×D411yescharacteristic
164C22×C4×C61—22no—
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61—82no—
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×Dic61—41no—
178C2×Dic122—22no—
179C2×Dic122—44no—
180C2×Dic122—41no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D412yes—
199C4⋊C4×C61C2×D412yes—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—21no—
211C22⋊C4×C61—21no—
212C22⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—21no—
215C42×C61—22no—
216C42×C61—22no—
217C42×C61—21no—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2×D411yescharacteristic
221C42×C61C2411yescharacteristic
222C4⋊Dic61—22no—
223C4⋊Dic61—22no—
224C4⋊Dic61—41no—
225C4⋊Dic61—41no—
226#11758194—42no—
227C2×#92418—22no—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—42no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—42no—
235C23×C61—42no—
236C23×C61—81no—
237C23×C61#12011yescharacteristic
238Q8×C61—22no—
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—42no—
242Q8×C61—82no—
243D4×C61—21no—
244D4×C61—21no—
245D4×C61—21no—
246D4×C61—22no—
247D4×C61—42no—
248D4×C61—44no—
249D4×C61—42no—
250D4×C61—82no—
251D4×C61#13712yes—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—22no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—22no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—44no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—81no—
269C2×C4×C61—81no—
270C2×C4×C61—81no—
271C2×C4×C61—82no—
272C2×C4×C61—82no—
273C2×C4×C61#12011yescharacteristic
274C2×C4×C61#12011yescharacteristic
275C2×C4×C61#12011yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×Dic61—42no—
280C2×Dic61—82no—
281Dic122—44no—
282Dic122—82no—
283#92418—42no—
284#92418—42no—
285#92418—42no—
286#18579—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—84no—
299C22×C61#45412yes—
300C22×C61C2×#12011yescharacteristic
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—41no—
306C4×C61—42no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61C2×#13711yescharacteristic
313C4×C61#57511yescharacteristic
314C4×C61#57712yes—
315Dic61—82no—
316#3520D6111yescharacteristic
317#3238—611no—
318#3139—611no—
319C2×#476—611no—
320#1638—612no—
321#1637—612no—
322#1635—612no—
323#1635—612no—
324#1552—611no—
325#1490—611no—
326#1483—611no—
327C2×C61—22no—
328C2×C61—42no—
329C2×C61—81no—
330C2×C61—81no—
331C2×C61#300711yescharacteristic
332C2×C61#300911yescharacteristic
333C2×C61C2×#45411yescharacteristic
334#556C2×D6111yescharacteristic
335#554—21no—
336#552—21no—
337#551—22no—
338D4×Q8C2×D6111yescharacteristic
339#548—21no—
340#547C2×D6111yescharacteristic
341#547C2×D6111yescharacteristic
342#546—21no—
343#542—21no—
344#539—22no—
345#533C2×D6111yescharacteristic
346C2×C4⋊Q8—611no—
347#529C2×D6111yescharacteristic
348#525—21no—
349C2×C22⋊Q8—21no—
350#520—21no—
351C2×C4×D4C2×D6111yescharacteristic
352#501—1221no—
353#491—1222no—
354#489—1221no—
355#487—612no—
356#487—612no—
357#483—1221no—
358#480—1221no—
359#477—612no—
360#476—612no—
361#476—612no—
362#476—612no—
363#476—1221no—
364#476—1221no—
365#432—611no—
366C2×#106—1221no—
367C2×C4⋊C8—611no—
368#419—1222no—
369C2×#102—1221no—
370C4×M16—611no—
371#332—612no—
372#332—612no—
373Q8⋊C8—612no—
374D4⋊C8—612no—
375D4⋊C8—612no—
376D4⋊C8—612no—
377C61#1857911yescharacteristic, complemented, Sylow
378C2×#41—21no—
379C2×#41—21no—
380C22×Q8—21no—
381C22×Q8—1221no—
382C22×D4C22⋊D6111yescharacteristic
383C23×C4—21no—
384C2×M16—1221no—
385C22×C8—1221no—
386C4⋊Q8—21no—
387C4⋊Q8—612no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390C4⋊Q8—1221no—
391C4⋊Q8C22×D6111yescharacteristic
392C4⋊Q8C22×D6111yescharacteristic
393#126—22no—
394#126—22no—
395#126—21no—
396#125—22no—
397#125C22⋊D6111yescharacteristic
398#125C22⋊D6112yes—
399#124—22no—
400#124—21no—
401#124C22⋊D6111yescharacteristic
402#123—22no—
403#123—22no—
404#123—21no—
405#123—22no—
406#123—22no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417C22⋊Q8C22⋊D6111yescharacteristic
418#121—22no—
419#121—21no—
420#121—21no—
421#121C22⋊D6111yescharacteristic
422#120—22no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8C22×D6111yescharacteristic
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4C22⋊D6111yescharacteristic
435C4×D4C22⋊D6111yescharacteristic
436C4×D4C22×D6111yescharacteristic
437C4×D4C22×D6111yescharacteristic
438C4×D4C22×D6111yescharacteristic
439#117—22no—
440#117—21no—
441#117—21no—
442#117C22⋊D6112yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4C22⋊D6111yescharacteristic
449C2×C22⋊C4—22no—
450C2×C22⋊C4—21no—
451C2×C42C22×D6111yescharacteristic
452#107—1222no—
453#106—1222no—
454#106—1222no—
455#106—1222no—
456C4⋊C8—612no—
457C4⋊C8—612no—
458C4⋊C8—1221no—
459#103—1222no—
460#103—1224no—
461#103—1224no—
462#102—1222no—
463#102—1222no—
464#102—1222no—
465C22⋊C8—1221no—
466#97—1221no—
467#97—1221no—
468C4×C8—612no—
469C24—21no—
470#41—42no—
471#41—42no—
472#41—82no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—82no—
477C2×Q8—1222no—
478C2×Q8—2444no—
479C2×Q8—2441no—
480C2×Q8C2×C22⋊D6111yescharacteristic
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—41no—
487C2×D4C2×C22⋊D6111yescharacteristic
488C2×D4C2×C22⋊D6111yescharacteristic
489C2×D4C2×C22⋊D6111yescharacteristic
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—82no—
500C22×C4—2441no—
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504M16—2442no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—2442no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4C2×C22⋊D6112yes—
527C4⋊C4C2×C22⋊D6112yes—
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C4⋊C4C2×C22⋊D6111yescharacteristic
530C4⋊C4C2×C22⋊D6111yescharacteristic
531C4⋊C4C2×C22⋊D6111yescharacteristic
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C42—22no—
548C42—21no—
549C42—22no—
550C42C2×C22⋊D6111yescharacteristic
551C42C2×C22⋊D6111yescharacteristic
552C42C2×C22⋊D6111yescharacteristic
553C42C23×D6111yescharacteristic
554C23—21no—
555C23—21no—
556C23—21no—
557C23—42no—
558C23—42no—
559C23—81no—
560C23#1451974611yescharacteristic
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—42no—
565Q8—82no—
566Q8—4882no—
567Q8—2444no—
568D4—21no—
569D4—21no—
570D4—21no—
571D4—22no—
572D4—42no—
573D4—44no—
574D4—42no—
575D4—82no—
576D4#1451980712yes—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—4882no—
599C2×C4—2442no—
600C2×C4#1451974611yescharacteristic
601C2×C4#1451974611yescharacteristic
602C2×C4#1451974611yescharacteristic
603C2×C4C22×C22⋊D6111yescharacteristic
604C2×C4C22×C22⋊D6111yescharacteristic
605C2×C4C22×C22⋊D6111yescharacteristic
606C8—2442no—
607C8—2442no—
608C8—2442no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—84no—
621C22#1683993512yes—
622C22C2×#1451974611yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—4882no—
635C4C2×#1451980711yescharacteristic
636C4#1684056011yescharacteristic
637C4#1684057312yes—
638C2—22no—
639C2—42no—
640C2—81no—
641C2—81no—
642C2#2145507911yescharacteristic
643C2#2145509611yescharacteristic
644C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3520C2
socleC22×C61C2×#120
hypercenter#3520D61
hypocenterC61#18579
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3520

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. #21456201C2semidirect
3. C2×#16840052C2semidirect
4. #21444991C2semidirect
5. #21444984C2semidirect
6. #21444970C2semidirect
7. #21444970C2semidirect
8. #21444519C2semidirect
9. #21444262C2semidirect
10. #21444232C2semidirect
16. #16840114C22semidirect
17. #16840114C22semidirect
20. #16840057C22semidirect
25. #16840052C22semidirect
32. #16839508C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
377. C61#18579semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214562012
C2×#168400522
#214449912
#214449842
#214449702
#214445192
#214442622
#214442322
C61×#35202
#1857961