Finite Group Databasebeta

← #29656472

Subgroups — #29656472

Order 15616.

1060 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456279C211yesmaximal, characteristic, complemented
3C2×#16840052C211yesmaximal, characteristic, complemented
4#21444964C212yesmaximal, complemented
5#21444953C212yesmaximal, complemented
6#21444946C212yesmaximal, complemented
7#21444935C212yesmaximal, complemented
8#21444514C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3519C211yesmaximal, characteristic
12C2×#14519807—22no—
13C2×#14519777C2211yescharacteristic, complemented
14#16840172—21no—
15#16840157—22no—
16#16840141C2212yescomplemented
17#16840134C2212yescomplemented
18#16840121—21no—
19#16840090—21no—
20#16840073—21no—
21#16840057C2212yescomplemented
22#16840052—21no—
23#16840052—21no—
24#16840052C2212yes—
25#16840052C2212yescomplemented
26#16840052C2212yescomplemented
27#16839819C2211yescharacteristic, complemented
28C2×#14519708—21no—
29C2×C4⋊#92418C2211yescharacteristic, complemented
30C2×#14519698—21no—
31C4×#11758194C2211yescharacteristic, complemented
32#16839511C2212yescomplemented
33#16839506C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35D4⋊#92418C2212yes—
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C61×#555C2211yescharacteristic
39C61×#554C2211yescharacteristic
40C61×#553—21no—
41C61×#553—21no—
42C61×#550C2211yescharacteristic
43C61×#546—22no—
44C61×#546—21no—
45D42×C61—21no—
46D42×C61C2211yescharacteristic
47C61×#540—21no—
48C61×#538—22no—
49C61×#532C2211yescharacteristic
50C61×#529C2211yescharacteristic
51C61×#525—21no—
52C2×#121×C61—21no—
53C61×#518—21no—
54C2×C4×D4×C61C2211yescharacteristic
55C22×Dic122—21no—
56#14519807—48no—
57C2×#11758202—24no—
58C2×#11758200—24no—
59C2×#11758194—21no—
60C22×#92418—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777—21no—
64#14519777C2312yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67#14519710—22no—
68#14519708—22no—
69#14519708—22no—
70#14519708—22no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—22no—
78C22⋊#92418—21no—
79#14519688—21no—
80#14519688—21no—
81C4×#92418C2312yes—
82C2×#41×C61—21no—
83C2×#41×C61D412yes—
84C22×D4×C61—22no—
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89C4⋊Q8×C61C2311yescharacteristic
90#127×C61—21no—
91#127×C61C2311yescharacteristic
92#126×C61—22no—
93#126×C61—21no—
94#125×C61—21no—
95#125×C61D411yescharacteristic
96#124×C61—22no—
97#124×C61D412yes—
98#124×C61D411yescharacteristic
99#123×C61—22no—
100#123×C61—22no—
101#123×C61—22no—
102#123×C61—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61D411yescharacteristic
106#121×C61—22no—
107#121×C61—22no—
108#121×C61—22no—
109#121×C61—22no—
110#121×C61—22no—
111#121×C61—21no—
112#121×C61—21no—
113#121×C61—21no—
114#121×C61—21no—
115#121×C61—21no—
116#121×C61—21no—
117#121×C61—21no—
118#121×C61D411yescharacteristic
119#120×C61—22no—
120#120×C61—22no—
121#120×C61—22no—
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×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61D411yescharacteristic
134C4×D4×C61D411yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137C4×D4×C61C2311yescharacteristic
138#117×C61—21no—
139#117×C61—21no—
140#117×C61—21no—
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61D411yescharacteristic
143C2×C22⋊C4×C61—22no—
144C2×C22⋊C4×C61—22no—
145C2×C22⋊C4×C61—21no—
146C2×C42×C61C2311yescharacteristic
147C24×C61—22no—
148C24×C61—21no—
149#41×C61—44no—
150#41×C61—82no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D412yes—
153C2×D4×C61—22no—
154C2×D4×C61—22no—
155C2×D4×C61—22no—
156C2×D4×C61—21no—
157C2×D4×C61—22no—
158C2×D4×C61—22no—
159C2×D4×C61—22no—
160C2×D4×C61—22no—
161C2×D4×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61—21no—
164C2×D4×C61—41no—
165C2×D4×C61—42no—
166C2×D4×C61—42no—
167C2×D4×C61—82no—
168C2×D4×C61—84no—
169C2×D4×C61C2×D411yescharacteristic
170C2×D4×C61C2×D411yescharacteristic
171C2×D4×C61C2×D411yescharacteristic
172C2×D4×C61C2×D412yes—
173C22×C4×C61—22no—
174C22×C4×C61—22no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—21no—
178C22×C4×C61—21no—
179C22×C4×C61—21no—
180C22×C4×C61—22no—
181C22×C4×C61—21no—
182C22×C4×C61—82no—
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D411yescharacteristic
185C22×C4×C61C2×D411yescharacteristic
186C22×Dic61—41no—
187C2×Dic122—22no—
188C2×Dic122—44no—
189C2×Dic122—41no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61C2×D411yescharacteristic
200C4⋊C4×C61C2×D411yescharacteristic
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—21no—
211C22⋊C4×C61—21no—
212C22⋊C4×C61—22no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—21no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—21no—
217C22⋊C4×C61—21no—
218C22⋊C4×C61—21no—
219C42×C61—22no—
220C42×C61—21no—
221C42×C61C2×D411yescharacteristic
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2411yescharacteristic
225#11758202—48no—
226#11758200—48no—
227C4⋊Dic61—22no—
228C4⋊Dic61—22no—
229C4⋊Dic61—41no—
230C4⋊Dic61—41no—
231#11758194—24no—
232C2×#92418—22no—
233C2×#92418—22no—
234C2×#92418—22no—
235C2×#92418—42no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61—22no—
240C23×C61—42no—
241C23×C61—44no—
242C23×C61—42no—
243C23×C61—81no—
244C23×C61—82no—
245C23×C61—82no—
246C23×C61#12011yescharacteristic
247Q8×C61—22no—
248Q8×C61—42no—
249D4×C61—21no—
250D4×C61—21no—
251D4×C61—21no—
252D4×C61—22no—
253D4×C61—42no—
254D4×C61—42no—
255D4×C61—44no—
256D4×C61—42no—
257D4×C61—44no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61—42no—
261D4×C61—82no—
262D4×C61—82no—
263D4×C61—82no—
264D4×C61—82no—
265D4×C61#13712yescomplemented
266C2×C4×C61—22no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—22no—
270C2×C4×C61—21no—
271C2×C4×C61—21no—
272C2×C4×C61—21no—
273C2×C4×C61—22no—
274C2×C4×C61—21no—
275C2×C4×C61—21no—
276C2×C4×C61—41no—
277C2×C4×C61—41no—
278C2×C4×C61—42no—
279C2×C4×C61—42no—
280C2×C4×C61—81no—
281C2×C4×C61—81no—
282C2×C4×C61—82no—
283C2×C4×C61—81no—
284C2×C4×C61—82no—
285C2×C4×C61—82no—
286C2×C4×C61#12011yescharacteristic
287C2×C4×C61#12011yescharacteristic
288C2×C4×C61#12011yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C22×D411yescharacteristic
291C2×C4×C61C22×D411yescharacteristic
292C2×Dic61—42no—
293C2×Dic61—82no—
294Dic122—44no—
295Dic122—82no—
296#92418—24no—
297#92418—42no—
298#92418—42no—
299#18563—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—22no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—42no—
308C22×C61—41no—
309C22×C61—42no—
310C22×C61—41no—
311C22×C61—42no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—84no—
315C22×C61—164no—
316C22×C61#45312yes—
317C22×C61C2×#12011yescharacteristic
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—42no—
328C4×C61C2×#13711yescharacteristic
329C4×C61#57511yescharacteristic
330C4×C61#57712yes—
331Dic61—82no—
332#3519D6111yescharacteristic
333#3238—611no—
334#3152—611no—
335C2×#476—611no—
336#1634—612no—
337#1632—612no—
338#1631—612no—
339#1629—612no—
340#1551—611no—
341#1490—611no—
342#1483—611no—
343C2×C61—22no—
344C2×C61—42no—
345C2×C61—81no—
346C2×C61—82no—
347C2×C61—81no—
348C2×C61#300711yescharacteristic
349C2×C61#300911yescharacteristic
350C2×C61C2×#45311yescharacteristic
351C2×#137—1222no—
352#555C2×D6111yescharacteristic
353#554C2×D6111yescharacteristic
354#553—21no—
355#553—21no—
356#550C2×D6111yescharacteristic
357#546—22no—
358#546—21no—
359D42—21no—
360D42C2×D6111yescharacteristic
361#540—21no—
362#538—22no—
363#532C2×D6111yescharacteristic
364C2×C4⋊Q8—611no—
365#529C2×D6111yescharacteristic
366#525—21no—
367C2×#121—21no—
368#518—21no—
369C2×C4×D4C2×D6111yescharacteristic
370#501—1221no—
371#497—1222no—
372#494—612no—
373#492—612no—
374#489—1221no—
375#483—1221no—
376#480—1221no—
377#477—612no—
378#476—612no—
379#476—612no—
380#476—612no—
381#476—1221no—
382#476—1221no—
383#432—611no—
384C2×#106—1221no—
385C2×C4⋊C8—611no—
386C2×#102—1221no—
387C4×M16—611no—
388#333—612no—
389#331—612no—
390Q8⋊C8—612no—
391D4⋊C8—612no—
392D4⋊C8—612no—
393D4⋊C8—612no—
394C61#1856311yescharacteristic, complemented, Sylow
395C2×#41—21no—
396C2×#41C22⋊D6112yes—
397C22×Q8—1221no—
398C22×D4—22no—
399C22×D4—21no—
400C22×D4—21no—
401C22×D4C22⋊D6111yescharacteristic
402C23×C4—21no—
403#137—2448no—
404C2×Q16—1224no—
405C2×SD16—1224no—
406C2×M16—1221no—
407C22×C8—1221no—
408C4⋊Q8—612no—
409C4⋊Q8—1221no—
410C4⋊Q8—1221no—
411C4⋊Q8—1221no—
412C4⋊Q8C22×D6111yescharacteristic
413#127—21no—
414#127C22×D6111yescharacteristic
415#126—22no—
416#126—21no—
417#125—21no—
418#125C22⋊D6111yescharacteristic
419#124—22no—
420#124C22⋊D6111yescharacteristic
421#124C22⋊D6112yes—
422#123—22no—
423#123—22no—
424#123—22no—
425#123—21no—
426C22⋊Q8—21no—
427C22⋊Q8—22no—
428C22⋊Q8C22⋊D6111yescharacteristic
429#121—22no—
430#121—22no—
431#121—22no—
432#121—22no—
433#121—21no—
434#121—22no—
435#121—21no—
436#121—21no—
437#121—21no—
438#121—21no—
439#121—21no—
440#121—21no—
441#121C22⋊D6111yescharacteristic
442#120—22no—
443#120—22no—
444#120—22no—
445C4×Q8C22×D6111yescharacteristic
446C4×D4—22no—
447C4×D4—22no—
448C4×D4—22no—
449C4×D4—21no—
450C4×D4—21no—
451C4×D4—21no—
452C4×D4—21no—
453C4×D4—21no—
454C4×D4—21no—
455C4×D4—21no—
456C4×D4C22⋊D6111yescharacteristic
457C4×D4C22⋊D6111yescharacteristic
458C4×D4C22×D6111yescharacteristic
459C4×D4C22×D6111yescharacteristic
460C4×D4C22×D6111yescharacteristic
461#117—21no—
462#117—21no—
463#117—21no—
464C2×C4⋊C4—21no—
465C2×C4⋊C4—1221no—
466C2×C4⋊C4—1221no—
467C2×C4⋊C4C22⋊D6111yescharacteristic
468C2×C22⋊C4—22no—
469C2×C22⋊C4—22no—
470C2×C22⋊C4—21no—
471C2×C42C22×D6111yescharacteristic
472#107—1222no—
473#106—1222no—
474#106—1222no—
475#106—1222no—
476C4⋊C8—612no—
477C4⋊C8—612no—
478C4⋊C8—1221no—
479#103—1222no—
480#102—1222no—
481#102—1222no—
482#102—1222no—
483C22⋊C8—1221no—
484#97—1221no—
485#97—1221no—
486C4×C8—612no—
487C24—22no—
488C24—21no—
489#41—44no—
490#41—82no—
491C2×Q8—1222no—
492C2×Q8—2444no—
493C2×Q8—2441no—
494C2×Q8C2×C22⋊D6111yescharacteristic
495C2×Q8C2×C22⋊D6112yes—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—21no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—41no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4—82no—
511C2×D4—84no—
512C2×D4C2×C22⋊D6111yescharacteristic
513C2×D4C2×C22⋊D6111yescharacteristic
514C2×D4C2×C22⋊D6111yescharacteristic
515C2×D4C2×C22⋊D6112yes—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—21no—
525C22×C4—82no—
526C22×C4—2441no—
527C22×C4C2×C22⋊D6111yescharacteristic
528C22×C4C2×C22⋊D6111yescharacteristic
529C22×C4C2×C22⋊D6111yescharacteristic
530Q16—2448no—
531SD16—2448no—
532M16—1224no—
533C2×C8—1222no—
534C2×C8—1222no—
535C2×C8—1222no—
536C2×C8—2442no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—2441no—
549C4⋊C4—2441no—
550C4⋊C4C2×C22⋊D6111yescharacteristic
551C4⋊C4C2×C22⋊D6111yescharacteristic
552C4⋊C4C2×C22⋊D6111yescharacteristic
553C4⋊C4C2×C22⋊D6111yescharacteristic
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—21no—
566C22⋊C4—22no—
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—22no—
570C42—22no—
571C42—21no—
572C42C2×C22⋊D6111yescharacteristic
573C42C2×C22⋊D6111yescharacteristic
574C42C2×C22⋊D6111yescharacteristic
575C42C23×D6111yescharacteristic
576C23—21no—
577C23—21no—
578C23—22no—
579C23—21no—
580C23—42no—
581C23—44no—
582C23—42no—
583C23—81no—
584C23—82no—
585C23—82no—
586C23#1451974611yescharacteristic
587Q8—22no—
588Q8—42no—
589Q8—4882no—
590Q8—2444no—
591D4—21no—
592D4—21no—
593D4—21no—
594D4—22no—
595D4—42no—
596D4—42no—
597D4—44no—
598D4—42no—
599D4—44no—
600D4—41no—
601D4—41no—
602D4—42no—
603D4—82no—
604D4—82no—
605D4—82no—
606D4—82no—
607D4#1451980712yescomplemented
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—82no—
625C2×C4—81no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—4882no—
629C2×C4—2442no—
630C2×C4#1451974611yescharacteristic
631C2×C4#1451974611yescharacteristic
632C2×C4#1451974611yescharacteristic
633C2×C4C22×C22⋊D6111yescharacteristic
634C2×C4C22×C22⋊D6111yescharacteristic
635C2×C4C22×C22⋊D6111yescharacteristic
636C8—1224no—
637C8—2442no—
638C8—2442no—
639C22—21no—
640C22—21no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—42no—
650C22—41no—
651C22—42no—
652C22—41no—
653C22—84no—
654C22—164no—
655C22#1683993012yes—
656C22C2×#1451974611yescharacteristic
657C4—21no—
658C4—21no—
659C4—21no—
660C4—21no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—4882no—
668C4C2×#1451980711yescharacteristic
669C4#1684056011yescharacteristic
670C4#1684057312yes—
671C2—22no—
672C2—42no—
673C2—81no—
674C2—81no—
675C2—82no—
676C2#2145507911yescharacteristic
677C2#2145509611yescharacteristic
678C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3519C2
socleC22×C61C2×#120
hypercenter#3519D61
hypocenterC61#18563
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3519

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. #21456279C2semidirect
3. C2×#16840052C2semidirect
4. #21444964C2semidirect
5. #21444953C2semidirect
6. #21444946C2semidirect
7. #21444935C2semidirect
8. #21444514C2semidirect
9. #21444262C2semidirect
10. #21444232C2semidirect
13. C2×#14519777C22semidirect
16. #16840141C22semidirect
17. #16840134C22semidirect
21. #16840057C22semidirect
25. #16840052C22semidirect
26. #16840052C22semidirect
27. #16839819C22semidirect
29. C2×C4⋊#92418C22semidirect
31. C4×#11758194C22semidirect
32. #16839511C22semidirect
33. #16839506C22semidirect
34. Q8⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
265. D4×C61#137semidirect
394. C61#18563semidirect
607. D4#14519807semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214562792
C2×#168400522
#214449642
#214449532
#214449462
#214449352
#214445142
#214442622
#214442322
C61×#35192
#1856361