Finite Group Databasebeta

← #29656637

Subgroups — #29656637

Order 15616.

1028 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456834C211yesmaximal, characteristic, complemented
2#21456195C211yesmaximal, characteristic, complemented
3C2×#16839957C211yesmaximal, characteristic, complemented
4#21444929C212yesmaximal, complemented
5#21444922C212yesmaximal, complemented
6#21444908C212yesmaximal, complemented
7#21444908C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3520C211yesmaximal, characteristic, complemented
12C22×#11758199—21no—
13C2×#14519774C2211yescharacteristic
14#16840152—21no—
15#16840128—22no—
16#16840128—21no—
17#16840110C2212yescomplemented
18#16840110C2212yescomplemented
19#16839962C2212yescomplemented
20#16839957—21no—
21#16839957—21no—
22#16839957C2212yes—
23#16839957C2212yes—
24#16839957C2212yescomplemented
25#16839912—21no—
26#16839895—21no—
27#16839819C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839766—22no—
30C2×#14519697—21no—
31C4×#11758194C2211yescharacteristic
32#16839508C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35D4⋊#92418C2212yescomplemented
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C61×#556C2211yescharacteristic, complemented
39C61×#554—21no—
40C61×#552—21no—
41C61×#551—22no—
42D4×Q8×C61C2211yescharacteristic, complemented
43C61×#548—21no—
44C61×#547C2211yescharacteristic, complemented
45C61×#547C2211yescharacteristic, complemented
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×D244—21no—
56C22×D244—21no—
57C22×D244—21no—
58C2×#11758201—22no—
59C2×#11758199—22no—
60C2×#11758199—22no—
61C2×#11758199—22no—
62C2×#11758199—44no—
63C2×#11758194—21no—
64C22×#92418—21no—
65#14519774—21no—
66#14519774—21no—
67#14519774—21no—
68#14519774C2312yes—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75#14519697—22no—
76#14519697—24no—
77#14519697—24no—
78C22⋊#92418—21no—
79#14519688—21no—
80#14519688—21no—
81C4×#92418C2312yes—
82C2×#41×C61—21no—
83C2×#41×C61—21no—
84C22×Q8×C61—21no—
85C22×D4×C61D411yescharacteristic
86C23×C4×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#126×C61—22no—
91#126×C61—22no—
92#126×C61—21no—
93#125×C61—22no—
94#125×C61D412yes—
95#125×C61D411yescharacteristic
96#124×C61—22no—
97#124×C61—21no—
98#124×C61D411yescharacteristic
99#123×C61—22no—
100#123×C61—22no—
101#123×C61—21no—
102#123×C61—22no—
103#123×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—21no—
109C22⋊Q8×C61—21no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61D411yescharacteristic
115#121×C61—22no—
116#121×C61—21no—
117#121×C61—21no—
118#121×C61D411yescharacteristic
119#120×C61—22no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—22no—
125C4×D4×C61—22no—
126C4×D4×C61—22no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61D411yescharacteristic
132C4×D4×C61D411yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#117×C61—22no—
137#117×C61—21no—
138#117×C61—21no—
139#117×C61D412yes—
140C2×C4⋊C4×C61—22no—
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61D411yescharacteristic
144C2×C22⋊C4×C61—22no—
145C2×C22⋊C4×C61—21no—
146C2×C42×C61C2311yescharacteristic
147C24×C61—21no—
148C23×D61—41no—
149#41×C61—42no—
150#41×C61—42no—
151#41×C61—82no—
152C2×Q8×C61—22no—
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—82no—
156C2×Q8×C61C2×D411yescharacteristic
157C2×D4×C61—22no—
158C2×D4×C61—22no—
159C2×D4×C61—21no—
160C2×D4×C61—22no—
161C2×D4×C61—21no—
162C2×D4×C61—41no—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D411yescharacteristic
165C2×D4×C61C2×D411yescharacteristic
166C22×C4×C61—22no—
167C22×C4×C61—22no—
168C22×C4×C61—22no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61—21no—
174C22×C4×C61—21no—
175C22×C4×C61—82no—
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C2×D244—22no—
180C2×D244—22no—
181C2×D244—22no—
182C2×D244—44no—
183C2×D244—42no—
184C2×D244—42no—
185C2×D244—41no—
186C2×D244—41no—
187C2×D244—41no—
188C2×D244—82no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61C2×D411yescharacteristic
203C4⋊C4×C61C2×D411yescharacteristic
204C4⋊C4×C61C2×D411yescharacteristic
205C4⋊C4×C61C2×D411yescharacteristic
206C4⋊C4×C61C2×D412yes—
207C4⋊C4×C61C2×D412yes—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C22⋊C4×C61—22no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—21no—
216C22⋊C4×C61—21no—
217C22⋊C4×C61—21no—
218C22⋊C4×C61—21no—
219C22⋊C4×C61—21no—
220C22⋊C4×C61—21no—
221C22⋊C4×C61—21no—
222C22⋊C4×C61—21no—
223C42×C61—22no—
224C42×C61—22no—
225C42×C61—21no—
226C42×C61C2×D411yescharacteristic
227C42×C61C2×D411yescharacteristic
228C42×C61C2×D411yescharacteristic
229C42×C61C2411yescharacteristic
230#11758201—82no—
231#11758199—44no—
232#11758199—44no—
233#11758199—82no—
234#11758194—42no—
235C2×#92418—22no—
236C2×#92418—22no—
237C2×#92418—22no—
238C2×#92418—42no—
239C23×C61—21no—
240C23×C61—21no—
241C23×C61—21no—
242C23×C61—42no—
243C23×C61—42no—
244C23×C61—81no—
245C23×C61#12011yescharacteristic
246C22×D61—42no—
247C22×D61—82no—
248C22×D61—82no—
249C22×D61—82no—
250Q8×C61—22no—
251Q8×C61—41no—
252Q8×C61—41no—
253Q8×C61—42no—
254Q8×C61—82no—
255D4×C61—21no—
256D4×C61—21no—
257D4×C61—21no—
258D4×C61—22no—
259D4×C61—42no—
260D4×C61—44no—
261D4×C61—42no—
262D4×C61—82no—
263D4×C61AGL(1,ℤ/8)12yes—
264C2×C4×C61—22no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—22no—
268C2×C4×C61—21no—
269C2×C4×C61—22no—
270C2×C4×C61—21no—
271C2×C4×C61—21no—
272C2×C4×C61—22no—
273C2×C4×C61—21no—
274C2×C4×C61—21no—
275C2×C4×C61—44no—
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—81no—
283C2×C4×C61—82no—
284C2×C4×C61—82no—
285C2×C4×C61#12011yescharacteristic
286C2×C4×C61#12011yescharacteristic
287C2×C4×C61#12011yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C22×D411yescharacteristic
291D244—44no—
292D244—42no—
293D244—42no—
294D244—42no—
295D244—42no—
296D244—82no—
297D244—82no—
298D244—82no—
299#92418—42no—
300#92418—42no—
301#92418—42no—
302#18578—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—22no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—84no—
315C22×C61#45412yes—
316C22×C61C2×#12011yescharacteristic
317C2×D61—82no—
318C2×D61—82no—
319C2×D61—82no—
320C2×D61—164no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—21no—
324C4×C61—21no—
325C4×C61—41no—
326C4×C61—42no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—42no—
330C4×C61—41no—
331C4×C61—41no—
332C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
333C4×C61#57511yescharacteristic
334C4×C61#57612yes—
335#3520D6111yescharacteristic, complemented
336#3217—611no—
337#3138—611no—
338C2×#459—611no—
339#1628—612no—
340#1627—612no—
341#1625—612no—
342#1625—612no—
343#1552—611no—
344#1490—611no—
345#1483—611no—
346C2×C61—22no—
347C2×C61—42no—
348C2×C61—81no—
349C2×C61—81no—
350C2×C61#300611yescharacteristic
351C2×C61#300811yescharacteristic
352C2×C61C2×#45411yescharacteristic
353D61—162no—
354C22×D8—1221no—
355#556C2×D6111yescharacteristic, complemented
356#554—21no—
357#552—21no—
358#551—22no—
359D4×Q8C2×D6111yescharacteristic, complemented
360#548—21no—
361#547C2×D6111yescharacteristic, complemented
362#547C2×D6111yescharacteristic, complemented
363#546—21no—
364#542—21no—
365#539—22no—
366#533C2×D6111yescharacteristic
367C2×#127—611no—
368#529C2×D6111yescharacteristic
369#525—21no—
370C2×C22⋊Q8—21no—
371#520—21no—
372C2×C4×D4C2×D6111yescharacteristic
373#496—1221no—
374#490—1222no—
375#490—1221no—
376#486—612no—
377#486—612no—
378#460—612no—
379#459—612no—
380#459—612no—
381#459—612no—
382#459—1221no—
383#459—1221no—
384#450—1221no—
385#447—1221no—
386#432—611no—
387C2×C4⋊C8—611no—
388#418—1222no—
389C2×#102—1221no—
390C4×M16—611no—
391#332—612no—
392#332—612no—
393Q8⋊C8—612no—
394D4⋊C8—612no—
395D4⋊C8—612no—
396D4⋊C8—612no—
397C61#1857811yescharacteristic, complemented, Sylow
398C2×#41—21no—
399C2×#41—21no—
400C22×Q8—21no—
401C22×D4—1221no—
402C22×D4—1221no—
403C22×D4—1221no—
404C22×D4C22⋊D6111yescharacteristic
405C23×C4—21no—
406C2×SD16—1222no—
407C2×D8—1222no—
408C2×D8—1222no—
409C2×D8—1222no—
410C2×D8—2444no—
411C2×M16—1221no—
412C22×C8—1221no—
413C4⋊Q8—21no—
414C4⋊Q8C22×D6111yescharacteristic
415C4⋊Q8C22×D6111yescharacteristic
416#127—612no—
417#127—1221no—
418#127—1221no—
419#127—1221no—
420#126—22no—
421#126—22no—
422#126—21no—
423#125—22no—
424#125C22⋊D6111yescharacteristic
425#125C22⋊D6112yes—
426#124—22no—
427#124—21no—
428#124C22⋊D6111yescharacteristic
429#123—22no—
430#123—22no—
431#123—21no—
432#123—22no—
433#123—22no—
434C22⋊Q8—22no—
435C22⋊Q8—22no—
436C22⋊Q8—22no—
437C22⋊Q8—21no—
438C22⋊Q8—21no—
439C22⋊Q8—21no—
440C22⋊Q8—21no—
441C22⋊Q8—21no—
442C22⋊Q8—22no—
443C22⋊Q8—21no—
444C22⋊Q8C22⋊D6111yescharacteristic
445#121—22no—
446#121—21no—
447#121—21no—
448#121C22⋊D6111yescharacteristic
449#120—22no—
450C4×Q8—21no—
451C4×Q8—21no—
452C4×Q8—21no—
453C4×Q8C22×D6111yescharacteristic
454C4×D4—22no—
455C4×D4—22no—
456C4×D4—22no—
457C4×D4—21no—
458C4×D4—21no—
459C4×D4—21no—
460C4×D4—21no—
461C4×D4C22⋊D6111yescharacteristic
462C4×D4C22⋊D6111yescharacteristic
463C4×D4C22×D6111yescharacteristic
464C4×D4C22×D6111yescharacteristic
465C4×D4C22×D6111yescharacteristic
466#117—22no—
467#117—21no—
468#117—21no—
469#117C22⋊D6112yes—
470C2×C4⋊C4—22no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4C22⋊D6111yescharacteristic
474C2×C22⋊C4—22no—
475C2×C22⋊C4—21no—
476C2×C42C22×D6111yescharacteristic
477C4⋊C8—612no—
478C4⋊C8—612no—
479C4⋊C8—1221no—
480#102—1222no—
481#102—1222no—
482#102—1224no—
483#102—1224no—
484#102—1222no—
485#102—1222no—
486C22⋊C8—1221no—
487#97—1221no—
488#97—1221no—
489C4×C8—612no—
490C24—21no—
491C24—2441no—
492#41—42no—
493#41—42no—
494#41—82no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—82no—
499C2×Q8C2×C22⋊D6111yescharacteristic
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—21no—
503C2×D4—22no—
504C2×D4—21no—
505C2×D4—41no—
506C2×D4—1222no—
507C2×D4—1222no—
508C2×D4—1222no—
509C2×D4—4882no—
510C2×D4—2442no—
511C2×D4—2444no—
512C2×D4—2442no—
513C2×D4—2441no—
514C2×D4—2441no—
515C2×D4—2441no—
516C2×D4C2×C22⋊D6111yescharacteristic
517C2×D4C2×C22⋊D6111yescharacteristic
518C2×D4C2×C22⋊D6111yescharacteristic
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—82no—
529C22×C4C2×C22⋊D6111yescharacteristic
530C22×C4C2×C22⋊D6111yescharacteristic
531C22×C4C2×C22⋊D6111yescharacteristic
532SD16—4882no—
533D8—4882no—
534D8—2444no—
535D8—2444no—
536M16—2442no—
537C2×C8—1222no—
538C2×C8—1222no—
539C2×C8—1222no—
540C2×C8—2442no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4C2×C22⋊D6111yescharacteristic
555C4⋊C4C2×C22⋊D6111yescharacteristic
556C4⋊C4C2×C22⋊D6111yescharacteristic
557C4⋊C4C2×C22⋊D6111yescharacteristic
558C4⋊C4C2×C22⋊D6112yes—
559C4⋊C4C2×C22⋊D6112yes—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—21no—
566C22⋊C4—22no—
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—21no—
571C22⋊C4—21no—
572C22⋊C4—21no—
573C22⋊C4—21no—
574C22⋊C4—21no—
575C42—22no—
576C42—21no—
577C42—22no—
578C42C2×C22⋊D6111yescharacteristic
579C42C2×C22⋊D6111yescharacteristic
580C42C2×C22⋊D6111yescharacteristic
581C42C23×D6111yescharacteristic
582C23—21no—
583C23—21no—
584C23—21no—
585C23—42no—
586C23—42no—
587C23—81no—
588C23—4882no—
589C23—4882no—
590C23—4882no—
591C23—2442no—
592C23#1451974611yescharacteristic
593Q8—22no—
594Q8—41no—
595Q8—41no—
596Q8—42no—
597Q8—82no—
598D4—21no—
599D4—21no—
600D4—21no—
601D4—22no—
602D4—42no—
603D4—44no—
604D4—42no—
605D4—82no—
606D4—4882no—
607D4—4882no—
608D4—4882no—
609D4—2444no—
610D4—2442no—
611D4—2442no—
612D4—2442no—
613D4—2442no—
614D4#1451980212yes—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4#1451974611yescharacteristic
637C2×C4#1451974611yescharacteristic
638C2×C4#1451974611yescharacteristic
639C2×C4C22×C22⋊D6111yescharacteristic
640C2×C4C22×C22⋊D6111yescharacteristic
641C2×C4C22×C22⋊D6111yescharacteristic
642C8—2442no—
643C8—2442no—
644C8—2442no—
645C22—21no—
646C22—21no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—84no—
657C22—9764no—
658C22—4882no—
659C22—4882no—
660C22—4882no—
661C22#1683993512yes—
662C22C2×#1451974611yescharacteristic
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—42no—
673C4—41no—
674C4C2×#1451980211yescharacteristic
675C4#1684056011yescharacteristic
676C4#1684056512yes—
677C2—22no—
678C2—42no—
679C2—81no—
680C2—81no—
681C2—9762no—
682C2#2145507111yescharacteristic
683C2#2145508711yescharacteristic
684C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3520C2
socleC22×C61C2×#120
hypercenter#3520D61
hypocenterC61#18578
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. #21456834C2semidirect
2. #21456195C2semidirect
3. C2×#16839957C2semidirect
4. #21444929C2semidirect
5. #21444922C2semidirect
6. #21444908C2semidirect
7. #21444908C2semidirect
8. #21444519C2semidirect
9. #21444262C2semidirect
10. #21444232C2semidirect
11. C61×#3520C2semidirect
17. #16840110C22semidirect
18. #16840110C22semidirect
19. #16839962C22semidirect
24. #16839957C22semidirect
32. #16839508C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
35. D4⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
38. C61×#556C22semidirect
42. D4×Q8×C61C22semidirect
44. C61×#547C22semidirect
45. C61×#547C22semidirect
335. #3520D61semidirect
355. #556C2×D61semidirect
359. D4×Q8C2×D61semidirect
361. #547C2×D61semidirect
362. #547C2×D61semidirect
397. C61#18578semidirect

Maximal subgroups

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

subgroupindex
#214568342
#214561952
C2×#168399572
#214449292
#214449222
#214449082
#214445192
#214442622
#214442322
C61×#35202
#1857861