Finite Group Databasebeta

← #29656483

Subgroups — #29656483

Order 15616.

1082 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456834C211yesmaximal, characteristic, complemented
2#21456272C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic, complemented
4#21444897C212yesmaximal, complemented
5#21444891C212yesmaximal, complemented
6#21444884C212yesmaximal, complemented
7#21444878C212yesmaximal, complemented
8#21444514C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3522C211yesmaximal, characteristic, complemented
12C2×#14519802—22no—
13C22×#11758201—21no—
14C2×#14519774C2211yescharacteristic
15#16840152—22no—
16#16840152—21no—
17#16840138C2212yes—
18#16840136C2212yes—
19#16840128—21no—
20#16839962—21no—
21#16839962—21no—
22#16839962C2212yes—
23#16839962C2212yes—
24#16839962C2212yes—
25#16839957C2212yes—
26#16839912—21no—
27#16839895—21no—
28#16839819C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519697—21no—
31C4×#11758194C2211yescharacteristic
32#16839511C2212yescomplemented
33#16839506C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36Q8⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C61×#559—21no—
39C61×#559C2211yescharacteristic, complemented
40Q82×C61C2211yescharacteristic, complemented
41C61×#554—21no—
42C61×#554C2211yescharacteristic, complemented
43C61×#550C2211yescharacteristic, complemented
44C61×#548—21no—
45C61×#547—22no—
46C61×#547—21no—
47C61×#542—22no—
48C61×#542—21no—
49C61×#532—21no—
50C61×#532C2211yescharacteristic
51C61×#529—21no—
52C61×#529C2211yescharacteristic
53C61×#520—21no—
54C2×C4×Q8×C61C2211yescharacteristic
55C22×D244—21no—
56C22×D244—21no—
57C22×D244—21no—
58#14519802—48no—
59C2×#11758201—22no—
60C2×#11758201—22no—
61C2×#11758201—24no—
62C2×#11758201—22no—
63C2×#11758201—44no—
64C2×#11758199—24no—
65C2×#11758199—22no—
66C2×#11758194—21no—
67C22×#92418—21no—
68#14519774—21no—
69#14519774—21no—
70#14519774—21no—
71#14519774C2312yes—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2312yes—
75#14519697—22no—
76#14519697—22no—
77#14519697—22no—
78#14519697—22no—
79C22⋊#92418—21no—
80#14519688—21no—
81#14519688—21no—
82C4×#92418C2312yes—
83C2×#41×C61—22no—
84C2×#41×C61—21no—
85C2×#41×C61D412yes—
86C22×Q8×C61D411yescharacteristic
87C4⋊Q8×C61—22no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61C2311yescharacteristic
91#127×C61—21no—
92#127×C61C2311yescharacteristic
93#126×C61—22no—
94#126×C61—21no—
95#125×C61—21no—
96#125×C61D411yescharacteristic
97#124×C61—22no—
98#124×C61—22no—
99#124×C61—21no—
100#124×C61—21no—
101#124×C61D412yes—
102#124×C61D411yescharacteristic
103#123×C61—22no—
104#123×C61—22no—
105#123×C61—22no—
106#123×C61—22no—
107#123×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—22no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61D411yescharacteristic
114#121×C61—22no—
115#121×C61—22no—
116#121×C61—21no—
117#121×C61—21no—
118#121×C61—21no—
119#121×C61D411yescharacteristic
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61D411yescharacteristic
125C4×Q8×C61D411yescharacteristic
126C4×Q8×C61C2311yescharacteristic
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—22no—
130C4×D4×C61—22no—
131C4×D4×C61—22no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61—21no—
138C4×D4×C61—21no—
139C4×D4×C61C2311yescharacteristic
140#117×C61—22no—
141#117×C61—21no—
142C2×C4⋊C4×C61—22no—
143C2×C4⋊C4×C61—21no—
144C2×C4⋊C4×C61—21no—
145C2×C4⋊C4×C61D411yescharacteristic
146C2×C42×C61—21no—
147C2×C42×C61C2311yescharacteristic
148C23×D61—41no—
149#41×C61—44no—
150#41×C61—42no—
151#41×C61—42no—
152#41×C61—84no—
153C2×Q8×C61—22no—
154C2×Q8×C61—21no—
155C2×Q8×C61—22no—
156C2×Q8×C61—41no—
157C2×Q8×C61C2×D411yescharacteristic
158C2×Q8×C61C2×D411yescharacteristic
159C2×Q8×C61C2×D411yescharacteristic
160C2×Q8×C61C2×D412yes—
161C2×D4×C61—22no—
162C2×D4×C61—22no—
163C2×D4×C61—21no—
164C2×D4×C61—22no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61C2×D411yescharacteristic
168C2×D4×C61C2×D412yes—
169C22×C4×C61—22no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—22no—
173C22×C4×C61—22no—
174C22×C4×C61—21no—
175C22×C4×C61—22no—
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C2×D244—22no—
180C2×D244—22no—
181C2×D244—22no—
182C2×D244—42no—
183C2×D244—42no—
184C2×D244—44no—
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—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61C2×D411yescharacteristic
205C4⋊C4×C61C2×D411yescharacteristic
206C4⋊C4×C61C2×D411yescharacteristic
207C4⋊C4×C61C2×D411yescharacteristic
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—22no—
218C22⋊C4×C61—21no—
219C22⋊C4×C61—21no—
220C42×C61—22no—
221C42×C61—21no—
222C42×C61—21no—
223C42×C61—21no—
224C42×C61—21no—
225C42×C61—21no—
226C42×C61C2×D411yescharacteristic
227C42×C61C2×D411yescharacteristic
228C42×C61C2×D411yescharacteristic
229C42×C61C2411yescharacteristic
230#11758201—48no—
231#11758201—44no—
232#11758201—44no—
233#11758201—82no—
234#11758199—48no—
235#11758199—82no—
236#11758194—24no—
237C2×#92418—22no—
238C2×#92418—22no—
239C2×#92418—22no—
240C2×#92418—42no—
241C23×C61—21no—
242C23×C61—22no—
243C23×C61#12011yescharacteristic
244C22×D61—42no—
245C22×D61—82no—
246C22×D61—82no—
247C22×D61—82no—
248Q8×C61—21no—
249Q8×C61—21no—
250Q8×C61—21no—
251Q8×C61—22no—
252Q8×C61—42no—
253Q8×C61—42no—
254Q8×C61—82no—
255Q8×C61AGL(1,ℤ/8)12yescomplemented
256D4×C61—22no—
257D4×C61—42no—
258D4×C61—44no—
259D4×C61—44no—
260D4×C61—42no—
261D4×C61—41no—
262D4×C61—41no—
263D4×C61—42no—
264D4×C61—82no—
265C2×C4×C61—21no—
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—21no—
274C2×C4×C61—22no—
275C2×C4×C61—21no—
276C2×C4×C61—21no—
277C2×C4×C61—42no—
278C2×C4×C61—44no—
279C2×C4×C61—42no—
280C2×C4×C61—41no—
281C2×C4×C61—41no—
282C2×C4×C61—82no—
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—24no—
300#92418—42no—
301#92418—42no—
302#18564—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—42no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—42no—
309C22×C61—42no—
310C22×C61—41no—
311C22×C61#45312yes—
312C22×C61C2×#12011yescharacteristic
313C2×D61—82no—
314C2×D61—82no—
315C2×D61—82no—
316C2×D61—164no—
317C4×C61—21no—
318C4×C61—22no—
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×C61—41no—
329C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
330C4×C61#57511yescharacteristic
331C4×C61#57712yes—
332#3522D6111yescharacteristic, complemented
333#3217—611no—
334#3151—611no—
335C2×#460—611no—
336#1623—612no—
337#1622—612no—
338#1621—612no—
339#1620—612no—
340#1551—611no—
341#1491—611no—
342#1483—611no—
343C2×C61—22no—
344C2×C61—81no—
345C2×C61—82no—
346C2×C61#300611yescharacteristic
347C2×C61#301011yescharacteristic
348C2×C61C2×#45311yescharacteristic
349D61—162no—
350C2×AGL(1,ℤ/8)—1222no—
351C22×SD16—1221no—
352#559—21no—
353#559C2×D6111yescharacteristic, complemented
354Q82C2×D6111yescharacteristic, complemented
355#554—21no—
356#554C2×D6111yescharacteristic, complemented
357#550C2×D6111yescharacteristic, complemented
358#548—21no—
359#547—22no—
360#547—21no—
361#542—22no—
362#542—21no—
363#532—21no—
364#532C2×D6111yescharacteristic
365C2×#127—611no—
366#529—21no—
367#529C2×D6111yescharacteristic
368#520—21no—
369C2×C4×Q8C2×D6111yescharacteristic
370#496—1222no—
371#496—1221no—
372#493—612no—
373#492—612no—
374#490—1221no—
375#460—612no—
376#460—612no—
377#460—612no—
378#460—1221no—
379#460—1221no—
380#459—612no—
381#450—1221no—
382#447—1221no—
383#432—611no—
384C2×C4⋊C8—611no—
385C2×#102—1221no—
386C4×M16—611no—
387#333—612no—
388#331—612no—
389Q8⋊C8—612no—
390Q8⋊C8—612no—
391Q8⋊C8—612no—
392D4⋊C8—612no—
393C61#1856411yescharacteristic, complemented, Sylow
394C2×#41—22no—
395C2×#41—21no—
396C2×#41C22⋊D6112yes—
397C22×Q8C22⋊D6111yescharacteristic
398C22×D4—1221no—
399C22×D4—1221no—
400C22×D4—1221no—
401AGL(1,ℤ/8)—2448no—
402C2×SD16—1222no—
403C2×SD16—1222no—
404C2×SD16—1222no—
405C2×SD16—1224no—
406C2×SD16—2444no—
407C2×D8—1222no—
408C2×D8—1224no—
409C2×M16—1221no—
410C22×C8—1221no—
411C4⋊Q8—22no—
412C4⋊Q8—21no—
413C4⋊Q8—21no—
414C4⋊Q8C22×D6111yescharacteristic
415#127—21no—
416#127—612no—
417#127—1221no—
418#127—1221no—
419#127—1221no—
420#127C22×D6111yescharacteristic
421#126—22no—
422#126—21no—
423#125—21no—
424#125C22⋊D6111yescharacteristic
425#124—22no—
426#124—22no—
427#124—21no—
428#124—21no—
429#124C22⋊D6111yescharacteristic
430#124C22⋊D6112yes—
431#123—22no—
432#123—22no—
433#123—22no—
434#123—22no—
435#123—21no—
436C22⋊Q8—22no—
437C22⋊Q8—22no—
438C22⋊Q8—22no—
439C22⋊Q8—22no—
440C22⋊Q8—21no—
441C22⋊Q8C22⋊D6111yescharacteristic
442#121—22no—
443#121—22no—
444#121—21no—
445#121—21no—
446#121—21no—
447#121C22⋊D6111yescharacteristic
448C4×Q8—21no—
449C4×Q8—21no—
450C4×Q8—21no—
451C4×Q8—21no—
452C4×Q8C22⋊D6111yescharacteristic
453C4×Q8C22⋊D6111yescharacteristic
454C4×Q8C22×D6111yescharacteristic
455C4×Q8C22×D6111yescharacteristic
456C4×Q8C22×D6111yescharacteristic
457C4×D4—22no—
458C4×D4—22no—
459C4×D4—22no—
460C4×D4—21no—
461C4×D4—21no—
462C4×D4—21no—
463C4×D4—21no—
464C4×D4—21no—
465C4×D4—21no—
466C4×D4—21no—
467C4×D4C22×D6111yescharacteristic
468#117—22no—
469#117—21no—
470C2×C4⋊C4—22no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4C22⋊D6111yescharacteristic
474C2×C42—21no—
475C2×C42C22×D6111yescharacteristic
476C4⋊C8—612no—
477C4⋊C8—612no—
478C4⋊C8—1221no—
479#102—1222no—
480#102—1222no—
481#102—1222no—
482#102—1222no—
483C22⋊C8—1221no—
484#97—1221no—
485#97—1221no—
486C4×C8—612no—
487C24—2441no—
488#41—44no—
489#41—42no—
490#41—42no—
491#41—84no—
492C2×Q8—22no—
493C2×Q8—21no—
494C2×Q8—22no—
495C2×Q8—41no—
496C2×Q8C2×C22⋊D6111yescharacteristic
497C2×Q8C2×C22⋊D6111yescharacteristic
498C2×Q8C2×C22⋊D6111yescharacteristic
499C2×Q8C2×C22⋊D6112yes—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—21no—
503C2×D4—22no—
504C2×D4—21no—
505C2×D4—21no—
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⋊D6112yes—
518C22×C4—22no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—21no—
524C22×C4—22no—
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C2×C22⋊D6111yescharacteristic
527C22×C4C2×C22⋊D6111yescharacteristic
528SD16—4882no—
529SD16—2448no—
530SD16—2444no—
531SD16—2444no—
532D8—4882no—
533D8—2448no—
534M16—1224no—
535C2×C8—1222no—
536C2×C8—1222no—
537C2×C8—1222no—
538C2×C8—2442no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4C2×C22⋊D6111yescharacteristic
555C4⋊C4C2×C22⋊D6111yescharacteristic
556C4⋊C4C2×C22⋊D6111yescharacteristic
557C4⋊C4C2×C22⋊D6111yescharacteristic
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—22no—
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—22no—
569C22⋊C4—21no—
570C42—22no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—21no—
576C42C2×C22⋊D6111yescharacteristic
577C42C2×C22⋊D6111yescharacteristic
578C42C2×C22⋊D6111yescharacteristic
579C42C23×D6111yescharacteristic
580C23—21no—
581C23—22no—
582C23—4882no—
583C23—4882no—
584C23—4882no—
585C23—2442no—
586C23#1451974611yescharacteristic
587Q8—21no—
588Q8—21no—
589Q8—21no—
590Q8—22no—
591Q8—42no—
592Q8—42no—
593Q8—82no—
594Q8#1451980212yescomplemented
595D4—22no—
596D4—42no—
597D4—44no—
598D4—44no—
599D4—42no—
600D4—41no—
601D4—41no—
602D4—42no—
603D4—82no—
604D4—4882no—
605D4—4882no—
606D4—4882no—
607D4—2444no—
608D4—2442no—
609D4—2442no—
610D4—2442no—
611D4—2442no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4#1451974611yescharacteristic
633C2×C4#1451974611yescharacteristic
634C2×C4#1451974611yescharacteristic
635C2×C4C22×C22⋊D6111yescharacteristic
636C2×C4C22×C22⋊D6111yescharacteristic
637C2×C4C22×C22⋊D6111yescharacteristic
638C8—1224no—
639C8—2442no—
640C8—2442no—
641C22—21no—
642C22—21no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22—9764no—
650C22—4882no—
651C22—4882no—
652C22—4882no—
653C22#1683993012yes—
654C22C2×#1451974611yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—22no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4C2×#1451980211yescharacteristic
668C4#1684056011yescharacteristic
669C4#1684057012yes—
670C2—22no—
671C2—81no—
672C2—82no—
673C2—9762no—
674C2#2145507111yescharacteristic
675C2#2145510511yescharacteristic
676C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3522C2
socleC22×C61C2×#120
hypercenter#3522D61
hypocenterC61#18564
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3522

Complements

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

NH ≅ G/Ntype
1. #21456834C2semidirect
2. #21456272C2semidirect
3. C2×#16839962C2semidirect
4. #21444897C2semidirect
5. #21444891C2semidirect
6. #21444884C2semidirect
7. #21444878C2semidirect
8. #21444514C2semidirect
9. #21444266C2semidirect
10. #21444232C2semidirect
11. C61×#3522C2semidirect
32. #16839511C22semidirect
33. #16839506C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
36. Q8⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
39. C61×#559C22semidirect
40. Q82×C61C22semidirect
42. C61×#554C22semidirect
43. C61×#550C22semidirect
255. Q8×C61AGL(1,ℤ/8)semidirect
332. #3522D61semidirect
353. #559C2×D61semidirect
354. Q82C2×D61semidirect
356. #554C2×D61semidirect
357. #550C2×D61semidirect
393. C61#18564semidirect
594. Q8#14519802semidirect

Maximal subgroups

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

subgroupindex
#214568342
#214562722
C2×#168399622
#214448972
#214448912
#214448842
#214448782
#214445142
#214442662
#214442322
C61×#35222
#1856461