Finite Group Databasebeta

← #29657921

Subgroups — #29657921

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456834C211yesmaximal, characteristic, complemented
2#21456834C211yesmaximal, characteristic, complemented
3C2×#16840110C211yesmaximal, characteristic, complemented
4#21444929C212yesmaximal, complemented
5#21444922C212yesmaximal, complemented
6#21444915C212yesmaximal, complemented
7#21444908C212yesmaximal, complemented
8#21444536C211yesmaximal, characteristic, complemented
9#21444279C211yesmaximal, characteristic, complemented
10#21444279C211yesmaximal, characteristic, complemented
11C61×#3552C211yesmaximal, characteristic, complemented
12C2×#14519774C2211yescharacteristic
13#16840152—21no—
14#16840152—21no—
15#16840128—21no—
16#16840128—21no—
17#16840110—22no—
18#16840110C2212yes—
19#16840110C2212yes—
20#16839962C2212yes—
21#16839962C2212yes—
22#16839957C2212yes—
23#16839957C2212yes—
24#16839912—21no—
25#16839912—21no—
26#16839895—21no—
27#16839895—21no—
28#16839819C2211yescharacteristic
29#16839819C2211yescharacteristic
30C2×#14519697—22no—
31C2×C4×#92418C2211yescharacteristic
32#16839508C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C61×#563—22no—
39C61×#557C2211yescharacteristic, complemented
40C61×#556C2211yescharacteristic, complemented
41C61×#554—21no—
42C61×#553—21no—
43D4×Q8×C61—21no—
44D4×Q8×C61C2211yescharacteristic, complemented
45C61×#548—21no—
46C61×#547C2211yescharacteristic, complemented
47C61×#544—21no—
48C61×#542—21no—
49C61×#541—21no—
50C61×#539—22no—
51C61×#536—21no—
52C2×C4⋊Q8×C61C2211yescharacteristic
53C61×#529C2211yescharacteristic
54C61×#529C2211yescharacteristic
55C22×D244—21no—
56C22×D244—21no—
57C22×D244—21no—
58C2×#11758201—22no—
59C2×#11758201—22no—
60C2×#11758199—22no—
61C2×#11758199—22no—
62C22×#92418—21no—
63#14519774—21no—
64#14519774—21no—
65#14519774—21no—
66#14519774C2312yes—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519697—22no—
70#14519697—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—24no—
74#14519697—24no—
75C22⋊#92418—21no—
76C22⋊#92418—21no—
77#14519688—21no—
78#14519688—21no—
79C4×#92418—21no—
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×Q8×C61—21no—
83C22×Q8×C61—21no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61D412yes—
90C4⋊Q8×C61C2311yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#126×C61—22no—
93#126×C61—22no—
94#126×C61—22no—
95#126×C61—21no—
96#126×C61—21no—
97#125×C61—22no—
98#125×C61—22no—
99#125×C61—22no—
100#125×C61—21no—
101#125×C61D411yescharacteristic
102#125×C61D411yescharacteristic
103#124×C61—22no—
104#124×C61—21no—
105#124×C61D411yescharacteristic
106#124×C61D411yescharacteristic
107#123×C61—22no—
108#123×C61—22no—
109#123×C61—21no—
110#123×C61—21no—
111#123×C61—22no—
112C22⋊Q8×C61—22no—
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—22no—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—21no—
118C22⋊Q8×C61—22no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61D411yescharacteristic
122C22⋊Q8×C61D411yescharacteristic
123#121×C61—22no—
124#121×C61—21no—
125#121×C61D411yescharacteristic
126#121×C61D411yescharacteristic
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61—22no—
133C4×D4×C61—22no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61C2311yescharacteristic
138C4×D4×C61C2311yescharacteristic
139#117×C61—21no—
140#117×C61—21no—
141#117×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61—21no—
144C2×C4⋊C4×C61D412yes—
145C2×C42×C61C2311yescharacteristic
146C23×D61—41no—
147#41×C61—82no—
148C2×Q8×C61—22no—
149C2×Q8×C61—22no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—82no—
153C2×Q8×C61—82no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×Q8×C61C2×D411yescharacteristic
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—22no—
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C22×C4×C61—22no—
162C22×C4×C61—22no—
163C22×C4×C61—22no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C2×D244—22no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244—44no—
173C2×D244—41no—
174C2×D244—41no—
175C2×D244—41no—
176C2×D244—82no—
177C2×D244—82no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D412yes—
198C4⋊C4×C61C2×D412yes—
199C4⋊C4×C61C2×D411yescharacteristic
200C4⋊C4×C61C2×D411yescharacteristic
201C4⋊C4×C61C2×D411yescharacteristic
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—21no—
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—
215C22⋊C4×C61—21no—
216C42×C61—22no—
217C42×C61—22no—
218C42×C61—21no—
219C42×C61—21no—
220C42×C61C2×D411yescharacteristic
221C42×C61C2×D411yescharacteristic
222C42×C61C2×D411yescharacteristic
223C42×C61C2411yescharacteristic
224#11758201—82no—
225#11758201—82no—
226#11758199—82no—
227#11758199—82no—
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—22no—
231C2×#92418—42no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61#12011yescharacteristic
235C22×D61—42no—
236C22×D61—82no—
237C22×D61—82no—
238C22×D61—82no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—42no—
242Q8×C61—42no—
243Q8×C61—82no—
244Q8×C61—82no—
245D4×C61—22no—
246D4×C61—22no—
247D4×C61—42no—
248D4×C61—42no—
249D4×C61—82no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—21no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—82no—
264C2×C4×C61—82no—
265C2×C4×C61—82no—
266C2×C4×C61#12011yescharacteristic
267C2×C4×C61#12011yescharacteristic
268C2×C4×C61#12011yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272D244—44no—
273D244—42no—
274D244—42no—
275D244—42no—
276D244—42no—
277D244—82no—
278D244—82no—
279D244—82no—
280#92418—42no—
281#92418—42no—
282#92418—42no—
283#18694—611nomaximal, Sylow
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61#45412yes—
293C22×C61C2×#12011yescharacteristic
294C2×D61—82no—
295C2×D61—82no—
296C2×D61—82no—
297C2×D61—164no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61#57612yes—
311C4×C61#57712yes—
312#3552D6111yescharacteristic, complemented
313#3217—611no—
314#3217—611no—
315C2×#486—611no—
316#1628—612no—
317#1627—612no—
318#1626—612no—
319#1625—612no—
320#1556—611no—
321#1494—611no—
322#1494—611no—
323C2×C61—22no—
324C2×C61—81no—
325C2×C61—81no—
326C2×C61#300811yescharacteristic
327C2×C61#301011yescharacteristic
328C2×C61C2×#45411yescharacteristic
329D61—162no—
330#563—22no—
331#557C2×D6111yescharacteristic, complemented
332#556C2×D6111yescharacteristic, complemented
333#554—21no—
334#553—21no—
335D4×Q8—21no—
336D4×Q8C2×D6111yescharacteristic, complemented
337#548—21no—
338#547C2×D6111yescharacteristic, complemented
339#544—21no—
340#542—21no—
341#541—21no—
342#539—22no—
343#536—21no—
344C2×C4⋊Q8C2×D6111yescharacteristic
345C2×#127—611no—
346#529C2×D6111yescharacteristic
347#529C2×D6111yescharacteristic
348#496—1221no—
349#496—1221no—
350#490—1221no—
351#490—1221no—
352#486—612no—
353#486—612no—
354#486—1222no—
355#460—612no—
356#460—612no—
357#459—612no—
358#459—612no—
359#450—1221no—
360#450—1221no—
361#447—1221no—
362#447—1221no—
363#432—611no—
364#432—611no—
365C2×#102—1222no—
366C2×C4×C8—611no—
367#332—612no—
368#332—612no—
369Q8⋊C8—612no—
370Q8⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373C61#1869411yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C22×Q8—21no—
376C22×Q8—21no—
377C22×D4—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380C2×SD16—1222no—
381C2×SD16—1222no—
382C2×D8—1222no—
383C2×D8—1222no—
384C22×C8—1221no—
385C4⋊Q8—22no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8C22⋊D6112yes—
391C4⋊Q8C22×D6111yescharacteristic
392C4⋊Q8C22×D6111yescharacteristic
393#127—612no—
394#127—1221no—
395#127—1221no—
396#127—1221no—
397#126—22no—
398#126—22no—
399#126—22no—
400#126—21no—
401#126—21no—
402#125—22no—
403#125—22no—
404#125—22no—
405#125—21no—
406#125C22⋊D6111yescharacteristic
407#125C22⋊D6111yescharacteristic
408#124—22no—
409#124—21no—
410#124C22⋊D6111yescharacteristic
411#124C22⋊D6111yescharacteristic
412#123—22no—
413#123—22no—
414#123—21no—
415#123—21no—
416#123—22no—
417C22⋊Q8—22no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421C22⋊Q8—22no—
422C22⋊Q8—21no—
423C22⋊Q8—22no—
424C22⋊Q8—21no—
425C22⋊Q8—21no—
426C22⋊Q8C22⋊D6111yescharacteristic
427C22⋊Q8C22⋊D6111yescharacteristic
428#121—22no—
429#121—21no—
430#121C22⋊D6111yescharacteristic
431#121C22⋊D6111yescharacteristic
432C4×Q8—21no—
433C4×Q8—21no—
434C4×Q8—21no—
435C4×Q8C22×D6111yescharacteristic
436C4×Q8C22×D6111yescharacteristic
437C4×D4—22no—
438C4×D4—22no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4C22×D6111yescharacteristic
443C4×D4C22×D6111yescharacteristic
444#117—21no—
445#117—21no—
446#117—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4C22⋊D6112yes—
450C2×C42C22×D6111yescharacteristic
451C4⋊C8—612no—
452C4⋊C8—612no—
453#102—1224no—
454#102—1224no—
455#102—1222no—
456#102—1222no—
457#102—1222no—
458#102—1222no—
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461#97—1221no—
462#97—1221no—
463C4×C8—612no—
464C4×C8—1221no—
465C24—2441no—
466#41—82no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—82no—
472C2×Q8—82no—
473C2×Q8C2×C22⋊D6111yescharacteristic
474C2×Q8C2×C22⋊D6111yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—1222no—
479C2×D4—1222no—
480C2×D4—1222no—
481C2×D4—4882no—
482C2×D4—4882no—
483C2×D4—2444no—
484C2×D4—2441no—
485C2×D4—2441no—
486C2×D4—2441no—
487C2×D4C2×C22⋊D6111yescharacteristic
488C2×D4C2×C22⋊D6111yescharacteristic
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497SD16—4882no—
498SD16—4882no—
499D8—4882no—
500D8—4882no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—2442no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4C2×C22⋊D6111yescharacteristic
524C4⋊C4C2×C22⋊D6112yes—
525C4⋊C4C2×C22⋊D6112yes—
526C4⋊C4C2×C22⋊D6111yescharacteristic
527C4⋊C4C2×C22⋊D6111yescharacteristic
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C42—22no—
544C42—22no—
545C42—21no—
546C42—21no—
547C42C2×C22⋊D6111yescharacteristic
548C42C2×C22⋊D6111yescharacteristic
549C42C2×C22⋊D6111yescharacteristic
550C42C23×D6111yescharacteristic
551C23—21no—
552C23—21no—
553C23—4882no—
554C23—4882no—
555C23—4882no—
556C23—2442no—
557C23#1451974611yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—42no—
561Q8—42no—
562Q8—82no—
563Q8—82no—
564D4—22no—
565D4—22no—
566D4—42no—
567D4—42no—
568D4—82no—
569D4—4882no—
570D4—4882no—
571D4—4882no—
572D4—2444no—
573D4—2442no—
574D4—2442no—
575D4—2442no—
576D4—2442no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4#1451974611yescharacteristic
594C2×C4#1451974611yescharacteristic
595C2×C4#1451974611yescharacteristic
596C2×C4C22×C22⋊D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C22×C22⋊D6111yescharacteristic
599C8—2442no—
600C8—2442no—
601C8—2442no—
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—9764no—
611C22—4882no—
612C22—4882no—
613C22—4882no—
614C22#1683993512yes—
615C22C2×#1451974611yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4#1684056512yes—
629C4#1684057012yes—
630C2—22no—
631C2—81no—
632C2—81no—
633C2—9762no—
634C2#2145508711yescharacteristic
635C2#2145510511yescharacteristic
636C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3552C2
socleC22×C61C2×#120
hypercenter#3552D61
hypocenterC61#18694
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3552

Complements

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

NH ≅ G/Ntype
1. #21456834C2semidirect
2. #21456834C2semidirect
3. C2×#16840110C2semidirect
4. #21444929C2semidirect
5. #21444922C2semidirect
6. #21444915C2semidirect
7. #21444908C2semidirect
8. #21444536C2semidirect
9. #21444279C2semidirect
10. #21444279C2semidirect
11. C61×#3552C2semidirect
32. #16839508C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
39. C61×#557C22semidirect
40. C61×#556C22semidirect
44. D4×Q8×C61C22semidirect
46. C61×#547C22semidirect
312. #3552D61semidirect
331. #557C2×D61semidirect
332. #556C2×D61semidirect
336. D4×Q8C2×D61semidirect
338. #547C2×D61semidirect
373. C61#18694semidirect

Maximal subgroups

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

subgroupindex
#214568342
C2×#168401102
#214449292
#214449222
#214449152
#214449082
#214445362
#214442792
C61×#35522
#1869461