Finite Group Databasebeta

← #29656659

Subgroups — #29656659

Order 15616.

962 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#21456834C211yesmaximal, characteristic
2#21456195C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic, complemented
4#21444929C212yesmaximal, complemented
5#21444922C212yesmaximal, complemented
6#21444915C212yesmaximal, complemented
7#21444915C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3523C211yesmaximal, characteristic, complemented
12C22×#11758201—21no—
13C2×#14519774C2211yescharacteristic
14#16840152—21no—
15#16840128—22no—
16#16840128—21no—
17#16840110C2212yes—
18#16840110C2212yes—
19#16839962—21no—
20#16839962—21no—
21#16839962C2212yes—
22#16839962C2212yes—
23#16839962C2212yes—
24#16839957C2212yes—
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
35Q8⋊#92418C2212yescomplemented
36Q8⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C61×#557—21no—
39C61×#557C2211yescharacteristic, complemented
40C61×#557C2211yescharacteristic, complemented
41C61×#556—21no—
42C61×#556C2211yescharacteristic, complemented
43C61×#552—22no—
44C61×#552—21no—
45D4×Q8×C61—21no—
46D4×Q8×C61C2211yescharacteristic, complemented
47C61×#541—22no—
48C61×#541—21no—
49C61×#533—21no—
50C61×#533C2211yescharacteristic
51C61×#529—21no—
52C61×#529C2211yescharacteristic
53C61×#519—21no—
54C2×C4×Q8×C61C2211yescharacteristic
55C22×D244—21no—
56C22×D244—21no—
57C22×D244—21no—
58C2×#11758201—22no—
59C2×#11758201—22no—
60C2×#11758201—22no—
61C2×#11758201—44no—
62C2×#11758199—22no—
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—
82C22×Q8×C61—21no—
83C22×Q8×C61D411yescharacteristic
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#126×C61—22no—
91#126×C61—22no—
92#126×C61—22no—
93#126×C61—22no—
94#126×C61—21no—
95#125×C61—22no—
96#125×C61—22no—
97#125×C61—22no—
98#125×C61—22no—
99#125×C61—22no—
100#125×C61—21no—
101#125×C61—21no—
102#125×C61D412yes—
103#125×C61D411yescharacteristic
104#124×C61—21no—
105#124×C61D411yescharacteristic
106#123×C61—22no—
107#123×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—21no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61D411yescharacteristic
116#121×C61D411yescharacteristic
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61D411yescharacteristic
129C4×Q8×C61D411yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61—22no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#117×C61—22no—
138#117×C61—22no—
139#117×C61—22no—
140#117×C61—21no—
141#117×C61—21no—
142#117×C61D412yes—
143C2×C4⋊C4×C61—21no—
144C2×C4⋊C4×C61D411yescharacteristic
145C2×C42×C61—21no—
146C2×C42×C61C2311yescharacteristic
147C23×D61—41no—
148C2×Q8×C61—22no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—22no—
153C2×Q8×C61—41no—
154C2×Q8×C61—42no—
155C2×Q8×C61—42no—
156C2×Q8×C61C2×D411yescharacteristic
157C2×Q8×C61C2×D411yescharacteristic
158C2×Q8×C61C2×D411yescharacteristic
159C2×D4×C61—21no—
160C2×D4×C61C2×D411yescharacteristic
161C22×C4×C61—22no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—22no—
171C2×D244—44no—
172C2×D244—42no—
173C2×D244—42no—
174C2×D244—41no—
175C2×D244—41no—
176C2×D244—41no—
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—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—22no—
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×D412yes—
202C4⋊C4×C61C2×D412yes—
203C4⋊C4×C61C2×D411yescharacteristic
204C4⋊C4×C61C2×D411yescharacteristic
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61—21no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—21no—
211C22⋊C4×C61—21no—
212C22⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C42×C61—22no—
215C42×C61—22no—
216C42×C61—22no—
217C42×C61—21no—
218C42×C61—21no—
219C42×C61—22no—
220C42×C61—21no—
221C42×C61—21no—
222C42×C61—21no—
223C42×C61C2×D411yescharacteristic
224C42×C61C2×D411yescharacteristic
225C42×C61C2×D411yescharacteristic
226C42×C61C2411yescharacteristic
227#11758201—44no—
228#11758201—44no—
229#11758201—82no—
230#11758199—82no—
231#11758194—42no—
232C2×#92418—22no—
233C2×#92418—22no—
234C2×#92418—22no—
235C2×#92418—42no—
236C23×C61—21no—
237C23×C61#12011yescharacteristic
238C22×D61—42no—
239C22×D61—82no—
240C22×D61—82no—
241C22×D61—82no—
242Q8×C61—21no—
243Q8×C61—21no—
244Q8×C61—21no—
245Q8×C61—22no—
246Q8×C61—42no—
247Q8×C61—44no—
248Q8×C61—41no—
249Q8×C61—41no—
250Q8×C61—42no—
251Q8×C61AGL(1,ℤ/8)12yes—
252D4×C61—22no—
253D4×C61—42no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—22no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—22no—
267C2×C4×C61—42no—
268C2×C4×C61—44no—
269C2×C4×C61—42no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61#12011yescharacteristic
273C2×C4×C61#12011yescharacteristic
274C2×C4×C61#12011yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278D244—44no—
279D244—42no—
280D244—42no—
281D244—42no—
282D244—42no—
283D244—82no—
284D244—82no—
285D244—82no—
286#92418—42no—
287#92418—42no—
288#92418—42no—
289#18580—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61#45412yes—
296C22×C61C2×#12011yescharacteristic
297C2×D61—82no—
298C2×D61—82no—
299C2×D61—82no—
300C2×D61—164no—
301C4×C61—21no—
302C4×C61—22no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
315C4×C61#57511yescharacteristic
316C4×C61#57712yes—
317#3523D6111yescharacteristic, complemented
318#3217—611no—
319#3138—611no—
320C2×#460—611no—
321#1628—612no—
322#1627—612no—
323#1626—612no—
324#1626—612no—
325#1552—611no—
326#1491—611no—
327#1483—611no—
328C2×C61—22no—
329C2×C61—81no—
330C2×C61#300611yescharacteristic
331C2×C61#301011yescharacteristic
332C2×C61C2×#45411yescharacteristic
333D61—162no—
334C22×SD16—1221no—
335#557—21no—
336#557C2×D6111yescharacteristic, complemented
337#557C2×D6111yescharacteristic, complemented
338#556—21no—
339#556C2×D6111yescharacteristic, complemented
340#552—22no—
341#552—21no—
342D4×Q8—21no—
343D4×Q8C2×D6111yescharacteristic, complemented
344#541—22no—
345#541—21no—
346#533—21no—
347#533C2×D6111yescharacteristic
348C2×#127—611no—
349#529—21no—
350#529C2×D6111yescharacteristic
351#519—21no—
352C2×C4×Q8C2×D6111yescharacteristic
353#496—1221no—
354#490—1222no—
355#490—1221no—
356#486—612no—
357#486—612no—
358#460—612no—
359#460—612no—
360#460—612no—
361#460—1221no—
362#460—1221no—
363#459—612no—
364#450—1221no—
365#447—1221no—
366#432—611no—
367C2×C4⋊C8—611no—
368#418—1222no—
369C2×#102—1221no—
370C4×M16—611no—
371#332—612no—
372#332—612no—
373Q8⋊C8—612no—
374Q8⋊C8—612no—
375Q8⋊C8—612no—
376D4⋊C8—612no—
377C61#1858011yescharacteristic, complemented, Sylow
378C22×Q8—21no—
379C22×Q8C22⋊D6111yescharacteristic
380C22×D4—1221no—
381C22×D4—1221no—
382C22×D4—1221no—
383C2×SD16—1222no—
384C2×SD16—1222no—
385C2×SD16—1222no—
386C2×SD16—2444no—
387C2×D8—1222no—
388C2×M16—1221no—
389C22×C8—1221no—
390C4⋊Q8—22no—
391C4⋊Q8—21no—
392C4⋊Q8—21no—
393C4⋊Q8—21no—
394C4⋊Q8C22×D6111yescharacteristic
395C4⋊Q8C22×D6111yescharacteristic
396#127—612no—
397#127—1221no—
398#127—1221no—
399#127—1221no—
400#126—22no—
401#126—22no—
402#126—22no—
403#126—22no—
404#126—21no—
405#125—22no—
406#125—22no—
407#125—22no—
408#125—21no—
409#125—22no—
410#125—22no—
411#125—21no—
412#125C22⋊D6111yescharacteristic
413#125C22⋊D6112yes—
414#124—21no—
415#124C22⋊D6111yescharacteristic
416#123—22no—
417#123—21no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—21no—
424C22⋊Q8—22no—
425C22⋊Q8C22⋊D6111yescharacteristic
426#121C22⋊D6111yescharacteristic
427C4×Q8—22no—
428C4×Q8—22no—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8—21no—
434C4×Q8—21no—
435C4×Q8—21no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×Q8C22⋊D6111yescharacteristic
439C4×Q8C22⋊D6111yescharacteristic
440C4×Q8C22×D6111yescharacteristic
441C4×Q8C22×D6111yescharacteristic
442C4×Q8C22×D6111yescharacteristic
443C4×D4—22no—
444C4×D4—21no—
445C4×D4—21no—
446C4×D4C22×D6111yescharacteristic
447#117—22no—
448#117—22no—
449#117—22no—
450#117—21no—
451#117—21no—
452#117C22⋊D6112yes—
453C2×C4⋊C4—21no—
454C2×C4⋊C4C22⋊D6111yescharacteristic
455C2×C42—21no—
456C2×C42C22×D6111yescharacteristic
457C4⋊C8—612no—
458C4⋊C8—612no—
459C4⋊C8—1221no—
460#102—1222no—
461#102—1222no—
462#102—1224no—
463#102—1224no—
464#102—1222no—
465#102—1222no—
466C22⋊C8—1221no—
467#97—1221no—
468#97—1221no—
469C4×C8—612no—
470C24—2441no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—22no—
476C2×Q8—41no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8C2×C22⋊D6111yescharacteristic
480C2×Q8C2×C22⋊D6111yescharacteristic
481C2×Q8C2×C22⋊D6111yescharacteristic
482C2×D4—21no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—4882no—
487C2×D4—2442no—
488C2×D4—2444no—
489C2×D4—2442no—
490C2×D4—2441no—
491C2×D4—2441no—
492C2×D4—2441no—
493C2×D4C2×C22⋊D6111yescharacteristic
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501SD16—4882no—
502SD16—2444no—
503SD16—2444no—
504D8—4882no—
505M16—2442no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—2442no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4C2×C22⋊D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C4⋊C4C2×C22⋊D6112yes—
534C4⋊C4C2×C22⋊D6112yes—
535C4⋊C4C2×C22⋊D6111yescharacteristic
536C4⋊C4C2×C22⋊D6111yescharacteristic
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—22no—
555C42C2×C22⋊D6111yescharacteristic
556C42C2×C22⋊D6111yescharacteristic
557C42C2×C22⋊D6111yescharacteristic
558C42C23×D6111yescharacteristic
559C23—21no—
560C23—4882no—
561C23—4882no—
562C23—4882no—
563C23—2442no—
564C23#1451974611yescharacteristic
565Q8—21no—
566Q8—21no—
567Q8—21no—
568Q8—22no—
569Q8—42no—
570Q8—44no—
571Q8—41no—
572Q8—41no—
573Q8—42no—
574Q8#1451980212yes—
575D4—22no—
576D4—42no—
577D4—4882no—
578D4—4882no—
579D4—4882no—
580D4—2444no—
581D4—2442no—
582D4—2442no—
583D4—2442no—
584D4—2442no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4#1451974611yescharacteristic
604C2×C4#1451974611yescharacteristic
605C2×C4#1451974611yescharacteristic
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4C22×C22⋊D6111yescharacteristic
609C8—2442no—
610C8—2442no—
611C8—2442no—
612C22—21no—
613C22—21no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—9764no—
618C22—4882no—
619C22—4882no—
620C22—4882no—
621C22#1683993512yes—
622C22C2×#1451974611yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4C2×#1451980211yescharacteristic
637C4#1684056011yescharacteristic
638C4#1684057012yes—
639C2—22no—
640C2—81no—
641C2—9762no—
642C2#2145507111yescharacteristic
643C2#2145510511yescharacteristic
644C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3523C2
socleC22×C61C2×#120
prime radical#21456834C2
hypercenter#3523D61
hypocenterC61#18580
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3523

Complements

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

NH ≅ G/Ntype
2. #21456195C2semidirect
3. C2×#16839962C2semidirect
4. #21444929C2semidirect
5. #21444922C2semidirect
6. #21444915C2semidirect
7. #21444915C2semidirect
8. #21444519C2semidirect
9. #21444266C2semidirect
10. #21444232C2semidirect
11. C61×#3523C2semidirect
32. #16839508C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
36. Q8⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
39. C61×#557C22semidirect
40. C61×#557C22semidirect
42. C61×#556C22semidirect
46. D4×Q8×C61C22semidirect
317. #3523D61semidirect
336. #557C2×D61semidirect
337. #557C2×D61semidirect
339. #556C2×D61semidirect
343. D4×Q8C2×D61semidirect
377. C61#18580semidirect

Maximal subgroups

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

subgroupindex
#214568342
#214561952
C2×#168399622
#214449292
#214449222
#214449152
#214445192
#214442662
#214442322
C61×#35232
#1858061