Finite Group Databasebeta

← #29656736

Subgroups — #29656736

Order 15616.

944 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2#21456366C211yesmaximal, characteristic, complemented
3C2×#16840042C211yesmaximal, characteristic, complemented
4#21445103C212yesmaximal, complemented
5#21445089C212yesmaximal, complemented
6#21445068C212yesmaximal, complemented
7#21445054C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3520C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—22no—
14#16840165C2212yescomplemented
15#16840158C2212yescomplemented
16#16840157—21no—
17#16840132—21no—
18#16840042—21no—
19#16840042—21no—
20#16840042C2212yes—
21#16840042C2212yes—
22#16839972C2212yescomplemented
23#16839967C2212yescomplemented
24#16839917—21no—
25#16839900—21no—
26#16839819C2211yescharacteristic
27#16839805—22no—
28C2×#14519710—21no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519698—21no—
31C4×#11758194C2211yescharacteristic
32#16839508C2212yescomplemented
33#16839508C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35D4⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yescomplemented
38C61×#556C2211yescharacteristic
39C61×#554—21no—
40C61×#552—21no—
41C61×#551—22no—
42D4×Q8×C61C2211yescharacteristic
43C61×#548—21no—
44C61×#547C2211yescharacteristic
45C61×#547C2211yescharacteristic
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×Dic122—21no—
56C2×#11758202—22no—
57C2×#11758200—22no—
58C2×#11758194—21no—
59C22×#92418—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777C2312yes—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66#14519710—22no—
67#14519710—22no—
68#14519710—24no—
69#14519708—24no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73#14519700—22no—
74#14519700—22no—
75#14519698—22no—
76#14519698—22no—
77C22⋊#92418—21no—
78#14519688—21no—
79#14519688—21no—
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C2×#41×C61—21no—
83C22×Q8×C61—21no—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#126×C61—22no—
90#126×C61—22no—
91#126×C61—21no—
92#125×C61—22no—
93#125×C61D412yes—
94#125×C61D411yescharacteristic
95#124×C61—22no—
96#124×C61—21no—
97#124×C61D411yescharacteristic
98#123×C61—22no—
99#123×C61—22no—
100#123×C61—21no—
101#123×C61—22no—
102#123×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—21no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61D411yescharacteristic
114#121×C61—22no—
115#121×C61—21no—
116#121×C61—21no—
117#121×C61D411yescharacteristic
118#120×C61—22no—
119C4×Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
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×C61D411yescharacteristic
131C4×D4×C61D411yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#117×C61—22no—
136#117×C61—21no—
137#117×C61—21no—
138#117×C61D412yes—
139C2×C4⋊C4×C61—22no—
140C2×C4⋊C4×C61—21no—
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61D411yescharacteristic
143C2×C22⋊C4×C61—22no—
144C2×C22⋊C4×C61—21no—
145C2×C42×C61C2311yescharacteristic
146C24×C61—21no—
147#41×C61—42no—
148#41×C61—42no—
149#41×C61—82no—
150C2×Q8×C61—22no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—82no—
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61—22no—
156C2×D4×C61—22no—
157C2×D4×C61—21no—
158C2×D4×C61—22no—
159C2×D4×C61—21no—
160C2×D4×C61—41no—
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C2×D4×C61C2×D411yescharacteristic
164C22×C4×C61—22no—
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61—82no—
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×Dic61—41no—
178C2×Dic122—22no—
179C2×Dic122—41no—
180C2×Dic122—82no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61C2×D411yescharacteristic
195C4⋊C4×C61C2×D411yescharacteristic
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D412yes—
199C4⋊C4×C61C2×D412yes—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—22no—
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—
215C42×C61—22no—
216C42×C61—22no—
217C42×C61—21no—
218C42×C61C2×D411yescharacteristic
219C42×C61C2×D411yescharacteristic
220C42×C61C2×D411yescharacteristic
221C42×C61C2411yescharacteristic
222#11758202—82no—
223#11758200—82no—
224C4⋊Dic61—22no—
225C4⋊Dic61—22no—
226C4⋊Dic61—41no—
227C4⋊Dic61—41no—
228#11758194—24no—
229C2×#92418—22no—
230C2×#92418—22no—
231C2×#92418—22no—
232C2×#92418—42no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—42no—
237C23×C61—42no—
238C23×C61—81no—
239C23×C61#12011yescharacteristic
240Q8×C61—22no—
241Q8×C61—41no—
242Q8×C61—41no—
243Q8×C61—42no—
244Q8×C61—82no—
245D4×C61—21no—
246D4×C61—21no—
247D4×C61—21no—
248D4×C61—22no—
249D4×C61—42no—
250D4×C61—44no—
251D4×C61—42no—
252D4×C61—82no—
253D4×C61#13712yes—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—22no—
258C2×C4×C61—21no—
259C2×C4×C61—22no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—44no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—42no—
269C2×C4×C61—42no—
270C2×C4×C61—81no—
271C2×C4×C61—81no—
272C2×C4×C61—81no—
273C2×C4×C61—82no—
274C2×C4×C61—82no—
275C2×C4×C61#12011yescharacteristic
276C2×C4×C61#12011yescharacteristic
277C2×C4×C61#12011yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×Dic61—42no—
282C2×Dic61—82no—
283Dic122—42no—
284Dic122—42no—
285Dic122—82no—
286#92418—24no—
287#92418—42no—
288#92418—42no—
289#18587—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—22no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—84no—
302C22×C61#45612yes—
303C22×C61C2×#12011yescharacteristic
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—41no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61—42no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61C2×#13711yescharacteristic
316C4×C61#57511yescharacteristic
317C4×C61#57612yes—
318Dic61—82no—
319#3520D6111yescharacteristic
320#3218—611no—
321#3165—611no—
322C2×#474—611no—
323#1654—612no—
324#1652—612no—
325#1649—612no—
326#1647—612no—
327#1552—611no—
328#1490—611no—
329#1483—611no—
330C2×C61—22no—
331C2×C61—42no—
332C2×C61—81no—
333C2×C61—81no—
334C2×C61#300711yescharacteristic
335C2×C61#300811yescharacteristic
336C2×C61C2×#45611yescharacteristic
337#556C2×D6111yescharacteristic
338#554—21no—
339#552—21no—
340#551—22no—
341D4×Q8C2×D6111yescharacteristic
342#548—21no—
343#547C2×D6111yescharacteristic
344#547C2×D6111yescharacteristic
345#546—21no—
346#542—21no—
347#539—22no—
348#533C2×D6111yescharacteristic
349C2×C4⋊Q8—611no—
350#529C2×D6111yescharacteristic
351#525—21no—
352C2×C22⋊Q8—21no—
353#520—21no—
354C2×C4×D4C2×D6111yescharacteristic
355#501—1222no—
356#500—612no—
357#498—612no—
358#497—1221no—
359#491—1221no—
360#474—612no—
361#474—612no—
362#474—1221no—
363#474—1221no—
364#462—612no—
365#461—612no—
366#451—1221no—
367#448—1221no—
368#432—611no—
369#428—1222no—
370C2×#107—1221no—
371C2×C4⋊C8—611no—
372C2×#102—1221no—
373C4×M16—611no—
374#332—612no—
375#332—612no—
376Q8⋊C8—612no—
377D4⋊C8—612no—
378D4⋊C8—612no—
379D4⋊C8—612no—
380C61#1858711yescharacteristic, complemented, Sylow
381C2×#41—21no—
382C2×#41—21no—
383C22×Q8—21no—
384C22×Q8—1221no—
385C22×D4C22⋊D6111yescharacteristic
386C23×C4—21no—
387C2×Q16—1222no—
388C2×SD16—1222no—
389C2×M16—1221no—
390C22×C8—1221no—
391C4⋊Q8—21no—
392C4⋊Q8—612no—
393C4⋊Q8—1221no—
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8C22×D6111yescharacteristic
397C4⋊Q8C22×D6111yescharacteristic
398#126—22no—
399#126—22no—
400#126—21no—
401#125—22no—
402#125C22⋊D6111yescharacteristic
403#125C22⋊D6112yes—
404#124—22no—
405#124—21no—
406#124C22⋊D6111yescharacteristic
407#123—22no—
408#123—22no—
409#123—21no—
410#123—22no—
411#123—22no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—21no—
420C22⋊Q8—22no—
421C22⋊Q8—21no—
422C22⋊Q8C22⋊D6111yescharacteristic
423#121—22no—
424#121—21no—
425#121—21no—
426#121C22⋊D6111yescharacteristic
427#120—22no—
428C4×Q8—21no—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×Q8C22×D6111yescharacteristic
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4C22⋊D6111yescharacteristic
440C4×D4C22⋊D6111yescharacteristic
441C4×D4C22×D6111yescharacteristic
442C4×D4C22×D6111yescharacteristic
443C4×D4C22×D6111yescharacteristic
444#117—22no—
445#117—21no—
446#117—21no—
447#117C22⋊D6112yes—
448C2×C4⋊C4—22no—
449C2×C4⋊C4—21no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4C22⋊D6111yescharacteristic
454C2×C22⋊C4—22no—
455C2×C22⋊C4—21no—
456C2×C42C22×D6111yescharacteristic
457#107—1222no—
458#107—1222no—
459#107—1224no—
460#106—1224no—
461C4⋊C8—612no—
462C4⋊C8—612no—
463C4⋊C8—1221no—
464#103—1222no—
465#103—1222no—
466#102—1222no—
467#102—1222no—
468C22⋊C8—1221no—
469#97—1221no—
470#97—1221no—
471C4×C8—612no—
472C24—21no—
473#41—42no—
474#41—42no—
475#41—82no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—82no—
480C2×Q8—1222no—
481C2×Q8—4882no—
482C2×Q8—2441no—
483C2×Q8C2×C22⋊D6111yescharacteristic
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—41no—
490C2×D4C2×C22⋊D6111yescharacteristic
491C2×D4C2×C22⋊D6111yescharacteristic
492C2×D4C2×C22⋊D6111yescharacteristic
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—82no—
503C22×C4—2441no—
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507Q16—4882no—
508SD16—4882no—
509M16—1224no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—2442no—
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—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—2441no—
530C4⋊C4—2441no—
531C4⋊C4C2×C22⋊D6111yescharacteristic
532C4⋊C4C2×C22⋊D6111yescharacteristic
533C4⋊C4C2×C22⋊D6112yes—
534C4⋊C4C2×C22⋊D6111yescharacteristic
535C4⋊C4C2×C22⋊D6111yescharacteristic
536C4⋊C4C2×C22⋊D6112yes—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C42—22no—
553C42—21no—
554C42—22no—
555C42C2×C22⋊D6111yescharacteristic
556C42C2×C22⋊D6111yescharacteristic
557C42C2×C22⋊D6111yescharacteristic
558C42C23×D6111yescharacteristic
559C23—21no—
560C23—21no—
561C23—21no—
562C23—42no—
563C23—42no—
564C23—81no—
565C23#1451974611yescharacteristic
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—42no—
570Q8—82no—
571Q8—4882no—
572Q8—2442no—
573Q8—2442no—
574D4—21no—
575D4—21no—
576D4—21no—
577D4—22no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—82no—
582D4#1451980712yes—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—4882no—
605C2×C4—2442no—
606C2×C4#1451974611yescharacteristic
607C2×C4#1451974611yescharacteristic
608C2×C4#1451974611yescharacteristic
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4C22×C22⋊D6111yescharacteristic
611C2×C4C22×C22⋊D6111yescharacteristic
612C8—1224no—
613C8—2442no—
614C8—2442no—
615C22—21no—
616C22—21no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—84no—
627C22#1683994512yes—
628C22C2×#1451974611yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—41no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—4882no—
641C4C2×#1451980711yescharacteristic
642C4#1684056011yescharacteristic
643C4#1684056512yes—
644C2—22no—
645C2—42no—
646C2—81no—
647C2—81no—
648C2#2145507911yescharacteristic
649C2#2145508711yescharacteristic
650C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3520C2
socleC22×C61C2×#120
hypercenter#3520D61
hypocenterC61#18587
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. #21456843C2semidirect
2. #21456366C2semidirect
3. C2×#16840042C2semidirect
4. #21445103C2semidirect
5. #21445089C2semidirect
6. #21445068C2semidirect
7. #21445054C2semidirect
8. #21444519C2semidirect
9. #21444262C2semidirect
10. #21444232C2semidirect
14. #16840165C22semidirect
15. #16840158C22semidirect
22. #16839972C22semidirect
23. #16839967C22semidirect
32. #16839508C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
380. C61#18587semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214563662
C2×#168400422
#214451032
#214450892
#214450682
#214450542
#214445192
#214442622
#214442322
C61×#35202
#1858761