Finite Group Databasebeta

← #29656725

Subgroups — #29656725

Order 15616.

964 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic, complemented
2#21456366C211yesmaximal, characteristic, complemented
3C2×#16839967C211yesmaximal, characteristic, complemented
4#21445103C212yesmaximal, complemented
5#21445082C212yesmaximal, complemented
6#21445075C212yesmaximal, complemented
7#21445054C212yesmaximal, complemented
8#21444519C211yesmaximal, characteristic, complemented
9#21444262C211yesmaximal, characteristic, complemented
10#21444232C211yesmaximal, characteristic, complemented
11C61×#3520C211yesmaximal, characteristic
12C22×#11758200—21no—
13C2×#14519777C2211yescharacteristic
14#16840172—21no—
15#16840172—22no—
16#16840165C2212yescomplemented
17#16840158C2212yescomplemented
18#16840121—21no—
19#16840095—21no—
20#16840078—21no—
21#16840047C2212yescomplemented
22#16840042C2212yescomplemented
23#16839967—21no—
24#16839967—21no—
25#16839967C2212yes—
26#16839967C2212yes—
27#16839819C2211yescharacteristic
28#16839805—22no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519700—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×#11758200—22no—
57C2×#11758200—22no—
58C2×#11758200—44no—
59C2×#11758194—21no—
60C22×#92418—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777—21no—
64#14519777C2312yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67#14519710—22no—
68#14519710—24no—
69#14519708—22no—
70#14519708—24no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—22no—
75#14519700—22no—
76#14519700—22no—
77#14519698—22no—
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—
148#41×C61—42no—
149#41×C61—42no—
150#41×C61—82no—
151C2×Q8×C61—22no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—82no—
155C2×Q8×C61C2×D411yescharacteristic
156C2×D4×C61—22no—
157C2×D4×C61—22no—
158C2×D4×C61—21no—
159C2×D4×C61—22no—
160C2×D4×C61—21no—
161C2×D4×C61—41no—
162C2×D4×C61C2×D411yescharacteristic
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D411yescharacteristic
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—22no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—21no—
172C22×C4×C61—21no—
173C22×C4×C61—21no—
174C22×C4×C61—82no—
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×Dic61—41no—
179C2×Dic122—22no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122—41no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C4⋊C4×C61C2×D411yescharacteristic
200C4⋊C4×C61C2×D412yes—
201C4⋊C4×C61C2×D412yes—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—22no—
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—
216C22⋊C4×C61—21no—
217C42×C61—22no—
218C42×C61—22no—
219C42×C61—21no—
220C42×C61C2×D411yescharacteristic
221C42×C61C2×D411yescharacteristic
222C42×C61C2×D411yescharacteristic
223C42×C61C2411yescharacteristic
224#11758200—44no—
225#11758200—44no—
226C4⋊Dic61—22no—
227C4⋊Dic61—22no—
228C4⋊Dic61—41no—
229C4⋊Dic61—41no—
230#11758194—24no—
231C2×#92418—22no—
232C2×#92418—22no—
233C2×#92418—22no—
234C2×#92418—42no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—42no—
239C23×C61—42no—
240C23×C61—81no—
241C23×C61#12011yescharacteristic
242Q8×C61—22no—
243Q8×C61—41no—
244Q8×C61—41no—
245Q8×C61—42no—
246Q8×C61—82no—
247D4×C61—21no—
248D4×C61—21no—
249D4×C61—21no—
250D4×C61—22no—
251D4×C61—42no—
252D4×C61—44no—
253D4×C61—42no—
254D4×C61—82no—
255D4×C61AGL(1,ℤ/8)12yes—
256C2×C4×C61—22no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—22no—
260C2×C4×C61—21no—
261C2×C4×C61—22no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—22no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—44no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—81no—
273C2×C4×C61—81no—
274C2×C4×C61—81no—
275C2×C4×C61—82no—
276C2×C4×C61—82no—
277C2×C4×C61#12011yescharacteristic
278C2×C4×C61#12011yescharacteristic
279C2×C4×C61#12011yescharacteristic
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×Dic61—42no—
284C2×Dic61—82no—
285Dic122—42no—
286Dic122—42no—
287Dic122—82no—
288#92418—24no—
289#92418—42no—
290#92418—42no—
291#18586—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—22no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—84no—
304C22×C61#45612yes—
305C22×C61C2×#12011yescharacteristic
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—42no—
315C4×C61—41no—
316C4×C61—41no—
317C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
318C4×C61#57511yescharacteristic
319C4×C61#57712yes—
320Dic61—82no—
321#3520D6111yescharacteristic
322#3239—611no—
323#3165—611no—
324C2×#461—611no—
325#1654—612no—
326#1651—612no—
327#1650—612no—
328#1647—612no—
329#1552—611no—
330#1490—611no—
331#1483—611no—
332C2×C61—22no—
333C2×C61—42no—
334C2×C61—81no—
335C2×C61—81no—
336C2×C61#300611yescharacteristic
337C2×C61#300911yescharacteristic
338C2×C61C2×#45611yescharacteristic
339C22×SD16—1221no—
340#556C2×D6111yescharacteristic
341#554—21no—
342#552—21no—
343#551—22no—
344D4×Q8C2×D6111yescharacteristic
345#548—21no—
346#547C2×D6111yescharacteristic
347#547C2×D6111yescharacteristic
348#546—21no—
349#542—21no—
350#539—22no—
351#533C2×D6111yescharacteristic
352C2×C4⋊Q8—611no—
353#529C2×D6111yescharacteristic
354#525—21no—
355C2×C22⋊Q8—21no—
356#520—21no—
357C2×C4×D4C2×D6111yescharacteristic
358#501—1222no—
359#501—1221no—
360#500—612no—
361#498—612no—
362#489—1221no—
363#484—1221no—
364#481—1221no—
365#475—612no—
366#474—612no—
367#461—612no—
368#461—612no—
369#461—1221no—
370#461—1221no—
371#432—611no—
372#428—1222no—
373C2×C4⋊C8—611no—
374C2×#103—1221no—
375C4×M16—611no—
376#332—612no—
377#332—612no—
378Q8⋊C8—612no—
379D4⋊C8—612no—
380D4⋊C8—612no—
381D4⋊C8—612no—
382C61#1858611yescharacteristic, complemented, Sylow
383C2×#41—21no—
384C2×#41—21no—
385C22×Q8—21no—
386C22×Q8—1221no—
387C22×D4C22⋊D6111yescharacteristic
388C23×C4—21no—
389C2×SD16—1222no—
390C2×SD16—1222no—
391C2×SD16—2444no—
392C2×M16—1221no—
393C22×C8—1221no—
394C4⋊Q8—21no—
395C4⋊Q8—612no—
396C4⋊Q8—1221no—
397C4⋊Q8—1221no—
398C4⋊Q8—1221no—
399C4⋊Q8C22×D6111yescharacteristic
400C4⋊Q8C22×D6111yescharacteristic
401#126—22no—
402#126—22no—
403#126—21no—
404#125—22no—
405#125C22⋊D6111yescharacteristic
406#125C22⋊D6112yes—
407#124—22no—
408#124—21no—
409#124C22⋊D6111yescharacteristic
410#123—22no—
411#123—22no—
412#123—21no—
413#123—22no—
414#123—22no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—21no—
419C22⋊Q8—21no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—22no—
424C22⋊Q8—21no—
425C22⋊Q8C22⋊D6111yescharacteristic
426#121—22no—
427#121—21no—
428#121—21no—
429#121C22⋊D6111yescharacteristic
430#120—22no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8—21no—
434C4×Q8C22×D6111yescharacteristic
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4C22⋊D6111yescharacteristic
443C4×D4C22⋊D6111yescharacteristic
444C4×D4C22×D6111yescharacteristic
445C4×D4C22×D6111yescharacteristic
446C4×D4C22×D6111yescharacteristic
447#117—22no—
448#117—21no—
449#117—21no—
450#117C22⋊D6112yes—
451C2×C4⋊C4—22no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4C22⋊D6111yescharacteristic
457C2×C22⋊C4—22no—
458C2×C22⋊C4—21no—
459C2×C42C22×D6111yescharacteristic
460#107—1224no—
461#107—1222no—
462#106—1224no—
463#106—1222no—
464C4⋊C8—612no—
465C4⋊C8—612no—
466C4⋊C8—1221no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470#102—1222no—
471C22⋊C8—1221no—
472#97—1221no—
473#97—1221no—
474C4×C8—612no—
475C24—21no—
476#41—42no—
477#41—42no—
478#41—82no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—82no—
483C2×Q8—1222no—
484C2×Q8—2442no—
485C2×Q8—2442no—
486C2×Q8—2441no—
487C2×Q8C2×C22⋊D6111yescharacteristic
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—41no—
494C2×D4C2×C22⋊D6111yescharacteristic
495C2×D4C2×C22⋊D6111yescharacteristic
496C2×D4C2×C22⋊D6111yescharacteristic
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—82no—
507C22×C4—2441no—
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511SD16—2444no—
512SD16—2444no—
513M16—1224no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8—1222no—
517C2×C8—2442no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
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⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—2441no—
534C4⋊C4—2441no—
535C4⋊C4C2×C22⋊D6111yescharacteristic
536C4⋊C4C2×C22⋊D6111yescharacteristic
537C4⋊C4C2×C22⋊D6111yescharacteristic
538C4⋊C4C2×C22⋊D6112yes—
539C4⋊C4C2×C22⋊D6111yescharacteristic
540C4⋊C4C2×C22⋊D6112yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C42—22no—
557C42—21no—
558C42—22no—
559C42C2×C22⋊D6111yescharacteristic
560C42C2×C22⋊D6111yescharacteristic
561C42C2×C22⋊D6111yescharacteristic
562C42C23×D6111yescharacteristic
563C23—21no—
564C23—21no—
565C23—21no—
566C23—42no—
567C23—42no—
568C23—81no—
569C23#1451974611yescharacteristic
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—42no—
574Q8—82no—
575Q8—4882no—
576Q8—2442no—
577Q8—2442no—
578D4—21no—
579D4—21no—
580D4—21no—
581D4—22no—
582D4—42no—
583D4—44no—
584D4—42no—
585D4—82no—
586D4#1451980212yes—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—4882no—
609C2×C4—2442no—
610C2×C4#1451974611yescharacteristic
611C2×C4#1451974611yescharacteristic
612C2×C4#1451974611yescharacteristic
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4C22×C22⋊D6111yescharacteristic
615C2×C4C22×C22⋊D6111yescharacteristic
616C8—1224no—
617C8—2442no—
618C8—2442no—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—84no—
631C22#1683994512yes—
632C22C2×#1451974611yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—41no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—41no—
644C4—4882no—
645C4C2×#1451980211yescharacteristic
646C4#1684056011yescharacteristic
647C4#1684057312yes—
648C2—22no—
649C2—42no—
650C2—81no—
651C2—81no—
652C2#2145507111yescharacteristic
653C2#2145509611yescharacteristic
654C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3520C2
socleC22×C61C2×#120
hypercenter#3520D61
hypocenterC61#18586
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. #21457046C2semidirect
2. #21456366C2semidirect
3. C2×#16839967C2semidirect
4. #21445103C2semidirect
5. #21445082C2semidirect
6. #21445075C2semidirect
7. #21445054C2semidirect
8. #21444519C2semidirect
9. #21444262C2semidirect
10. #21444232C2semidirect
16. #16840165C22semidirect
17. #16840158C22semidirect
21. #16840047C22semidirect
22. #16840042C22semidirect
32. #16839508C22semidirect
33. #16839508C22semidirect
34. Q8⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
382. C61#18586semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214563662
C2×#168399672
#214451032
#214450822
#214450752
#214450542
#214445192
#214442622
#214442322
C61×#35202
#1858661