Finite Group Databasebeta

← #29648425

Subgroups — #29648425

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3C2×#16839814C211yesmaximal, characteristic, complemented
4#21444528C212yesmaximal, complemented
5#21444523C212yesmaximal, complemented
6#21444509C211yesmaximal, characteristic, complemented
7#21444497C211yesmaximal, characteristic, complemented
8#21444494C211yesmaximal, characteristic, complemented
9#21444482C212yesmaximal, complemented
10#21444480C212yesmaximal, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12#16840317C2211yescharacteristic
13#16840317C2211yescharacteristic
14#16840313C2211yescharacteristic
15#16840313C2211yescharacteristic
16#16840312C2211yescharacteristic
17#16840299—21no—
18#16840299C412yes—
19#16840298—21no—
20#16840298C412yes—
21#16840297—21no—
22#16840297—21no—
23#16840275—21no—
24#16840275C412yes—
25#16840274—21no—
26C4×#11758226—21no—
27C4×#11758226C412yes—
28#16840230—21no—
29C2×C42×D61C2211yescharacteristic
30#16839818—21no—
31#16839818C2212yes—
32#16839814—21no—
33#16839814—21no—
34#16839814—21no—
35#16839814C2212yes—
36#16839814C2212yes—
37#16839814C2212yes—
38#16839786C2211yescharacteristic
39C2×C4⋊#92418C2211yescharacteristic
40#16839725—21no—
41C2×#14519689—21no—
42M16×Dic61—21no—
43C2×C8×Dic61—21no—
44#16839511C2212yescomplemented
45#16839509C2212yescomplemented
46#16839508C2212yescomplemented
47#16839507C2212yescomplemented
48#16839505C2212yescomplemented
49#16839505C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#333C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×#11758226—21no—
56C2×#11758226C2×C412yes—
57C2×#11758222—21no—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C2×#11758194—21no—
62C22×#92418—21no—
63#14519777C2311yescharacteristic
64#14519774C2311yescharacteristic
65#14519772—22no—
66#14519772—22no—
67#14519768—21no—
68#14519768C2×C412yes—
69#14519766—21no—
70#14519766C2×C412yes—
71#14519765—21no—
72#14519762—22no—
73#14519762—22no—
74#14519761—22no—
75#14519757—21no—
76#14519757—21no—
77#14519753—21no—
78#14519753C2×C412yes—
79#14519752—21no—
80#14519752—21no—
81#14519752—21no—
82#14519752C2×C412yes—
83#14519751—21no—
84#14519751C2×C412yes—
85#14519747—22no—
86Q8×Dic61—21no—
87Q8×Dic61C2×C412yes—
88#14519742—21no—
89#14519742—21no—
90C4×D244—22no—
91C4×D244—22no—
92C4×D244—21no—
93C4×D244C2×C412yes—
94#14519739—22no—
95#14519739—22no—
96#14519739—21no—
97#14519739—21no—
98#14519739—21no—
99#14519739C2×C412yes—
100#14519737—22no—
101#14519737—22no—
102#14519735—21no—
103#14519735—22no—
104#14519735—22no—
105#14519735C2×C412yes—
106#14519734—22no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊#92418—21no—
120C4⋊#92418—21no—
121C4⋊#92418C2312yes—
122C4⋊#92418C2312yes—
123#14519689—22no—
124#14519689—22no—
125#14519689—22no—
126#14519689—22no—
127#14519687—22no—
128C8×Dic61—22no—
129C8×Dic61—22no—
130C8×Dic61—22no—
131C2×M16×C61—21no—
132C22×C8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134C4⋊Q8×C61C2311yescharacteristic
135#125×C61—21no—
136C22⋊Q8×C61—21no—
137C22⋊Q8×C61C2×C412yes—
138C4×Q8×C61—21no—
139C4×Q8×C61C2×C412yes—
140#117×C61—21no—
141C2×C42×C61C2311yescharacteristic
142C4⋊C8×C61—21no—
143C4⋊C8×C61—21no—
144C4⋊C8×C61C2312yes—
145C4⋊C8×C61C2312yes—
146C23×D61C22⋊C411yescharacteristic
147C2×Q8×C61—21no—
148C2×Q8×C61C22×C412yes—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C22×Dic61C22⋊C411yescharacteristic
156#11758226—42no—
157#11758226—42no—
158#11758226—44no—
159#11758222—42no—
160#11758222—42no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244C22×C412yes—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—42no—
172C2×C4×D61—41no—
173C2×C4×D61—41no—
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22⋊C411yescharacteristic
180C2×C4×D61C22×C411yescharacteristic
181C2×C4×D61C22×C411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×C4×D61C2×D411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×Dic122—21no—
187M16×C61—42no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C2×C8×C61—42no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C22×C412yes—
197C22⋊C4×C61—22no—
198C42×C61—22no—
199C42×C61C22×C411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2411yescharacteristic
203#11758198—22no—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207#11758198—22no—
208#11758198—22no—
209#11758198—22no—
210#11758198—22no—
211C4⋊Dic61—21no—
212C4⋊Dic61—22no—
213C4⋊Dic61—22no—
214C4⋊Dic61C22×C412yes—
215#11758196—22no—
216#11758196—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22×C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231#11758194—42no—
232C2×#92418—22no—
233C2×#92418—22no—
234C2×#92418—42no—
235C23×C61C2×C22⋊C411yescharacteristic
236C22×D61—22no—
237C22×D61—22no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61C2×C22⋊C411yescharacteristic
241C22×D61C2×C22⋊C411yescharacteristic
242Q8×C61—42no—
243Q8×C61—41no—
244Q8×C61—41no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61#13512yes—
254C2×C4×C61#13512yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C22⋊D61—44no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269D244—42no—
270D244—44no—
271D244—44no—
272D244—41no—
273D244—41no—
274D244—41no—
275D244—41no—
276C4×D61—21no—
277C4×D61—21no—
278C4×D61—21no—
279C4×D61—21no—
280C4×D61—21no—
281C4×D61—21no—
282C4×D61—21no—
283C4×D61—21no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—41no—
289C4×D61—41no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—44no—
293C4×D61—44no—
294C4×D61—44no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—42no—
298C8×C61—42no—
299#92418—42no—
300#92418—42no—
301#17716—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61#41612yes—
305C22×C61C22×C22⋊C411yescharacteristic
306C2×D61—21no—
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—21no—
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—44no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—42no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61#41612yes—
322C4×C61C2×#13511yescharacteristic
323C4×C61C2×#13511yescharacteristic
324C4×C61#57511yescharacteristic
325C4×C61#57511yescharacteristic
326Dic61—21no—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61—21no—
331Dic61—42no—
332Dic61#41312yes—
333#3444—611no—
334#2920—611no—
335C2×#431—611no—
336#1554—612no—
337#1553—612no—
338#1550—611no—
339#1550D6111yescharacteristic, complemented
340#1547—611no—
341#1547—611no—
342#1544—612no—
343#1543—612no—
344C2×C61—22no—
345C2×C61C2×#41311yescharacteristic
346C2×C61C2×#41611yescharacteristic
347C2×C61#289611yescharacteristic
348D61—41no—
349D61—41no—
350D61—82no—
351#533—611no—
352#533—611no—
353#533C2×D6111yescharacteristic
354#532—611no—
355#532—611no—
356#532—611no—
357#529—612no—
358#529—612no—
359#529—1221no—
360#529—1221no—
361#529—1221no—
362#529—1221no—
363#525—612no—
364#525—1221no—
365#525—1221no—
366C4×#41—612no—
367C4×#41—1221no—
368C4×#41—1221no—
369C22×C42—611no—
370#432—612no—
371#432—1221no—
372#431—612no—
373#431—612no—
374#431—612no—
375#431—1221no—
376#431—1221no—
377#431—1221no—
378#423—611no—
379#423C2×D6111yescharacteristic
380C2×C4⋊C8—611no—
381C2×C4⋊C8C2×D6111yescharacteristic
382#407—1221no—
383C2×C22⋊C8—1221no—
384C4×M16—1221no—
385C2×C4×C8—1221no—
386#333—612no—
387#333C2×D6112yescomplemented
388#332—612no—
389#332—612no—
390#332—612no—
391#332C2×D6112yescomplemented
392#331—612no—
393#331—612no—
394C61#1771611yescharacteristic, complemented, Sylow
395C2×#41—612no—
396C2×#41—1221no—
397C2×#41—1221no—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C2×M16—21no—
402C2×M16—1221no—
403C22×C8—21no—
404C22×C8—1221no—
405C4⋊Q8—611no—
406C4⋊Q8C22×D6111yescharacteristic
407C4⋊Q8C22×D6111yescharacteristic
408#127—611no—
409#126—1222no—
410#126—1222no—
411#125—21no—
412#125—612no—
413#125—1221no—
414#124—612no—
415#124—1221no—
416#124—1221no—
417#123—1222no—
418#123—1222no—
419#123—1222no—
420C22⋊Q8—21no—
421C22⋊Q8—612no—
422C22⋊Q8—1221no—
423C22⋊Q8—1221no—
424C22⋊Q8—1221no—
425C22⋊Q8C4×D6112yes—
426#121—612no—
427#121—612no—
428#121—1221no—
429#121—1221no—
430#121—1221no—
431#121—1221no—
432#120—1222no—
433C4×Q8—21no—
434C4×Q8—612no—
435C4×Q8—1221no—
436C4×Q8—1221no—
437C4×Q8—1221no—
438C4×Q8C4×D6112yes—
439C4×D4—612no—
440C4×D4—612no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1221no—
444C4×D4—1221no—
445C4×D4—1221no—
446C4×D4—1222no—
447C4×D4—1222no—
448C4×D4—1222no—
449C4×D4—1222no—
450C4×D4—1221no—
451#117—21no—
452#117—612no—
453#117—1222no—
454#117—1222no—
455#117—1222no—
456#117—1221no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—1222no—
469C2×C42C22×D6111yescharacteristic
470C4⋊C8—21no—
471C4⋊C8—21no—
472C4⋊C8—612no—
473C4⋊C8—612no—
474C4⋊C8—1221no—
475C4⋊C8—1221no—
476C4⋊C8C22×D6112yes—
477C4⋊C8C22×D6112yes—
478C22⋊C8—1222no—
479C22⋊C8—1222no—
480C22⋊C8—1222no—
481C22⋊C8—1222no—
482#97—1222no—
483C4×C8—1222no—
484C4×C8—1222no—
485C4×C8—1222no—
486C24—611no—
487#41—2442no—
488#41—2442no—
489#41—2444no—
490#41—2442no—
491#41—2442no—
492C2×Q8—21no—
493C2×Q8—1221no—
494C2×Q8C2×C4×D6112yes—
495C2×D4—612no—
496C2×D4—1222no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—1221no—
500C2×D4—1221no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—2441no—
522C22×C4—2441no—
523C22×C4—2442no—
524C22×C4C2×C4×D6111yescharacteristic
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527M16—42no—
528M16—2442no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—42no—
532C2×C8—1222no—
533C2×C8—1222no—
534C2×C8—2442no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—612no—
541C4⋊C4—1222no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4—1221no—
546C4⋊C4C2×C4×D6112yes—
547C22⋊C4—22no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C42—22no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—1222no—
570C42—1222no—
571C42C2×C4×D6111yescharacteristic
572C42D4×D6111yescharacteristic
573C42D4×D6111yescharacteristic
574C42C23×D6111yescharacteristic
575C23—611no—
576C23—611no—
577C23—1222no—
578C23—1222no—
579C23—2441no—
580C23—2441no—
581C23C22⋊C4×D6111yescharacteristic
582Q8—42no—
583Q8—41no—
584Q8—41no—
585Q8—2441no—
586Q8—2441no—
587D4—2442no—
588D4—2444no—
589D4—2444no—
590D4—2444no—
591D4—2441no—
592D4—2441no—
593D4—2441no—
594D4—2441no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—611no—
601C2×C4—611no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2444no—
626C2×C4—2444no—
627C2×C4—2441no—
628C2×C4—2444no—
629C2×C4C22⋊C4×D6111yescharacteristic
630C2×C4C22⋊C4×D6111yescharacteristic
631C2×C4C22⋊C4×D6111yescharacteristic
632C2×C4#1451979812yes—
633C2×C4#1451980012yes—
634C2×C4C22×C4×D6111yescharacteristic
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×D4×D6111yescharacteristic
637C8—42no—
638C8—42no—
639C8—2442no—
640C8—2442no—
641C22—21no—
642C22—21no—
643C22—1221no—
644C22—1221no—
645C22—1221no—
646C22—1221no—
647C22—1221no—
648C22—1221no—
649C22—2444no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22#1683975712yes—
654C22C2×C22⋊C4×D6111yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—42no—
659C4—42no—
660C4—612no—
661C4—1221no—
662C4—1221no—
663C4—1221no—
664C4—1221no—
665C4—1221no—
666C4—2442no—
667C4#1683975712yes—
668C4C2×#1451979811yescharacteristic
669C4C2×#1451980011yescharacteristic
670C4#1684055811yescharacteristic
671C4#1684056111yescharacteristic
672C2—22no—
673C2—4882no—
674C2—2441no—
675C2—2441no—
676C2#413×D6111yescharacteristic
677C2C2×#1683975711yescharacteristic
678C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1550D61
hypocenterC61#17716
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. C2×#16839814C2semidirect
4. #21444528C2semidirect
5. #21444523C2semidirect
6. #21444509C2semidirect
7. #21444497C2semidirect
8. #21444494C2semidirect
9. #21444482C2semidirect
10. #21444480C2semidirect
11. C61×#1550C2semidirect
44. #16839511C22semidirect
45. #16839509C22semidirect
46. #16839508C22semidirect
47. #16839507C22semidirect
48. #16839505C22semidirect
49. #16839505C22semidirect
53. C61×#333C22semidirect
54. C61×#332C22semidirect
339. #1550D61semidirect
387. #333C2×D61semidirect
391. #332C2×D61semidirect
394. C61#17716semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544512
C2×#168398142
#214445282
#214445232
#214445092
#214444972
#214444942
#214444822
#214444802
C61×#15502
#1771661