Finite Group Databasebeta

← #29647653

Subgroups — #29647653

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459548C211yesmaximal, characteristic
2#21454479C211yesmaximal, characteristic, complemented
3#21448710C211yesmaximal, characteristic, complemented
4#21444393C212yesmaximal, complemented
5#21444387C211yesmaximal, characteristic, complemented
6#21444364C212yesmaximal, complemented
7#21444363C211yesmaximal, characteristic, complemented
8#21444341C212yesmaximal, complemented
9#21444312C212yesmaximal, complemented
10#21444308C211yesmaximal, characteristic, complemented
11C61×#1507C211yesmaximal, characteristic, complemented
12#16840476—21no—
13#16840464—21no—
14#16840459—21no—
15#16840438—22no—
16#16840412—21no—
17#16840406—22no—
18#16840393C411yescharacteristic, complemented
19#16840379C412yescomplemented
20#16840365C412yescomplemented
21#16840329C411yescharacteristic, complemented
22#16840307C2212yes—
23#16840293C2212yes—
24#16840273C2211yescharacteristic
25#16840243C411yescharacteristic, complemented
26#16840237C411yescharacteristic, complemented
27C2×#14519736C2211yescharacteristic
28#16839820—22no—
29#16839816C2212yescomplemented
30#16839812C2212yescomplemented
31#16839790—21no—
32C2×C22⋊#92418C2211yescharacteristic
33#16839720—21no—
34#16839660—21no—
35#16839656—22no—
36#16839629C2212yescomplemented
37#16839622C2211yescharacteristic
38C4×C4⋊Dic61—21no—
39#16839613C2212yescomplemented
40#16839500C2212yescomplemented
41#16839498C2211yescharacteristic, complemented
42#16839498C2211yescharacteristic, complemented
43#16839497C2212yescomplemented
44#16839496C2212yes—
45#16839496C2212yes—
46#16839495C2211yescharacteristic, complemented
47#16839495C2211yescharacteristic, complemented
48C2×#121×C61C2211yescharacteristic
49C61×#407C2211yescharacteristic
50C4×C22⋊C4×C61C2211yescharacteristic
51C61×#327C2212yes—
52C61×#327C2212yes—
53C2×#11758222C2×C411yescharacteristic
54C2×#11758224—22no—
55C22×#92418—21no—
56#14519776—21no—
57#14519776C2×C412yes—
58#14519773C2×C412yes—
59#14519770—22no—
60#14519770—22no—
61#14519768—21no—
62#14519768C2×C412yes—
63#14519764—22no—
64#14519764C2×C412yes—
65#14519763—22no—
66#14519761—22no—
67#14519760—21no—
68#14519759—22no—
69#14519758—22no—
70#14519758—22no—
71#14519756—22no—
72#14519756C2×C411yescharacteristic
73#14519756C2311yescharacteristic
74#14519754—22no—
75#14519751C2×C411yescharacteristic
76#14519751C2311yescharacteristic
77#14519749—22no—
78#14519749—22no—
79#14519748—22no—
80#14519748—21no—
81#14519748—21no—
82#14519746—21no—
83#14519746—22no—
84#14519742C2×C412yes—
85D4×Dic61—22no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61C2×C412yes—
89D4×Dic61C2×C412yes—
90#14519739C2×C412yes—
91C4×C22⋊D61—21no—
92C4×C22⋊D61—21no—
93C4×C22⋊D61—22no—
94#14519737—22no—
95#14519736C2×C411yescharacteristic
96#14519736C2×C411yescharacteristic
97#14519736C2×C411yescharacteristic
98#14519736C2×C412yes—
99#14519736D411yescharacteristic
100#14519736D411yescharacteristic
101#14519736D412yes—
102#14519735—21no—
103#14519733—22no—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×C4⋊Dic61D411yescharacteristic
107C2×#11758196—22no—
108C4⋊C4×D61—21no—
109C2×C22⋊Dic61—22no—
110C2×C22⋊Dic61—21no—
111C2×C22⋊Dic61D412yes—
112C2×C4×Dic61—22no—
113C2×C4×Dic61C2312yes—
114#14519706—22no—
115#14519706—22no—
116C22⋊#92418—21no—
117C22⋊#92418C2311yescharacteristic
118C22⋊#92418C2311yescharacteristic
119#14519687—22no—
120C8×Dic61—22no—
121#14519684—22no—
122#14519684—22no—
123#14519684C2312yes—
124C22×D4×C61—21no—
125C22×D4×C61C2×C411yescharacteristic
126C23×C4×C61C2311yescharacteristic
127C2×M16×C61—21no—
128#121×C61—21no—
129#121×C61—21no—
130#121×C61C2×C412yes—
131#121×C61C2312yes—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C22⋊C4×C61—21no—
134C2×C22⋊C4×C61—21no—
135C2×C42×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138#95×C61C2311yescharacteristic
139#95×C61C2311yescharacteristic
140C24×C61—21no—
141C24×C61C2×D411yescharacteristic
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—42no—
146C2×D4×C61—81no—
147C2×D4×C61—81no—
148C2×D4×C61C22×C412yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—42no—
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C22×C411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C2×C22⋊D61—21no—
160C2×C22⋊D61—21no—
161C2×C22⋊D61—21no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169#11758224—82no—
170#11758224—82no—
171#11758222—42no—
172C2×D244C22×C411yescharacteristic
173C2×C4×D61—21no—
174C2×C4×D61—22no—
175C2×Dic122—22no—
176C2×Dic122C22×C411yescharacteristic
177M16×C61—42no—
178C2×C8×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C412yes—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—42no—
184C42×C61—42no—
185C22⋊Dic61—21no—
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—81no—
193C22⋊Dic61C22⋊C412yes—
194C22⋊Dic61C22⋊C412yes—
195C22⋊Dic61C22⋊C412yes—
196C22⋊Dic61C22⋊C412yes—
197#11758198—21no—
198#11758198—21no—
199#11758198—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—42no—
202C4⋊Dic61C22⋊C411yescharacteristic
203C4⋊Dic61C22⋊C411yescharacteristic
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22⋊C411yescharacteristic
206C4⋊Dic61C22⋊C412yes—
207C4⋊Dic61C22⋊C412yes—
208C4⋊Dic61C22×C411yescharacteristic
209C4⋊Dic61C22×C411yescharacteristic
210#11758196—21no—
211#11758196—21no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—44no—
216C4×Dic61—21no—
217C4×Dic61—22no—
218C4×Dic61—82no—
219C4×Dic61C22×C412yes—
220C4×Dic61C22×C412yes—
221C4×Dic61C2×D412yes—
222C4×Dic61C2×D412yes—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—42no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—22no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—42no—
232C23×C61—81no—
233C23×C61—81no—
234C23×C61C2×C22⋊C411yescharacteristic
235C23×C61C2×C22⋊C411yescharacteristic
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238D4×C61—42no—
239D4×C61—42no—
240D4×C61—42no—
241D4×C61—42no—
242D4×C61—82no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—81no—
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C22⋊D61—42no—
262C22⋊D61—41no—
263C22⋊D61—41no—
264C22⋊D61—82no—
265C22⋊D61—82no—
266C22⋊D61—81no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—82no—
277C2×Dic61—82no—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280C2×Dic61C2×C22⋊C412yes—
281C2×Dic61C2×C22⋊C412yes—
282D244—41no—
283C4×D61—42no—
284C4×D61—82no—
285Dic122—41no—
286Dic122—82no—
287C8×C61—42no—
288#92418—41no—
289#92418—41no—
290#17607—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—162no—
305C22×C61C22×C22⋊C411yescharacteristic
306C22×C61#57611yescharacteristic
307C22×C61#57711yescharacteristic
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—42no—
315C4×C61—42no—
316C4×C61—81no—
317C4×C61#41012yes—
318Dic61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—22no—
322Dic61—41no—
323Dic61—41no—
324Dic61—41no—
325Dic61—82no—
326#3494—611no—
327#2924—611no—
328#2347—611no—
329#1521—612no—
330#1519—611no—
331#1514—612no—
332#1513—611no—
333#1509—612no—
334#1507D6111yescharacteristic, complemented
335#1502—612no—
336#1501—611no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—41no—
340C2×C61—41no—
341C2×C61—82no—
342C2×C61C2×#41011yescharacteristic
343C2×C61#289711yescharacteristic
344C2×C61#290111yescharacteristic
345D61—81no—
346#555—1221no—
347#553—1221no—
348#553—1221no—
349#551—1222no—
350#547—1221no—
351#546—1222no—
352#544—611no—
353#542—612no—
354#540—612no—
355#535—611no—
356#532—612no—
357#529—612no—
358#525—611no—
359C2×#121C2×D6111yescharacteristic
360#520—611no—
361#518—611no—
362C2×#117—611no—
363#433—1222no—
364#432—612no—
365#431—612no—
366#424—1221no—
367#407C2×D6111yescharacteristic
368C2×C22⋊C8—611no—
369#405—1221no—
370#389—1221no—
371#388—1222no—
372#383—612no—
373#380—611no—
374C4×C4⋊C4—612no—
375C4×C4⋊C4—1221no—
376C4×C22⋊C4C2×D6111yescharacteristic
377#328—612no—
378#328—611no—
379#328—611no—
380#327—612no—
381#327—612no—
382#327—612no—
383#327—611no—
384#327—611no—
385#327C2×D6112yes—
386#327C2×D6112yes—
387C61#1760711yescharacteristic, complemented, Sylow
388C2×#41—611no—
389C2×#41—1222no—
390C22×D4—21no—
391C22×D4C4×D6111yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C2×M16—21no—
394C22×C8—1221no—
395C4⋊Q8—612no—
396C4⋊Q8—1221no—
397#127—612no—
398#126—1222no—
399#126—1222no—
400#125—612no—
401#125—1221no—
402#124—612no—
403#124—1222no—
404#124—1222no—
405#123—1222no—
406#123—1222no—
407#123—1222no—
408#123—1222no—
409#123—1221no—
410C22⋊Q8—611no—
411C22⋊Q8—611no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414#121—21no—
415#121—21no—
416#121—611no—
417#121—611no—
418#121—1222no—
419#121—1222no—
420#121—1221no—
421#121—1221no—
422#121—1222no—
423#121C4×D6112yes—
424#121C22×D6112yes—
425#120—1222no—
426#120—1221no—
427C4×Q8—612no—
428C4×D4—612no—
429C4×D4—612no—
430C4×D4—612no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1221no—
434C4×D4—1221no—
435C4×D4—1222no—
436C4×D4—1221no—
437C4×D4—1221no—
438#117—611no—
439#117—611no—
440#117—612no—
441#117—612no—
442#117—611no—
443#117—611no—
444#117—611no—
445#117—1222no—
446#117—1221no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C4×D6111yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—612no—
456C2×C22⋊C4—1222no—
457C2×C22⋊C4—1221no—
458C2×C42—21no—
459C2×C42—612no—
460C2×C42—1222no—
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463C22⋊C8—21no—
464C22⋊C8—611no—
465C22⋊C8—611no—
466C22⋊C8—1221no—
467C22⋊C8C22×D6112yes—
468#97—1222no—
469C4×C8—1222no—
470#95—612no—
471#95—1222no—
472#95—1222no—
473#95C22×D6111yescharacteristic
474#95C22×D6111yescharacteristic
475C24—21no—
476C24D4×D6111yescharacteristic
477#41—4882no—
478#41—4882no—
479#41—2442no—
480C2×Q8—611no—
481C2×Q8—1222no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4—611no—
489C2×D4—1222no—
490C2×D4—1222no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4C2×C4×D6112yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1221no—
505C22×C4—1222no—
506C22×C4C2×C4×D6111yescharacteristic
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512M16—42no—
513C2×C8—22no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—2442no—
517C4⋊C4—21no—
518C4⋊C4—611no—
519C4⋊C4—611no—
520C4⋊C4—611no—
521C4⋊C4—611no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2444no—
533C4⋊C4—2442no—
534C4⋊C4C2×C4×D6112yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—42no—
538C22⋊C4—612no—
539C22⋊C4—612no—
540C22⋊C4—612no—
541C22⋊C4—612no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1221no—
548C22⋊C4—1221no—
549C22⋊C4—1221no—
550C22⋊C4—1221no—
551C22⋊C4—1221no—
552C22⋊C4—4881no—
553C42—42no—
554C42—612no—
555C42—612no—
556C42—612no—
557C42—612no—
558C42—1222no—
559C42—1221no—
560C42—4882no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—42no—
567C23—81no—
568C23—81no—
569C23—1221no—
570C23C22⋊C4×D6111yescharacteristic
571C23C22⋊C4×D6111yescharacteristic
572C23C2×D4×D6111yescharacteristic
573Q8—4882no—
574Q8—2441no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—82no—
580D4—4882no—
581D4—4882no—
582D4—4881no—
583D4—2441no—
584D4—2442no—
585D4—2441no—
586D4—2441no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—81no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—4882no—
611C2×C4—4882no—
612C2×C4—4882no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4C22⋊C4×D6111yescharacteristic
619C2×C4C22⋊C4×D6111yescharacteristic
620C2×C4C22×C4×D6111yescharacteristic
621C2×C4C2×D4×D6111yescharacteristic
622C8—42no—
623C8—2441no—
624C8—2441no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—162no—
639C22—2441no—
640C22—2441no—
641C22—2441no—
642C22C2×C22⋊C4×D6111yescharacteristic
643C22#1684056611yescharacteristic
644C22#1684057211yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—1222no—
652C4—1222no—
653C4—1222no—
654C4—1222no—
655C4—4882no—
656C4—2441no—
657C4—2441no—
658C4—2441no—
659C4#1683973412yes—
660C2—21no—
661C2—21no—
662C2—41no—
663C2—41no—
664C2—82no—
665C2—4881no—
666C2C2×#1683973411yescharacteristic
667C2#2145434411yescharacteristic
668C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1507C2
socleC22×C61C22×C22⋊C4
prime radical#21459548C2
cosocleC22×C4×C61C24
hypercenter#1507D61
hypocenterC61#17607
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #21454479C2semidirect
3. #21448710C2semidirect
4. #21444393C2semidirect
5. #21444387C2semidirect
6. #21444364C2semidirect
7. #21444363C2semidirect
8. #21444341C2semidirect
9. #21444312C2semidirect
10. #21444308C2semidirect
11. C61×#1507C2semidirect
18. #16840393C4semidirect
19. #16840379C4semidirect
20. #16840365C4semidirect
21. #16840329C4semidirect
25. #16840243C4semidirect
26. #16840237C4semidirect
29. #16839816C22semidirect
30. #16839812C22semidirect
36. #16839629C22semidirect
39. #16839613C22semidirect
40. #16839500C22semidirect
41. #16839498C22semidirect
42. #16839498C22semidirect
43. #16839497C22semidirect
46. #16839495C22semidirect
47. #16839495C22semidirect
334. #1507D61semidirect
387. C61#17607semidirect

Maximal subgroups

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

subgroupindex
#214595482
#214544792
#214487102
#214443932
#214443872
#214443642
#214443632
#214443412
#214443122
#214443082
C61×#15072
#1760761