Finite Group Databasebeta

← #29647032

Subgroups — #29647032

Order 15616.

1008 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#21459429C211yesmaximal, characteristic
2#21454463C211yesmaximal, characteristic, complemented
3#21444282C212yesmaximal, complemented
4#21444269C211yesmaximal, characteristic, complemented
5#21444256C211yesmaximal, characteristic, complemented
6#21444243C212yesmaximal, complemented
7#21444235C211yesmaximal, characteristic, complemented
8#21444225C212yesmaximal, complemented
9#21444209C212yesmaximal, complemented
10#21444199C211yesmaximal, characteristic, complemented
11C61×#2991C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840472C411yescharacteristic
14#16840469C411yescharacteristic
15#16840458C412yes—
16#16840456—21no—
17#16840436—22no—
18#16840408C412yes—
19#16840400—21no—
20D4×D244C411yescharacteristic
21#16840312C2211yescharacteristic
22#16840299—21no—
23#16840296C2212yes—
24#16840271—22no—
25C4×#11758222C2211yescharacteristic
26C2×C4×D244C2212yes—
27C4×D4×D61—21no—
28#16839823—22no—
29#16839815C2212yescomplemented
30#16839815C2212yescomplemented
31#16839789—21no—
32#16839729—21no—
33C4×#11758194C2211yescharacteristic, complemented
34C2×C4×#92418C2211yescharacteristic, complemented
35#16839492C2211yescharacteristic, complemented
36#16839492C2211yescharacteristic, complemented
37#16839492C2212yescomplemented
38#16839489C2211yescharacteristic
39#16839489C2211yescharacteristic
40#16839489C2212yes—
41#16839489C2212yescomplemented
42#16839489C2212yescomplemented
43#16839489C2212yescomplemented
44#16839489C2212yescomplemented
45C4×#41×C61C2211yescharacteristic
46C8×Q8×C61—21no—
47C61×#436—21no—
48C61×#436—22no—
49C61×#431C2211yescharacteristic, complemented
50C61×#431C2212yescomplemented
51C61×#423C2212yescomplemented
52C61×#423C2211yescharacteristic, complemented
53C2×#11758222—21no—
54C2×#11758222C2×C411yescharacteristic
55C22×D244C2×C412yes—
56C2×D4×D61—21no—
57C22×C4×D61—22no—
58C2×#11758194—21no—
59C22×#92418—21no—
60#14519777C2×C411yescharacteristic
61#14519774C2×C411yescharacteristic
62#14519772—22no—
63#14519771—22no—
64#14519769C2×C412yes—
65#14519768—21no—
66#14519766—21no—
67#14519765—21no—
68#14519765C2×C412yes—
69#14519762—22no—
70#14519762—22no—
71#14519760—22no—
72#14519759—22no—
73#14519756C2×C412yes—
74#14519755—22no—
75#14519754—22no—
76#14519753—21no—
77#14519752—21no—
78#14519751—21no—
79#14519751C2×C412yes—
80#14519751C2×C412yes—
81#14519750—22no—
82#14519749—22no—
83#14519747—22no—
84#14519745—22no—
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122C2311yescharacteristic
87C4×D244—22no—
88C4×D244—21no—
89C4×D244—22no—
90C4×D244C2×C412yes—
91C4×D244C2×C412yes—
92C4×D244C2×C411yescharacteristic
93C4×D244C2311yescharacteristic
94C4×D244C2312yes—
95D4×Dic61—21no—
96#14519739—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—21no—
100#14519737—22no—
101#14519736C2×C411yescharacteristic
102#14519734—21no—
103#14519734—22no—
104C2×C4⋊Dic61C2×C412yes—
105C4⋊C4×D61—21no—
106C2×#11758198—22no—
107C22⋊C4×D61—22no—
108C42×D61—21no—
109C4⋊#92418—22no—
110C4⋊#92418—22no—
111C22⋊#92418—22no—
112C22⋊#92418—22no—
113#14519688—21no—
114C4×#92418—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2312yes—
118C2×#41×C61D411yescharacteristic
119C2×M16×C61—22no—
120C2×M16×C61—21no—
121C4×Q8×C61C2×C411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61C2×C412yes—
124C4×D4×C61C2×C411yescharacteristic
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127#117×C61D411yescharacteristic
128#117×C61D412yes—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2312yes—
131C4⋊C8×C61—21no—
132C4⋊C8×C61—22no—
133C4⋊C8×C61C2311yescharacteristic
134C4⋊C8×C61C2312yes—
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—22no—
137C22⋊C8×C61—21no—
138C4×C8×C61—21no—
139C4×C8×C61—22no—
140C23×D61—22no—
141#41×C61—42no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—22no—
154C22×Dic61—22no—
155D4×D61—44no—
156#11758222—22no—
157#11758222—41no—
158#11758222—81no—
159#11758222—81no—
160C2×D244—24no—
161C2×D244—21no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244—42no—
165C2×D244C22×C412yes—
166C2×D244C22×C411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—82no—
174C2×C4×D61—82no—
175C2×Dic122—21no—
176C2×Dic122C22×C411yescharacteristic
177M16×C61—82no—
178M16×C61—81no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—22no—
182C4⋊C4×C61C22⋊C411yescharacteristic
183C4⋊C4×C61C22⋊C412yes—
184C4⋊C4×C61C22⋊C411yescharacteristic
185C4⋊C4×C61C22⋊C412yes—
186C22⋊C4×C61C22⋊C412yes—
187C22⋊C4×C61C22⋊C412yes—
188C22⋊C4×C61C22⋊C412yes—
189C42×C61C22×C412yes—
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D412yes—
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D412yes—
195C42×C61C2411yescharacteristic
196C22⋊Dic61—22no—
197C22⋊Dic61—21no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—21no—
203#11758198—22no—
204#11758198—22no—
205#11758198—21no—
206#11758198—22no—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61C22×C411yescharacteristic
210C4⋊Dic61C22×C412yes—
211C4⋊Dic61C22×C411yescharacteristic
212#11758196—22no—
213#11758196—21no—
214#11758196—22no—
215#11758196—22no—
216#11758196—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220#11758194—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—22no—
224C2×#92418—42no—
225C23×C61C2×C22⋊C412yes—
226C23×C61C2×C22⋊C411yescharacteristic
227C22×D61—21no—
228C22×D61—22no—
229C22×D61—44no—
230C22×D61—44no—
231C22×D61—82no—
232Q8×C61—22no—
233D4×C61—22no—
234D4×C61—42no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—42no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C412yes—
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C22⋊D61—42no—
255C22⋊D61—44no—
256C22⋊D61—82no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—82no—
262D244—22no—
263D244—21no—
264D244—22no—
265D244—44no—
266D244—42no—
267D244—81no—
268D244#13112yes—
269C4×D61—42no—
270C4×D61—41no—
271C4×D61—42no—
272C4×D61—41no—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—82no—
276C4×D61—82no—
277Dic122—21no—
278Dic122—21no—
279Dic122—81no—
280C8×C61—41no—
281C8×C61—42no—
282C8×C61—41no—
283#92418—41no—
284#92418—41no—
285#92418—42no—
286#17522—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—21no—
291C22×C61—22no—
292C22×C61—21no—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—22no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—88no—
301C4×C61—21no—
302C4×C61—22no—
303C4×C61—21no—
304C4×C61—22no—
305C4×C61—21no—
306C4×C61—22no—
307C4×C61#42112yes—
308C4×C61C2×#13111yescharacteristic
309C4×C61#56811yescharacteristic
310C4×C61#57611yescharacteristic
311C4×C61#57711yescharacteristic
312Dic61—41no—
313Dic61—42no—
314Dic61—41no—
315Dic61—41no—
316#3481—611no—
317#2991D6111yescharacteristic, complemented
318#2922—611no—
319#1495—612no—
320#1492—611no—
321#1489—611no—
322#1486—612no—
323#1484—611no—
324#1482—612no—
325#1478—612no—
326#1476—611no—
327C2×C61—42no—
328C2×C61—41no—
329C2×C61#288011yescharacteristic
330C2×C61#289711yescharacteristic
331C2×C61C2×#42111yescharacteristic
332D61—42no—
333D61—82no—
334#559—611no—
335#554—611no—
336#554—611no—
337#553—612no—
338#553—1221no—
339#551—1222no—
340#546—612no—
341#546—1221no—
342D42—611no—
343#532—611no—
344#529—612no—
345#529—1221no—
346#525—1222no—
347C4×#41—611no—
348C4×#41C2×D6111yescharacteristic
349C2×C4×D4—612no—
350C2×C4×D4—1221no—
351C8×Q8—21no—
352#436—21no—
353#436—22no—
354#433—1222no—
355#431—612no—
356#431—612no—
357#431C2×D6111yescharacteristic, complemented
358#431C2×D6112yescomplemented
359#424—1221no—
360#423C2×D6112yescomplemented
361#423C2×D6111yescharacteristic, complemented
362#408—1221no—
363C4×M16—611no—
364C2×C4×C8—611no—
365Q8⋊C8—611no—
366Q8⋊C8—611no—
367Q8⋊C8—612no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374D4⋊C8—612no—
375C61#1752211yescharacteristic, complemented, Sylow
376C2×#41—611no—
377C2×#41—1221no—
378C2×#41D24411yescharacteristic
379C22×D4—612no—
380C22×D4—1221no—
381C23×C4—1222no—
382C2×M16—22no—
383C2×M16—21no—
384C2×M16—1221no—
385C22×C8—1221no—
386C4⋊Q8—611no—
387#127—611no—
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1221no—
392#124—612no—
393#124—1221no—
394#124—1221no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—1221no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403#121—612no—
404#121—612no—
405#121—1222no—
406#121—1221no—
407#121—1221no—
408#121—1222no—
409#120—1222no—
410#120—1222no—
411C4×Q8—611no—
412C4×Q8—611no—
413C4×Q8C4×D6111yescharacteristic
414C4×Q8C22⋊D6111yescharacteristic
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—611no—
418C4×D4—611no—
419C4×D4—612no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1221no—
429C4×D4C4×D6111yescharacteristic
430C4×D4C4×D6112yes—
431C4×D4C22⋊D6111yescharacteristic
432C4×D4C22⋊D6112yes—
433#117—611no—
434#117—1221no—
435#117—1222no—
436#117D24412yes—
437#117D24411yescharacteristic
438C2×C4⋊C4—612no—
439C2×C4⋊C4—1221no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C42—1221no—
443C2×C42C22×D6112yes—
444C2×C42C22×D6111yescharacteristic
445C4⋊C8—21no—
446C4⋊C8—22no—
447C4⋊C8—1222no—
448C4⋊C8—1222no—
449C4⋊C8C22×D6111yescharacteristic
450C4⋊C8C22×D6112yes—
451C4⋊C8C22×D6111yescharacteristic
452C22⋊C8—22no—
453C22⋊C8—21no—
454C22⋊C8—1222no—
455C22⋊C8—1222no—
456#97—1221no—
457C4×C8—21no—
458C4×C8—22no—
459C4×C8—611no—
460C4×C8—611no—
461C4×C8—612no—
462C4×C8—1221no—
463C24—1222no—
464#41—42no—
465#41—1222no—
466#41—4881no—
467#41—4881no—
468#41—2441no—
469C2×Q8—611no—
470C2×Q8—1221no—
471C2×Q8#1175819811yescharacteristic
472C2×D4—611no—
473C2×D4—612no—
474C2×D4—1224no—
475C2×D4—1222no—
476C2×D4—1221no—
477C2×D4—1222no—
478C2×D4—1221no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—2442no—
482C2×D4—2444no—
483C2×D4#1175819811yescharacteristic
484C2×D4#1175819812yes—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—4882no—
493C22×C4—4882no—
494C22×C4C2×C4×D6112yes—
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C2×D24412yes—
498C22×C4C2×C22⋊D6112yes—
499C22×C4C2×C22⋊D6111yescharacteristic
500M16—82no—
501M16—81no—
502M16—2442no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1222no—
509C2×C8—2442no—
510C4⋊C4—611no—
511C4⋊C4—612no—
512C4⋊C4—611no—
513C4⋊C4—1221no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4#1175819811yescharacteristic
521C4⋊C4#1175819811yescharacteristic
522C4⋊C4#1175819812yes—
523C4⋊C4#1175819812yes—
524C22⋊C4—1222no—
525C22⋊C4—1221no—
526C22⋊C4—1221no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1221no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4#1175819812yes—
536C22⋊C4#1175819812yes—
537C22⋊C4#1175819812yes—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42C2×C4×D6112yes—
542C42C2×C4×D6111yescharacteristic
543C42C2×D24411yescharacteristic
544C42C2×D24412yes—
545C42C2×C22⋊D6111yescharacteristic
546C42C2×C22⋊D6112yes—
547C42C23×D6111yescharacteristic
548C23—1221no—
549C23—1222no—
550C23—4882no—
551C23—2444no—
552C23—2444no—
553C23C2×#1175819812yes—
554C23C2×#1175819811yescharacteristic
555Q8—22no—
556Q8—1221no—
557Q8—1221no—
558Q8—4881no—
559D4—22no—
560D4—42no—
561D4—612no—
562D4—1222no—
563D4—1221no—
564D4—1222no—
565D4—4881no—
566D4—4882no—
567D4—2444no—
568D4—2442no—
569D4—2444no—
570D4—2442no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—42no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—4882no—
586C2×C4—4882no—
587C2×C4—4882no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2441no—
591C2×C4—2442no—
592C2×C4—2441no—
593C2×C4—2442no—
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819812yes—
596C2×C4C2×#1175819811yescharacteristic
597C2×C4C2×#1175819812yes—
598C2×C4C2×#1175819812yes—
599C2×C4C2×#1175819811yescharacteristic
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C2×C4C22×D24411yescharacteristic
603C8—41no—
604C8—42no—
605C8—41no—
606C8—2441no—
607C8—2441no—
608C8—2442no—
609C22—22no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—21no—
614C22—22no—
615C22—1222no—
616C22—4888no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2441no—
621C22—2441no—
622C22C22×#1175819811yescharacteristic
623C4—22no—
624C4—21no—
625C4—21no—
626C4—22no—
627C4—21no—
628C4—22no—
629C4—2441no—
630C4—2442no—
631C4—2441no—
632C4—2441no—
633C4#1683977812yes—
634C4#131×D6111yescharacteristic
635C4#1684052711yescharacteristic
636C4#1684056511yescharacteristic
637C4#1684057011yescharacteristic
638C2—42no—
639C2—41no—
640C2—4882no—
641C2—2442no—
642C2#2145426611yescharacteristic
643C2#2145434211yescharacteristic
644C2C2×#1683977811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2991C2
socleC22×C61C22×C22⋊C4
prime radical#21459429C2
cosocleC42×C61C24
hypercenter#2991D61
hypocenterC61#17522
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. #21454463C2semidirect
3. #21444282C2semidirect
4. #21444269C2semidirect
5. #21444256C2semidirect
6. #21444243C2semidirect
7. #21444235C2semidirect
8. #21444225C2semidirect
9. #21444209C2semidirect
10. #21444199C2semidirect
11. C61×#2991C2semidirect
29. #16839815C22semidirect
30. #16839815C22semidirect
33. C4×#11758194C22semidirect
34. C2×C4×#92418C22semidirect
35. #16839492C22semidirect
36. #16839492C22semidirect
37. #16839492C22semidirect
41. #16839489C22semidirect
42. #16839489C22semidirect
43. #16839489C22semidirect
44. #16839489C22semidirect
49. C61×#431C22semidirect
50. C61×#431C22semidirect
51. C61×#423C22semidirect
52. C61×#423C22semidirect
317. #2991D61semidirect
357. #431C2×D61semidirect
358. #431C2×D61semidirect
360. #423C2×D61semidirect
361. #423C2×D61semidirect
375. C61#17522semidirect

Maximal subgroups

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

subgroupindex
#214594292
#214544632
#214442822
#214442692
#214442562
#214442432
#214442352
#214442252
#214442092
#214441992
C61×#29912
#1752261