Finite Group Databasebeta

← #29646829

Subgroups — #29646829

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic, complemented
3C4×M16×D61C211yesmaximal, characteristic, complemented
4#21444278C214yesmaximal, complemented
5#21444257C214yesmaximal, complemented
6#21444240C211yesmaximal, characteristic, complemented
7#21444237C211yesmaximal, characteristic, complemented
8#21444235C211yesmaximal, characteristic, complemented
9C61×#1485C211yesmaximal, characteristic, complemented
10C2×M16×D61—21no—
11#16840317C411yescharacteristic
12#16840317C411yescharacteristic
13#16840313C411yescharacteristic
14#16840313C411yescharacteristic
15#16840312C2211yescharacteristic
16#16840299—21no—
17#16840299C2212yes—
18#16840298—21no—
19#16840298C412yes—
20#16840297—21no—
21#16840297—21no—
22#16840275—21no—
23#16840275C412yes—
24#16840274—21no—
25C4×#11758226—21no—
26C4×#11758226C2212yes—
27#16840230—21no—
28C2×C42×D61C2211yescharacteristic
29#16839818C2212yes—
30#16839816—21no—
31#16839815C2211yescharacteristic
32#16839813—21no—
33#16839784C2212yes—
34#16839724—21no—
35C4×#11758194C2211yescharacteristic
36#16839714—21no—
37M16×Dic61—21no—
38C4×#11758189—21no—
39C4×#11758189C2212yes—
40#97×D61—21no—
41C4×C8×D61C2212yes—
42Q8⋊#92418C2214yescomplemented
43#16839491C2214yescomplemented
44#16839489C2214yescomplemented
45C61×#533C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C61×Q8⋊C8C2214yescomplemented
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758222—21no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C2×#11758194—21no—
56C2×#11758189—22no—
57C2×#11758189—22no—
58M16×D61—28no—
59C2×C8×D61—22no—
60#14519777C2×C411yescharacteristic
61#14519774C2×C411yescharacteristic
62#14519772—22no—
63#14519772—22no—
64#14519768—21no—
65#14519768C2×C412yes—
66#14519766—21no—
67#14519766C2×C412yes—
68#14519765—21no—
69#14519762—22no—
70#14519762—22no—
71#14519761—22no—
72#14519757—21no—
73#14519757—21no—
74#14519753—21no—
75#14519753C2×C412yes—
76#14519752—21no—
77#14519752—21no—
78#14519752—21no—
79#14519752C2×C412yes—
80#14519751—21no—
81#14519751C2×C412yes—
82#14519747—22no—
83Q8×Dic61—21no—
84Q8×Dic61C2×C412yes—
85#14519742—21no—
86#14519742—21no—
87C4×D244—22no—
88C4×D244—22no—
89C4×D244—21no—
90C4×D244C2312yes—
91#14519739—22no—
92#14519739—22no—
93#14519739—21no—
94#14519739—21no—
95#14519739—21no—
96#14519739C2×C412yes—
97#14519737—22no—
98#14519737—22no—
99#14519735—21no—
100#14519735—22no—
101#14519735—22no—
102#14519735C2×C412yes—
103#14519734—22no—
104C2×C4×Dic61C2×C411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C42×D61—22no—
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61C2311yescharacteristic
115C42×D61C2311yescharacteristic
116C4⋊#92418C2312yes—
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519688—21no—
121#14519687—22no—
122#14519687—22no—
123C4×#92418—21no—
124C4×#92418C2312yes—
125C8×Dic61—22no—
126C2×M16×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128C4⋊Q8×C61C2×C411yescharacteristic
129#125×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61C2×C412yes—
132C4×Q8×C61—21no—
133C4×Q8×C61C2312yes—
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2312yes—
137C22⋊C8×C61—21no—
138#97×C61—21no—
139C4×C8×C61—21no—
140C4×C8×C61C2312yes—
141C23×D61C22⋊C411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—22no—
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151#11758226—44no—
152#11758226—44no—
153#11758222—44no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244C22×C412yes—
159C2×C4×D61—24no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22×C411yescharacteristic
172C2×C4×D61C22×C411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×Dic122—21no—
178M16×C61—24no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C22×C412yes—
187C22⋊C4×C61—22no—
188C42×C61—22no—
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2411yescharacteristic
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201C4⋊Dic61—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61C22×C412yes—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22×C411yescharacteristic
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221#11758194—24no—
222C2×#92418—22no—
223C2×#92418—22no—
224#11758189—28no—
225#11758189—44no—
226C8×D61—28no—
227C23×C61C2×C22⋊C411yescharacteristic
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61C2×C22⋊C411yescharacteristic
233C22×D61C2×C22⋊C411yescharacteristic
234Q8×C61—24no—
235Q8×C61—42no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C22⋊D61—44no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258D244—24no—
259D244—42no—
260D244—44no—
261D244—44no—
262D244—42no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—44no—
275C4×D61—44no—
276C4×D61—44no—
277C4×D61#10414yes—
278Dic122—42no—
279C8×C61—24no—
280C8×C61—42no—
281#92418—24no—
282#92418—42no—
283#17503—611nomaximal, Sylow
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61C22×C22⋊C411yescharacteristic
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—44no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61C2×#10412yes—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61C2×#10412yes—
304C4×C61#56812yes—
305C4×C61#57512yes—
306Dic61—21no—
307Dic61—21no—
308Dic61—21no—
309Dic61—21no—
310Dic61—21no—
311Dic61—42no—
312Dic61C2×#10412yes—
313#3444—611no—
314#2920—611no—
315C2×C4×M16—611no—
316#1494—614no—
317#1489—614no—
318#1485—611no—
319#1485D6111yescharacteristic, complemented
320#1484—611no—
321#1484—611no—
322C2×C61—41no—
323C2×C61#287911yescharacteristic
324C2×C61#289611yescharacteristic
325C2×C61C22×#10411yescharacteristic
326D61—22no—
327D61—41no—
328D61—82no—
329C22×M16—1221no—
330#533—611no—
331#533—611no—
332#533C2×D6111yescharacteristic
333#532—611no—
334#532—611no—
335#532—611no—
336#529—612no—
337#529—612no—
338#529—1221no—
339#529—1221no—
340#529—1221no—
341#529—1221no—
342#525—612no—
343#525—1221no—
344#525—1221no—
345C4×#41—612no—
346C4×#41—1221no—
347C4×#41—1221no—
348C22×C42—611no—
349#432—612no—
350#432—1221no—
351#431—611no—
352#431—1221no—
353#431C2×D6111yescharacteristic
354#423—612no—
355#407—1221no—
356C4×M16—611no—
357C4×M16—612no—
358C4×M16—1221no—
359C4×M16—1221no—
360C4×M16—1221no—
361C4×M16C2×D6111yescharacteristic
362C2×#97—1221no—
363C2×C4×C8—612no—
364Q8⋊C8—614no—
365Q8⋊C8C2×D6114yescomplemented
366D4⋊C8—614no—
367D4⋊C8—614no—
368C61#1750311yescharacteristic, complemented, Sylow
369C2×#41—612no—
370C2×#41—1221no—
371C2×#41—1221no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C2×M16—21no—
376C2×M16—1222no—
377C2×M16—1222no—
378C2×M16—1228no—
379C2×M16—1221no—
380C22×C8—1222no—
381C4⋊Q8—611no—
382C4⋊Q8C4×D6111yescharacteristic
383C4⋊Q8C4×D6111yescharacteristic
384#127—611no—
385#126—1222no—
386#126—1222no—
387#125—21no—
388#125—612no—
389#125—1221no—
390#124—612no—
391#124—1221no—
392#124—1221no—
393#123—1222no—
394#123—1222no—
395#123—1222no—
396C22⋊Q8—21no—
397C22⋊Q8—612no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C22⋊Q8C4×D6112yes—
402#121—612no—
403#121—612no—
404#121—1221no—
405#121—1221no—
406#121—1221no—
407#121—1221no—
408#120—1222no—
409C4×Q8—21no—
410C4×Q8—612no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×Q8C22×D6112yes—
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1221no—
427#117—21no—
428#117—612no—
429#117—1222no—
430#117—1222no—
431#117—1222no—
432#117—1221no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—1222no—
445C2×C42C22×D6111yescharacteristic
446C4⋊C8—612no—
447C4⋊C8—1222no—
448C4⋊C8C22×D6112yes—
449C22⋊C8—21no—
450C22⋊C8—1222no—
451C22⋊C8—1221no—
452#97—21no—
453#97—1222no—
454#97—1222no—
455#97—1221no—
456C4×C8—21no—
457C4×C8—612no—
458C4×C8—1222no—
459C4×C8—1221no—
460C4×C8C22×D6112yes—
461C24—611no—
462#41—2444no—
463#41—2444no—
464#41—2444no—
465C2×Q8—21no—
466C2×Q8—1221no—
467C2×Q8C2×C4×D6112yes—
468C2×D4—612no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1221no—
473C2×D4—1221no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1224no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498M16—24no—
499M16—1228no—
500M16—1224no—
501M16—2444no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—1228no—
505C2×C8—1222no—
506C2×C8—1222no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—612no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4C2×C4×D6112yes—
519C22⋊C4—22no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C42—22no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—1222no—
542C42—1222no—
543C42C2×C4×D6111yescharacteristic
544C42D4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42C23×D6111yescharacteristic
547C23—611no—
548C23—611no—
549C23—1222no—
550C23—1222no—
551C23—2441no—
552C23—2441no—
553C23C22⋊C4×D6111yescharacteristic
554Q8—24no—
555Q8—42no—
556Q8—2442no—
557D4—1224no—
558D4—2442no—
559D4—2444no—
560D4—2444no—
561D4—2444no—
562D4—2442no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—611no—
569C2×C4—614no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2444no—
593C2×C4C22⋊C4×D6111yescharacteristic
594C2×C4C22⋊C4×D6111yescharacteristic
595C2×C4C22⋊C4×D6111yescharacteristic
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C8—24no—
600C8—42no—
601C8—1224no—
602C8—2442no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—612no—
607C22—1221no—
608C22—1221no—
609C22—1221no—
610C22—1221no—
611C22—1221no—
612C22—2444no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22C2×C22⋊C4×D6111yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—42no—
621C4—42no—
622C4—612no—
623C4—1221no—
624C4—1221no—
625C4—1221no—
626C4—1221no—
627C4—1221no—
628C4—2442no—
629C4#104×D6112yes—
630C4#1684052712yes—
631C4#1684056112yes—
632C2—41no—
633C2—1222no—
634C2—4882no—
635C2—2441no—
636C2#2145426111yescharacteristic
637C2#2145433711yescharacteristic
638C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1485C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1485D61
hypocenterC61#17503
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
2. #21454447C2semidirect
3. C4×M16×D61C2semidirect
4. #21444278C2semidirect
5. #21444257C2semidirect
6. #21444240C2semidirect
7. #21444237C2semidirect
8. #21444235C2semidirect
9. C61×#1485C2semidirect
42. Q8⋊#92418C22semidirect
43. #16839491C22semidirect
44. #16839489C22semidirect
48. C61×Q8⋊C8C22semidirect
319. #1485D61semidirect
365. Q8⋊C8C2×D61semidirect
368. C61#17503semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544472
C4×M16×D612
#214442782
#214442572
#214442402
#214442372
#214442352
C61×#14852
#1750361