Finite Group Databasebeta

← #29647974

Subgroups — #29647974

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic, complemented
3C4×M16×D61C211yesmaximal, characteristic, complemented
4#21444438C214yesmaximal, complemented
5#21444425C214yesmaximal, complemented
6#21444414C211yesmaximal, characteristic, complemented
7#21444410C211yesmaximal, characteristic, complemented
8#21444410C211yesmaximal, characteristic, complemented
9C61×#1526C211yesmaximal, characteristic, complemented
10C2×M16×D61—21no—
11#16840317C411yescharacteristic
12#16840317C411yescharacteristic
13#16840313C411yescharacteristic
14#16840313C411yescharacteristic
15#16840312C2211yescharacteristic
16#16840299—21no—
17#16840299—21no—
18#16840298—21no—
19#16840298—21no—
20#16840297C2212yes—
21#16840297C2212yes—
22#16840275—21no—
23#16840275—21no—
24#16840274C412yes—
25C4×#11758226—21no—
26C4×#11758226—21no—
27#16840230C412yes—
28C2×C42×D61C2211yescharacteristic
29#16839817C2212yes—
30#16839817C2212yes—
31#16839813—21no—
32#16839813—21no—
33#16839785—21no—
34#16839725—21no—
35C4×#11758194C2211yescharacteristic
36C4×#11758194C2211yescharacteristic
37#16839714C2212yes—
38M16×Dic61—21no—
39C4×#11758189—21no—
40#97×D61C2212yes—
41C4×C8×D61—21no—
42#16839504C2214yescomplemented
43#16839501C2214yescomplemented
44#16839501C2214yescomplemented
45C61×#533C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C61×#330C2214yescomplemented
49C2×#11758226—21no—
50C2×#11758226—21no—
51C2×#11758222C2×C412yes—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C2×#11758194—21no—
56C2×#11758194—21no—
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#14519768—21no—
66#14519766—21no—
67#14519766—21no—
68#14519765C2312yes—
69#14519762—22no—
70#14519762—22no—
71#14519761—22no—
72#14519757C2×C412yes—
73#14519757C2×C412yes—
74#14519753—21no—
75#14519753—21no—
76#14519752—21no—
77#14519752—21no—
78#14519752C2×C412yes—
79#14519752C2×C412yes—
80#14519751—21no—
81#14519751—21no—
82#14519747—22no—
83Q8×Dic61—21no—
84Q8×Dic61—21no—
85#14519742C2×C412yes—
86#14519742C2×C412yes—
87C4×D244—22no—
88C4×D244—22no—
89C4×D244—21no—
90C4×D244—21no—
91#14519739—22no—
92#14519739—22no—
93#14519739—21no—
94#14519739—21no—
95#14519739C2×C412yes—
96#14519739C2×C412yes—
97#14519737—22no—
98#14519737—22no—
99#14519735—21no—
100#14519735—21no—
101#14519735—22no—
102#14519735—22no—
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
116#14519706—22no—
117#14519706—22no—
118#14519689—22no—
119#14519689—22no—
120#14519688C2312yes—
121#14519688C2312yes—
122#14519687—22no—
123C4×#92418—21no—
124C4×#92418—21no—
125C8×Dic61—22no—
126C2×M16×C61—21no—
127C2×M16×C61—21no—
128C4⋊Q8×C61C2×C411yescharacteristic
129C4⋊Q8×C61C2×C411yescharacteristic
130#125×C61C2312yes—
131C22⋊Q8×C61—21no—
132C22⋊Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×Q8×C61—21no—
135#117×C61C2×C412yes—
136C2×C42×C61C2311yescharacteristic
137#97×C61C2312yes—
138#97×C61C2312yes—
139C4×C8×C61—21no—
140C4×C8×C61—21no—
141C23×D61C22⋊C411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
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×Dic122C22×C412yes—
178M16×C61—24no—
179M16×C61—42no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C22×C412yes—
187C4⋊C4×C61C22×C412yes—
188C22⋊C4×C61—22no—
189C42×C61—22no—
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2411yescharacteristic
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206#11758196—22no—
207#11758196—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C22×C411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222#11758194—24no—
223#11758194—42no—
224C2×#92418—22no—
225C2×#92418—22no—
226#11758189—28no—
227C8×D61—28no—
228C23×C61C2×C22⋊C411yescharacteristic
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61C2×C22⋊C411yescharacteristic
234C22×D61C2×C22⋊C411yescharacteristic
235Q8×C61—42no—
236Q8×C61—42no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C22⋊D61—44no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259D244—42no—
260D244—44no—
261D244—44no—
262D244—42no—
263D244—42no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—44no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61#10414yes—
279Dic122—42no—
280C8×C61—24no—
281C8×C61—42no—
282#92418—24no—
283#92418—42no—
284#17649—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61C22×C22⋊C411yescharacteristic
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—44no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61C2×#10412yes—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61#41312yes—
305C4×C61C2×#10412yes—
306C4×C61#42112yes—
307Dic61—21no—
308Dic61—21no—
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—42no—
313Dic61C2×#10412yes—
314#3444—611no—
315#2920—611no—
316C2×C4×M16—611no—
317#1532—614no—
318#1529—614no—
319#1526—611no—
320#1526D6111yescharacteristic, complemented
321#1525—611no—
322#1525—611no—
323C2×C61—41no—
324C2×C61C2×#41311yescharacteristic
325C2×C61C22×#10411yescharacteristic
326C2×C61C2×#42111yescharacteristic
327D61—22no—
328D61—41no—
329D61—82no—
330C22×M16—1221no—
331#533—611no—
332#533—611no—
333#533C2×D6111yescharacteristic
334#532—611no—
335#532—611no—
336#532—611no—
337#529—612no—
338#529—612no—
339#529—1221no—
340#529—1221no—
341#529—1221no—
342#529—1221no—
343#525—612no—
344#525—1221no—
345#525—1221no—
346C4×#41—612no—
347C4×#41—1221no—
348C4×#41—1221no—
349C22×C42—611no—
350#432—612no—
351#432—612no—
352#431—1221no—
353#431—1221no—
354#423—1221no—
355#407—1221no—
356C4×M16—611no—
357C4×M16—611no—
358C4×M16—612no—
359C4×M16—1221no—
360C4×M16—1221no—
361C4×M16C2×D6111yescharacteristic
362C4×M16C2×D6111yescharacteristic
363C2×#97—612no—
364C2×C4×C8—1221no—
365#330—614no—
366#330C2×D6114yescomplemented
367#329—614no—
368#329—614no—
369C61#1764911yescharacteristic, complemented, Sylow
370C2×#41—612no—
371C2×#41—1221no—
372C2×#41—1221no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—611no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—1222no—
379C2×M16—1228no—
380C2×M16—1221no—
381C2×M16—1221no—
382C22×C8—1222no—
383C4⋊Q8—611no—
384C4⋊Q8C4×D6111yescharacteristic
385C4⋊Q8C4×D6111yescharacteristic
386#127—611no—
387#126—1222no—
388#126—1222no—
389#125—1221no—
390#125—1221no—
391#125C22×D6112yes—
392#124—612no—
393#124—1221no—
394#124—1221no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—612no—
401C22⋊Q8—612no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404#121—612no—
405#121—612no—
406#121—1221no—
407#121—1221no—
408#121—1221no—
409#121—1221no—
410#120—1222no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—612no—
414C4×Q8—612no—
415C4×Q8—1221no—
416C4×Q8—1221no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4—1221no—
422C4×D4—1221no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1221no—
428C4×D4—1221no—
429#117—1222no—
430#117—1222no—
431#117—1222no—
432#117—1221no—
433#117—1221no—
434#117C4×D6112yes—
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—611no—
445C2×C42—611no—
446C2×C42—1222no—
447C2×C42C22×D6111yescharacteristic
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450C22⋊C8—1222no—
451C22⋊C8—1222no—
452#97—612no—
453#97—612no—
454#97—1222no—
455#97C22×D6112yes—
456#97C22×D6112yes—
457C4×C8—21no—
458C4×C8—21no—
459C4×C8—1222no—
460C4×C8—1221no—
461C4×C8—1221no—
462C24—611no—
463#41—2444no—
464#41—2444no—
465#41—2444no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—612no—
469C2×D4—612no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1221no—
474C2×D4—1221no—
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—611no—
490C22×C4—1224no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4C2×C4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499M16—24no—
500M16—42no—
501M16—1228no—
502M16—1224no—
503M16—2442no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—1228no—
507C2×C8—1222no—
508C2×C8—1222no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4C2×C4×D6112yes—
520C4⋊C4C2×C4×D6112yes—
521C22⋊C4—22no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C42—22no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—1222no—
544C42—1222no—
545C42C2×C4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C42D4×D6111yescharacteristic
548C42C23×D6111yescharacteristic
549C23—611no—
550C23—611no—
551C23—1222no—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23C22⋊C4×D6111yescharacteristic
556Q8—42no—
557Q8—42no—
558Q8—2442no—
559D4—2442no—
560D4—2444no—
561D4—2444no—
562D4—2444no—
563D4—2442no—
564D4—2442no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—611no—
571C2×C4—614no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2444no—
595C2×C4C22⋊C4×D6111yescharacteristic
596C2×C4C22⋊C4×D6111yescharacteristic
597C2×C4C22⋊C4×D6111yescharacteristic
598C2×C4C22×C4×D6111yescharacteristic
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C8—24no—
602C8—42no—
603C8—1224no—
604C8—2442no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—612no—
609C22—1221no—
610C22—1221no—
611C22—1221no—
612C22—1221no—
613C22—1221no—
614C22—2444no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22C2×C22⋊C4×D6111yescharacteristic
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—612no—
625C4—1221no—
626C4—1221no—
627C4—1221no—
628C4—1221no—
629C4—1221no—
630C4—2442no—
631C4#1683974512yes—
632C4#104×D6112yes—
633C4#1683977712yes—
634C2—41no—
635C2—1222no—
636C2—4882no—
637C2—2441no—
638C2C2×#1683974511yescharacteristic
639C2C2×#104×D6111yescharacteristic
640C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1526C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1526D61
hypocenterC61#17649
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. C4×M16×D61C2semidirect
4. #21444438C2semidirect
5. #21444425C2semidirect
6. #21444414C2semidirect
7. #21444410C2semidirect
8. #21444410C2semidirect
9. C61×#1526C2semidirect
42. #16839504C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
48. C61×#330C22semidirect
320. #1526D61semidirect
366. #330C2×D61semidirect
369. C61#17649semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544502
C4×M16×D612
#214444382
#214444252
#214444142
#214444102
C61×#15262
#1764961