Finite Group Databasebeta

← #29647516

Subgroups — #29647516

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic, complemented
3#21448211C211yesmaximal, characteristic, complemented
4#21444395C214yesmaximal, complemented
5#21444361C214yesmaximal, complemented
6#21444331C211yesmaximal, characteristic, complemented
7#21444328C211yesmaximal, characteristic, complemented
8#21444326C211yesmaximal, characteristic, complemented
9C61×#1506C211yesmaximal, characteristic, complemented
10#16840317C411yescharacteristic, complemented
11#16840317C411yescharacteristic, complemented
12#16840313C411yescharacteristic, complemented
13#16840313C411yescharacteristic, complemented
14#16840312C2211yescharacteristic
15#16840299—21no—
16#16840299C412yescomplemented
17#16840298—21no—
18#16840298C2212yes—
19#16840297—21no—
20#16840297—21no—
21#16840275—21no—
22#16840275C2212yes—
23#16840274—21no—
24C4×#11758226—21no—
25C4×#11758226C412yescomplemented
26#16840230—21no—
27C2×C42×D61C2211yescharacteristic
28#16839819C2211yescharacteristic
29#16839817—21no—
30#16839814—21no—
31#16839812C2212yes—
32#16839784—21no—
33#16839724C2212yes—
34#16839714—21no—
35#16839629C2212yes—
36#16839626—21no—
37#16839624—21no—
38#16839621C2212yes—
39C4×#11758196—21no—
40C4×#11758198—21no—
41#16839608C2211yescharacteristic
42#16839499C2214yes—
43#16839497C2214yes—
44#16839495C2214yes—
45C61×#533C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C61×#376C2211yescharacteristic
48C61×#328C2214yes—
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758222—21no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758189—22no—
56#14519777C2×C411yescharacteristic
57#14519774C2×C411yescharacteristic
58#14519772—22no—
59#14519772—22no—
60#14519768—21no—
61#14519768C2×C412yes—
62#14519766—21no—
63#14519766C2×C412yes—
64#14519765—21no—
65#14519762—22no—
66#14519762—22no—
67#14519761—22no—
68#14519757—21no—
69#14519757—21no—
70#14519753—21no—
71#14519753C2×C412yes—
72#14519752—21no—
73#14519752—21no—
74#14519752—21no—
75#14519752C2×C412yes—
76#14519751—21no—
77#14519751C2312yes—
78#14519747—22no—
79Q8×Dic61—21no—
80Q8×Dic61C2×C412yes—
81#14519742—21no—
82#14519742—21no—
83C4×D244—22no—
84C4×D244—22no—
85C4×D244—21no—
86C4×D244C2×C412yes—
87#14519739—22no—
88#14519739—22no—
89#14519739—21no—
90#14519739—21no—
91#14519739—21no—
92#14519739C2×C412yes—
93#14519737—22no—
94#14519737—22no—
95#14519735—21no—
96#14519735—22no—
97#14519735—22no—
98#14519735C2×C412yes—
99#14519734—22no—
100C2×#11758196—22no—
101C2×#11758198—22no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C42×D61—22no—
107C42×D61C2×C411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C4⋊#92418—21no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118#14519689—22no—
119#14519688—21no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C4⋊Q8×C61C2×C411yescharacteristic
126C4⋊Q8×C61C2×C411yescharacteristic
127#125×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61C2312yes—
130C4×Q8×C61—21no—
131C4×Q8×C61C2×C412yes—
132#117×C61—21no—
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C22⋊C8×C61C2312yes—
137#97×C61—21no—
138#95×C61—21no—
139#95×C61C2312yes—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—22no—
148C22×Dic61—22no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758226—44no—
153#11758226—44no—
154#11758222—44no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C22×C412yes—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
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×D61C22⋊C411yescharacteristic
174C2×C4×D61C22×C411yescharacteristic
175C2×C4×D61C22×C411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×Dic122—21no—
179C2×C8×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C42×C61—22no—
188C42×C61—42no—
189C42×C61C22×C411yescharacteristic
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—44no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C412yes—
206#11758196—22no—
207#11758196—22no—
208#11758196—44no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—42no—
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C2×#92418—22no—
225#11758189—44no—
226C23×C61C22×D411yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61C2×C22⋊C411yescharacteristic
232C22×D61C2×C22⋊C411yescharacteristic
233Q8×C61—42no—
234Q8×C61—42no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C22⋊D61—44no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263D244—42no—
264D244—42no—
265D244—44no—
266D244—44no—
267D244—42no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—44no—
281C4×D61—44no—
282C4×D61—44no—
283Dic122—42no—
284C8×C61—42no—
285#92418—42no—
286#17594—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61C22×C22⋊C411yescharacteristic
290C22×C61#57512yes—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—44no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61—82no—
309C4×C61#42112yes—
310Dic61—21no—
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—42no—
316Dic61—82no—
317Dic61#41012yes—
318#3444—611no—
319#2920—611no—
320#2283—611no—
321#1521—614no—
322#1513—614no—
323#1506—611no—
324#1506D6111yescharacteristic, complemented
325#1505—611no—
326#1505—611no—
327C2×C61—22no—
328C2×C61C2×#41011yescharacteristic
329C2×C61#289611yescharacteristic
330C2×C61C2×#42111yescharacteristic
331D61—41no—
332D61—41no—
333D61—82no—
334#533—611no—
335#533—611no—
336#533C2×D6111yescharacteristic
337#532—611no—
338#532—611no—
339#532—611no—
340#529—612no—
341#529—612no—
342#529—1221no—
343#529—1221no—
344#529—1221no—
345#529—1221no—
346#525—612no—
347#525—1221no—
348#525—1221no—
349C4×#41—612no—
350C4×#41—1221no—
351C4×#41—1221no—
352C22×C42—611no—
353#432—611no—
354#432—1221no—
355#432C2×D6111yescharacteristic
356#431—612no—
357#431—1221no—
358#423—1221no—
359#407—612no—
360C4×M16—1221no—
361#383—612no—
362#382—1221no—
363#381—1221no—
364#380—612no—
365C4×C4⋊C4—1221no—
366C4×C22⋊C4—1221no—
367#376—611no—
368#376C2×D6111yescharacteristic
369#328—614no—
370#328C2×D6114yes—
371#327—614no—
372#327—614no—
373C61#1759411yescharacteristic, complemented, Sylow
374C2×#41—612no—
375C2×#41—1221no—
376C2×#41—1221no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C2×M16—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⋊Q8C22×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×Q8C4×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×C4⋊C4—1222no—
434C2×C22⋊C4—1222no—
435C2×C42—21no—
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—611no—
447C2×C42—1222no—
448C2×C42—1221no—
449C2×C42C22×D6111yescharacteristic
450C4⋊C8—21no—
451C4⋊C8—1222no—
452C4⋊C8—1221no—
453C22⋊C8—612no—
454C22⋊C8—1222no—
455C22⋊C8C22×D6112yes—
456#97—21no—
457#97—1221no—
458C4×C8—1222no—
459#95—21no—
460#95—612no—
461#95—1222no—
462#95—1222no—
463#95—1221no—
464#95C22×D6112yes—
465C24—611no—
466#41—2444no—
467#41—2444no—
468#41—2444no—
469C2×Q8—21no—
470C2×Q8—1221no—
471C2×Q8C2×C4×D6112yes—
472C2×D4—612no—
473C2×D4—1222no—
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—1221no—
477C2×D4—1221no—
478C22×C4—22no—
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—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504M16—2444no—
505C2×C8—22no—
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⋊C4—2444no—
519C4⋊C4C2×C4×D6112yes—
520C22⋊C4—22no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—2444no—
530C42—22no—
531C42—42no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—1222no—
545C42—1222no—
546C42—2442no—
547C42C2×C4×D6111yescharacteristic
548C42C2×C4×D6111yescharacteristic
549C42D4×D6111yescharacteristic
550C42D4×D6111yescharacteristic
551C23—611no—
552C23—611no—
553C23—1222no—
554C23—1222no—
555C23—2441no—
556C23—2441no—
557C23C2×D4×D6111yescharacteristic
558Q8—42no—
559Q8—42no—
560Q8—2442no—
561D4—2442no—
562D4—2442no—
563D4—2444no—
564D4—2444no—
565D4—2444no—
566D4—2442no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2444no—
599C2×C4—2441no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4C2×D4×D6111yescharacteristic
609C8—42no—
610C8—2442no—
611C22—21no—
612C22—21no—
613C22—1221no—
614C22—1221no—
615C22—1221no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—2444no—
620C22—2442no—
621C22—2442no—
622C22—2442no—
623C22C2×C22⋊C4×D6111yescharacteristic
624C22#1684056112yes—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—42no—
631C4—42no—
632C4—82no—
633C4—612no—
634C4—1221no—
635C4—1221no—
636C4—1221no—
637C4—1221no—
638C4—1221no—
639C4—4882no—
640C4—2442no—
641C4#1683977712yes—
642C2—22no—
643C2—4882no—
644C2—2441no—
645C2—2441no—
646C2#410×D6111yescharacteristic
647C2#2145433711yescharacteristic
648C2C2×#1683977711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1506C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC22×C4×C61C24
hypercenter#1506D61
hypocenterC61#17594
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
2. #21454449C2semidirect
3. #21448211C2semidirect
4. #21444395C2semidirect
5. #21444361C2semidirect
6. #21444331C2semidirect
7. #21444328C2semidirect
8. #21444326C2semidirect
9. C61×#1506C2semidirect
10. #16840317C4semidirect
11. #16840317C4semidirect
12. #16840313C4semidirect
13. #16840313C4semidirect
16. #16840299C4semidirect
25. C4×#11758226C4semidirect
324. #1506D61semidirect
373. C61#17594semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544492
#214482112
#214443952
#214443612
#214443312
#214443282
#214443262
C61×#15062
#1759461