Finite Group Databasebeta

← #29646918

Subgroups — #29646918

Order 15616.

1004 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#21459383C211yesmaximal, characteristic
2#21454786C211yesmaximal, characteristic, complemented
3#21444282C212yesmaximal, complemented
4#21444273C211yesmaximal, characteristic, complemented
5#21444256C212yesmaximal, complemented
6#21444243C211yesmaximal, characteristic, complemented
7#21444230C212yesmaximal, complemented
8#21444221C211yesmaximal, characteristic, complemented
9#21444204C212yesmaximal, complemented
10C2×#16839489C211yesmaximal, characteristic, complemented
11C61×#2906C211yesmaximal, characteristic, complemented
12#16840485C411yescharacteristic
13#16840477C411yescharacteristic
14#16840472C411yescharacteristic
15#16840462—21no—
16#16840453—21no—
17#16840452C412yes—
18#16840441—22no—
19#16840417C411yescharacteristic
20#16840408C412yes—
21#16840319C2211yescharacteristic
22#16840296C2212yes—
23#16840290—22no—
24#16840269—21no—
25C4×#11758222C2212yes—
26C2×C4×D244C2211yescharacteristic
27#117×D61—21no—
28#16839893—21no—
29#16839840—21no—
30#16839834—22no—
31#16839815C2211yescharacteristic
32#16839815C2212yescomplemented
33C4×#11758194C2212yescomplemented
34C2×C4×#92418C2211yescharacteristic
35#16839492C2212yescomplemented
36#16839492C2212yescomplemented
37#16839492C2212yescomplemented
38#16839492C2212yescomplemented
39#16839489C2211yescharacteristic
40#16839489C2211yescharacteristic
41#16839489C2211yescharacteristic
42#16839489C2211yescharacteristic
43#16839489C2212yes—
44#16839489C2212yes—
45C2×#131×C61—21no—
46C4×#41×C61C2211yescharacteristic
47C61×#424—22no—
48C61×#424—21no—
49C61×#423C2211yescharacteristic
50C61×#423C2212yescomplemented
51C2×C4⋊C8×C61C2211yescharacteristic
52C2×C4⋊C8×C61C2212yescomplemented
53C2×#11758222C2×C412yes—
54C22×D244C2×C411yescharacteristic
55C22×C4×D61—21no—
56C2×#11758194—22no—
57C22×#92418—21no—
58#14519777C2×C411yescharacteristic
59#14519777C2×C411yescharacteristic
60#14519772—22no—
61#14519771—22no—
62#14519770—22no—
63#14519769—21no—
64#14519769C2×C412yes—
65#14519767—22no—
66#14519765C2×C412yes—
67#14519763—22no—
68#14519762—22no—
69#14519760—22no—
70#14519759—22no—
71#14519758—22no—
72#14519756—21no—
73#14519756C2×C412yes—
74#14519756C2×C412yes—
75#14519755—22no—
76#14519753—21no—
77#14519753—21no—
78#14519751C2×C412yes—
79#14519750—22no—
80#14519748—21no—
81#14519748—21no—
82#14519745—22no—
83C4×Dic122—21no—
84C4×Dic122C2×C412yes—
85C4×Dic122C2312yes—
86C4×D244—21no—
87C4×D244—21no—
88C4×D244C2×C411yescharacteristic
89C4×D244C2×C412yes—
90C4×D244C2×C411yescharacteristic
91C4×D244C2311yescharacteristic
92C4×D244C2311yescharacteristic
93C4×C22⋊D61—22no—
94C4×C22⋊D61—22no—
95C4×C22⋊D61—22no—
96#14519736—21no—
97#14519736C2×C412yes—
98#14519735—21no—
99#14519735—21no—
100#14519734—22no—
101#14519734—22no—
102#14519733—22no—
103C2×C4⋊Dic61C2×C411yescharacteristic
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106C2×#11758198—21no—
107C22⋊C4×D61—22no—
108C42×D61—22no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—21no—
112C22⋊#92418—22no—
113#14519688—22no—
114C4×#92418—21no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2312yes—
118C2×#41×C61D411yescharacteristic
119#131×C61—44no—
120C2×M16×C61—21no—
121C2×M16×C61—22no—
122C22×C8×C61—21no—
123C22×C8×C61—22no—
124C4×Q8×C61C2×C411yescharacteristic
125C4×Q8×C61D411yescharacteristic
126C4×D4×C61C2×C412yes—
127C4×D4×C61C2×C411yescharacteristic
128C4×D4×C61D411yescharacteristic
129C4×D4×C61D412yes—
130#117×C61D411yescharacteristic
131#117×C61D412yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2312yes—
134C4⋊C8×C61—21no—
135C4⋊C8×C61—22no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61—22no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2312yes—
141C23×D61—21no—
142#41×C61—22no—
143#41×C61—22no—
144C2×Q8×C61C22⋊C411yescharacteristic
145C2×D4×C61C22⋊C412yes—
146C2×D4×C61C22⋊C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C22×Dic61—21no—
156#11758222—24no—
157#11758222—42no—
158C2×D244—22no—
159C2×D244—21no—
160C2×D244—41no—
161C2×D244C22×C411yescharacteristic
162C2×D244C22×C411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—22no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—22no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173M16×C61—44no—
174M16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—22no—
178C2×C8×C61—44no—
179C2×C8×C61—42no—
180C4⋊C4×C61C22⋊C411yescharacteristic
181C4⋊C4×C61C22⋊C412yes—
182C4⋊C4×C61C22⋊C411yescharacteristic
183C4⋊C4×C61C22⋊C412yes—
184C22⋊C4×C61C22⋊C412yes—
185C22⋊C4×C61C22⋊C412yes—
186C22⋊C4×C61C22⋊C412yes—
187C42×C61C22×C411yescharacteristic
188C42×C61C22×C412yes—
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D412yes—
191C42×C61C2×D412yes—
192C42×C61C2×D411yescharacteristic
193C42×C61C2411yescharacteristic
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—21no—
200#11758198—21no—
201#11758198—22no—
202#11758198—22no—
203#11758198—21no—
204#11758198—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61C22×C411yescharacteristic
210C4⋊Dic61C22×C411yescharacteristic
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221#11758194—44no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—22no—
225C2×#92418—42no—
226C23×C61C2×C22⋊C412yes—
227C23×C61C2×C22⋊C411yescharacteristic
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—42no—
231C22×D61—42no—
232C22×D61—81no—
233Q8×C61—22no—
234D4×C61—24no—
235D4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C412yes—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61AGL(1,ℤ/8)12yes—
251C2×C4×C61#13712yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C22⋊D61—44no—
256C22⋊D61—82no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—42no—
262D244—21no—
263D244—21no—
264D244—21no—
265D244—42no—
266D244#13112yescomplemented
267C4×D61—44no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—82no—
275Dic122—22no—
276Dic122—22no—
277Dic122—81no—
278C8×C61—41no—
279C8×C61—42no—
280C8×C61—41no—
281#92418—22no—
282#92418—41no—
283#92418—42no—
284#17512—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61#41712yes—
291C22×C61C22×C22⋊C411yescharacteristic
292C2×D61—22no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—84no—
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—22no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—22no—
305C4×C61#41712yes—
306C4×C61C2×#13111yescharacteristic
307C4×C61#56811yescharacteristic
308C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
309C4×C61C2×#13711yescharacteristic
310Dic61—42no—
311Dic61—41no—
312Dic61—41no—
313Dic61—42no—
314#3475—611no—
315#2965—611no—
316#2906D6111yescharacteristic, complemented
317#1495—612no—
318#1493—611no—
319#1489—612no—
320#1486—611no—
321#1483—612no—
322#1481—611no—
323#1477—612no—
324C2×D4⋊C8—611no—
325C2×C61—22no—
326C2×C61—42no—
327C2×C61#288011yescharacteristic
328C2×C61C2×#41711yescharacteristic
329C2×C61#290111yescharacteristic
330D61—42no—
331D61—81no—
332C2×#131—21no—
333#556—611no—
334#555—611no—
335#554—611no—
336#553—1221no—
337#552—612no—
338#552—1221no—
339#551—1222no—
340#547—611no—
341#546—612no—
342#533—611no—
343#529—612no—
344#529—1222no—
345#525—1221no—
346C4×#41—612no—
347C4×#41C2×D6111yescharacteristic
348C2×C4×D4—611no—
349C2×#117—1221no—
350#446—1221no—
351#436—1221no—
352#435—1222no—
353#431—611no—
354#431—612no—
355#424—22no—
356#424—21no—
357#423C2×D6111yescharacteristic
358#423C2×D6112yescomplemented
359C2×C4⋊C8C2×D6111yescharacteristic
360C2×C4⋊C8C2×D6112yescomplemented
361C4×M16—612no—
362C2×C4×C8—611no—
363Q8⋊C8—612no—
364Q8⋊C8—612no—
365Q8⋊C8—612no—
366Q8⋊C8—612no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373C61#1751211yescharacteristic, complemented, Sylow
374C2×#41—612no—
375C2×#41C22⋊D6111yescharacteristic
376C22×D4—611no—
377C23×C4—1221no—
378#131—44no—
379C2×M16—21no—
380C2×M16—22no—
381C2×M16—1222no—
382C22×C8—21no—
383C22×C8—22no—
384C22×C8—1221no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387#126—1222no—
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1222no—
392#125—1221no—
393#124—612no—
394#124—1222no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—612no—
401C22⋊Q8—1221no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C22⋊Q8—1222no—
405#121—612no—
406#121—1221no—
407#121—1221no—
408#121—1222no—
409#120—1222no—
410C4×Q8—612no—
411C4×Q8—612no—
412C4×Q8—1221no—
413C4×Q8C4×D6111yescharacteristic
414C4×Q8D24411yescharacteristic
415C4×D4—611no—
416C4×D4—611no—
417C4×D4—612no—
418C4×D4—611no—
419C4×D4—611no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4C4×D6112yes—
426C4×D4C4×D6111yescharacteristic
427C4×D4D24412yes—
428C4×D4D24411yescharacteristic
429#117—612no—
430#117—1221no—
431#117—1221no—
432#117—1221no—
433#117—1222no—
434#117—1222no—
435#117—1222no—
436#117C22⋊D6111yescharacteristic
437#117C22⋊D6112yes—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1222no—
443C2×C42—1222no—
444C2×C42C22×D6111yescharacteristic
445C2×C42C22×D6112yes—
446C4⋊C8—21no—
447C4⋊C8—22no—
448C4⋊C8—21no—
449C4⋊C8—22no—
450C4⋊C8—1222no—
451C4⋊C8—1221no—
452C4⋊C8C22×D6111yescharacteristic
453C4⋊C8C22×D6111yescharacteristic
454C4⋊C8C22×D6112yes—
455C22⋊C8—1221no—
456C22⋊C8—1222no—
457#97—1222no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8—612no—
461C4×C8—1221no—
462C24—1221no—
463#41—22no—
464#41—22no—
465#41—1224no—
466#41—2442no—
467C2×Q8—612no—
468C2×Q8—1221no—
469C2×Q8#1175819811yescharacteristic
470C2×D4—611no—
471C2×D4—611no—
472C2×D4—1222no—
473C2×D4—1222no—
474C2×D4—1222no—
475C2×D4—1221no—
476C2×D4—2441no—
477C2×D4#1175819812yes—
478C2×D4#1175819811yescharacteristic
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4C2×C4×D6111yescharacteristic
489C22×C4C2×C4×D6112yes—
490C22×C4C2×D24411yescharacteristic
491C22×C4C2×D24412yes—
492C22×C4C2×C22⋊D6112yes—
493C22×C4C2×C22⋊D6111yescharacteristic
494M16—44no—
495M16—42no—
496M16—2444no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—44no—
501C2×C8—42no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1222no—
505C2×C8—2442no—
506C4⋊C4—611no—
507C4⋊C4—611no—
508C4⋊C4—612no—
509C4⋊C4—1222no—
510C4⋊C4—1221no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1221no—
519C4⋊C4#1175819811yescharacteristic
520C4⋊C4#1175819811yescharacteristic
521C4⋊C4#1175819812yes—
522C4⋊C4#1175819812yes—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1221no—
527C22⋊C4—1221no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1221no—
531C22⋊C4—1221no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4#1175819812yes—
535C22⋊C4#1175819812yes—
536C22⋊C4#1175819812yes—
537C42—1222no—
538C42—1222no—
539C42—1222no—
540C42C2×C4×D6111yescharacteristic
541C42C2×C4×D6112yes—
542C42C2×D24411yescharacteristic
543C42C2×D24412yes—
544C42C2×C22⋊D6112yes—
545C42C2×C22⋊D6111yescharacteristic
546C42C23×D6111yescharacteristic
547C23—1221no—
548C23—1221no—
549C23—4881no—
550C23—2442no—
551C23—2442no—
552C23C2×#1175819812yes—
553C23C2×#1175819811yescharacteristic
554Q8—22no—
555Q8—1222no—
556Q8—1222no—
557Q8—4881no—
558D4—24no—
559D4—22no—
560D4—612no—
561D4—1221no—
562D4—1221no—
563D4—1221no—
564D4—4882no—
565D4—2442no—
566D4—2444no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—4882no—
580C2×C4—2444no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4C2×#1175819812yes—
589C2×C4C2×#1175819812yes—
590C2×C4C2×#1175819812yes—
591C2×C4C2×#1175819811yescharacteristic
592C2×C4C2×#1175819811yescharacteristic
593C2×C4C2×#1175819811yescharacteristic
594C2×C4#1451980312yes—
595C2×C4#1451980812yes—
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C22×D24411yescharacteristic
599C8—41no—
600C8—42no—
601C8—41no—
602C8—1222no—
603C8—2441no—
604C8—2442no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—1222no—
611C22—4884no—
612C22—2441no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22—2441no—
617C22#1683976112yes—
618C22C22×#1175819811yescharacteristic
619C4—21no—
620C4—22no—
621C4—21no—
622C4—22no—
623C4—21no—
624C4—22no—
625C4—2442no—
626C4—2442no—
627C4—2441no—
628C4—2441no—
629C4#1683976112yes—
630C4#131×D6111yescharacteristic
631C4#1684052711yescharacteristic
632C4C2×#1451980311yescharacteristic
633C4C2×#1451980811yescharacteristic
634C2—22no—
635C2—42no—
636C2—4881no—
637C2—2442no—
638C2#2145426611yescharacteristic
639C2C2×#1683976111yescharacteristic
640C2#2145436411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2906C2
socleC22×C61C22×C22⋊C4
prime radical#21459383C2
cosocleC42×C61C24
hypercenter#2906D61
hypocenterC61#17512
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #21454786C2semidirect
3. #21444282C2semidirect
4. #21444273C2semidirect
5. #21444256C2semidirect
6. #21444243C2semidirect
7. #21444230C2semidirect
8. #21444221C2semidirect
9. #21444204C2semidirect
10. C2×#16839489C2semidirect
11. C61×#2906C2semidirect
32. #16839815C22semidirect
33. C4×#11758194C22semidirect
35. #16839492C22semidirect
36. #16839492C22semidirect
37. #16839492C22semidirect
38. #16839492C22semidirect
50. C61×#423C22semidirect
52. C2×C4⋊C8×C61C22semidirect
266. D244#131semidirect
316. #2906D61semidirect
358. #423C2×D61semidirect
360. C2×C4⋊C8C2×D61semidirect
373. C61#17512semidirect

Maximal subgroups

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

subgroupindex
#214593832
#214547862
#214442822
#214442732
#214442562
#214442432
#214442302
#214442212
#214442042
C2×#168394892
C61×#29062
#1751261