Finite Group Databasebeta

← #29647999

Subgroups — #29647999

Order 15616.

1028 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459780C211yesmaximal, characteristic
2#21454247C211yesmaximal, characteristic, complemented
3#21444440C212yesmaximal, complemented
4#21444431C211yesmaximal, characteristic, complemented
5#21444427C211yesmaximal, characteristic, complemented
6#21444418C212yesmaximal, complemented
7#21444410C211yesmaximal, characteristic, complemented
8#21444408C211yesmaximal, characteristic, complemented
9#21444405C212yesmaximal, complemented
10C2×#16839501C212yesmaximal, complemented
11C61×#2970C211yesmaximal, characteristic, complemented
12C2×#14519784—21no—
13#16840497C411yescharacteristic
14#16840485C411yescharacteristic
15#16840475—21no—
16#16840436—22no—
17D4×Dic122C411yescharacteristic
18#16840421C412yes—
19#16840408C412yes—
20#16840400—21no—
21D4×D244C411yescharacteristic
22#16840384—21no—
23#16840345—22no—
24#16840312C2211yescharacteristic
25#16840296C2212yes—
26#16840296C2211yescharacteristic
27C2×#14519765C2212yes—
28#16840234—21no—
29#16839718—21no—
30#16839718—22no—
31C4×#11758194C2212yescomplemented
32C4×#11758194C2211yescharacteristic, complemented
33C2×#14519688C2212yescomplemented
34C2×#14519688C2211yescharacteristic, complemented
35#16839503C2211yescharacteristic, complemented
36#16839503C2212yescomplemented
37#16839503C2211yescharacteristic, complemented
38#16839501C2211yescharacteristic
39#16839501C2212yes—
40#16839501C2211yescharacteristic
41#16839501C2212yescomplemented
42#16839501C2212yescomplemented
43#16839501C2212yescomplemented
44#16839501C2212yescomplemented
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2212yescomplemented
50C61×#432C2211yescharacteristic, complemented
51C4×M16×C61C2211yescharacteristic, complemented
52C2×#97×C61C2212yescomplemented
53C2×#11758222C2×C412yes—
54C22×Dic122C2×C412yes—
55C22×D244C2×C412yes—
56C2×D4×D61—21no—
57#14519784—28no—
58C2×#11758194—22no—
59C2×#11758194—21no—
60C22×#92418—22no—
61C22×#92418—21no—
62#14519777C2×C411yescharacteristic
63#14519776—21no—
64#14519774C2×C411yescharacteristic
65#14519772—22no—
66#14519771—22no—
67#14519771—21no—
68#14519769C2311yescharacteristic
69#14519766—21no—
70#14519765—22no—
71#14519765—22no—
72#14519765C2×C412yes—
73#14519765C2×C412yes—
74#14519765C2311yescharacteristic
75#14519765C2312yes—
76#14519763—22no—
77#14519762—22no—
78#14519762—22no—
79#14519760—21no—
80#14519760—22no—
81#14519756C2×C412yes—
82#14519756C2×C411yescharacteristic
83#14519755—22no—
84#14519754—22no—
85#14519753—21no—
86#14519752—21no—
87#14519751—21no—
88#14519751—21no—
89#14519751C2×C412yes—
90#14519751C2×C411yescharacteristic
91#14519750—22no—
92#14519749—22no—
93#14519747—22no—
94#14519745—22no—
95C4×Dic122C2×C412yes—
96C4×Dic122C2×C411yescharacteristic
97C4×D244—21no—
98C4×D244—21no—
99C4×D244C2×C411yescharacteristic
100C4×D244C2×C412yes—
101D4×Dic61—21no—
102#14519739—21no—
103C4×C22⋊D61—21no—
104C4×C22⋊D61—22no—
105#14519737—22no—
106#14519735—21no—
107#14519734—22no—
108#14519734—21no—
109C2×#11758198—22no—
110C2×#11758198—22no—
111C22⋊C4×D61—22no—
112#14519688—22no—
113#14519688—21no—
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116#14519688C2311yescharacteristic
117C4×#92418—22no—
118C4×#92418—21no—
119C2×#41×C61D411yescharacteristic
120C2×M16×C61—21no—
121C22×C8×C61—22no—
122C4×Q8×C61C2×C411yescharacteristic
123C4×Q8×C61D411yescharacteristic
124C4×D4×C61C2×C411yescharacteristic
125C4×D4×C61C2×C412yes—
126C4×D4×C61D411yescharacteristic
127C4×D4×C61D412yes—
128#117×C61D411yescharacteristic
129#117×C61D412yes—
130C2×C42×C61C2312yes—
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—21no—
133C4⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135C22⋊C8×C61—21no—
136#97×C61—22no—
137#97×C61C2311yescharacteristic
138#97×C61C2312yes—
139#97×C61C2311yescharacteristic
140C4×C8×C61—21no—
141C23×D61—22no—
142#41×C61—24no—
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C2×D4×C61C22⋊C411yescharacteristic
146C22×C4×C61C22×C412yes—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—21no—
155C22×Dic61—22no—
156D4×D61—44no—
157#11758222—82no—
158C2×D244—21no—
159C2×D244—22no—
160C2×D244—21no—
161C2×D244—82no—
162C2×D244C22×C411yescharacteristic
163C2×D244C22×C412yes—
164C2×C4×D61—21no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×Dic122—22no—
168C2×Dic122—82no—
169C2×Dic122C22×C411yescharacteristic
170C2×Dic122C22×C412yes—
171M16×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—22no—
175C2×C8×C61—44no—
176C4⋊C4×C61C22⋊C411yescharacteristic
177C4⋊C4×C61C22⋊C412yes—
178C4⋊C4×C61C22⋊C411yescharacteristic
179C4⋊C4×C61C22⋊C412yes—
180C22⋊C4×C61C22⋊C412yes—
181C22⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61C22⋊C412yes—
183C42×C61C22×C412yes—
184C42×C61C22×C411yescharacteristic
185C42×C61C2×D412yes—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D412yes—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—22no—
191C22⋊Dic61—21no—
192#11758198—21no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—21no—
200#11758198—21no—
201#11758198—21no—
202#11758198—22no—
203#11758198—22no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C22×C411yescharacteristic
207C4⋊Dic61C22×C411yescharacteristic
208#11758196—21no—
209#11758196—22no—
210#11758196—22no—
211#11758196—21no—
212#11758196—21no—
213#11758196—21no—
214C4×Dic61—22no—
215C4×Dic61—21no—
216#11758194—28no—
217#11758194—24no—
218C2×#92418—28no—
219C2×#92418—24no—
220C2×#92418—21no—
221C2×#92418—22no—
222C2×#92418—21no—
223C23×C61C2×C22⋊C411yescharacteristic
224C23×C61C2×C22⋊C412yes—
225C22×D61—21no—
226C22×D61—22no—
227C22×D61—44no—
228C22×D61—82no—
229C22×D61—82no—
230Q8×C61—22no—
231D4×C61—24no—
232D4×C61#10414yes—
233C2×C4×C61—24no—
234C2×C4×C61—22no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C412yes—
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—
249C22⋊D61—82no—
250C22⋊D61—81no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—82no—
256D244—41no—
257D244—42no—
258D244—42no—
259D244—81no—
260D244—82no—
261C4×D61—42no—
262C4×D61—82no—
263C4×D61—82no—
264Dic122—41no—
265Dic122—42no—
266Dic122—82no—
267C8×C61—22no—
268C8×C61—42no—
269C8×C61—41no—
270#92418—22no—
271#92418—24no—
272#92418—22no—
273#17652—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61C2×#10414yes—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—42no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—41no—
287C2×D61—164no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—22no—
293C4×C61—21no—
294C4×C61—22no—
295C4×C61—21no—
296C4×C61C2×#10412yes—
297Dic61—41no—
298Dic61—42no—
299Dic61—41no—
300Dic61—41no—
301#3514—611no—
302#2970D6111yescharacteristic, complemented
303#2875—611no—
304#1532—612no—
305#1530—611no—
306#1529—611no—
307#1527—612no—
308#1525—611no—
309#1524—611no—
310#1524—612no—
311C2×#329—612no—
312C2×C61—24no—
313C2×C61—41no—
314C2×C61#288811yescharacteristic
315C2×C61C22×#10411yescharacteristic
316C2×C61#290111yescharacteristic
317D61—82no—
318D61—81no—
319C2×#131—1221no—
320#559—611no—
321#556—611no—
322#555—1221no—
323#551—1222no—
324D4×Q8—611no—
325#548—612no—
326#546—612no—
327#546—1221no—
328D42—611no—
329#542—1221no—
330#538—1222no—
331#532—611no—
332#529—611no—
333#529—612no—
334C2×#124—612no—
335#518—1221no—
336C4×#41C2×D6111yescharacteristic
337#446—21no—
338#435—22no—
339C8×D4—21no—
340#432C2×D6112yescomplemented
341#432C2×D6111yescharacteristic, complemented
342#405—1222no—
343#405—1221no—
344C4×M16—612no—
345C4×M16—611no—
346C4×M16C2×D6111yescharacteristic, complemented
347C2×#97—612no—
348C2×#97—611no—
349C2×#97C2×D6112yescomplemented
350#330—611no—
351#330—612no—
352#330—611no—
353#329—611no—
354#329—612no—
355#329—611no—
356#329—612no—
357#329—612no—
358#329—612no—
359#329—612no—
360C61#1765211yescharacteristic, complemented, Sylow
361C2×#41—612no—
362C2×#41D24411yescharacteristic
363C22×Q8—612no—
364C22×D4—612no—
365C22×D4—1221no—
366#131—1228no—
367C2×M16—21no—
368C2×M16—1222no—
369C2×M16—1221no—
370C22×C8—22no—
371C22×C8—1222no—
372C22×C8—1221no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375#127—611no—
376#126—1222no—
377#126—1221no—
378#126—1222no—
379#125—611no—
380#124—612no—
381#124—612no—
382#124—611no—
383#124—612no—
384#124—1221no—
385#124—1222no—
386#124—1222no—
387#124—1222no—
388#123—1222no—
389#123—1221no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—612no—
393C22⋊Q8—611no—
394C22⋊Q8—1221no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397#121—612no—
398#121—611no—
399#121—1222no—
400#121—1221no—
401#121—1221no—
402#121—1221no—
403#121—1222no—
404#120—1222no—
405#120—1222no—
406C4×Q8—612no—
407C4×Q8—611no—
408C4×Q8C4×D6111yescharacteristic
409C4×Q8C22⋊D6111yescharacteristic
410C4×D4—611no—
411C4×D4—612no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4—1222no—
415C4×D4—1221no—
416C4×D4—1222no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4C4×D6111yescharacteristic
420C4×D4C4×D6112yes—
421C4×D4C22⋊D6111yescharacteristic
422C4×D4C22⋊D6112yes—
423#117—1221no—
424#117—1222no—
425#117—1221no—
426#117D24412yes—
427#117D24411yescharacteristic
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1222no—
430C2×C22⋊C4—1222no—
431C2×C42C22×D6112yes—
432C2×C42C22×D6111yescharacteristic
433C4⋊C8—21no—
434C4⋊C8—22no—
435C22⋊C8—22no—
436C22⋊C8—21no—
437#97—22no—
438#97—611no—
439#97—612no—
440#97—611no—
441#97—1222no—
442#97—1221no—
443#97C22×D6111yescharacteristic
444#97C22×D6112yes—
445#97C22×D6111yescharacteristic
446C4×C8—21no—
447C4×C8—1222no—
448C4×C8—1221no—
449C24—1222no—
450#41—24no—
451#41—4882no—
452C2×Q8—611no—
453C2×Q8—612no—
454C2×Q8—1222no—
455C2×Q8—4882no—
456C2×Q8#1175819811yescharacteristic
457C2×D4—611no—
458C2×D4—612no—
459C2×D4—1221no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1221no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—4882no—
466C2×D4—2444no—
467C2×D4#1175819812yes—
468C2×D4#1175819811yescharacteristic
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4C2×C4×D6112yes—
474C22×C4C2×C4×D6111yescharacteristic
475C22×C4C2×D24412yes—
476C22×C4C2×D24411yescharacteristic
477C22×C4C2×C22⋊D6112yes—
478C22×C4C2×C22⋊D6111yescharacteristic
479M16—42no—
480M16—1228no—
481M16—1224no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—44no—
486C2×C8—1228no—
487C2×C8—1224no—
488C2×C8—1221no—
489C2×C8—1222no—
490C2×C8—1221no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4#1175819811yescharacteristic
502C4⋊C4#1175819811yescharacteristic
503C4⋊C4#1175819812yes—
504C4⋊C4#1175819812yes—
505C22⋊C4—1222no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1222no—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1221no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1221no—
518C22⋊C4—1222no—
519C22⋊C4#1175819812yes—
520C22⋊C4#1175819812yes—
521C22⋊C4#1175819812yes—
522C42—1222no—
523C42—1221no—
524C42C2×C4×D6112yes—
525C42C2×C4×D6111yescharacteristic
526C42C2×D24412yes—
527C42C2×D24411yescharacteristic
528C42C2×C22⋊D6111yescharacteristic
529C42C2×C22⋊D6112yes—
530C42C23×D6111yescharacteristic
531C23—1221no—
532C23—1222no—
533C23—4882no—
534C23—4882no—
535C23—2444no—
536C23C2×#1175819811yescharacteristic
537C23C2×#1175819812yes—
538Q8—22no—
539Q8—4882no—
540Q8—2441no—
541Q8—2442no—
542D4—24no—
543D4—4881no—
544D4—4882no—
545D4—4881no—
546D4—4882no—
547D4—2441no—
548D4—2442no—
549D4—2444no—
550D4—2442no—
551D4#1451970314yes—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—4882no—
563C2×C4—4882no—
564C2×C4—4882no—
565C2×C4—2442no—
566C2×C4C2×#1175819812yes—
567C2×C4C2×#1175819811yescharacteristic
568C2×C4C2×#1175819812yes—
569C2×C4C2×#1175819811yescharacteristic
570C2×C4C2×#1175819812yes—
571C2×C4C2×#1175819811yescharacteristic
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C22×D24411yescharacteristic
575C8—22no—
576C8—42no—
577C8—41no—
578C8—1222no—
579C8—1224no—
580C8—1222no—
581C22—22no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—9764no—
587C22—2441no—
588C22—2442no—
589C22—2442no—
590C22—2441no—
591C22—2442no—
592C22—2441no—
593C22C2×#1451970314yes—
594C22C22×#1175819811yescharacteristic
595C4—21no—
596C4—22no—
597C4—21no—
598C4—22no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—2441no—
604C4—2441no—
605C4—2442no—
606C4—2441no—
607C4C2×#1451970312yes—
608C2—24no—
609C2—41no—
610C2—4881no—
611C2—4882no—
612C2#2145429711yescharacteristic
613C2C22×#1451970311yescharacteristic
614C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2970C2
socleC22×C61C22×C22⋊C4
prime radical#21459780C2
cosocleC42×C61C24
hypercenter#2970D61
hypocenterC61#17652
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #21454247C2semidirect
3. #21444440C2semidirect
4. #21444431C2semidirect
5. #21444427C2semidirect
6. #21444418C2semidirect
7. #21444410C2semidirect
8. #21444408C2semidirect
9. #21444405C2semidirect
10. C2×#16839501C2semidirect
11. C61×#2970C2semidirect
31. C4×#11758194C22semidirect
32. C4×#11758194C22semidirect
33. C2×#14519688C22semidirect
34. C2×#14519688C22semidirect
35. #16839503C22semidirect
36. #16839503C22semidirect
37. #16839503C22semidirect
41. #16839501C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
49. C61×#432C22semidirect
50. C61×#432C22semidirect
51. C4×M16×C61C22semidirect
52. C2×#97×C61C22semidirect
302. #2970D61semidirect
340. #432C2×D61semidirect
341. #432C2×D61semidirect
346. C4×M16C2×D61semidirect
349. C2×#97C2×D61semidirect
360. C61#17652semidirect

Maximal subgroups

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

subgroupindex
#214597802
#214542472
#214444402
#214444312
#214444272
#214444182
#214444102
#214444082
#214444052
C2×#168395012
C61×#29702
#1765261