Finite Group Databasebeta

← #29647993

Subgroups — #29647993

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#21454833C211yesmaximal, characteristic, complemented
3#21444437C212yesmaximal, complemented
4#21444429C211yesmaximal, characteristic, complemented
5#21444424C211yesmaximal, characteristic, complemented
6#21444416C212yesmaximal, complemented
7#21444410C211yesmaximal, characteristic, complemented
8#21444408C212yesmaximal, complemented
9#21444405C211yesmaximal, characteristic, complemented
10C2×#16839501C212yesmaximal, complemented
11C61×#2875C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840485C411yescharacteristic
14#16840475—21no—
15#16840436—22no—
16D4×Dic122C411yescharacteristic
17#16840421C412yes—
18#16840408C412yes—
19#16840400—21no—
20D4×D244C411yescharacteristic
21#16840384—21no—
22#16840345—22no—
23#16840312C2211yescharacteristic
24#16840296C2211yescharacteristic
25#16840296C2212yes—
26C2×#14519765C2212yes—
27#16840234—21no—
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2212yescomplemented
32#16839819C2211yescharacteristic, complemented
33C4×#11758194C2211yescharacteristic, complemented
34C2×#14519688C2212yescomplemented
35#16839503C2211yescharacteristic, complemented
36#16839503C2211yescharacteristic, complemented
37#16839503C2212yescomplemented
38#16839501C2211yescharacteristic
39#16839501C2212yes—
40#16839501C2211yescharacteristic
41#16839501C2212yescomplemented
42#16839501C2212yescomplemented
43#16839501C2212yescomplemented
44#16839501C2212yescomplemented
45C2×#131×C61—21no—
46C4×#41×C61C2211yescharacteristic
47C61×#405—21no—
48C61×#405—22no—
49C4×M16×C61C2212yescomplemented
50C4×M16×C61C2211yescharacteristic, complemented
51C2×#97×C61C2212yescomplemented
52C2×#97×C61C2211yescharacteristic, complemented
53C2×#11758222C2×C412yes—
54C22×Dic122C2×C412yes—
55C22×D244C2×C412yes—
56C2×D4×D61—21no—
57C2×#11758194—21no—
58C22×#92418—22no—
59#14519777C2×C411yescharacteristic
60#14519776—21no—
61#14519774C2×C411yescharacteristic
62#14519772—22no—
63#14519771—22no—
64#14519771—21no—
65#14519769C2311yescharacteristic
66#14519766—21no—
67#14519765—22no—
68#14519765—22no—
69#14519765C2×C412yes—
70#14519765C2×C412yes—
71#14519765C2311yescharacteristic
72#14519765C2312yes—
73#14519763—22no—
74#14519762—22no—
75#14519762—22no—
76#14519760—21no—
77#14519760—22no—
78#14519756C2×C412yes—
79#14519756C2×C411yescharacteristic
80#14519755—22no—
81#14519754—22no—
82#14519753—21no—
83#14519752—21no—
84#14519751—21no—
85#14519751—21no—
86#14519751C2×C412yes—
87#14519751C2×C411yescharacteristic
88#14519750—22no—
89#14519749—22no—
90#14519747—22no—
91#14519745—22no—
92C4×Dic122C2×C412yes—
93C4×Dic122C2×C411yescharacteristic
94C4×D244—21no—
95C4×D244—21no—
96C4×D244C2×C411yescharacteristic
97C4×D244C2×C412yes—
98D4×Dic61—21no—
99#14519739—21no—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519735—21no—
104#14519734—22no—
105#14519734—21no—
106C2×#11758198—22no—
107C2×#11758198—22no—
108C22⋊C4×D61—22no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—22no—
112C22⋊#92418—21no—
113#14519688—22no—
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116#14519688C2311yescharacteristic
117C4×#92418—21no—
118C2×#41×C61D411yescharacteristic
119#131×C61—28no—
120C2×M16×C61—22no—
121C2×M16×C61—21no—
122C22×C8×C61—22no—
123C22×C8×C61—21no—
124C4×Q8×C61C2×C411yescharacteristic
125C4×Q8×C61D411yescharacteristic
126C4×D4×C61C2×C411yescharacteristic
127C4×D4×C61C2×C412yes—
128C4×D4×C61D411yescharacteristic
129C4×D4×C61D412yes—
130#117×C61D411yescharacteristic
131#117×C61D412yes—
132C2×C42×C61C2312yes—
133C2×C42×C61C2311yescharacteristic
134#97×C61—22no—
135#97×C61—21no—
136#97×C61C2311yescharacteristic
137#97×C61C2312yes—
138#97×C61C2311yescharacteristic
139C4×C8×C61—22no—
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—28no—
172M16×C61—24no—
173C2×C8×C61—28no—
174C2×C8×C61—24no—
175C2×C8×C61—21no—
176C2×C8×C61—22no—
177C2×C8×C61—21no—
178C4⋊C4×C61C22⋊C411yescharacteristic
179C4⋊C4×C61C22⋊C412yes—
180C4⋊C4×C61C22⋊C411yescharacteristic
181C4⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61C22⋊C412yes—
183C22⋊C4×C61C22⋊C412yes—
184C22⋊C4×C61C22⋊C412yes—
185C42×C61C22×C412yes—
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D412yes—
189C42×C61C2×D412yes—
190C42×C61C2×D411yescharacteristic
191C42×C61C2411yescharacteristic
192C22⋊Dic61—22no—
193C22⋊Dic61—21no—
194#11758198—21no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—21no—
202#11758198—21no—
203#11758198—21no—
204#11758198—22no—
205#11758198—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C22×C411yescharacteristic
209C4⋊Dic61C22×C411yescharacteristic
210#11758196—21no—
211#11758196—22no—
212#11758196—22no—
213#11758196—21no—
214#11758196—21no—
215#11758196—21no—
216C4×Dic61—22no—
217C4×Dic61—21no—
218#11758194—42no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—22no—
222C2×#92418—44no—
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—24no—
269C8×C61—22no—
270#92418—22no—
271#92418—42no—
272#92418—41no—
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#2970—611no—
303#2875D6111yescharacteristic, complemented
304#1532—612no—
305#1530—611no—
306#1529—611no—
307#1527—612no—
308#1525—611no—
309#1524—612no—
310#1524—611no—
311C2×#329—612no—
312C2×C61—24no—
313C2×C61—41no—
314C2×C61#288811yescharacteristic
315C2×C61C22×#10411yescharacteristic
316C2×C61#290111yescharacteristic
317D61—82no—
318D61—81no—
319C2×#131—21no—
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—1221no—
338#435—1222no—
339C8×D4—1221no—
340#432—612no—
341#432—611no—
342#405—22no—
343#405—21no—
344C4×M16—611no—
345C4×M16C2×D6112yescomplemented
346C4×M16C2×D6111yescharacteristic, complemented
347C2×#97—612no—
348C2×#97C2×D6112yescomplemented
349C2×#97C2×D6111yescharacteristic, complemented
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×#41C22⋊D6111yescharacteristic
363C22×Q8—612no—
364C22×D4—612no—
365C22×D4—1221no—
366#131—28no—
367C2×M16—22no—
368C2×M16—21no—
369C2×M16—1221no—
370C22×C8—22no—
371C22×C8—21no—
372C22×C8—1222no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375#127—611no—
376#126—1222no—
377#126—1222no—
378#126—1221no—
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—1221no—
389#123—1222no—
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×Q8D24411yescharacteristic
410C4×D4—611no—
411C4×D4—612no—
412C4×D4—1221no—
413C4×D4—1222no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4—1222no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4C4×D6111yescharacteristic
420C4×D4C4×D6112yes—
421C4×D4D24411yescharacteristic
422C4×D4D24412yes—
423#117—1221no—
424#117—1221no—
425#117—1222no—
426#117C22⋊D6112yes—
427#117C22⋊D6111yescharacteristic
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1222no—
430C2×C22⋊C4—1222no—
431C2×C42C22×D6112yes—
432C2×C42C22×D6111yescharacteristic
433C4⋊C8—1221no—
434C4⋊C8—1222no—
435C22⋊C8—1222no—
436C22⋊C8—1221no—
437#97—22no—
438#97—21no—
439#97—611no—
440#97—612no—
441#97—611no—
442#97—1222no—
443#97C22×D6111yescharacteristic
444#97C22×D6112yes—
445#97C22×D6111yescharacteristic
446C4×C8—22no—
447C4×C8—21no—
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—28no—
480M16—24no—
481M16—2442no—
482C2×C8—28no—
483C2×C8—24no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1222no—
490C2×C8—2444no—
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×D24411yescharacteristic
527C42C2×D24412yes—
528C42C2×C22⋊D6112yes—
529C42C2×C22⋊D6111yescharacteristic
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#1451970514yes—
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×#1175819812yes—
568C2×C4C2×#1175819811yescharacteristic
569C2×C4C2×#1175819811yescharacteristic
570C2×C4C2×#1175819812yes—
571C2×C4C2×#1175819811yescharacteristic
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C22×D24411yescharacteristic
575C8—22no—
576C8—24no—
577C8—22no—
578C8—1222no—
579C8—2442no—
580C8—2441no—
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×#1451970514yes—
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×#1451970512yes—
608C2—24no—
609C2—41no—
610C2—4881no—
611C2—4882no—
612C2#2145429711yescharacteristic
613C2C22×#1451970511yescharacteristic
614C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2875C2
socleC22×C61C22×C22⋊C4
prime radical#21459780C2
cosocleC42×C61C24
hypercenter#2875D61
hypocenterC61#17652
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #21454833C2semidirect
3. #21444437C2semidirect
4. #21444429C2semidirect
5. #21444424C2semidirect
6. #21444416C2semidirect
7. #21444410C2semidirect
8. #21444408C2semidirect
9. #21444405C2semidirect
10. C2×#16839501C2semidirect
11. C61×#2875C2semidirect
31. #16839819C22semidirect
32. #16839819C22semidirect
33. C4×#11758194C22semidirect
34. C2×#14519688C22semidirect
35. #16839503C22semidirect
36. #16839503C22semidirect
37. #16839503C22semidirect
41. #16839501C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
49. C4×M16×C61C22semidirect
50. C4×M16×C61C22semidirect
51. C2×#97×C61C22semidirect
52. C2×#97×C61C22semidirect
303. #2875D61semidirect
345. C4×M16C2×D61semidirect
346. C4×M16C2×D61semidirect
348. C2×#97C2×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
#214548332
#214444372
#214444292
#214444242
#214444162
#214444102
#214444082
#214444052
C2×#168395012
C61×#28752
#1765261