Finite Group Databasebeta

← #29647992

Subgroups — #29647992

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#21459782C211yesmaximal, characteristic
2#21454834C211yesmaximal, characteristic, complemented
3#21454244C211yesmaximal, characteristic, complemented
4#21444438C212yesmaximal, complemented
5#21444429C211yesmaximal, characteristic, complemented
6#21444426C211yesmaximal, characteristic, complemented
7#21444415C212yesmaximal, complemented
8#21444410C211yesmaximal, characteristic, complemented
9#21444406C212yesmaximal, complemented
10#329×D61C212yesmaximal, complemented
11C61×#1524C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840497C411yescharacteristic
14#16840482C411yescharacteristic
15#16840477—21no—
16#16840437—22no—
17Q8×D244C411yescharacteristic
18#16840422C412yes—
19#16840409C412yes—
20#16840408—21no—
21D4×D244C411yescharacteristic
22#16840383—21no—
23#16840348—22no—
24#16840312C2211yescharacteristic
25#16840297C2212yes—
26#124×D61C2212yes—
27#16840236—21no—
28C4×#11758222C2211yescharacteristic
29#16839892—21no—
30#16839831—22no—
31C8×D244—21no—
32#16839819C2211yescharacteristic, complemented
33#16839817C2212yescomplemented
34#16839719—22no—
35#16839717—21no—
36C4×#11758194C2211yescharacteristic, complemented
37#16839714C2212yescomplemented
38#97×D61C2212yescomplemented
39#97×D61C2212yescomplemented
40#16839504C2212yescomplemented
41#16839502C2212yes—
42#16839501C2212yescomplemented
43#16839501C2212yescomplemented
44#16839501C2212yescomplemented
45#16839501C2212yescomplemented
46C61×#529C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic, complemented
48C2×#97×C61C2211yescharacteristic, complemented
49C61×#330C2211yescharacteristic, complemented
50C61×#330C2211yescharacteristic, complemented
51C61×#329C2211yescharacteristic
52C61×#329C2211yescharacteristic
53C2×#11758222C2×C412yes—
54C2×#11758222D411yescharacteristic
55C2×Q8×D61C2×C412yes—
56C22×D244—21no—
57C2×D4×D61C2×C412yes—
58#14519783—28no—
59C2×#11758194—21no—
60C2×#11758189—22no—
61C2×C8×D61—22no—
62C2×C8×D61—22no—
63#14519777—21no—
64#14519777C2×C411yescharacteristic
65#14519774C2×C411yescharacteristic
66#14519772—22no—
67#14519770—22no—
68#14519766C2×C412yes—
69#14519765—21no—
70#14519765C2312yes—
71#14519764C2×C412yes—
72#14519763—22no—
73#14519763—22no—
74#14519763—22no—
75#14519762—22no—
76#14519762—22no—
77#14519759—22no—
78#14519757—22no—
79#14519757C2×C412yes—
80#14519756—21no—
81#14519755—22no—
82#14519752C2×C412yes—
83#14519751—21no—
84#14519751—21no—
85#14519751—21no—
86#14519750—22no—
87#14519748—22no—
88#14519747—22no—
89#14519745—22no—
90C4×Dic122C2×C411yescharacteristic
91C4×Dic122D411yescharacteristic
92#14519742C2×C412yes—
93C4×D244—21no—
94C4×D244—21no—
95C4×D244—21no—
96C4×D244—21no—
97C4×D244C2×C411yescharacteristic
98C4×D244D411yescharacteristic
99#14519739—22no—
100#14519739—22no—
101#14519739C2×C412yes—
102C4×C22⋊D61C2×C412yes—
103C4×C22⋊D61D412yes—
104#14519736—21no—
105#14519736D411yescharacteristic
106#14519735—22no—
107#14519734D412yes—
108C22⋊C4×D61—22no—
109C22⋊C4×D61—22no—
110C22⋊C4×D61—22no—
111C42×D61C2312yes—
112C4⋊#92418—21no—
113#14519706—22no—
114C22⋊#92418—21no—
115#14519689—22no—
116#14519688C2312yes—
117#14519688C2312yes—
118#14519687—22no—
119#14519687—22no—
120C4×#92418—21no—
121C8×Dic61—22no—
122C2×M16×C61—21no—
123C22×C8×C61—21no—
124#126×C61—21no—
125#125×C61C2311yescharacteristic
126#124×C61C2311yescharacteristic
127#123×C61—21no—
128C22⋊Q8×C61C2×C411yescharacteristic
129#121×C61C2×C411yescharacteristic
130C4×Q8×C61C2×C411yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2×C411yescharacteristic
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#97×C61—21no—
136#97×C61C2311yescharacteristic
137#97×C61C2311yescharacteristic
138#97×C61C2311yescharacteristic
139#97×C61C2311yescharacteristic
140C4×C8×C61—21no—
141C23×D61—22no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61C22×C411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61C22⋊C412yes—
152Q8×D61—82no—
153D4×D61—82no—
154#11758222—24no—
155#11758222—82no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—44no—
160C2×D244C22⋊C411yescharacteristic
161C2×D244C22×C412yes—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D412yes—
168C2×Dic122—22no—
169C2×Dic122C22⋊C411yescharacteristic
170C2×Dic122C22×C412yes—
171M16×C61—42no—
172C2×C8×C61—24no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C22×C411yescharacteristic
182C4⋊C4×C61C22×C411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C42×C61—21no—
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2411yescharacteristic
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61C22⋊C412yes—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22⋊C411yescharacteristic
209C4⋊Dic61C22⋊C411yescharacteristic
210#11758196—22no—
211#11758196C22⋊C412yes—
212#11758196C22⋊C412yes—
213C4×Dic61—22no—
214C4×Dic61C22×C412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D412yes—
217#11758194—24no—
218C2×#92418—22no—
219C2×#92418—22no—
220#11758189—28no—
221C8×D61—28no—
222C8×D61—44no—
223C23×C61—21no—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61—22no—
226C22×D61—44no—
227C22×D61—82no—
228C22×D61—82no—
229C22×D61C2×C22⋊C412yes—
230Q8×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C22⋊D61—24no—
247C22⋊D61—82no—
248C22⋊D61—82no—
249C2×Dic61—22no—
250C2×Dic61C2×C22⋊C412yes—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C412yes—
253D244—42no—
254D244—44no—
255D244—42no—
256D244—81no—
257D244#10414yes—
258C4×D61—24no—
259C4×D61—42no—
260C4×D61—42no—
261C4×D61—82no—
262C4×D61—82no—
263C4×D61—82no—
264Dic122—22no—
265Dic122—42no—
266Dic122—82no—
267C8×C61—22no—
268C8×C61—22no—
269C8×C61—22no—
270C8×C61—41no—
271#92418—24no—
272#92418—42no—
273#17652—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—164no—
287C2×D61C2×#10414yes—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61C2×#10412yes—
297Dic61—22no—
298Dic61—22no—
299Dic61—22no—
300Dic61—42no—
301#3514—611no—
302#2970—611no—
303#2875—611no—
304#1532—612no—
305#1530—611no—
306#1529—611no—
307#1527—612no—
308#1525—611no—
309#1524—612no—
310#1524D6111yescharacteristic, complemented
311C2×#329—612no—
312C2×C61—41no—
313C2×C61—81no—
314C2×C61#288811yescharacteristic
315C2×C61C22×#10411yescharacteristic
316C2×C61#290111yescharacteristic
317D61—24no—
318D61—82no—
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—612no—
333#529C2×D6111yescharacteristic
334C2×#124—612no—
335#518—1221no—
336C4×#41—611no—
337#446—1221no—
338#435—1222no—
339C8×D4—1221no—
340#432—611no—
341#432—612no—
342#405—1222no—
343#405—1221no—
344C4×M16—611no—
345C4×M16—612no—
346C4×M16C2×D6111yescharacteristic, complemented
347C2×#97—612no—
348C2×#97—612no—
349C2×#97C2×D6111yescharacteristic, complemented
350#330—612no—
351#330C2×D6111yescharacteristic, complemented
352#330C2×D6111yescharacteristic, complemented
353#329—612no—
354#329—612no—
355#329—612no—
356#329—612no—
357#329—612no—
358#329C2×D6111yescharacteristic
359#329C2×D6111yescharacteristic
360C61#1765211yescharacteristic, complemented, Sylow
361C2×#41—611no—
362C2×#41—612no—
363C22×Q8—612no—
364C22×D4—612no—
365C22×D4—1221no—
366#131—1228no—
367C2×M16—21no—
368C2×M16—1222no—
369C2×M16—1221no—
370C22×C8—21no—
371C22×C8—1222no—
372C22×C8—1222no—
373C4⋊Q8—611no—
374C4⋊Q8—1221no—
375#127—611no—
376#126—21no—
377#126—1222no—
378#126—1222no—
379#125C22×D6111yescharacteristic
380#124—612no—
381#124—612no—
382#124—612no—
383#124—1221no—
384#124—1222no—
385#124—1222no—
386#124—1222no—
387#124C22×D6111yescharacteristic
388#123—21no—
389#123—1222no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—612no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—1221no—
396C22⋊Q8C4×D6111yescharacteristic
397#121—612no—
398#121—1222no—
399#121—1222no—
400#121—1221no—
401#121—1221no—
402#121—1221no—
403#121C4×D6111yescharacteristic
404#120—1222no—
405#120—1222no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—612no—
409C4×Q8C4×D6111yescharacteristic
410C4×D4—21no—
411C4×D4—612no—
412C4×D4—612no—
413C4×D4—612no—
414C4×D4—611no—
415C4×D4—611no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1222no—
419C4×D4—1222no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4C4×D6111yescharacteristic
423#117—21no—
424#117—611no—
425#117—612no—
426#117—1222no—
427#117—1221no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1222no—
430C2×C22⋊C4—1222no—
431C2×C42—612no—
432C2×C42C22×D6111yescharacteristic
433C4⋊C8—1222no—
434C4⋊C8—1221no—
435C22⋊C8—1222no—
436C22⋊C8—1221no—
437#97—21no—
438#97—612no—
439#97—612no—
440#97—1222no—
441#97—1222no—
442#97C22×D6111yescharacteristic
443#97C22×D6111yescharacteristic
444#97C22×D6111yescharacteristic
445#97C22×D6111yescharacteristic
446C4×C8—21no—
447C4×C8—1222no—
448C4×C8—1221no—
449C24—1222no—
450#41—1224no—
451#41—4882no—
452C2×Q8—611no—
453C2×Q8—612no—
454C2×Q8—1222no—
455C2×Q8—4882no—
456C2×Q8C2×C4×D6111yescharacteristic
457C2×D4—21no—
458C2×D4—612no—
459C2×D4—612no—
460C2×D4—611no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—4882no—
467C2×D4—2444no—
468C2×D4C2×C4×D6111yescharacteristic
469C22×C4—21no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4C2×C4×D6111yescharacteristic
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479M16—42no—
480M16—1228no—
481M16—1224no—
482C2×C8—24no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—1228no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—2444no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—612no—
496C4⋊C4—612no—
497C4⋊C4—611no—
498C4⋊C4—611no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4C2×C4×D6111yescharacteristic
504C4⋊C4C2×C4×D6111yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—612no—
511C22⋊C4—612no—
512C22⋊C4—612no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C42—21no—
523C42—612no—
524C42—612no—
525C42—612no—
526C42—1222no—
527C42C2×C4×D6111yescharacteristic
528C42D4×D6111yescharacteristic
529C42D4×D6111yescharacteristic
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—612no—
533C23—1222no—
534C23—4882no—
535C23—4882no—
536C23—2444no—
537C23C22⋊C4×D6111yescharacteristic
538Q8—41no—
539Q8—1222no—
540Q8—4882no—
541Q8—2442no—
542D4—41no—
543D4—81no—
544D4—614no—
545D4—1224no—
546D4—4881no—
547D4—4882no—
548D4—4882no—
549D4—2442no—
550D4—2444no—
551D4—2442no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—1224no—
563C2×C4—1222no—
564C2×C4—4882no—
565C2×C4—4882no—
566C2×C4—4882no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4C22⋊C4×D6111yescharacteristic
570C2×C4C22⋊C4×D6111yescharacteristic
571C2×C4C22⋊C4×D6111yescharacteristic
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C8—22no—
576C8—22no—
577C8—22no—
578C8—41no—
579C8—1224no—
580C8—2442no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—614no—
588C22—1222no—
589C22—1222no—
590C22—9764no—
591C22—2442no—
592C22—2442no—
593C22—2442no—
594C22C2×C22⋊C4×D6111yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—1222no—
604C4—1222no—
605C4—1222no—
606C4—2442no—
607C4#104×D6112yes—
608C2—41no—
609C2—81no—
610C2—1224no—
611C2—4882no—
612C2#2145429611yescharacteristic
613C2C2×#104×D6111yescharacteristic
614C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1524C2
socleC22×C61C22×C22⋊C4
prime radical#21459782C2
cosocleC42×C61C24
hypercenter#1524D61
hypocenterC61#17652
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
2. #21454834C2semidirect
3. #21454244C2semidirect
4. #21444438C2semidirect
5. #21444429C2semidirect
6. #21444426C2semidirect
7. #21444415C2semidirect
8. #21444410C2semidirect
9. #21444406C2semidirect
10. #329×D61C2semidirect
11. C61×#1524C2semidirect
32. #16839819C22semidirect
33. #16839817C22semidirect
36. C4×#11758194C22semidirect
37. #16839714C22semidirect
38. #97×D61C22semidirect
39. #97×D61C22semidirect
40. #16839504C22semidirect
42. #16839501C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
45. #16839501C22semidirect
47. C4×M16×C61C22semidirect
48. C2×#97×C61C22semidirect
49. C61×#330C22semidirect
50. C61×#330C22semidirect
310. #1524D61semidirect
346. C4×M16C2×D61semidirect
349. C2×#97C2×D61semidirect
351. #330C2×D61semidirect
352. #330C2×D61semidirect
360. C61#17652semidirect

Maximal subgroups

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

subgroupindex
#214597822
#214548342
#214542442
#214444382
#214444292
#214444262
#214444152
#214444102
#214444062
#329×D612
C61×#15242
#1765261