Finite Group Databasebeta

← #30571115

Subgroups — #30571115

Order 15616.

1706 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459624C211yesmaximal, characteristic, complemented
2C2×#16840486C211yesmaximal, characteristic
3#21458855C214yesmaximal, complemented
4#21458849C214yesmaximal, complemented
5#21458564C214yesmaximal, complemented
6#21458563C214yesmaximal, complemented
7#21456821C214yesmaximal
8#21456767C211yesmaximal, characteristic, complemented
9#21455944C214yesmaximal, complemented
10#21455828C211yesmaximal, characteristic
11#21455823C211yesmaximal, characteristic
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3209C211yesmaximal, characteristic, complemented
14Q8×Dic122C2212yescomplemented
15#16840492C2212yescomplemented
16#16840490—22no—
17#16840490C2212yescomplemented
18#16840489—22no—
19#16840487C2212yescomplemented
20#16840486—22no—
21#16840486—22no—
22#16840486C2214yes—
23#16840486C2214yes—
24#16840391—22no—
25#16840390—21no—
26#16840319C2211yescharacteristic
27#16840314C2214yes—
28C2×#14519772—22no—
29C2×#14519766C2212yes—
30C2×#14519757—21no—
31#16840239—21no—
32C2×Q8×Dic61C2212yes—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244C2211yescharacteristic
35C2×#14519735—21no—
36#16840155C2218yes—
37#16840141C2212yescomplemented
38#16840123C2218yes—
39#16840112C2212yescomplemented
40#16840093C2212yes—
41#16840090C2212yes—
42#16840053C2212yes—
43#16840049C2212yes—
44#16840045C2218yescomplemented
45#16839987C2218yes—
46#16839986C2218yes—
47#16839970C2218yescomplemented
48#16839969C2214yescomplemented
49#16839968C2212yes—
50#16839964C2212yes—
51#16839918C2214yes—
52#16839915C2212yes—
53#16839912C2212yes—
54#16839892C2214yescomplemented
55#16839889C2214yescomplemented
56#16839869C2218yescomplemented
57#16839866C2218yescomplemented
58C4×#11758190C2214yes—
59C4×#11758201C2214yes—
60#16839816C2214yes—
61#16839815C2211yescharacteristic
62C61×#533C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#494C2212yescomplemented
65C61×#487C2212yescomplemented
66C61×#462C2214yescomplemented
67C61×#451C2214yes—
68C61×#431C2211yescharacteristic
69C22×Dic122D411yescharacteristic
70C22×D244C2311yescharacteristic
71C22×C4×D61—21no—
72C2×#11758202C2318yes—
73C2×#11758190C2314yes—
74C2×#11758201C2314yes—
75#14519777—22no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—22no—
79#14519776—24no—
80#14519776C2314yes—
81#14519775—24no—
82#14519775C2314yes—
83#14519772—28no—
84#14519772—28no—
85#14519769—21no—
86#14519769—22no—
87#14519768—22no—
88#14519767—24no—
89#14519767—24no—
90#14519767—24no—
91#14519766—24no—
92#14519766C2318yes—
93#14519757—24no—
94#14519757—24no—
95#14519756—21no—
96#14519756C2312yes—
97#14519754—24no—
98Q8×Dic61—24no—
99Q8×Dic61C2318yes—
100Q8×Dic61C2314yes—
101C4×Dic122—22no—
102C4×Dic122—22no—
103C4×Dic122—21no—
104C4×Dic122D412yes—
105C4×Dic122D412yes—
106C4×Dic122C2312yes—
107C4×Dic122C2312yes—
108C4×Dic122C2312yes—
109#14519742—22no—
110#14519742—22no—
111#14519742—24no—
112#14519742C2314yes—
113C4×D244—21no—
114C4×D244—21no—
115C4×D244C2312yes—
116C4×D244C2312yes—
117#14519736—21no—
118#14519735—24no—
119#14519735—24no—
120#14519733—24no—
121C2×C4⋊Dic61—22no—
122C2×C4⋊Dic61D411yescharacteristic
123C2×C4⋊Dic61C2311yescharacteristic
124C2×#11758196D412yes—
125C2×#11758198—21no—
126C2×#11758198—22no—
127C2×#11758198—22no—
128C2×C4×Dic61—22no—
129C2×C4×Dic61C2312yes—
130#14519709C2314yes—
131#14519708C2314yes—
132C4⋊#92418C2312yes—
133#14519706C2318yes—
134#14519702C2314yes—
135#14519701C2314yes—
136#14519701C2318yes—
137#14519699C2314yes—
138#14519697C2314yes—
139C22⋊#92418C2312yes—
140#14519687C2318yes—
141C4×#92418C2312yes—
142C22×Q8×C61C2312yes—
143C2×Q16×C61C2318yes—
144C4⋊Q8×C61—21no—
145C4⋊Q8×C61—21no—
146C4⋊Q8×C61C2312yes—
147C4⋊Q8×C61C2312yes—
148C4⋊Q8×C61C2311yescharacteristic
149C4⋊Q8×C61C2311yescharacteristic
150#125×C61—21no—
151C22⋊Q8×C61—21no—
152C22⋊Q8×C61C2312yes—
153C4×Q8×C61—21no—
154C4×Q8×C61C2312yes—
155#117×C61—21no—
156C2×C4⋊C4×C61—21no—
157C2×C4⋊C4×C61—21no—
158C2×C42×C61C2311yescharacteristic
159C4⋊C8×C61C2312yes—
160#103×C61C2318yes—
161C22⋊C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C23×D61—21no—
164C2×Q8×C61—22no—
165C2×Q8×C61—21no—
166C2×Q8×C61—44no—
167C2×Q8×C61C2414yes—
168C2×Q8×C61C2412yes—
169C22×C4×C61—22no—
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2411yescharacteristic
173C22×Dic61—21no—
174C22×Dic61C2×D412yes—
175C22×Dic61C2×D412yes—
176C2×D244—21no—
177C2×D244—42no—
178C2×D244C2412yes—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—44no—
182C2×Dic122—24no—
183C2×Dic122—21no—
184C2×Dic122—24no—
185C2×Dic122C2×D412yes—
186C2×Dic122C2×D412yes—
187C2×Dic122C2412yes—
188Q16×C61—48no—
189C2×C8×C61C2414yes—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61C2412yes—
200C22⋊C4×C61—22no—
201C42×C61—22no—
202C42×C61C2×D411yescharacteristic
203C42×C61C2×D411yescharacteristic
204C42×C61C2411yescharacteristic
205C42×C61C2411yescharacteristic
206C22⋊Dic61—22no—
207#11758202—48no—
208#11758201—44no—
209#11758198—22no—
210#11758198—22no—
211#11758198—28no—
212#11758198—28no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—28no—
218C4⋊Dic61—24no—
219C4⋊Dic61C2×D412yes—
220C4⋊Dic61C2×D412yes—
221C4⋊Dic61C2412yes—
222C4⋊Dic61C2412yes—
223#11758196—22no—
224#11758196—22no—
225#11758196—24no—
226#11758196—24no—
227#11758196—24no—
228#11758196—24no—
229#11758196C2×D414yes—
230#11758196C2×D414yes—
231C4×Dic61—22no—
232C4×Dic61—28no—
233C4×Dic61—24no—
234C4×Dic61—24no—
235C4×Dic61C2×D414yes—
236C4×Dic61C2414yes—
237C2×#92418C2414yes—
238#11758190—44no—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241C22×D61—42no—
242C22×D61—42no—
243C22×D61—42no—
244Q8×C61—24no—
245Q8×C61—28no—
246Q8×C61—44no—
247Q8×C61—81no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—24no—
252C2×C4×C61—41no—
253C2×C4×C61—44no—
254C2×C4×C61#13714yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2511yescharacteristic
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61C22×D414yes—
268C2×Dic61C22×D414yes—
269D244—22no—
270D244—22no—
271D244—42no—
272C4×D61—44no—
273C4×D61—44no—
274Dic122—24no—
275Dic122—24no—
276Dic122—24no—
277Dic122—84no—
278Dic122—81no—
279C8×C61—24no—
280#92418—24no—
281#61889—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61C2×#13712yes—
285C22×C61C23×D411yescharacteristic
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—84no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—42no—
294C4×C61—44no—
295C4×C61—42no—
296C4×C61C2×#13712yes—
297C4×C61C2×#13712yes—
298C4×C61#58512yes—
299Dic61—24no—
300Dic61—24no—
301Dic61—42no—
302Dic61—42no—
303Dic61—42no—
304#3502—611no—
305C2×#556—611no—
306#3398—614no—
307#3398—614no—
308#3369—614no—
309#3369—614no—
310#3215—614no—
311#3209—611no—
312#3209D6111yescharacteristic, complemented
313#3113—614no—
314#3101—611no—
315#3101—611no—
316#2983—611no—
317C2×C61—22no—
318C2×C61#348711yescharacteristic
319C2×C61C22×#13711yescharacteristic
320C2×C61#358611yescharacteristic
321D61—82no—
322Q82—612no—
323Q82—612no—
324#557—612no—
325#557—612no—
326#557—1222no—
327#557—1222no—
328#556—614no—
329#556—614no—
330#556—1222no—
331#556—1222no—
332#544—1221no—
333#544—1222no—
334#533—614no—
335#533—611no—
336#533C2×D6111yescharacteristic
337C2×C4⋊Q8C2×D6111yescharacteristic
338C2×#126—1222no—
339C2×#124—612no—
340C2×C22⋊Q8—1221no—
341#519—1221no—
342C2×C4×Q8—611no—
343C2×C4×Q8—612no—
344C2×C4×D4—611no—
345C2×#117—1221no—
346#497—618no—
347#494—612no—
348#494C2×D6112yescomplemented
349#489—618no—
350#487—612no—
351#487C2×D6112yescomplemented
352#483—612no—
353#483—612no—
354#476—612no—
355#476—612no—
356#475—618no—
357#464—618no—
358#464—618no—
359#462—618no—
360#462—614no—
361#462C2×D6114yescomplemented
362#461—612no—
363#461—612no—
364#451—614no—
365#451C2×D6114yes—
366#450—612no—
367#450—612no—
368#446—614no—
369#446—614no—
370#441—618no—
371#441—618no—
372C4×SD16—614no—
373C4×SD16—614no—
374#432—614no—
375#431—611no—
376#431C2×D6111yescharacteristic
377C61#6188911yescharacteristic, complemented, Sylow
378C22×Q8—611no—
379C22×Q8C22×D6112yes—
380C22×D4—611no—
381C23×C4—1221no—
382C2×Q16—618no—
383C2×Q16C22×D6118yes—
384C2×SD16—614no—
385C2×SD16—614no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—614no—
389C4⋊Q8—614no—
390C4⋊Q8—611no—
391C4⋊Q8—611no—
392C4⋊Q8—1222no—
393C4⋊Q8—1224no—
394C4⋊Q8—1222no—
395C4⋊Q8—1224no—
396C4⋊Q8C22×D6112yes—
397C4⋊Q8C22×D6112yes—
398C4⋊Q8C22×D6111yescharacteristic
399C4⋊Q8C22×D6111yescharacteristic
400#126—1228no—
401#126—1228no—
402#125—21no—
403#125—1222no—
404#125—1224no—
405#125—1224no—
406#125—1224no—
407#125—1222no—
408#125—1221no—
409#124—618no—
410#124—1224no—
411C22⋊Q8—21no—
412C22⋊Q8—612no—
413C22⋊Q8—1224no—
414C22⋊Q8—1224no—
415C22⋊Q8—1224no—
416C22⋊Q8—1221no—
417C22⋊Q8C22×D6112yes—
418C4×Q8—21no—
419C4×Q8—612no—
420C4×Q8—612no—
421C4×Q8—612no—
422C4×Q8—612no—
423C4×Q8—618no—
424C4×Q8—614no—
425C4×Q8—614no—
426C4×Q8—612no—
427C4×Q8—1222no—
428C4×Q8—1222no—
429C4×Q8—1224no—
430C4×Q8—1224no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×Q8—1221no—
434C4×Q8C22×D6112yes—
435C4×D4—612no—
436C4×D4—612no—
437C4×D4—1221no—
438C4×D4—1221no—
439#117—21no—
440#117—1224no—
441#117—1224no—
442#117—1224no—
443#117—1221no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—612no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—1222no—
450C2×C22⋊C4—1222no—
451C2×C22⋊C4—1222no—
452C2×C22⋊C4—1221no—
453C2×C42—612no—
454C2×C42—1222no—
455C2×C42C22×D6111yescharacteristic
456#106—614no—
457#106—614no—
458C4⋊C8—618no—
459C4⋊C8—612no—
460C4⋊C8C22×D6112yes—
461#103—614no—
462#103—618no—
463#103—614no—
464#103C22×D6118yes—
465#102—614no—
466#102—614no—
467C22⋊C8—612no—
468C22⋊C8C22×D6112yes—
469#97—618no—
470C4×C8—612no—
471C4×C8C22×D6112yes—
472C24—1221no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—44no—
476C2×Q8—612no—
477C2×Q8—612no—
478C2×Q8—612no—
479C2×Q8—1224no—
480C2×Q8—1224no—
481C2×Q8—1221no—
482C2×Q8C23×D6114yes—
483C2×Q8C23×D6112yes—
484C2×D4—612no—
485C2×D4—1221no—
486C2×D4—2442no—
487C22×C4—22no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—2444no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497Q16—48no—
498Q16—2448no—
499SD16—2444no—
500SD16—2444no—
501C2×C8—614no—
502C2×C8C23×D6114yes—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—614no—
513C4⋊C4—614no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—1228no—
519C4⋊C4—1224no—
520C4⋊C4—1224no—
521C4⋊C4—1224no—
522C4⋊C4—1224no—
523C4⋊C4—1224no—
524C4⋊C4—1221no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4C23×D6112yes—
531C22⋊C4—22no—
532C22⋊C4—1228no—
533C22⋊C4—1228no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C42—22no—
538C42—614no—
539C42—614no—
540C42—1228no—
541C42—1224no—
542C42—1224no—
543C42—1222no—
544C42D4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42C23×D6111yescharacteristic
547C42C23×D6111yescharacteristic
548C23—1222no—
549C23—2442no—
550C23—2442no—
551C23—2442no—
552C23C2×D4×D6111yescharacteristic
553Q8—24no—
554Q8—28no—
555Q8—44no—
556Q8—81no—
557Q8—1224no—
558Q8—1224no—
559Q8—1224no—
560Q8—4881no—
561Q8—4884no—
562D4—1222no—
563D4—1222no—
564D4—2442no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—614no—
572C2×C4—614no—
573C2×C4—1224no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2444no—
580C2×C4—2444no—
581C2×C4#1451980914yes—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C2×D4×D6111yescharacteristic
586C2×C4C2×D4×D6111yescharacteristic
587C2×C4C24×D6111yescharacteristic
588C8—24no—
589C8—1224no—
590C22—21no—
591C22—21no—
592C22—4884no—
593C22—2442no—
594C22—2442no—
595C22—2442no—
596C22C2×#1451980912yes—
597C22C22×D4×D6111yescharacteristic
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—42no—
603C4—42no—
604C4—1224no—
605C4—1224no—
606C4—2442no—
607C4—2442no—
608C4—2442no—
609C4C2×#1451980912yes—
610C4C2×#1451980912yes—
611C4#1684059712yes—
612C2—22no—
613C2—4882no—
614C2#2145947511yescharacteristic
615C2C22×#1451980911yescharacteristic
616C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3209C2
socleC22×C61C23×D4
prime radicalC22×D244C23
hypercenter#3209D61
hypocenterC61#61889
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. #21459624C2semidirect
3. #21458855C2semidirect
4. #21458849C2semidirect
5. #21458564C2semidirect
6. #21458563C2semidirect
8. #21456767C2semidirect
9. #21455944C2semidirect
12. #21454902C2semidirect
13. C61×#3209C2semidirect
14. Q8×Dic122C22semidirect
15. #16840492C22semidirect
17. #16840490C22semidirect
19. #16840487C22semidirect
37. #16840141C22semidirect
39. #16840112C22semidirect
44. #16840045C22semidirect
47. #16839970C22semidirect
48. #16839969C22semidirect
54. #16839892C22semidirect
55. #16839889C22semidirect
56. #16839869C22semidirect
57. #16839866C22semidirect
64. C61×#494C22semidirect
65. C61×#487C22semidirect
66. C61×#462C22semidirect
312. #3209D61semidirect
348. #494C2×D61semidirect
351. #487C2×D61semidirect
361. #462C2×D61semidirect
377. C61#61889semidirect

Maximal subgroups

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

subgroupindex
#214596242
C2×#168404862
#214588552
#214588492
#214585642
#214585632
#214568212
#214567672
#214559442
#214558282
#214558232
#214549022
C61×#32092
#6188961