Finite Group Databasebeta

← #30571037

Subgroups — #30571037

Order 15616.

1750 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459681C211yesmaximal, characteristic, complemented
2C2×#16840486C211yesmaximal, characteristic, complemented
3#21458820C214yesmaximal, complemented
4#21458806C214yesmaximal, complemented
5#21458509C214yesmaximal, complemented
6#21458472C214yesmaximal, complemented
7#21456783C214yesmaximal, complemented
8#21456767C211yesmaximal, characteristic, complemented
9#21455894C214yesmaximal, complemented
10#21455828C211yesmaximal, characteristic, complemented
11#21455815C211yesmaximal, characteristic
12#21454899C211yesmaximal, characteristic, complemented
13C61×#3206C211yesmaximal, characteristic, complemented
14#16840498C2212yescomplemented
15#16840487C2212yescomplemented
16#16840486—22no—
17#16840486—22no—
18#16840486C2214yes—
19#16840486C2214yes—
20#16840482C2212yescomplemented
21#16840481—22no—
22#16840469C2212yescomplemented
23#16840452—22no—
24#16840357—22no—
25#16840333—21no—
26#16840319C2211yescharacteristic
27#16840293C2214yes—
28C2×#14519772—22no—
29C2×#14519766C2212yes—
30C2×#14519757—21no—
31#16840245—21no—
32C2×Q8×Dic61C2212yes—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244C2211yescharacteristic
35C2×#14519735—21no—
36#16840143C2218yes—
37#16840141C2212yescomplemented
38#16840118C2218yes—
39#16840112C2212yescomplemented
40#16840093C2212yes—
41#16840073C2212yes—
42#16840067C2218yescomplemented
43#16840053C2212yes—
44#16840039C2212yes—
45#16839987C2218yes—
46#16839982C2218yescomplemented
47#16839979C2218yes—
48#16839969C2214yescomplemented
49#16839968C2212yes—
50#16839954C2212yes—
51#16839918C2214yes—
52#16839915C2212yes—
53#16839895C2212yes—
54C8×Dic122C2214yescomplemented
55#16839886C2214yescomplemented
56#16839854C2218yescomplemented
57C4×#11758190C2214yes—
58SD16×Dic61C2218yescomplemented
59C4×#11758199C2214yes—
60#16839815C2211yescharacteristic
61#16839812C2214yes—
62C61×#532C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#492C2212yescomplemented
65C61×#487C2212yescomplemented
66C61×#461C2214yescomplemented
67C61×#448C2214yes—
68C61×#431C2211yescharacteristic
69C22×Dic122D411yescharacteristic
70C22×D244C2311yescharacteristic
71C22×C4×D61—21no—
72C2×#11758202C2318yes—
73C2×#11758190C2314yes—
74C2×#11758199C2314yes—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519775—22no—
78#14519772—28no—
79#14519772—28no—
80#14519770—24no—
81#14519770—24no—
82#14519769—21no—
83#14519768C2314yes—
84#14519767—24no—
85#14519766—24no—
86#14519766C2318yes—
87#14519764—22no—
88#14519764C2314yes—
89#14519758—24no—
90#14519758—24no—
91#14519757—24no—
92#14519757—24no—
93#14519756—21no—
94#14519756—22no—
95#14519756—22no—
96#14519756C2312yes—
97#14519754—24no—
98Q8×Dic61—22no—
99Q8×Dic61—24no—
100Q8×Dic61C2318yes—
101C4×Dic122—21no—
102C4×Dic122D412yes—
103C4×Dic122D412yes—
104C4×Dic122C2312yes—
105C4×Dic122C2312yes—
106C4×Dic122C2312yes—
107#14519742C2314yes—
108C4×D244—21no—
109C4×D244—21no—
110C4×D244C2312yes—
111C4×D244C2312yes—
112D4×Dic61—24no—
113D4×Dic61—22no—
114D4×Dic61C2314yes—
115#14519736—21no—
116#14519736—22no—
117#14519735—24no—
118#14519735—24no—
119#14519733—24no—
120C2×C4⋊Dic61—22no—
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#14519710C2314yes—
131#14519709C2314yes—
132C4⋊#92418C2312yes—
133#14519706C2318yes—
134#14519702C2314yes—
135#14519701C2318yes—
136#14519699C2314yes—
137#14519698C2314yes—
138#14519697C2314yes—
139C22⋊#92418C2312yes—
140C4×#92418C2312yes—
141C8×Dic61C2318yes—
142C2×#41×C61—21no—
143C22×Q8×C61C2312yes—
144C2×SD16×C61C2318yes—
145C4⋊Q8×C61—21no—
146C4⋊Q8×C61—21no—
147C4⋊Q8×C61C2312yes—
148C4⋊Q8×C61C2312yes—
149C4⋊Q8×C61C2311yescharacteristic
150#127×C61C2311yescharacteristic
151#124×C61—21no—
152#121×C61—21no—
153#121×C61C2312yes—
154C4×D4×C61—21no—
155C4×D4×C61C2312yes—
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—
164#41×C61—44no—
165C2×Q8×C61—21no—
166C2×Q8×C61—22no—
167C2×Q8×C61—44no—
168C2×Q8×C61C2414yes—
169C2×D4×C61—22no—
170C2×D4×C61—21no—
171C2×D4×C61—21no—
172C2×D4×C61C2412yes—
173C22×C4×C61—22no—
174C22×C4×C61—22no—
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2411yescharacteristic
178C22×Dic61—21no—
179C22×Dic61—24no—
180C22×Dic61C2×D412yes—
181C22×Dic61C2×D412yes—
182C2×D244—21no—
183C2×D244—42no—
184C2×D244C2412yes—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—44no—
188C2×Dic122—24no—
189C2×Dic122—21no—
190C2×Dic122C2×D412yes—
191C2×Dic122C2×D412yes—
192C2×Dic122C2412yes—
193SD16×C61—48no—
194C2×C8×C61C2414yes—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61C2412yes—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C42×C61C2×D411yescharacteristic
204C42×C61C2×D411yescharacteristic
205C42×C61C2411yescharacteristic
206C42×C61C2411yescharacteristic
207C22⋊Dic61—22no—
208C22⋊Dic61—24no—
209C22⋊Dic61—24no—
210C22⋊Dic61—24no—
211#11758202—48no—
212#11758199—44no—
213#11758198—22no—
214#11758198—22no—
215#11758198—28no—
216#11758198—28no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—24no—
222C4⋊Dic61—28no—
223C4⋊Dic61C2×D412yes—
224C4⋊Dic61C2×D412yes—
225C4⋊Dic61C2412yes—
226C4⋊Dic61C2412yes—
227#11758196—22no—
228#11758196—22no—
229#11758196—24no—
230#11758196—24no—
231#11758196C2×D414yes—
232#11758196C2×D414yes—
233C4×Dic61—22no—
234C4×Dic61—28no—
235C4×Dic61—24no—
236C4×Dic61C2×D414yes—
237C4×Dic61C2414yes—
238C2×#92418C2414yes—
239#11758190—44no—
240C23×C61—22no—
241C23×C61C22×D411yescharacteristic
242C22×D61—22no—
243C22×D61—42no—
244C22×D61—42no—
245C22×D61—42no—
246Q8×C61—28no—
247Q8×C61—44no—
248Q8×C61—42no—
249D4×C61—24no—
250D4×C61—44no—
251D4×C61—42no—
252D4×C61—81no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—24no—
256C2×C4×C61—41no—
257C2×C4×C61—44no—
258C2×C4×C61—44no—
259C2×C4×C61#13514yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2511yescharacteristic
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—84no—
273C2×Dic61C22×D414yes—
274C2×Dic61C22×D414yes—
275D244—22no—
276D244—22no—
277D244—42no—
278C4×D61—44no—
279C4×D61—44no—
280Dic122—24no—
281Dic122—24no—
282Dic122—24no—
283Dic122—81no—
284C8×C61—24no—
285#92418—24no—
286#61882—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61C2×#13512yes—
293C22×C61C23×D411yescharacteristic
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—84no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—44no—
302C4×C61—42no—
303C4×C61C2×#13512yes—
304C4×C61C2×#13512yes—
305C4×C61#58512yes—
306Dic61—24no—
307Dic61—24no—
308Dic61—42no—
309Dic61—42no—
310Dic61—42no—
311#3507—611no—
312C2×#556—611no—
313#3394—614no—
314#3393—614no—
315#3364—614no—
316#3360—614no—
317#3211—614no—
318#3209—611no—
319#3206D6111yescharacteristic, complemented
320#3108—614no—
321#3101—611no—
322#3100—611no—
323#2982—611no—
324C2×C61—22no—
325C2×C61—82no—
326C2×C61#348711yescharacteristic
327C2×C61C22×#13511yescharacteristic
328C2×C61#358611yescharacteristic
329D61—82no—
330#559—612no—
331#557—612no—
332#556—614no—
333#556—614no—
334#556—612no—
335#556—1222no—
336#556—1222no—
337#556—1222no—
338#554—612no—
339#552—1222no—
340#539—1222no—
341#536—1221no—
342#533—611no—
343#532C2×D6111yescharacteristic
344C2×C4⋊Q8C2×D6111yescharacteristic
345#529—614no—
346C2×#126—1222no—
347C2×#124—612no—
348C2×C22⋊Q8—1221no—
349#520—1221no—
350C2×C4×Q8—611no—
351C2×C4×Q8—612no—
352C2×C4×D4—611no—
353C2×#117—1221no—
354#495—618no—
355#494—612no—
356#492C2×D6112yescomplemented
357#488—618no—
358#487—612no—
359#487C2×D6112yescomplemented
360#483—612no—
361#480—612no—
362#479—618no—
363#476—612no—
364#474—612no—
365#464—618no—
366#464—618no—
367#463—618no—
368#462—614no—
369#461—612no—
370#461C2×D6114yescomplemented
371#459—612no—
372#451—614no—
373#450—612no—
374#448C2×D6114yes—
375#447—612no—
376C8×Q8—614no—
377C8×Q8—614no—
378C4×Q16—618no—
379C4×SD16—618no—
380C4×SD16—614no—
381C4×D8—614no—
382#431—611no—
383#431—614no—
384#431C2×D6111yescharacteristic
385C61#6188211yescharacteristic, complemented, Sylow
386C2×#41—21no—
387C22×Q8—611no—
388C22×Q8C22×D6112yes—
389C22×D4—611no—
390C23×C4—1221no—
391C2×Q16—618no—
392C2×SD16—614no—
393C2×SD16C22×D6118yes—
394C2×D8—614no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—611no—
398C4⋊Q8—611no—
399C4⋊Q8—1222no—
400C4⋊Q8C22×D6112yes—
401C4⋊Q8C22×D6112yes—
402C4⋊Q8C22×D6111yescharacteristic
403#127C22×D6111yescharacteristic
404#126—1224no—
405#126—1224no—
406#126—1228no—
407#126—1228no—
408#125—614no—
409#125—1224no—
410#125—1221no—
411#124—21no—
412#124—618no—
413#124—614no—
414#124—1224no—
415#124—1222no—
416#123—1224no—
417#123—1224no—
418C22⋊Q8—612no—
419C22⋊Q8—1224no—
420C22⋊Q8—1224no—
421C22⋊Q8—1224no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C22⋊Q8—1221no—
425#121—21no—
426#121C22×D6112yes—
427C4×Q8—612no—
428C4×Q8—612no—
429C4×Q8—612no—
430C4×Q8—612no—
431C4×Q8—618no—
432C4×Q8—614no—
433C4×Q8—612no—
434C4×Q8—1224no—
435C4×Q8—1222no—
436C4×Q8—1221no—
437C4×D4—21no—
438C4×D4—614no—
439C4×D4—612no—
440C4×D4—612no—
441C4×D4—1222no—
442C4×D4—1224no—
443C4×D4—1221no—
444C4×D4—1221no—
445C4×D4C22×D6112yes—
446#117—1224no—
447#117—1224no—
448#117—1224no—
449#117—1222no—
450#117—1221no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—1222no—
457C2×C4⋊C4—1222no—
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1221no—
461C2×C42—612no—
462C2×C42—1222no—
463C2×C42C22×D6111yescharacteristic
464#107—614no—
465#106—614no—
466C4⋊C8—618no—
467C4⋊C8—612no—
468C4⋊C8C22×D6112yes—
469#103—618no—
470#103—614no—
471#103C22×D6118yes—
472#102—614no—
473#102—614no—
474#102—614no—
475C22⋊C8—612no—
476C22⋊C8C22×D6112yes—
477C4×C8—618no—
478C4×C8—612no—
479C4×C8C22×D6112yes—
480C24—1221no—
481#41—44no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—44no—
485C2×Q8—612no—
486C2×Q8—612no—
487C2×Q8—612no—
488C2×Q8—1221no—
489C2×Q8—1224no—
490C2×Q8C23×D6114yes—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—612no—
495C2×D4—1221no—
496C2×D4—2442no—
497C2×D4C23×D6112yes—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—1224no—
503C22×C4—1221no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—2444no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—2448no—
511SD16—48no—
512SD16—2444no—
513D8—2444no—
514C2×C8—614no—
515C2×C8C23×D6114yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—612no—
522C4⋊C4—614no—
523C4⋊C4—614no—
524C4⋊C4—612no—
525C4⋊C4—612no—
526C4⋊C4—612no—
527C4⋊C4—1224no—
528C4⋊C4—1228no—
529C4⋊C4—1224no—
530C4⋊C4—1224no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4C23×D6112yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—1228no—
541C22⋊C4—1228no—
542C22⋊C4—1224no—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C42—614no—
549C42—614no—
550C42—1228no—
551C42—1224no—
552C42—1222no—
553C42D4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C42C23×D6111yescharacteristic
556C42C23×D6111yescharacteristic
557C23—22no—
558C23—1222no—
559C23—2442no—
560C23—2442no—
561C23—2442no—
562C23C2×D4×D6111yescharacteristic
563Q8—28no—
564Q8—44no—
565Q8—42no—
566Q8—1224no—
567Q8—1224no—
568Q8—1224no—
569Q8—4881no—
570D4—24no—
571D4—44no—
572D4—42no—
573D4—81no—
574D4—1222no—
575D4—1222no—
576D4—2442no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—614no—
584C2×C4—614no—
585C2×C4—1224no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—4884no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2444no—
593C2×C4—2444no—
594C2×C4#1451979814yes—
595C2×C4C2×D4×D6111yescharacteristic
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C24×D6111yescharacteristic
601C8—24no—
602C8—1224no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—4884no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22C2×#1451979812yes—
613C22C22×D4×D6111yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—44no—
618C4—42no—
619C4—1224no—
620C4—1224no—
621C4—2442no—
622C4—2442no—
623C4—2442no—
624C4C2×#1451979812yes—
625C4C2×#1451979812yes—
626C4#1684059712yes—
627C2—22no—
628C2—82no—
629C2—4882no—
630C2#2145947511yescharacteristic
631C2C22×#1451979811yescharacteristic
632C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3206C2
socleC22×C61C23×D4
prime radical#21455815C2
hypercenter#3206D61
hypocenterC61#61882
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
1. #21459681C2semidirect
2. C2×#16840486C2semidirect
3. #21458820C2semidirect
4. #21458806C2semidirect
5. #21458509C2semidirect
6. #21458472C2semidirect
7. #21456783C2semidirect
8. #21456767C2semidirect
9. #21455894C2semidirect
10. #21455828C2semidirect
12. #21454899C2semidirect
13. C61×#3206C2semidirect
14. #16840498C22semidirect
15. #16840487C22semidirect
20. #16840482C22semidirect
22. #16840469C22semidirect
37. #16840141C22semidirect
39. #16840112C22semidirect
42. #16840067C22semidirect
46. #16839982C22semidirect
48. #16839969C22semidirect
54. C8×Dic122C22semidirect
55. #16839886C22semidirect
56. #16839854C22semidirect
58. SD16×Dic61C22semidirect
64. C61×#492C22semidirect
65. C61×#487C22semidirect
66. C61×#461C22semidirect
319. #3206D61semidirect
356. #492C2×D61semidirect
359. #487C2×D61semidirect
370. #461C2×D61semidirect
385. C61#61882semidirect

Maximal subgroups

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

subgroupindex
#214596812
C2×#168404862
#214588202
#214588062
#214585092
#214584722
#214567832
#214567672
#214558942
#214558282
#214558152
#214548992
C61×#32062
#6188261