Finite Group Databasebeta

← #30571068

Subgroups — #30571068

Order 15616.

1814 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459686C211yesmaximal, characteristic, complemented
2C2×#16840497C211yesmaximal, characteristic, complemented
3#21458829C214yesmaximal, complemented
4#21458814C214yesmaximal, complemented
5#21458513C214yesmaximal, complemented
6#21458478C214yesmaximal, complemented
7#21456789C214yesmaximal, complemented
8#21456757C211yesmaximal, characteristic, complemented
9#21456739C211yesmaximal, characteristic
10#21455869C214yesmaximal, complemented
11#21455850C211yesmaximal, characteristic, complemented
12#21454899C211yesmaximal, characteristic, complemented
13C61×#3104C211yesmaximal, characteristic, complemented
14#16840497—22no—
15#16840497—22no—
16#16840497C2214yes—
17#16840497C2214yes—
18#16840497C2212yescomplemented
19#16840491C2212yescomplemented
20#16840486C2212yescomplemented
21#16840485—22no—
22#16840473C2212yescomplemented
23#16840450—22no—
24#16840361—22no—
25#16840338—21no—
26#16840312C2211yescharacteristic
27C2×#14519774C2211yescharacteristic
28#16840298C2214yes—
29C2×#14519766C2212yes—
30C2×#14519765—21no—
31C2×#14519762—22no—
32#16840242—21no—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×D244—21no—
35C2×#14519739C2212yes—
36#16840147C2218yescomplemented
37#16840139C2212yes—
38#16840136C2212yes—
39#16840118C2218yescomplemented
40#16840110C2212yes—
41#16840109C2212yes—
42#16840080C2212yes—
43#16840065C2218yescomplemented
44#16840047C2212yescomplemented
45#16839986C2218yescomplemented
46#16839980C2218yescomplemented
47#16839980C2218yescomplemented
48#16839962C2214yes—
49#16839962C2212yescomplemented
50#16839958C2214yes—
51#16839912C2214yes—
52#16839901C2212yes—
53#16839898C2214yes—
54C8×Dic122C2214yescomplemented
55#16839886C2214yescomplemented
56C4×#11758201C2214yescomplemented
57#16839847C2218yescomplemented
58#16839844C2218yescomplemented
59#16839815C2211yescharacteristic
60#16839812C2214yes—
61C61×#533C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C61×#484C2212yes—
64C61×#477C2212yescomplemented
65C61×#462C2212yescomplemented
66C61×#451C2212yes—
67C4×Q16×C61C2214yescomplemented
68C61×#431C2211yescharacteristic
69C2×#11758222—21no—
70C22×Dic122D411yescharacteristic
71C22×D244—21no—
72C22×D244—21no—
73C22×D244C2312yes—
74C22×C4×D61—22no—
75C2×#11758201C2318yes—
76C2×#11758201C2314yes—
77C2×D488C2318yes—
78#14519777—22no—
79#14519777C2311yescharacteristic
80#14519774—21no—
81#14519774—21no—
82#14519774C2312yes—
83#14519774C2312yes—
84#14519774C2311yescharacteristic
85#14519772—24no—
86#14519772—24no—
87#14519768—24no—
88#14519768C2314yes—
89#14519766—24no—
90#14519766C2314yes—
91#14519766C2318yes—
92#14519765—22no—
93#14519765—24no—
94#14519765—24no—
95#14519765—21no—
96#14519762—24no—
97#14519762—28no—
98#14519762—28no—
99#14519761—24no—
100#14519757—22no—
101#14519757—22no—
102#14519753—24no—
103#14519751—21no—
104#14519751C2312yes—
105Q8×Dic61C2314yes—
106C4×Dic122—22no—
107C4×Dic122D412yes—
108C4×Dic122D412yes—
109C4×Dic122C2312yes—
110C4×Dic122C2312yes—
111C4×D244—21no—
112C4×D244—24no—
113C4×D244—24no—
114C4×D244—22no—
115C4×D244C2312yes—
116#14519739—24no—
117#14519739C2314yes—
118#14519739C2318yes—
119#14519737—24no—
120#14519735—22no—
121#14519735—24no—
122C2×C4⋊Dic61D411yescharacteristic
123C2×#11758196D412yes—
124C4⋊C4×D61—22no—
125C2×#11758198—22no—
126C2×#11758198—22no—
127C2×#11758198—22no—
128C2×C4×Dic61C2312yes—
129#14519708C2314yes—
130C4⋊#92418C2312yes—
131#14519706C2318yes—
132#14519701C2314yes—
133#14519699C2318yes—
134#14519697C2318yes—
135#14519697C2314yes—
136C22⋊#92418C2312yes—
137C4×#92418C2312yes—
138C8×Dic61C2318yes—
139C22×Q8×C61C2311yescharacteristic
140C2×Q16×C61C2314yes—
141C4⋊Q8×C61C2311yescharacteristic
142C4⋊Q8×C61C2311yescharacteristic
143#125×C61—21no—
144C22⋊Q8×C61—21no—
145C22⋊Q8×C61C2312yes—
146C4×Q8×C61—21no—
147C4×Q8×C61—21no—
148C4×Q8×C61—21no—
149C4×Q8×C61C2312yes—
150C4×Q8×C61C2312yes—
151C4×Q8×C61C2312yes—
152#117×C61—21no—
153C2×C4⋊C4×C61—21no—
154C2×C4⋊C4×C61C2311yescharacteristic
155C2×C42×C61—21no—
156C2×C42×C61C2311yescharacteristic
157#107×C61C2314yes—
158C4⋊C8×C61C2312yes—
159#103×C61C2314yes—
160#103×C61C2314yes—
161C22⋊C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C23×D61—22no—
164C2×Q8×C61—21no—
165C2×Q8×C61—21no—
166C2×Q8×C61—42no—
167C2×Q8×C61C2412yes—
168C2×Q8×C61C2412yes—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2411yescharacteristic
174C2×C22⋊D61—22no—
175C22×Dic61C2×D412yes—
176C22×Dic61C2×D412yes—
177#11758222—44no—
178C2×D244—22no—
179C2×D244—22no—
180C2×D244—22no—
181C2×D244—21no—
182C2×D244—22no—
183C2×D244—22no—
184C2×D244—21no—
185C2×D244—44no—
186C2×D244—44no—
187C2×D244—82no—
188C2×D244C2414yes—
189C2×D244C2412yes—
190C2×C4×D61—22no—
191C2×C4×D61—28no—
192C2×C4×D61—24no—
193C2×C4×D61—84no—
194C2×Dic122—24no—
195C2×Dic122—21no—
196C2×Dic122C2×D412yes—
197C2×Dic122C2×D412yes—
198Q16×C61—44no—
199C2×C8×C61C2414yes—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61—21no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61—22no—
208C4⋊C4×C61C2412yes—
209C4⋊C4×C61C2412yes—
210C22⋊C4×C61—22no—
211C42×C61—22no—
212C42×C61—22no—
213C42×C61—22no—
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C42×C61C2411yescharacteristic
217C42×C61C2411yescharacteristic
218#11758201—44no—
219#11758201—48no—
220#11758198—22no—
221#11758198—22no—
222#11758198—28no—
223#11758198—28no—
224#11758198—28no—
225#11758198—24no—
226#11758198—24no—
227#11758198—24no—
228C4⋊Dic61—21no—
229C4⋊Dic61—24no—
230C4⋊Dic61—24no—
231C4⋊Dic61C2×D412yes—
232C4⋊Dic61C2×D412yes—
233C4⋊Dic61C2412yes—
234#11758196—24no—
235#11758196C2×D414yes—
236#11758196C2×D414yes—
237C4×Dic61—24no—
238C4×Dic61C2×D414yes—
239C4×Dic61C2414yes—
240C2×#92418C2414yes—
241D488—48no—
242C23×C61C22×D411yescharacteristic
243C22×D61—22no—
244C22×D61—24no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61—44no—
248Q8×C61—24no—
249Q8×C61—22no—
250Q8×C61—22no—
251Q8×C61—42no—
252Q8×C61—81no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—41no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61#13514yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2511yescharacteristic
268C22⋊D61—44no—
269C2×Dic61—24no—
270C2×Dic61—22no—
271C2×Dic61—42no—
272C2×Dic61C22×D414yes—
273C2×Dic61C22×D414yes—
274D244—24no—
275D244—28no—
276D244—44no—
277D244—44no—
278D244—44no—
279D244—42no—
280D244—81no—
281D244—82no—
282D244—82no—
283C4×D61—44no—
284C4×D61—84no—
285C4×D61—88no—
286Dic122—24no—
287Dic122—24no—
288Dic122—42no—
289C8×C61—24no—
290#92418—24no—
291#61885—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61C2×#13512yes—
295C22×C61C23×D411yescharacteristic
296C2×D61—44no—
297C2×D61—42no—
298C2×D61—44no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—44no—
302C2×D61—88no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61C2×#13512yes—
311C4×C61C2×#13512yes—
312C4×C61#58512yes—
313Dic61—24no—
314Dic61—24no—
315Dic61—42no—
316#3507—611no—
317C2×#559—611no—
318#3396—614no—
319#3394—614no—
320#3365—614no—
321#3361—614no—
322#3211—614no—
323#3208—611no—
324#3205—611no—
325#3105—614no—
326#3104D6111yescharacteristic, complemented
327#3103—611no—
328#2982—611no—
329C2×C61—22no—
330C2×C61#348711yescharacteristic
331C2×C61C22×#13511yescharacteristic
332C2×C61#358511yescharacteristic
333D61—82no—
334D61—84no—
335#559—614no—
336#559—614no—
337#559—612no—
338#559—1222no—
339#559—1222no—
340#557—612no—
341#556—612no—
342#556—1222no—
343#554—612no—
344#552—1222no—
345#539—1222no—
346#536—1221no—
347#533C2×D6111yescharacteristic
348#532—611no—
349C2×#127—611no—
350#529—614no—
351C2×#124—612no—
352C2×#124—1221no—
353C2×#123—1222no—
354#520—1221no—
355C2×C4×Q8—611no—
356C2×C4×Q8C2×D6111yescharacteristic
357C2×C4×D4—612no—
358C2×C4×D4—1221no—
359#495—618no—
360#493—612no—
361#492—612no—
362#488—618no—
363#486—612no—
364#486—612no—
365#484C2×D6112yes—
366#481—612no—
367#478—618no—
368#477C2×D6112yescomplemented
369#475—612no—
370#464—618no—
371#463—618no—
372#463—618no—
373#462C2×D6112yescomplemented
374#460—614no—
375#460—612no—
376#459—614no—
377#451C2×D6112yes—
378#450—614no—
379#448—612no—
380#447—614no—
381C8×Q8—614no—
382C8×Q8—614no—
383C4×Q16C2×D6114yescomplemented
384C4×SD16—614no—
385C4×SD16—618no—
386C4×D8—618no—
387#431—611no—
388#431—614no—
389#431C2×D6111yescharacteristic
390C61#6188511yescharacteristic, complemented, Sylow
391C2×#41—1221no—
392C22×Q8—611no—
393C22×Q8C22×D6111yescharacteristic
394C22×D4—612no—
395C22×D4—1221no—
396C22×D4—1221no—
397C23×C4—1222no—
398C2×Q16C22×D6114yes—
399C2×SD16—614no—
400C2×SD16—618no—
401C2×D8—618no—
402C4⋊Q8—611no—
403C4⋊Q8—1222no—
404C4⋊Q8C22×D6111yescharacteristic
405C4⋊Q8C22×D6111yescharacteristic
406#127—612no—
407#127—612no—
408#127—611no—
409#127—1221no—
410#127—1221no—
411#126—1224no—
412#126—1224no—
413#125—21no—
414#125—614no—
415#125—1224no—
416#124—618no—
417#124—614no—
418#124—1224no—
419#124—1224no—
420#124—1222no—
421#124—1224no—
422#124—1221no—
423#123—1224no—
424#123—1228no—
425#123—1228no—
426#123—1224no—
427C22⋊Q8—21no—
428C22⋊Q8—1222no—
429C22⋊Q8—1222no—
430C22⋊Q8—1224no—
431C22⋊Q8C22×D6112yes—
432#121—612no—
433#121—1221no—
434C4×Q8—21no—
435C4×Q8—21no—
436C4×Q8—21no—
437C4×Q8—612no—
438C4×Q8—612no—
439C4×Q8—612no—
440C4×Q8—612no—
441C4×Q8—614no—
442C4×Q8—1222no—
443C4×Q8C22×D6112yes—
444C4×Q8C22×D6112yes—
445C4×Q8C22×D6112yes—
446C4×D4—618no—
447C4×D4—614no—
448C4×D4—612no—
449C4×D4—1224no—
450C4×D4—1224no—
451C4×D4—1222no—
452C4×D4—1224no—
453C4×D4—1224no—
454C4×D4—1221no—
455#117—21no—
456#117—1224no—
457#117—1222no—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—611no—
460C2×C4⋊C4—612no—
461C2×C4⋊C4—1222no—
462C2×C4⋊C4C22×D6111yescharacteristic
463C2×C22⋊C4—1222no—
464C2×C22⋊C4—1222no—
465C2×C22⋊C4—1222no—
466C2×C42—21no—
467C2×C42—612no—
468C2×C42C22×D6111yescharacteristic
469#107C22×D6114yes—
470#106—614no—
471C4⋊C8—618no—
472C4⋊C8—612no—
473C4⋊C8C22×D6112yes—
474#103—614no—
475#103C22×D6114yes—
476#103C22×D6114yes—
477#102—618no—
478#102—614no—
479#102—618no—
480C22⋊C8—612no—
481C22⋊C8C22×D6112yes—
482C4×C8—618no—
483C4×C8—612no—
484C4×C8C22×D6112yes—
485C24—1222no—
486#41—2444no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—42no—
490C2×Q8—612no—
491C2×Q8—612no—
492C2×Q8—1221no—
493C2×Q8—1224no—
494C2×Q8C23×D6112yes—
495C2×Q8C23×D6112yes—
496C2×D4—614no—
497C2×D4—612no—
498C2×D4—1222no—
499C2×D4—1221no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1221no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—1222no—
506C2×D4—4882no—
507C2×D4—2444no—
508C2×D4—2444no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—612no—
512C22×C4—612no—
513C22×C4—1224no—
514C22×C4—1228no—
515C22×C4—1222no—
516C22×C4—4884no—
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520Q16—44no—
521SD16—2444no—
522SD16—2448no—
523D8—2448no—
524C2×C8—614no—
525C2×C8C23×D6114yes—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—612no—
535C4⋊C4—614no—
536C4⋊C4—614no—
537C4⋊C4—612no—
538C4⋊C4—612no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4C23×D6112yes—
544C4⋊C4C23×D6112yes—
545C22⋊C4—22no—
546C22⋊C4—1228no—
547C22⋊C4—1228no—
548C22⋊C4—1228no—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—614no—
558C42—614no—
559C42—1224no—
560C42D4×D6111yescharacteristic
561C42D4×D6111yescharacteristic
562C42C23×D6111yescharacteristic
563C42C23×D6111yescharacteristic
564C23—1222no—
565C23—1224no—
566C23—2444no—
567C23—2444no—
568C23—2444no—
569C23C2×D4×D6111yescharacteristic
570Q8—24no—
571Q8—22no—
572Q8—22no—
573Q8—42no—
574Q8—81no—
575Q8—1224no—
576Q8—1224no—
577Q8—2442no—
578D4—1224no—
579D4—1228no—
580D4—4881no—
581D4—4882no—
582D4—4882no—
583D4—2444no—
584D4—2444no—
585D4—2444no—
586D4—2444no—
587D4—2442no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—614no—
597C2×C4—614no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—4884no—
601C2×C4—4888no—
602C2×C4—2442no—
603C2×C4—2444no—
604C2×C4#1451980014yes—
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C24×D6111yescharacteristic
611C8—24no—
612C8—1224no—
613C22—21no—
614C22—21no—
615C22—4888no—
616C22—2444no—
617C22—2442no—
618C22—2442no—
619C22—2444no—
620C22—2444no—
621C22—2442no—
622C22C2×#1451980012yes—
623C22C22×D4×D6111yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—1224no—
632C4—1224no—
633C4—2442no—
634C4C2×#1451980012yes—
635C4C2×#1451980012yes—
636C4#1684059812yes—
637C2—22no—
638C2—4884no—
639C2—4882no—
640C2#2145947311yescharacteristic
641C2C22×#1451980011yescharacteristic
642C2#2146059611yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3104C2
socleC22×C61C23×D4
prime radical#21456739C2
hypercenter#3104D61
hypocenterC61#61885
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
1. #21459686C2semidirect
2. C2×#16840497C2semidirect
3. #21458829C2semidirect
4. #21458814C2semidirect
5. #21458513C2semidirect
6. #21458478C2semidirect
7. #21456789C2semidirect
8. #21456757C2semidirect
10. #21455869C2semidirect
11. #21455850C2semidirect
12. #21454899C2semidirect
13. C61×#3104C2semidirect
18. #16840497C22semidirect
19. #16840491C22semidirect
20. #16840486C22semidirect
22. #16840473C22semidirect
36. #16840147C22semidirect
39. #16840118C22semidirect
43. #16840065C22semidirect
44. #16840047C22semidirect
45. #16839986C22semidirect
46. #16839980C22semidirect
47. #16839980C22semidirect
49. #16839962C22semidirect
54. C8×Dic122C22semidirect
55. #16839886C22semidirect
56. C4×#11758201C22semidirect
57. #16839847C22semidirect
58. #16839844C22semidirect
64. C61×#477C22semidirect
65. C61×#462C22semidirect
67. C4×Q16×C61C22semidirect
326. #3104D61semidirect
368. #477C2×D61semidirect
373. #462C2×D61semidirect
383. C4×Q16C2×D61semidirect
390. C61#61885semidirect

Maximal subgroups

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

subgroupindex
#214596862
C2×#168404972
#214588292
#214588142
#214585132
#214584782
#214567892
#214567572
#214567392
#214558692
#214558502
#214548992
C61×#31042
#6188561