Finite Group Databasebeta

← #30616724

Subgroups — #30616724

Order 15616.

1430 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#21462496C211yesmaximal, characteristic, complemented
2C2×C42×#6923C211yesmaximal, characteristic, complemented
3#21462004C212yesmaximal, complemented
4#21461952C212yesmaximal, complemented
5#21461836C214yesmaximal, complemented
6C4×#14519842C214yesmaximal, complemented
7#21459116C211yesmaximal, characteristic
8C22×C4×#6923—21no—
9C22×C4×#6923—21no—
10C22×C4×#6923C2212yes—
11#16840784—21no—
12#16840783—21no—
13#16840781—21no—
14#16840778—21no—
15#16840766C2212yes—
16C2×#14519842C2218yes—
17#16840733C2212yes—
18C4×#11758213C2212yes—
19C42×#6923—21no—
20C42×#6923—21no—
21C42×#6923C2212yes—
22C42×#6923C2212yes—
23#16840632C2218yes—
24#16840317C411yescharacteristic, complemented
25#16840317C2211yescharacteristic
26#16840313C411yescharacteristic, complemented
27#16840313C2211yescharacteristic
28#16840312C411yescharacteristic, complemented
29#16840299—21no—
30#16840299C412yescomplemented
31#16840298—21no—
32#16840298C412yescomplemented
33#16840297—21no—
34#16840297—21no—
35#16840275—21no—
36#16840275C2212yes—
37#16840274—21no—
38C4×#11758226—21no—
39C4×#11758226C2212yes—
40#16840230—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C23×#6923—22no—
44C2×#11758213C2314yes—
45C2×C4×#6923—28no—
46C2×C4×#6923—28no—
47C2×C4×#6923—22no—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50C2×C4×#6923—22no—
51C2×C4×#6923—22no—
52C2×C4×#6923—28no—
53C2×C4×#6923C2314yes—
54C4⋊#92420—22no—
55C4⋊#92420—22no—
56#14519842—216no—
57#14519834—22no—
58C22⋊#92420—22no—
59#14519832—22no—
60C4×#92420—22no—
61C2×#11758226—21no—
62C2×#11758226C2312yes—
63C2×#11758222—21no—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61C2311yescharacteristic
67#14519777C2×C411yescharacteristic, complemented
68#14519774C2×C411yescharacteristic, complemented
69#14519772—22no—
70#14519772—22no—
71#14519768—21no—
72#14519768C2×C412yescomplemented
73#14519766—21no—
74#14519766C2×C412yescomplemented
75#14519765—21no—
76#14519762—22no—
77#14519762—22no—
78#14519761—22no—
79#14519757—21no—
80#14519757—21no—
81#14519753—21no—
82#14519753C2×C412yes—
83#14519752—21no—
84#14519752—21no—
85#14519752—21no—
86#14519752C2×C412yes—
87#14519751—21no—
88#14519751C2×C412yes—
89#14519747—22no—
90Q8×Dic61—21no—
91Q8×Dic61C2×C412yescomplemented
92#14519742—21no—
93#14519742—21no—
94C4×D244—22no—
95C4×D244—22no—
96C4×D244—21no—
97C4×D244C2×C412yescomplemented
98#14519739—22no—
99#14519739—22no—
100#14519739—21no—
101#14519739—21no—
102#14519739—21no—
103#14519739C2×C412yescomplemented
104#14519737—22no—
105#14519737—22no—
106#14519735—21no—
107#14519735—22no—
108#14519735—22no—
109#14519735C2312yes—
110#14519734—22no—
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C42×D61—22no—
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61C2311yescharacteristic
122C42×D61C2311yescharacteristic
123C4⋊Q8×C61C2×C411yescharacteristic, complemented
124C4⋊Q8×C61C2×C411yescharacteristic, complemented
125#125×C61—21no—
126C22⋊Q8×C61—21no—
127C22⋊Q8×C61C2×C412yes—
128C4×Q8×C61—21no—
129C4×Q8×C61C2×C412yescomplemented
130#117×C61—21no—
131C2×C42×C61C2×C411yescharacteristic
132C23×D61C2×D411yescharacteristic
133C22×#6923—28no—
134C22×#6923—28no—
135C22×#6923—28no—
136C22×#6923—24no—
137C2×Q8×C61—21no—
138C2×Q8×C61C22×C412yes—
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C2×C22⋊D61—22no—
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C22⋊C411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146#11758226—28no—
147#11758226—44no—
148#11758222—44no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×C4×D61—24no—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22×C411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2411yescharacteristic
172C2×Dic122—21no—
173C2×#92420—24no—
174C4×#6923—28no—
175C4×#6923—28no—
176C4×#6923—28no—
177C4×#6923—28no—
178C4×#6923—28no—
179C4×#6923—24no—
180C4×#6923#4118yescomplemented
181#11758213—28no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61C22×C412yes—
188C22⋊C4×C61—22no—
189C42×C61—22no—
190C42×C61C22⋊C411yescharacteristic
191C42×C61C22⋊C411yescharacteristic
192C42×C61C22×C411yescharacteristic
193C42×C61C22×C411yescharacteristic
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C412yes—
206#11758196—22no—
207#11758196—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C22×C411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C23×C61C2×C22⋊C411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61C2×C22⋊C411yescharacteristic
228C22×D61C22×D411yescharacteristic
229C2×#6923—28no—
230C2×#6923—28no—
231C2×#6923—232no—
232C2×#6923—28no—
233Q8×C61—24no—
234Q8×C61—42no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61#10414yescomplemented
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C23×C411yescharacteristic
246C22⋊D61—44no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C22×D411yescharacteristic
257D244—24no—
258D244—44no—
259D244—44no—
260D244—42no—
261D244—42no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—44no—
273C4×D61—44no—
274C4×D61—44no—
275C4×D61#10414yescomplemented
276C4×D61C2×#4114yes—
277Dic122—42no—
278#92420—44no—
279#21415—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61C2×#10412yes—
283C22×C61C22×C22⋊C411yescharacteristic
284C2×D61—28no—
285C2×D61—21no—
286C2×D61—21no—
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61C2×#10412yes—
293C2×D61#52512yes—
294#6923—216no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61C2×#10412yes—
300C4×C61C2×#10412yes—
301C4×C61C2×#10412yes—
302C4×C61C4×#4112yes—
303Dic61—21no—
304Dic61—21no—
305Dic61—21no—
306Dic61—21no—
307Dic61—42no—
308Dic61C2×#10412yes—
309Dic61#53212yes—
310#3444—611no—
311#2920—611no—
312C2×C43—611no—
313#1994—612no—
314#1962—612no—
315#1896—614no—
316C4×#104—614no—
317C2×C61—22no—
318C2×C61#229711yescharacteristic
319C2×C61C22×#10411yescharacteristic
320C2×C61C22×#10411yescharacteristic
321D61—22no—
322D61—22no—
323D61—82no—
324#533—611no—
325#533—611no—
326#533#692311yescharacteristic, complemented
327#532—611no—
328#532—611no—
329#532—611no—
330#529—612no—
331#529—612no—
332#529—1221no—
333#529—1221no—
334#529—1221no—
335#529—1221no—
336#525—612no—
337#525—1221no—
338#525—1221no—
339C4×#41—612no—
340C4×#41—1221no—
341C4×#41—1221no—
342C22×C42—611no—
343C22×C42—612no—
344C22×C42—1221no—
345C22×C42—1221no—
346#432—1221no—
347#432—1221no—
348#431—1221no—
349#431—1221no—
350#423—612no—
351C2×#104—618no—
352#407—612no—
353C4×M16—612no—
354C43—612no—
355C43—612no—
356C43—1221no—
357C43—1221no—
358#339—618no—
359C61#2141511yescharacteristic, complemented, Sylow
360C2×#41—612no—
361C2×#41—1221no—
362C2×#41—1221no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—1222no—
367C2×M16—614no—
368C4⋊Q8—611no—
369C4⋊Q8C2×#692311yescharacteristic, complemented
370C4⋊Q8C2×#692311yescharacteristic, complemented
371#127—611no—
372#126—1222no—
373#126—1222no—
374#125—21no—
375#125—612no—
376#125—1221no—
377#124—612no—
378#124—1221no—
379#124—1221no—
380#123—1222no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—21no—
384C22⋊Q8—612no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8C2×#692312yes—
389#121—612no—
390#121—612no—
391#121—1221no—
392#121—1221no—
393#121—1221no—
394#121—1221no—
395#120—1222no—
396C4×Q8—21no—
397C4×Q8—612no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8C2×#692312yescomplemented
402C4×D4—612no—
403C4×D4—612no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1221no—
414#117—21no—
415#117—612no—
416#117—1222no—
417#117—1222no—
418#117—1222no—
419#117—1221no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—614no—
432C2×C42—1222no—
433C2×C42—1228no—
434C2×C42—1222no—
435C2×C42—1222no—
436C2×C42—1222no—
437C2×C42—1222no—
438C2×C42—1222no—
439C2×C42—1228no—
440C2×C42—1228no—
441C2×C42C2×#692311yescharacteristic
442C4⋊C8—1222no—
443C4⋊C8—1222no—
444#104—12216no—
445C22⋊C8—1222no—
446C22⋊C8—1222no—
447#97—1222no—
448C4×C8—1222no—
449C24—611no—
450#41—1228no—
451#41—2444no—
452#41—2444no—
453C2×Q8—21no—
454C2×Q8—1221no—
455C2×Q8C22×#692312yes—
456C2×D4—612no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1221no—
461C2×D4—1221no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1224no—
482C22×C4—1228no—
483C22×C4—1228no—
484C22×C4—1228no—
485C22×C4—1224no—
486C22×C4—1222no—
487C22×C4C22⋊#692311yescharacteristic
488C22×C4C22⋊#692311yescharacteristic
489C22×C4C22×#692311yescharacteristic
490M16—1228no—
491C2×C8—1224no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—612no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4C22×#692312yes—
504C22⋊C4—22no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C42—22no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—618no—
527C42—1222no—
528C42—1222no—
529C42—1224no—
530C42—1228no—
531C42—1228no—
532C42—1228no—
533C42—1228no—
534C42—1228no—
535C42C22⋊#692311yescharacteristic
536C42C22⋊#692311yescharacteristic
537C42C22×#692311yescharacteristic
538C42C22×#692311yescharacteristic
539C23—611no—
540C23—611no—
541C23—1222no—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23C2×C22⋊#692311yescharacteristic
546Q8—24no—
547Q8—42no—
548Q8—2442no—
549D4—1224no—
550D4—2444no—
551D4—2444no—
552D4—2444no—
553D4—2442no—
554D4—2442no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—611no—
560C2×C4—611no—
561C2×C4—614no—
562C2×C4—614no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1228no—
574C2×C4—1228no—
575C2×C4—1228no—
576C2×C4—12232no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—2444no—
586C2×C4—2444no—
587C2×C4—2444no—
588C2×C4#1451984214yescomplemented
589C2×C4C2×C22⋊#692311yescharacteristic
590C2×C4C2×C22⋊#692311yescharacteristic
591C2×C4C2×C22⋊#692311yescharacteristic
592C2×C4C2×C22⋊#692311yescharacteristic
593C2×C4C2×C22⋊#692311yescharacteristic
594C2×C4C23×#692311yescharacteristic
595C8—2444no—
596C22—21no—
597C22—21no—
598C22—612no—
599C22—612no—
600C22—1228no—
601C22—1221no—
602C22—1221no—
603C22—1221no—
604C22—1221no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22C2×#1451984212yes—
609C22C22×C22⋊#692311yescharacteristic
610C4—21no—
611C4—21no—
612C4—42no—
613C4—42no—
614C4—612no—
615C4—612no—
616C4—12216no—
617C4—1221no—
618C4—1221no—
619C4—1221no—
620C4—1221no—
621C4—2442no—
622C4D61⋊#10412yes—
623C4C2×#1451984212yes—
624C4C2×#1451984212yes—
625C4#41×#692312yes—
626C2—22no—
627C2—1222no—
628C2—1222no—
629C2—4882no—
630C2#2146228011yescharacteristic
631C2C2×D61⋊#10411yescharacteristic
632C2C22×#1451984211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#21415
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #21462496C2semidirect
2. C2×C42×#6923C2semidirect
3. #21462004C2semidirect
4. #21461952C2semidirect
5. #21461836C2semidirect
6. C4×#14519842C2semidirect
24. #16840317C4semidirect
26. #16840313C4semidirect
28. #16840312C4semidirect
30. #16840299C4semidirect
32. #16840298C4semidirect
42. C61×#533C4semidirect
67. #14519777C2×C4semidirect
68. #14519774C2×C4semidirect
72. #14519768C2×C4semidirect
74. #14519766C2×C4semidirect
91. Q8×Dic61C2×C4semidirect
97. C4×D244C2×C4semidirect
103. #14519739C2×C4semidirect
123. C4⋊Q8×C61C2×C4semidirect
124. C4⋊Q8×C61C2×C4semidirect
129. C4×Q8×C61C2×C4semidirect
180. C4×#6923#41semidirect
239. C2×C4×C61#104semidirect
275. C4×D61#104semidirect
326. #533#6923semidirect
359. C61#21415semidirect
369. C4⋊Q8C2×#6923semidirect
370. C4⋊Q8C2×#6923semidirect
401. C4×Q8C2×#6923semidirect
588. C2×C4#14519842semidirect

Maximal subgroups

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

subgroupindex
#214624962
C2×C42×#69232
#214620042
#214619522
#214618362
C4×#145198422
#214591162
#2141561