Finite Group Databasebeta

← #30616725

Subgroups — #30616725

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#21461953C212yesmaximal, complemented
5#21461837C214yesmaximal, complemented
6C4×#14519841C214yesmaximal, complemented
7#21459114C211yesmaximal, 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×#14519841C2218yes—
17#16840733C2212yes—
18C4×#11758213C2212yes—
19C42×#6923—21no—
20C42×#6923—21no—
21C42×#6923C2212yes—
22C42×#6923C2212yes—
23#16840632C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840317C411yescharacteristic, complemented
26#16840317C2211yescharacteristic
27#16840310C411yescharacteristic, complemented
28#16840310C2211yescharacteristic
29#16840294—21no—
30#16840294C412yescomplemented
31#16840293—21no—
32#16840293C412yescomplemented
33#16840291—21no—
34#16840291—21no—
35#16840272—21no—
36#16840272C2212yes—
37#16840269—21no—
38#41×Dic61—21no—
39C4×#11758224—21no—
40C4×#11758224C2212yes—
41C2×C42×D61C2211yescharacteristic
42C61×#532C411yescharacteristic, 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#14519841—216no—
57#14519834—22no—
58C22⋊#92420—22no—
59#14519832—22no—
60C4×#92420—22no—
61C2×#11758224—21no—
62C2×#11758224C2312yes—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66#14519777C2×C411yescharacteristic, complemented
67#14519777C2×C411yescharacteristic, complemented
68#14519770—22no—
69#14519770—22no—
70#14519769—21no—
71#14519768—21no—
72#14519768C2×C412yescomplemented
73#14519764—21no—
74#14519764C2×C412yescomplemented
75#14519759—22no—
76#14519759—22no—
77#14519758—22no—
78#14519757—21no—
79#14519757C2×C412yes—
80#14519756—21no—
81#14519756C2×C412yes—
82#14519753—21no—
83#14519753—21no—
84#14519748—21no—
85#14519748—21no—
86#14519748—21no—
87#14519748C2×C412yes—
88#14519745—22no—
89Q8×Dic61—21no—
90Q8×Dic61—21no—
91C4×Dic122—21no—
92C4×Dic122C2×C412yescomplemented
93#14519742—21no—
94#14519742C2×C412yescomplemented
95D4×Dic61—22no—
96D4×Dic61—22no—
97D4×Dic61—21no—
98D4×Dic61—21no—
99D4×Dic61—21no—
100D4×Dic61C2×C412yescomplemented
101C4×C22⋊D61—22no—
102C4×C22⋊D61—22no—
103#14519737—22no—
104#14519737—22no—
105#14519736—21no—
106#14519736—22no—
107#14519735—21no—
108#14519735C2312yes—
109#14519733—22no—
110#14519733—22no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61C2×C411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61C2311yescharacteristic
122C42×D61C2311yescharacteristic
123C2×#41×C61—21no—
124C4⋊Q8×C61C2×C411yescharacteristic, complemented
125#127×C61C2×C411yescharacteristic, complemented
126#124×C61—21no—
127#121×C61—21no—
128#121×C61C2×C412yes—
129C4×D4×C61—21no—
130C4×D4×C61C2×C412yescomplemented
131C2×C42×C61C2×C411yescharacteristic
132C23×D61C2×D411yescharacteristic
133C22×#6923—28no—
134C22×#6923—28no—
135C22×#6923—28no—
136C22×#6923—24no—
137#41×C61—44no—
138C2×Q8×C61—21no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C22×C412yes—
143C22×C4×C61—22no—
144C22×C4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22⋊C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61C22⋊C411yescharacteristic
152C22×Dic61C22⋊C411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154#11758224—28no—
155#11758224—44no—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2411yescharacteristic
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
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×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C42×C61C22⋊C411yescharacteristic
187C42×C61C22⋊C411yescharacteristic
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195#11758198—22no—
196#11758198—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61—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×C61—22no—
223C23×C61C2×C22⋊C411yescharacteristic
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—42no—
234D4×C61—24no—
235D4×C61—44no—
236D4×C61—42no—
237D4×C61—42no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—44no—
242C2×C4×C61#10414yescomplemented
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C23×C411yescharacteristic
249C22⋊D61—44no—
250C22⋊D61—44no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C22×D411yescharacteristic
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—22no—
273C4×D61—22no—
274C4×D61#10414yescomplemented
275C4×D61C2×#4114yes—
276Dic122—24no—
277Dic122—42no—
278#92420—44no—
279#21415—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61C2×#10412yes—
286C22×C61C22×C22⋊C411yescharacteristic
287C2×D61—28no—
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61C2×#10412yes—
293C2×D61#52512yes—
294#6923—216no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—42no—
298C4×C61C2×#10412yes—
299C4×C61C2×#10412yes—
300C4×C61C2×#10412yes—
301C4×C61C4×#4112yes—
302Dic61—21no—
303Dic61—21no—
304Dic61—21no—
305Dic61—21no—
306Dic61—42no—
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—82no—
319C2×C61#229711yescharacteristic
320C2×C61C22×#10411yescharacteristic
321C2×C61C22×#10411yescharacteristic
322D61—22no—
323D61—22no—
324#533—611no—
325#533—611no—
326#533—611no—
327#532—611no—
328#532—611no—
329#532#692311yescharacteristic, complemented
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—21no—
361C2×#41—612no—
362C2×#41—1221no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—1222no—
367C2×M16—614no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370C4⋊Q8C2×#692311yescharacteristic, complemented
371#127C2×#692311yescharacteristic, complemented
372#126—1222no—
373#126—1222no—
374#125—612no—
375#125—1221no—
376#125—1221no—
377#124—21no—
378#124—612no—
379#124—1221no—
380#123—1222no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—612no—
384C22⋊Q8—612no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389#121—21no—
390#121—612no—
391#121—1221no—
392#121—1221no—
393#121—1221no—
394#121C2×#692312yes—
395#120—1222no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×D4—21no—
403C4×D4—612no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4C2×#692312yescomplemented
414#117—612no—
415#117—1222no—
416#117—1222no—
417#117—1222no—
418#117—1221no—
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—44no—
451#41—1228no—
452#41—2444no—
453C2×Q8—21no—
454C2×Q8—612no—
455C2×Q8—1221no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4C22×#692312yes—
462C22×C4—22no—
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—611no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1224no—
483C22×C4—1228no—
484C22×C4—1228no—
485C22×C4—1228no—
486C22×C4—1224no—
487C22×C4C22⋊#692311yescharacteristic
488C22×C4C22⋊#692311yescharacteristic
489C22×C4C22×#692311yescharacteristic
490M16—1228no—
491C2×C8—1224no—
492C4⋊C4—21no—
493C4⋊C4—612no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4C22×#692312yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—618no—
526C42—1222no—
527C42—1222no—
528C42—1224no—
529C42—1228no—
530C42—1228no—
531C42—1228no—
532C42—1228no—
533C42—1228no—
534C42—1222no—
535C42C22⋊#692311yescharacteristic
536C42C22⋊#692311yescharacteristic
537C42C22×#692311yescharacteristic
538C42C22×#692311yescharacteristic
539C23—22no—
540C23—611no—
541C23—611no—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23C2×C22⋊#692311yescharacteristic
546Q8—42no—
547Q8—1224no—
548Q8—2442no—
549D4—24no—
550D4—44no—
551D4—42no—
552D4—42no—
553D4—2444no—
554D4—2444no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—44no—
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—1222no—
586C2×C4—2444no—
587C2×C4—2444no—
588C2×C4#1451984114yescomplemented
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—42no—
599C22—42no—
600C22—42no—
601C22—612no—
602C22—612no—
603C22—1228no—
604C22—1221no—
605C22—1221no—
606C22—1221no—
607C22—1221no—
608C22C2×#1451984112yes—
609C22C22×C22⋊#692311yescharacteristic
610C4—21no—
611C4—21no—
612C4—42no—
613C4—612no—
614C4—612no—
615C4—12216no—
616C4—1221no—
617C4—1221no—
618C4—1221no—
619C4—1221no—
620C4—2442no—
621C4—2442no—
622C4C2×#1451984112yes—
623C4C2×#1451984112yes—
624C4D61⋊#10412yes—
625C4#41×#692312yes—
626C2—22no—
627C2—82no—
628C2—1222no—
629C2—1222no—
630C2#2146228011yescharacteristic
631C2C22×#1451984111yescharacteristic
632C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#532C4
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC2×C4×D61C24
hypercenter#532#6923
hypocenterC61#21415
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #21462496C2semidirect
2. C2×C42×#6923C2semidirect
3. #21462004C2semidirect
4. #21461953C2semidirect
5. #21461837C2semidirect
6. C4×#14519841C2semidirect
24. #16840319C4semidirect
25. #16840317C4semidirect
27. #16840310C4semidirect
30. #16840294C4semidirect
32. #16840293C4semidirect
42. C61×#532C4semidirect
66. #14519777C2×C4semidirect
67. #14519777C2×C4semidirect
72. #14519768C2×C4semidirect
74. #14519764C2×C4semidirect
92. C4×Dic122C2×C4semidirect
94. #14519742C2×C4semidirect
100. D4×Dic61C2×C4semidirect
124. C4⋊Q8×C61C2×C4semidirect
125. #127×C61C2×C4semidirect
130. C4×D4×C61C2×C4semidirect
180. C4×#6923#41semidirect
242. C2×C4×C61#104semidirect
274. C4×D61#104semidirect
329. #532#6923semidirect
359. C61#21415semidirect
370. C4⋊Q8C2×#6923semidirect
371. #127C2×#6923semidirect
413. C4×D4C2×#6923semidirect
588. C2×C4#14519841semidirect

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
#214619532
#214618372
C4×#145198412
#214591142
#2141561