Finite Group Databasebeta

← #30611236

Subgroups — #30611236

Order 15616.

1042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462498C211yesmaximal, characteristic, complemented
2C2×#16840707C211yesmaximal, characteristic, complemented
3#21461263C214yesmaximal, complemented
4#21461243C212yesmaximal, complemented
5#21461240C212yesmaximal, complemented
6#21461206C214yesmaximal, complemented
7#21459116C211yesmaximal, characteristic
8#16840785C2212yes—
9#16840782—21no—
10#16840780C2212yes—
11#16840779—21no—
12#16840767—21no—
13#16840733C2212yes—
14C4×#11758213—21no—
15#16840707—21no—
16#16840707—21no—
17#16840707C2212yes—
18#16840707C2212yes—
19C2×(C2×#6923)⋊C4—21no—
20C2×(C2×#6923)⋊C4—21no—
21C2×(C2×#6923)⋊C4C2212yes—
22#16840616C2218yes—
23#16840614C2218yes—
24#16840317C411yescharacteristic, complemented
25#16840317C2211yescharacteristic
26#16840313C411yescharacteristic, complemented
27#16840313C2211yescharacteristic
28#16840312C411yescharacteristic, complemented
29#16840299—21no—
30#16840299C2212yes—
31#16840298—21no—
32#16840298C412yescomplemented
33#16840297—21no—
34#16840297—21no—
35#16840275—21no—
36#16840275C2212yes—
37#16840274—21no—
38C4×#11758226—21no—
39C4×#11758226C412yescomplemented
40#16840230—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C23×#6923—22no—
44C2×#11758213—22no—
45#14519844—22no—
46#14519844—22no—
47#14519834C2314yes—
48C22⋊#92420—22no—
49#14519832—22no—
50C4×#92420—22no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4C2314yes—
57C2×#11758226—21no—
58C2×#11758226C2×C412yes—
59C2×#11758222—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63#14519777C2×C411yescharacteristic, complemented
64#14519774C2×C411yescharacteristic, complemented
65#14519772—22no—
66#14519772—22no—
67#14519768—21no—
68#14519768C2×C412yescomplemented
69#14519766—21no—
70#14519766C2×C412yescomplemented
71#14519765—21no—
72#14519762—22no—
73#14519762—22no—
74#14519761—22no—
75#14519757—21no—
76#14519757—21no—
77#14519753—21no—
78#14519753C2312yes—
79#14519752—21no—
80#14519752—21no—
81#14519752—21no—
82#14519752C2312yes—
83#14519751—21no—
84#14519751C2×C412yes—
85#14519747—22no—
86Q8×Dic61—21no—
87Q8×Dic61C2×C412yescomplemented
88#14519742—21no—
89#14519742—21no—
90C4×D244—22no—
91C4×D244—22no—
92C4×D244—21no—
93C4×D244C2×C412yescomplemented
94#14519739—22no—
95#14519739—22no—
96#14519739—21no—
97#14519739—21no—
98#14519739—21no—
99#14519739C2×C412yescomplemented
100#14519737—22no—
101#14519737—22no—
102#14519735—21no—
103#14519735—22no—
104#14519735—22no—
105#14519735C2×C412yes—
106#14519734—22no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊Q8×C61C2×C411yescharacteristic, complemented
120C4⋊Q8×C61C2×C411yescharacteristic, complemented
121#125×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61C2×C412yes—
124C4×Q8×C61—21no—
125C4×Q8×C61C2×C412yescomplemented
126#117×C61—21no—
127C2×C42×C61C2×C411yescharacteristic
128C23×D61C2×D411yescharacteristic
129C22×#6923—24no—
130C22×#6923—44no—
131C22×#6923—44no—
132C22×#6923—44no—
133C2×Q8×C61—21no—
134C2×Q8×C61C22×C412yes—
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C2×C22⋊D61—22no—
139C22×Dic61C22⋊C411yescharacteristic
140C22×Dic61C22⋊C411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142#11758226—44no—
143#11758226—44no—
144#11758222—44no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244C22×C412yes—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2411yescharacteristic
168C2×Dic122—21no—
169C2×#92420—24no—
170#11758213—44no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—22no—
178C42×C61—22no—
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22×C411yescharacteristic
182C42×C61C22×C411yescharacteristic
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61C22×C412yes—
195#11758196—22no—
196#11758196—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22×C411yescharacteristic
206C4×Dic61C22×C411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C23×C61C2×C22⋊C411yescharacteristic
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61C2×C22⋊C411yescharacteristic
217C22×D61C22×D411yescharacteristic
218C2×#6923—44no—
219C2×#6923—44no—
220C2×#6923—44no—
221C2×#6923—88no—
222Q8×C61—42no—
223Q8×C61—42no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C22⋊D61—44no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C22×D411yescharacteristic
246D244—42no—
247D244—42no—
248D244—42no—
249D244—44no—
250D244—44no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—44no—
264C4×D61—44no—
265C4×D61—44no—
266Dic122—42no—
267#92420—44no—
268#17593—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61#42112yes—
272C22×C61C22×C22⋊C411yescharacteristic
273C2×D61—28no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61#41012yes—
282C2×D61#57512yes—
283#6923—84no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—42no—
291C4×C61—42no—
292Dic61—21no—
293Dic61—21no—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—42no—
299#3444—611no—
300#2920—611no—
301C2×#383—611no—
302#1521—614no—
303#1515—612no—
304#1514—612no—
305#1504—614no—
306C2×C61—22no—
307C2×C61C2×#41011yescharacteristic
308C2×C61#289611yescharacteristic
309C2×C61C2×#42111yescharacteristic
310D61—22no—
311D61—22no—
312D61—82no—
313#533—611no—
314#533—611no—
315#533#692311yescharacteristic, complemented
316#532—611no—
317#532—611no—
318#532—611no—
319#529—612no—
320#529—612no—
321#529—1221no—
322#529—1221no—
323#529—1221no—
324#529—1221no—
325#525—612no—
326#525—1221no—
327#525—1221no—
328C4×#41—612no—
329C4×#41—1221no—
330C4×#41—1221no—
331C22×C42—611no—
332#432—612no—
333#432—1221no—
334#431—612no—
335#431—1221no—
336#423—1221no—
337#407—612no—
338C4×M16—1221no—
339#383—612no—
340#383—612no—
341#383—1221no—
342#383—1221no—
343C2×#95—612no—
344C2×#95—1221no—
345C2×#95—1221no—
346#328—618no—
347#327—618no—
348C61#1759311yescharacteristic, complemented, Sylow
349C2×#41—612no—
350C2×#41—1221no—
351C2×#41—1221no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1222no—
356C2×M16—1222no—
357C4⋊Q8—611no—
358C4⋊Q8C2×#692311yescharacteristic, complemented
359C4⋊Q8C2×#692311yescharacteristic, complemented
360#127—611no—
361#126—1222no—
362#126—1222no—
363#125—21no—
364#125—612no—
365#125—1221no—
366#124—612no—
367#124—1221no—
368#124—1221no—
369#123—1222no—
370#123—1222no—
371#123—1222no—
372C22⋊Q8—21no—
373C22⋊Q8—612no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377C22⋊Q8C2×#692312yes—
378#121—612no—
379#121—612no—
380#121—1221no—
381#121—1221no—
382#121—1221no—
383#121—1221no—
384#120—1222no—
385C4×Q8—21no—
386C4×Q8—612no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390C4×Q8C2×#692312yescomplemented
391C4×D4—612no—
392C4×D4—612no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1221no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1221no—
403#117—21no—
404#117—612no—
405#117—1222no—
406#117—1222no—
407#117—1222no—
408#117—1221no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—1222no—
421C2×C42C2×#692311yescharacteristic
422C4⋊C8—1222no—
423C4⋊C8—1222no—
424C22⋊C8—614no—
425C22⋊C8—1222no—
426#97—1222no—
427C4×C8—1222no—
428#95—614no—
429#95—1222no—
430#95—1222no—
431#95—1222no—
432#95—1222no—
433#95—1222no—
434C24—611no—
435#41—2444no—
436#41—2444no—
437#41—2444no—
438C2×Q8—21no—
439C2×Q8—1221no—
440C2×Q8C22×#692312yes—
441C2×D4—612no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1221no—
446C2×D4—1221no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1224no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2444no—
470C22×C4—2444no—
471C22×C4—2444no—
472C22×C4C22⋊#692311yescharacteristic
473C22×C4C22⋊#692311yescharacteristic
474C22×C4C22×#692311yescharacteristic
475M16—2444no—
476C2×C8—1224no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—612no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4C22×#692312yes—
489C22⋊C4—22no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C42—22no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—1222no—
512C42—1222no—
513C42C22⋊#692311yescharacteristic
514C42C22⋊#692311yescharacteristic
515C42C22×#692311yescharacteristic
516C42C22×#692311yescharacteristic
517C23—611no—
518C23—611no—
519C23—1222no—
520C23—1222no—
521C23—1222no—
522C23—1222no—
523C23C2×C22⋊#692311yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—2442no—
527D4—2442no—
528D4—2442no—
529D4—2442no—
530D4—2444no—
531D4—2444no—
532D4—2444no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—4888no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2444no—
560C2×C4—2444no—
561C2×C4—2444no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2444no—
565C2×C4—2444no—
566C2×C4—2444no—
567C2×C4C2×C22⋊#692311yescharacteristic
568C2×C4C2×C22⋊#692311yescharacteristic
569C2×C4C2×C22⋊#692311yescharacteristic
570C2×C4C2×C22⋊#692311yescharacteristic
571C2×C4C2×C22⋊#692311yescharacteristic
572C2×C4C23×#692311yescharacteristic
573C8—2444no—
574C22—21no—
575C22—21no—
576C22—612no—
577C22—612no—
578C22—1228no—
579C22—1221no—
580C22—1221no—
581C22—1221no—
582C22—1221no—
583C22—2442no—
584C22—2442no—
585C22—2442no—
586C22#1684076112yes—
587C22C22×C22⋊#692311yescharacteristic
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—42no—
596C4—1221no—
597C4—1221no—
598C4—1221no—
599C4—1221no—
600C4—1221no—
601C4—1221no—
602C4—4884no—
603C4—2442no—
604C2—22no—
605C2—1222no—
606C2—1222no—
607C2—4882no—
608C2D61⋊#41011yescharacteristic
609C2#2146243911yescharacteristic
610C2C2×#1684076111yescharacteristic
subgroupG/H
centerC22C22×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#17593
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #21462498C2semidirect
2. C2×#16840707C2semidirect
3. #21461263C2semidirect
4. #21461243C2semidirect
5. #21461240C2semidirect
6. #21461206C2semidirect
24. #16840317C4semidirect
26. #16840313C4semidirect
28. #16840312C4semidirect
32. #16840298C4semidirect
39. C4×#11758226C4semidirect
42. C61×#533C4semidirect
63. #14519777C2×C4semidirect
64. #14519774C2×C4semidirect
68. #14519768C2×C4semidirect
70. #14519766C2×C4semidirect
87. Q8×Dic61C2×C4semidirect
93. C4×D244C2×C4semidirect
99. #14519739C2×C4semidirect
119. C4⋊Q8×C61C2×C4semidirect
120. C4⋊Q8×C61C2×C4semidirect
125. C4×Q8×C61C2×C4semidirect
315. #533#6923semidirect
348. C61#17593semidirect
358. C4⋊Q8C2×#6923semidirect
359. C4⋊Q8C2×#6923semidirect
390. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624982
C2×#168407072
#214612632
#214612432
#214612402
#214612062
#214591162
#1759361