Finite Group Databasebeta

← #30616733

Subgroups — #30616733

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462496C211yesmaximal, characteristic, complemented
2C4×#14519856C211yesmaximal, characteristic, complemented
3#21462004C212yesmaximal, complemented
4#21461952C212yesmaximal, complemented
5#21461838C214yesmaximal, complemented
6#21461583C214yesmaximal, complemented
7#21459116C211yesmaximal, characteristic
8C2×#14519856C2212yes—
9#16840784—21no—
10#16840783—21no—
11#16840781—21no—
12#16840778—21no—
13#16840766C2212yes—
14C2×#14519842C2218yes—
15#16840733C2212yes—
16C4×#11758213C2212yes—
17C4×C4⋊#6923—21no—
18C4×C4⋊#6923—21no—
19C4×C22⋊#6923—21no—
20C4×C22⋊#6923—21no—
21#16840698C2212yes—
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
43C2×#11758213C2314yes—
44#14519856—28no—
45C2×C4⋊#6923—22no—
46C2×C22⋊#6923—22no—
47C2×C4×#6923—22no—
48C2×C4×#6923—22no—
49C2×C4×#6923C2314yes—
50C4⋊#92420—22no—
51C4⋊#92420—22no—
52#14519842—216no—
53#14519834—22no—
54C22⋊#92420—22no—
55#14519832—22no—
56C4×#92420—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4—22no—
59C2×#11758226—21no—
60C2×#11758226C2312yes—
61C2×#11758222—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65#14519777C2×C411yescharacteristic
66#14519774C2×C411yescharacteristic
67#14519772—22no—
68#14519772—22no—
69#14519768—21no—
70#14519768C2×C412yes—
71#14519766—21no—
72#14519766C2×C412yes—
73#14519765—21no—
74#14519762—22no—
75#14519762—22no—
76#14519761—22no—
77#14519757—21no—
78#14519757—21no—
79#14519753—21no—
80#14519753C2×C412yes—
81#14519752—21no—
82#14519752—21no—
83#14519752—21no—
84#14519752C2×C412yes—
85#14519751—21no—
86#14519751C2×C412yes—
87#14519747—22no—
88Q8×Dic61—21no—
89Q8×Dic61C2×C412yes—
90#14519742—21no—
91#14519742—21no—
92C4×D244—22no—
93C4×D244—22no—
94C4×D244—21no—
95C4×D244C2×C412yes—
96#14519739—22no—
97#14519739—22no—
98#14519739—21no—
99#14519739—21no—
100#14519739—21no—
101#14519739C2×C412yes—
102#14519737—22no—
103#14519737—22no—
104#14519735—21no—
105#14519735—22no—
106#14519735—22no—
107#14519735C2312yes—
108#14519734—22no—
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C42×D61—22no—
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61C2311yescharacteristic
120C42×D61C2311yescharacteristic
121C4⋊Q8×C61C2×C411yescharacteristic
122C4⋊Q8×C61C2×C411yescharacteristic
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61C2×C412yes—
126C4×Q8×C61—21no—
127C4×Q8×C61C2×C412yes—
128#117×C61—21no—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61C2×D411yescharacteristic
131C22×#6923—24no—
132C2×Q8×C61—21no—
133C2×Q8×C61C22×C412yes—
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C2×C22⋊D61—22no—
138C22×Dic61C22⋊C411yescharacteristic
139C22×Dic61C22⋊C411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141#11758226—28no—
142#11758226—44no—
143#11758222—44no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244C22×C412yes—
149C2×C4×D61—24no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22×C411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2411yescharacteristic
167C2×Dic122—21no—
168C22⋊#6923—28no—
169C2×#92420—24no—
170C4⋊#6923—28no—
171C4×#6923—24no—
172C4×#6923—44no—
173C4×#6923—44no—
174C4×#6923#4118yescomplemented
175#11758213—28no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C22×C412yes—
182C22⋊C4×C61—22no—
183C42×C61—22no—
184C42×C61C22⋊C411yescharacteristic
185C42×C61C22⋊C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C42×C61C22×C411yescharacteristic
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22×C412yes—
200#11758196—22no—
201#11758196—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22×C411yescharacteristic
211C4×Dic61C22×C411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C23×C61C2×C22⋊C411yescharacteristic
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61C2×C22⋊C411yescharacteristic
222C22×D61C22×D411yescharacteristic
223C2×#6923—28no—
224C2×#6923—44no—
225C2×#6923—44no—
226Q8×C61—24no—
227Q8×C61—42no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C23×C411yescharacteristic
239C22⋊D61—44no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C22×D411yescharacteristic
250D244—24no—
251D244—44no—
252D244—44no—
253D244—42no—
254D244—42no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—44no—
267C4×D61—44no—
268C4×D61—44no—
269C4×D61C2×#4114yes—
270Dic122—42no—
271#92420—44no—
272#21418—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61C22×C22⋊C411yescharacteristic
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—44no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61#52512yes—
287#6923—48no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61#42112yes—
294C4×C61#42112yes—
295C4×C61C4×#4112yes—
296Dic61—21no—
297Dic61—21no—
298Dic61—21no—
299Dic61—21no—
300Dic61—21no—
301Dic61—42no—
302Dic61#53212yes—
303#3444—611no—
304#2920—611no—
305C4×#117—611no—
306#1994—612no—
307#1962—612no—
308#1897—614no—
309#1759—614no—
310C2×C61—41no—
311C2×C61#229711yescharacteristic
312C2×C61C2×#42111yescharacteristic
313C2×C61C2×#42111yescharacteristic
314D61—22no—
315D61—41no—
316D61—82no—
317#533—611no—
318#533—611no—
319#533#692311yescharacteristic, complemented
320#532—611no—
321#532—611no—
322#532—611no—
323#529—612no—
324#529—612no—
325#529—1221no—
326#529—1221no—
327#529—1221no—
328#529—1221no—
329#525—612no—
330#525—1221no—
331#525—1221no—
332C4×#41—612no—
333C4×#41—1221no—
334C4×#41—1221no—
335C2×#117—612no—
336C22×C42—611no—
337#432—1221no—
338#432—1221no—
339#431—1221no—
340#431—1221no—
341#423—612no—
342C2×#104—618no—
343#407—612no—
344C4×M16—612no—
345C4×C4⋊C4—1221no—
346C4×C4⋊C4—1221no—
347C4×C22⋊C4—1221no—
348C4×C22⋊C4—1221no—
349#376—612no—
350C43—612no—
351#339—618no—
352C61#2141811yescharacteristic, complemented, Sylow
353C2×#41—612no—
354C2×#41—1221no—
355C2×#41—1221no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C2×M16—614no—
360C4⋊Q8—611no—
361C4⋊Q8C2×#692311yescharacteristic
362C4⋊Q8C2×#692311yescharacteristic
363#127—611no—
364#126—1222no—
365#126—1222no—
366#125—21no—
367#125—612no—
368#125—1221no—
369#124—612no—
370#124—1221no—
371#124—1221no—
372#123—1222no—
373#123—1222no—
374#123—1222no—
375C22⋊Q8—21no—
376C22⋊Q8—612no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379C22⋊Q8—1221no—
380C22⋊Q8C2×#692312yes—
381#121—612no—
382#121—612no—
383#121—1221no—
384#121—1221no—
385#121—1221no—
386#121—1221no—
387#120—1222no—
388C4×Q8—21no—
389C4×Q8—612no—
390C4×Q8—1221no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8C2×#692312yes—
394C4×D4—612no—
395C4×D4—612no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1221no—
399C4×D4—1221no—
400C4×D4—1221no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1221no—
406#117—21no—
407#117—612no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117—1228no—
412#117—1221no—
413C2×C4⋊C4—1222no—
414C2×C22⋊C4—1222no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—614no—
427C2×C42—1222no—
428C2×C42—1222no—
429C2×C42—1222no—
430C2×C42C2×#692311yescharacteristic
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433#104—12216no—
434C22⋊C8—1222no—
435C22⋊C8—1222no—
436#97—1222no—
437C4×C8—1222no—
438#95—1222no—
439#95—1222no—
440C24—611no—
441#41—1228no—
442#41—2444no—
443#41—2444no—
444C2×Q8—21no—
445C2×Q8—1221no—
446C2×Q8C22×#692312yes—
447C2×D4—612no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—1221no—
452C2×D4—1221no—
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—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1224no—
472C22×C4—1222no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4C22⋊#692311yescharacteristic
476C22×C4C22⋊#692311yescharacteristic
477C22×C4C22×#692311yescharacteristic
478M16—1228no—
479C2×C8—1224no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—612no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1228no—
491C4⋊C4—1221no—
492C4⋊C4C22×#692312yes—
493C22⋊C4—22no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1228no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C42—22no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—618no—
517C42—1222no—
518C42—1222no—
519C42—1224no—
520C42—2444no—
521C42—2444no—
522C42C22⋊#692311yescharacteristic
523C42C22⋊#692311yescharacteristic
524C42C22×#692311yescharacteristic
525C42C22×#692311yescharacteristic
526C23—611no—
527C23—611no—
528C23—1222no—
529C23—1222no—
530C23—2441no—
531C23—2441no—
532C23C2×C22⋊#692311yescharacteristic
533Q8—24no—
534Q8—42no—
535Q8—2442no—
536D4—1224no—
537D4—2444no—
538D4—2444no—
539D4—2444no—
540D4—2442no—
541D4—2442no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—611no—
548C2×C4—614no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1228no—
563C2×C4—1222no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4C2×C22⋊#692311yescharacteristic
576C2×C4C2×C22⋊#692311yescharacteristic
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C2×C22⋊#692311yescharacteristic
579C2×C4C2×C22⋊#692311yescharacteristic
580C2×C4C23×#692311yescharacteristic
581C8—2444no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—612no—
586C22—1221no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—2444no—
592C22—2442no—
593C22—2442no—
594C22—2442no—
595C22C22×C22⋊#692311yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—42no—
600C4—42no—
601C4—612no—
602C4—1221no—
603C4—1221no—
604C4—1221no—
605C4—1221no—
606C4—1221no—
607C4—2448no—
608C4—2442no—
609C4#1684075912yes—
610C4#1684076112yes—
611C4#41×#692312yes—
612C2—41no—
613C2—1222no—
614C2—4882no—
615C2—2441no—
616C2#2146228011yescharacteristic
617C2C2×#1684075911yescharacteristic
618C2C2×#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#21418
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #21462496C2semidirect
2. C4×#14519856C2semidirect
3. #21462004C2semidirect
4. #21461952C2semidirect
5. #21461838C2semidirect
6. #21461583C2semidirect
24. #16840317C4semidirect
26. #16840313C4semidirect
28. #16840312C4semidirect
30. #16840299C4semidirect
32. #16840298C4semidirect
42. C61×#533C4semidirect
174. C4×#6923#41semidirect
319. #533#6923semidirect
352. C61#21418semidirect

Maximal subgroups

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

subgroupindex
#214624962
C4×#145198562
#214620042
#214619522
#214618382
#214615832
#214591162
#2141861