Finite Group Databasebeta

← #29730765

Subgroups — #29730765

Order 15616.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839992C212yesmaximal, complemented
2C2×#16839963C211yesmaximal, characteristic, complemented
3C2×#16839827C211yesmaximal, characteristic, complemented
4#21450761C211yesmaximal, characteristic
5#21450742C211yesmaximal, characteristic, complemented
6#21447010C212yesmaximal, complemented
7#21446825C212yesmaximal, complemented
8#21446624C211yesmaximal, characteristic, complemented
9#21446246C212yesmaximal, complemented
10C4×#14519700C211yesmaximal, characteristic, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C22×C8×D61C2212yes—
14C2×#14519774—21no—
15C2×#14519774C2211yescharacteristic
16C2×#14519757C2212yes—
17C2×#14519752C2212yes—
18C2×#14519752C2212yes—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16839992—28no—
22#16839963—24no—
23#16839827—24no—
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29C2×#14519689C2212yes—
30C2×C4×#92418C2211yescharacteristic
31#16839684—22no—
32#16839672—22no—
33#16839664—21no—
34#16839663—21no—
35#16839661—21no—
36#16839619C2212yes—
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39C2×#106×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122C2311yescharacteristic
45C22×D244—21no—
46C22×D244—22no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244D411yescharacteristic
50C22×D244C2312yes—
51C22×C4×D61C2312yes—
52C2×#11758190—24no—
53C2×#11758190—24no—
54C2×#11758190—42no—
55C22×#92418C2311yescharacteristic
56C22×#92418C2311yescharacteristic
57C2×C8×D61—24no—
58C2×C8×D61—24no—
59C2×C8×D61—42no—
60#14519774—22no—
61#14519774—22no—
62#14519774—44no—
63#14519757—24no—
64#14519757—42no—
65#14519752—24no—
66#14519752—24no—
67#14519752—42no—
68#14519752—42no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71C4×D244—22no—
72C4×D244—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198D412yes—
82C2×C4×Dic61—21no—
83#14519700—22no—
84#14519700D414yescomplemented
85#14519699—24no—
86#14519697—24no—
87#14519697—42no—
88#14519689—24no—
89C4×#92418—22no—
90#14519684—22no—
91C22×C8×C61C2312yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#106×C61—24no—
100C4⋊C8×C61D414yescomplemented
101#95×C61—21no—
102C23×D61—22no—
103C23×D61C2×D412yes—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61C2×D412yes—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—44no—
122C2×D244—44no—
123C2×D244—44no—
124C2×D244—44no—
125C2×D244—42no—
126C2×D244—42no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244—81no—
130C2×D244—81no—
131C2×D244—82no—
132C2×D244—82no—
133C2×D244SD1614yescomplemented
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122C2×D412yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61C2×D414yes—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—81no—
155C4⋊C4×C61C2×D412yes—
156C4⋊C4×C61#4112yes—
157C42×C61—42no—
158C42×C61C2×D412yes—
159C42×C61#4112yes—
160#11758198—24no—
161#11758198—44no—
162#11758198—44no—
163#11758198—44no—
164#11758198—82no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—44no—
169#11758196—42no—
170C4×Dic61—42no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418C2×D412yes—
177#11758190—48no—
178C8×D61—48no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61C2×SD1614yes—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61#12112yes—
206C2×C4×C61#12112yes—
207C2×C4×C61C2×SD1612yes—
208C2×C4×C61#13512yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61H(5,𝔽2)11yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224D244—28no—
225D244—42no—
226D244—42no—
227D244—42no—
228D244—42no—
229D244—84no—
230D244—88no—
231D244—88no—
232D244—84no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237Dic122—22no—
238Dic122—22no—
239Dic122—41no—
240Dic122—41no—
241C8×C61—24no—
242#92418—22no—
243#92418—41no—
244#25387—611nomaximal, Sylow
245C22×C61C2×#12111yescharacteristic
246C22×C61D4211yescharacteristic
247C22×C61#54611yescharacteristic
248C22×C61C22×SD1611yescharacteristic
249C22×C61C2×#13511yescharacteristic
250C22×C61#57611yescharacteristic
251C22×C61#57711yescharacteristic
252C2×D61—24no—
253C2×D61—24no—
254C2×D61—24no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C2×D61—82no—
264C2×D61—82no—
265C2×D61—82no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—82no—
270C4×C61#46514yescomplemented
271C4×C61#53211yescharacteristic
272C4×C61#54011yescharacteristic
273C4×C61D4211yescharacteristic
274C4×C61#55011yescharacteristic
275Dic61—42no—
276Dic61—42no—
277Dic61—81no—
278C2×#465—612no—
279C2×#460—611no—
280C2×C8×D4—611no—
281#2541—611no—
282#2538—611no—
283#2034—612no—
284#2000—612no—
285#1957—611no—
286#1948D6111yescharacteristic, complemented
287#1893—612no—
288C4×#103—611no—
289C2×C61#242811yescharacteristic
290C2×C61C2×#46511yescharacteristic
291C2×C61#306211yescharacteristic
292C2×C61D4×SD1611yescharacteristic
293C2×C61#328711yescharacteristic
294C2×C61#329511yescharacteristic
295C2×C61#330411yescharacteristic
296D61—44no—
297D61—162no—
298C22×SD16—612no—
299C23×C8—612no—
300C2×#127—611no—
301C2×#127—1221no—
302C2×C22⋊Q8—612no—
303C2×#121—612no—
304C2×#121—612no—
305C2×C4×Q8—611no—
306C2×C4×D4—611no—
307#465—1228no—
308#460—1224no—
309C8×D4—1224no—
310C2×#106C2×D6112yes—
311C2×C4⋊C8C2×D6111yescharacteristic
312C2×#103—611no—
313C2×#103—611no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8—612no—
318C2×C4×C8—611no—
319#396—1222no—
320#392—1222no—
321#390—1221no—
322#390—1221no—
323#389—1221no—
324#384C2×D6111yescharacteristic
325#380—612no—
326C4×C4⋊C4C2×D6111yescharacteristic
327#340—611no—
328#340—611no—
329#340C2×D6112yes—
330C61#2538711yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×D4—611no—
333C22×D4—612no—
334C22×D4—1222no—
335C22×D4—1222no—
336C22×D4—1221no—
337C22×D4—1221no—
338C23×C4—612no—
339C2×SD16—1224no—
340C2×SD16—1224no—
341C2×SD16—2442no—
342C22×C8—611no—
343C22×C8—611no—
344C22×C8—1224no—
345C22×C8—1224no—
346C22×C8—2442no—
347C22×C8C22×D6112yes—
348#127—1222no—
349#127—1222no—
350#127—2444no—
351C22⋊Q8—1224no—
352C22⋊Q8—2442no—
353#121—1224no—
354#121—1224no—
355#121—2442no—
356#121—2442no—
357C4×Q8—1222no—
358C4×Q8—1222no—
359C4×D4—1222no—
360C4×D4—1222no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4C22×D6112yes—
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C22⋊C4—612no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C42—21no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378#106—24no—
379C4⋊C8D24414yescomplemented
380#103—614no—
381#103—1222no—
382#102—1224no—
383#102—1224no—
384#102—2442no—
385C22⋊C8—1224no—
386C4×C8—1222no—
387#95—21no—
388#95—1222no—
389C24—612no—
390C24—1222no—
391C2×Q8—612no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×D4—614no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—1222no—
406C2×D4—4881no—
407C2×D4—4881no—
408C2×D4—4882no—
409C2×D4—4882no—
410C2×D4—2444no—
411C2×D4—2444no—
412C2×D4—2444no—
413C2×D4—2444no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4—2442no—
417C2×D4—2442no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—612no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1221no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—2448no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—612no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—2448no—
443C2×C8C2×D24414yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—81no—
450C4⋊C4—1222no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—2444no—
454C4⋊C4—2442no—
455C4⋊C4D4×D6112yes—
456C4⋊C4#1175822612yes—
457C22⋊C4—1224no—
458C22⋊C4—4882no—
459C22⋊C4—2444no—
460C22⋊C4—2444no—
461C22⋊C4—2444no—
462C42—42no—
463C42—2442no—
464C42C2×D24412yes—
465C42#1175822612yes—
466C23—614no—
467C23—1222no—
468C23—1222no—
469C23—1222no—
470C23—1222no—
471C23—1222no—
472C23—1222no—
473C23—2442no—
474C23—2442no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2442no—
479C23—2442no—
480C23C2×D4×D6111yescharacteristic
481Q8—1222no—
482Q8—1222no—
483Q8—2441no—
484Q8—2441no—
485D4—1228no—
486D4—4884no—
487D4—4888no—
488D4—4888no—
489D4—4884no—
490D4—2442no—
491D4—2442no—
492D4—2442no—
493D4—2442no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4#1451975212yes—
519C2×C4#1451975212yes—
520C2×C4SD16×D6112yes—
521C2×C4#1451979812yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×D24411yescharacteristic
524C2×C4C2×#1175822611yescharacteristic
525C2×C4C2×#1175822611yescharacteristic
526C2×C4#1451982311yescharacteristic
527C2×C4#1451982311yescharacteristic
528C8—24no—
529C8—1222no—
530C8—2441no—
531C22—1224no—
532C22—1224no—
533C22—1224no—
534C22—4882no—
535C22—4882no—
536C22—4882no—
537C22—4882no—
538C22—4882no—
539C22—4882no—
540C22—4882no—
541C22—2442no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22C2×#1451975211yescharacteristic
546C22#1684039611yescharacteristic
547C22#1684040311yescharacteristic
548C22C2×SD16×D6111yescharacteristic
549C22C2×#1451979811yescharacteristic
550C22#1684056611yescharacteristic
551C22#1684057111yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—4881no—
557C4—2442no—
558C4—2442no—
559C4#1683999214yescomplemented
560C4#1684031311yescharacteristic
561C4#1684036811yescharacteristic
562C4D4×D24411yescharacteristic
563C4#1684043411yescharacteristic
564C2—9762no—
565C2—2444no—
566C2#2144940511yescharacteristic
567C2C2×#1683999211yescharacteristic
568C2#2145550011yescharacteristic
569C2#2145751811yescharacteristic
570C2#2145757411yescharacteristic
571C2#2145764011yescharacteristic
572C2#2145775511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radical#21450761C2
hypercenter#1948D61
hypocenterC61#25387
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. C2×#16839992C2semidirect
2. C2×#16839963C2semidirect
3. C2×#16839827C2semidirect
5. #21450742C2semidirect
6. #21447010C2semidirect
7. #21446825C2semidirect
8. #21446624C2semidirect
9. #21446246C2semidirect
10. C4×#14519700C2semidirect
11. C61×#1948C2semidirect
84. #14519700D4semidirect
100. C4⋊C8×C61D4semidirect
133. C2×D244SD16semidirect
270. C4×C61#465semidirect
286. #1948D61semidirect
330. C61#25387semidirect
379. C4⋊C8D244semidirect
559. C4#16839992semidirect

Maximal subgroups

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

subgroupindex
C2×#168399922
C2×#168399632
C2×#168398272
#214507612
#214507422
#214470102
#214468252
#214466242
#214462462
C4×#145197002
C61×#19482
#2538761