Finite Group Databasebeta

← #29730761

Subgroups — #29730761

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×#16839991C212yesmaximal, complemented
2C2×#16839963C211yesmaximal, characteristic, complemented
3#21450765C211yesmaximal, characteristic
4#21450741C211yesmaximal, characteristic, complemented
5#21447009C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446583C211yesmaximal, characteristic, complemented
9#21446247C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519773—21no—
15C2×#14519756C2212yes—
16C2×#14519751C2212yes—
17C2×#14519751C2212yes—
18C2×C4×Dic122C2211yescharacteristic
19#16839991—28no—
20#16839963—24no—
21C2×#14519709C2212yes—
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27#16839685—22no—
28#16839671—22no—
29#16839665—21no—
30#16839663—21no—
31#16839658—21no—
32#16839632C2211yescharacteristic
33#16839622C2212yes—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2212yes—
38C23×C8×C61C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C8×D4×C61—24no—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C22×Dic122C2311yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48C22×C22⋊D61—22no—
49C22×C22⋊D61—22no—
50C2×#11758190—24no—
51C2×#11758190—24no—
52C2×#11758190—42no—
53#14519774—22no—
54#14519774—22no—
55#14519773—44no—
56#14519756—24no—
57#14519756—42no—
58#14519751—24no—
59#14519751—24no—
60#14519751—42no—
61#14519751—42no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78#14519709—24no—
79#14519702—22no—
80#14519702D414yescomplemented
81#14519699—24no—
82#14519699—24no—
83#14519699—42no—
84#14519684—22no—
85#14519684—21no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61C2312yes—
88C22×C8×C61—24no—
89C22×C8×C61—24no—
90C22×C8×C61—42no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C22⋊C4×C61D412yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61D414yescomplemented
100C22⋊C8×C61—24no—
101C4×C8×C61—22no—
102C24×C61C2×D412yes—
103C23×D61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61SD1614yescomplemented
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—44no—
118C2×C22⋊D61—44no—
119C2×C22⋊D61—44no—
120C2×C22⋊D61—44no—
121C2×C22⋊D61—82no—
122C2×C22⋊D61—82no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—42no—
137C2×D244—42no—
138C2×D244—81no—
139C2×D244—81no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122C2×D412yes—
146C2×C8×C61—21no—
147C2×C8×C61—22no—
148C2×C8×C61—21no—
149C2×C8×C61—22no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—48no—
154C2×C8×C61C2×D412yes—
155C2×C8×C61C2×D414yes—
156C4⋊C4×C61—22no—
157C22⋊C4×C61—24no—
158C42×C61C2×D412yes—
159C42×C61#4112yes—
160C22⋊Dic61—44no—
161C22⋊Dic61—82no—
162#11758198—44no—
163#11758198—44no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—22no—
170C4⋊Dic61—42no—
171C4⋊Dic61—81no—
172C4⋊Dic61C2×D412yes—
173C4⋊Dic61#4112yes—
174#11758196—44no—
175#11758196—42no—
176C4×Dic61—42no—
177C4×Dic61—42no—
178#11758190—48no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61C2×SD1614yes—
186C23×C61C22×D411yescharacteristic
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194D4×C61—28no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61#12112yes—
205C2×C4×C61#12112yes—
206C2×C4×C61C2×SD1612yes—
207C2×C4×C61#13512yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)11yescharacteristic
213C2×C4×C61H(5,𝔽2)11yescharacteristic
214C22⋊D61—88no—
215C22⋊D61—88no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—42no—
232D244—42no—
233D244—42no—
234D244—42no—
235D244—84no—
236D244—84no—
237Dic122—22no—
238Dic122—22no—
239Dic122—41no—
240Dic122—41no—
241C8×C61—24no—
242C8×C61—22no—
243C8×C61—41no—
244#25387—611nomaximal, Sylow
245C22×C61—24no—
246C22×C61—24no—
247C22×C61—24no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61C2×#12111yescharacteristic
253C22×C61D4211yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61C22×SD1611yescharacteristic
256C22×C61C2×#13511yescharacteristic
257C22×C61#57611yescharacteristic
258C22×C61#57711yescharacteristic
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#46514yescomplemented
269C4×C61#53211yescharacteristic
270C4×C61#54011yescharacteristic
271C4×C61D4211yescharacteristic
272C4×C61#55011yescharacteristic
273Dic61—42no—
274Dic61—42no—
275Dic61—42no—
276Dic61—82no—
277Dic61—81no—
278C2×#465—612no—
279C2×#460—611no—
280C2×C8×D4D6111yescharacteristic, complemented
281#2541—611no—
282#2538—611no—
283#2034—612no—
284#2000—612no—
285#1957—611no—
286#1948—611no—
287#1893—612no—
288C4×#103—611no—
289C2×C61—44no—
290C2×C61#242811yescharacteristic
291C2×C61C2×#46511yescharacteristic
292C2×C61#306211yescharacteristic
293C2×C61D4×SD1611yescharacteristic
294C2×C61#328711yescharacteristic
295C2×C61#329511yescharacteristic
296C2×C61#330411yescharacteristic
297D61—162no—
298C22×SD16—612no—
299C23×C8C2×D6112yes—
300C2×#127—611no—
301C2×#127—1221no—
302C2×C22⋊Q8—612no—
303C2×#121—612no—
304C2×#121—612no—
305C2×C4×Q8—611no—
306C2×C4×D4C2×D6111yescharacteristic
307#465—1228no—
308#460—1224no—
309C8×D4—24no—
310C2×#106—612no—
311C2×C4⋊C8C2×D6111yescharacteristic
312C2×#103—611no—
313C2×#103—611no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8C2×D6112yes—
318C2×C4×C8C2×D6111yescharacteristic
319#396—1222no—
320#392—1222no—
321#390—1221no—
322#390—1221no—
323#389—1221no—
324#384—611no—
325#380—612no—
326C4×C4⋊C4—611no—
327#340—611no—
328#340—612no—
329#340—611no—
330C61#2538711yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×D4—612no—
333C22×D4—1222no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4D24411yescharacteristic
338C23×C4C22×D6112yes—
339C2×SD16—1224no—
340C2×SD16—1224no—
341C2×SD16—2442no—
342C22×C8—24no—
343C22×C8—24no—
344C22×C8—42no—
345C22×C8C22×D6111yescharacteristic
346C22×C8C22×D6111yescharacteristic
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—22no—
360C4×D4—22no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4D24411yescharacteristic
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4D24412yes—
375C2×C42—1221no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378#106—1224no—
379C4⋊C8C22⋊D6114yescomplemented
380#103—614no—
381#103—1222no—
382#102—1224no—
383#102—1224no—
384#102—2442no—
385C22⋊C8—24no—
386C4×C8—22no—
387#95—1221no—
388#95—1222no—
389C24—1222no—
390C24C2×D24412yes—
391C2×Q8—612no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—4881no—
406C2×D4—4881no—
407C2×D4—4882no—
408C2×D4—4882no—
409C2×D4—2444no—
410C2×D4—2444no—
411C2×D4—2444no—
412C2×D4—2444no—
413C2×D4—2442no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4—2442no—
417C2×D4#1175819014yescomplemented
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4C2×D24412yes—
430C22×C4C2×D24411yescharacteristic
431C22×C4C2×D24411yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—2448no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—48no—
442C2×C8D4×D6112yes—
443C2×C8C2×C22⋊D6114yes—
444C4⋊C4—22no—
445C4⋊C4—612no—
446C4⋊C4—612no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—1222no—
453C4⋊C4—4881no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C4⋊C4—2442no—
457C22⋊C4—24no—
458C22⋊C4—4882no—
459C22⋊C4—2444no—
460C22⋊C4—2444no—
461C22⋊C4—2444no—
462C42—2442no—
463C42—2442no—
464C42#1175822212yes—
465C42C2×C22⋊D6112yes—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—2442no—
473C23—2442no—
474C23—2442no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2442no—
479C23C2×#1175819014yes—
480C23C22×D24411yescharacteristic
481Q8—1222no—
482Q8—1222no—
483Q8—2441no—
484Q8—2441no—
485D4—28no—
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—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—2441no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4—2442no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4#1451975112yes—
519C2×C4#1451975112yes—
520C2×C4C2×#1175819012yes—
521C2×C4#1451979912yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4#41×D6111yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C2×C4#1451982211yescharacteristic
527C2×C4#1451982311yescharacteristic
528C8—22no—
529C8—24no—
530C8—41no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—4882no—
539C22—4882no—
540C22—4882no—
541C22—4882no—
542C22—4882no—
543C22—4882no—
544C22—4882no—
545C22C2×#1451975111yescharacteristic
546C22D4×D24411yescharacteristic
547C22#1684040711yescharacteristic
548C22C22×#1175819011yescharacteristic
549C22C2×#1451979911yescharacteristic
550C22#1684056711yescharacteristic
551C22#1684057411yescharacteristic
552C4—22no—
553C4—42no—
554C4—4881no—
555C4—4882no—
556C4—2442no—
557C4—2442no—
558C4—2442no—
559C4#1683999114yescomplemented
560C4#1684030911yescharacteristic
561C4#1684036311yescharacteristic
562C4D4×C22⋊D6111yescharacteristic
563C4#1684043111yescharacteristic
564C2—44no—
565C2—9762no—
566C2#2144940111yescharacteristic
567C2C2×#1683999111yescharacteristic
568C2#2145549911yescharacteristic
569C2D4×#1175819011yescharacteristic
570C2#2145757911yescharacteristic
571C2#2145764411yescharacteristic
572C2#2145775911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radical#21450765C2
hypercenterC2×C8×D4D61
hypocenterC61#25387
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16839991C2semidirect
2. C2×#16839963C2semidirect
4. #21450741C2semidirect
5. #21447009C2semidirect
6. #21446824C2semidirect
7. #21446625C2semidirect
8. #21446583C2semidirect
9. #21446247C2semidirect
10. C4×#14519702C2semidirect
11. C2×C8×D4×C61C2semidirect
80. #14519702D4semidirect
99. C4⋊C8×C61D4semidirect
106. C2×D4×C61SD16semidirect
268. C4×C61#465semidirect
280. C2×C8×D4D61semidirect
330. C61#25387semidirect
379. C4⋊C8C22⋊D61semidirect
417. C2×D4#11758190semidirect
559. C4#16839991semidirect

Maximal subgroups

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

subgroupindex
C2×#168399912
C2×#168399632
#214507652
#214507412
#214470092
#214468242
#214466252
#214465832
#214462472
C4×#145197022
C2×C8×D4×C612
#2538761