Finite Group Databasebeta

← #29730769

Subgroups — #29730769

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×#16839993C212yesmaximal, complemented
2C2×#16839962C211yesmaximal, characteristic, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21450765C211yesmaximal, characteristic
5#21447008C212yesmaximal, complemented
6#21446820C212yesmaximal, complemented
7#21446624C211yesmaximal, characteristic, complemented
8#21446580C211yesmaximal, characteristic, complemented
9#21446245C212yesmaximal, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C61×#2538C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C23×#92418C2212yes—
14C2×#14519774C2211yescharacteristic
15C2×#14519773—21no—
16C2×#14519751C2212yes—
17C2×#14519751C2212yes—
18#16839993—28no—
19#16839962—24no—
20D4×#92418—24no—
21C2×#14519708C2212yes—
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519697C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×C22⋊#92418C2212yes—
29C2×C4×#92418C2211yescharacteristic
30#16839671—22no—
31#16839663—21no—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839524C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2212yes—
36C2×C22⋊Q8×C61C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C61×#396—22no—
40C61×#390—21no—
41C61×#389—21no—
42C61×#384C2211yescharacteristic
43C61×#380C2212yes—
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C2×#11758201—24no—
50C2×#11758201—24no—
51C2×#11758201—42no—
52C22×#92418—24no—
53C22×#92418—24no—
54C22×#92418—42no—
55C22×#92418C2311yescharacteristic
56C22×#92418C2311yescharacteristic
57C22×#92418C2312yes—
58#14519774—22no—
59#14519774—22no—
60#14519773—44no—
61#14519751—24no—
62#14519751—24no—
63#14519751—42no—
64#14519751—42no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×C4×Dic61—21no—
70#14519708—24no—
71C4⋊#92418D414yescomplemented
72#14519701—22no—
73#14519701D414yescomplemented
74#14519697—24no—
75#14519697—24no—
76#14519697—42no—
77C22⋊#92418—24no—
78C4×#92418—22no—
79#14519684—21no—
80C22×Q8×C61C2311yescharacteristic
81C22×D4×C61D411yescharacteristic
82C23×C4×C61C2312yes—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—42no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61D412yes—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#95×C61—22no—
102C24×C61C2×D412yes—
103C23×D61—22no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61C2×D412yes—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61SD1614yescomplemented
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—44no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—82no—
131C2×C22⋊D61—82no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244—42no—
144C2×D244—81no—
145C2×D244—81no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—44no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—81no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—44no—
157C22⋊C4×C61—82no—
158C42×C61—42no—
159C42×C61C2×D412yes—
160C42×C61#4112yes—
161#11758201—48no—
162#11758198—44no—
163#11758198—44no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61#4112yes—
168C4×Dic61—42no—
169C2×#92418—21no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—48no—
177C2×#92418C2×D414yes—
178C2×#92418C2×D412yes—
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—
194Q8×C61—22no—
195Q8×C61—22no—
196Q8×C61—41no—
197Q8×C61—41no—
198D4×C61—28no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61#12112yes—
218C2×C4×C61#12112yes—
219C2×C4×C61C2×SD1612yes—
220C2×C4×C61#13512yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C22⋊D61—88no—
228C22⋊D61—88no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235D244—42no—
236D244—42no—
237D244—42no—
238D244—42no—
239D244—84no—
240D244—84no—
241#92418—22no—
242#92418—24no—
243#92418—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—42no—
269C4×C61—82no—
270C4×C61—81no—
271C4×C61#46514yescomplemented
272C4×C61#53211yescharacteristic
273C4×C61#54011yescharacteristic
274C4×C61D4211yescharacteristic
275C4×C61#55011yescharacteristic
276Dic61—42no—
277Dic61—42no—
278C2×#465—612no—
279C2×#460—611no—
280C2×C8×D4—611no—
281#2541—611no—
282#2538D6111yescharacteristic, complemented
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×C8—612no—
300C2×#127—611no—
301C2×#127—1221no—
302C2×C22⋊Q8C2×D6112yes—
303C2×#121—612no—
304C2×#121—612no—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4C2×D6111yescharacteristic
307#465—1228no—
308#460—1224no—
309C8×D4—1224no—
310C2×#106—612no—
311C2×C4⋊C8—611no—
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—22no—
320#392—1222no—
321#390—21no—
322#390—1221no—
323#389—21no—
324#384C2×D6111yescharacteristic
325#380C2×D6112yes—
326C4×C4⋊C4—611no—
327#340—611no—
328#340—612no—
329#340—611no—
330C61#2538711yescharacteristic, complemented, Sylow
331C22×Q8C22×D6111yescharacteristic
332C22×D4—612no—
333C22×D4—1222no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4C22⋊D6111yescharacteristic
338C23×C4C22×D6112yes—
339C2×SD16—1224no—
340C2×SD16—1224no—
341C2×SD16—2442no—
342C22×C8—611no—
343C22×C8—612no—
344C22×C8—611no—
345C22×C8—1224no—
346C22×C8—1224no—
347C22×C8—2442no—
348#127—1222no—
349#127—1222no—
350#127—2444no—
351C22⋊Q8—24no—
352C22⋊Q8—42no—
353#121—1224no—
354#121—1224no—
355#121—2442no—
356#121—2442no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×D4—22no—
360C4×D4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4C22⋊D6111yescharacteristic
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6112yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4C22⋊D6112yes—
375C2×C42—21no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378#106—1224no—
379C4⋊C8—614no—
380#103—614no—
381#103—1222no—
382#102—1224no—
383#102—1224no—
384#102—2442no—
385C22⋊C8—1224no—
386C4×C8—1222no—
387#95—22no—
388#95—1221no—
389C24—1222no—
390C24C2×C22⋊D6112yes—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8D4×D6112yes—
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#1175820114yescomplemented
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—2448no—
434C2×C8—614no—
435C2×C8—612no—
436C2×C8—1221no—
437C2×C8—1222no—
438C2×C8—1221no—
439C2×C8—1222no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—2448no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—81no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C22⋊C4—24no—
458C22⋊C4—44no—
459C22⋊C4—82no—
460C22⋊C4—2444no—
461C22⋊C4—2444no—
462C42—42no—
463C42—2442no—
464C42D4×D6112yes—
465C42#1175822412yes—
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×#1175820114yes—
480C23C22×C22⋊D6111yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
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—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4#1451974912yes—
519C2×C4#1451974912yes—
520C2×C4C2×#1175820112yes—
521C2×C4#1451979712yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4#41×D6111yescharacteristic
525C2×C4C2×#1175822411yescharacteristic
526C2×C4#1451982211yescharacteristic
527C2×C4#1451982311yescharacteristic
528C8—1222no—
529C8—1224no—
530C8—2441no—
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×#1451974911yescharacteristic
546C22D4×C22⋊D6111yescharacteristic
547C22#1684040011yescharacteristic
548C22C22×#1175820111yescharacteristic
549C22C2×#1451979711yescharacteristic
550C22#1684056511yescharacteristic
551C22#1684057011yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—82no—
557C4—2442no—
558C4—2442no—
559C4#1683999314yescomplemented
560C4#1684030711yescharacteristic
561C4#1684036511yescharacteristic
562C4#1684039611yescharacteristic
563C4#1684043011yescharacteristic
564C2—44no—
565C2—9762no—
566C2#2144940911yescharacteristic
567C2C2×#1683999311yescharacteristic
568C2#2145550111yescharacteristic
569C2D4×#1175820111yescharacteristic
570C2#2145757511yescharacteristic
571C2#2145763711yescharacteristic
572C2#2145775211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2538C2
socleC23×C61C22×D4
prime radical#21450765C2
hypercenter#2538D61
hypocenterC61#25387
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#16839993C2semidirect
2. C2×#16839962C2semidirect
3. C2×D4×#92418C2semidirect
5. #21447008C2semidirect
6. #21446820C2semidirect
7. #21446624C2semidirect
8. #21446580C2semidirect
9. #21446245C2semidirect
10. C4×#14519701C2semidirect
11. C61×#2538C2semidirect
71. C4⋊#92418D4semidirect
73. #14519701D4semidirect
112. C2×D4×C61SD16semidirect
271. C4×C61#465semidirect
282. #2538D61semidirect
330. C61#25387semidirect
417. C2×D4#11758201semidirect
559. C4#16839993semidirect

Maximal subgroups

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

subgroupindex
C2×#168399932
C2×#168399622
C2×D4×#924182
#214507652
#214470082
#214468202
#214466242
#214465802
#214462452
C4×#145197012
C61×#25382
#2538761