Finite Group Databasebeta

← #29730767

Subgroups — #29730767

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×#16839994C212yesmaximal, complemented
2C2×#16839962C211yesmaximal, characteristic, complemented
3C2×C8×D244C211yesmaximal, characteristic, complemented
4#21450761C211yesmaximal, characteristic
5#21450737C211yesmaximal, characteristic, complemented
6#21447011C212yesmaximal, complemented
7#21446821C212yesmaximal, complemented
8#21446625C211yesmaximal, characteristic, complemented
9#21446582C211yesmaximal, characteristic, complemented
10#21446246C212yesmaximal, complemented
11C4×#103×C61C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×C8×D61C2212yes—
14C2×#14519774—21no—
15C2×#14519774C2211yescharacteristic
16C2×#14519753C2212yes—
17C2×#14519752C2212yes—
18C2×#14519752C2212yes—
19C2×C4×D244C2211yescharacteristic
20#16839994—28no—
21#16839962—24no—
22C8×D244—24no—
23C2×#14519708C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27C2×#14519697C2212yes—
28C2×#14519689C2212yes—
29#16839684—22no—
30#16839672—22no—
31#16839664—21no—
32#16839664—21no—
33#16839660—21no—
34#16839632C2211yescharacteristic
35#16839619C2212yes—
36#16839524C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×#103×C61C2211yescharacteristic
39C2×#103×C61C2211yescharacteristic
40C2×C4×C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2211yescharacteristic
43C61×#340C2211yescharacteristic
44C22×D244—21no—
45C22×D244—22no—
46C22×D244—22no—
47C22×D244—21no—
48C22×D244D411yescharacteristic
49C22×D244C2312yes—
50C22×C4×D61C2312yes—
51C2×#11758201—24no—
52C2×#11758201—24no—
53C2×#11758201—42no—
54C22×#92418C2312yes—
55C2×C8×D61—24no—
56C2×C8×D61—24no—
57C2×C8×D61—42no—
58#14519774—22no—
59#14519774—22no—
60#14519774—44no—
61#14519753—24no—
62#14519753—42no—
63#14519752—24no—
64#14519752—24no—
65#14519752—42no—
66#14519752—42no—
67C4×D244—22no—
68C4×D244—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198D412yes—
79#14519708—24no—
80C4⋊#92418D414yescomplemented
81#14519699—24no—
82#14519699—42no—
83#14519697—24no—
84#14519689—24no—
85#14519684—22no—
86C22×Q8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C42×C61—21no—
96C2×C42×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#103×C61—22no—
99#103×C61D414yescomplemented
100C4×C8×C61—22no—
101#95×C61—21no—
102C23×D61—22no—
103C23×D61C2×D412yes—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61C2×D412yes—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61C2×D412yes—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—44no—
128C2×D244—44no—
129C2×D244—44no—
130C2×D244—44no—
131C2×D244—42no—
132C2×D244—42no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244—81no—
136C2×D244—81no—
137C2×D244—82no—
138C2×D244—82no—
139C2×D244SD1614yescomplemented
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61C2×D412yes—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61C2×D412yes—
158C4⋊C4×C61#4112yes—
159C42×C61—42no—
160C42×C61—42no—
161C42×C61C2×D412yes—
162C42×C61#4112yes—
163#11758201—48no—
164#11758198—24no—
165#11758198—44no—
166#11758198—44no—
167#11758198—44no—
168#11758198—82no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—42no—
173C4⋊Dic61—81no—
174#11758196—44no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418C2×D414yes—
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—
194Q8×C61—22no—
195Q8×C61—22no—
196Q8×C61—41no—
197Q8×C61—41no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61#12112yes—
213C2×C4×C61#12112yes—
214C2×C4×C61C2×SD1612yes—
215C2×C4×C61#13512yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228D244—28no—
229D244—42no—
230D244—42no—
231D244—42no—
232D244—42no—
233D244—84no—
234D244—88no—
235D244—88no—
236D244—84no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C8×C61—22no—
242C8×C61—41no—
243#92418—24no—
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—42no—
270C4×C61—81no—
271C4×C61#46514yescomplemented
272C4×C61#53211yescharacteristic
273C4×C61#54011yescharacteristic
274C4×C61D4211yescharacteristic
275C4×C61#55011yescharacteristic
276Dic61—42no—
277Dic61—82no—
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#1948—611no—
287#1893—612no—
288C4×#103D6111yescharacteristic, complemented
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×Q8C2×D6111yescharacteristic
306C2×C4×D4—611no—
307#465—1228no—
308#460—1224no—
309C8×D4—1224no—
310C2×#106—612no—
311C2×C4⋊C8—611no—
312C2×#103C2×D6111yescharacteristic
313C2×#103C2×D6111yescharacteristic
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8—612no—
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⋊C4C2×D6111yescharacteristic
327#340—612no—
328#340C2×D6111yescharacteristic
329#340C2×D6111yescharacteristic
330C61#2538711yescharacteristic, complemented, Sylow
331C22×Q8C22×D6111yescharacteristic
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—612no—
343C22×C8—1224no—
344C22×C8—1224no—
345C22×C8—2442no—
346C22×C8C22×D6111yescharacteristic
347C22×C8C22×D6111yescharacteristic
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—22no—
358C4×Q8—22no—
359C4×D4—1222no—
360C4×D4—1222no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4C22×D6111yescharacteristic
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—21no—
377C2×C42C22×D6111yescharacteristic
378#106—1224no—
379C4⋊C8—614no—
380#103—22no—
381#103D24414yescomplemented
382#102—1224no—
383#102—1224no—
384#102—2442no—
385C22⋊C8—1224no—
386C4×C8—22no—
387#95—21no—
388#95—1222no—
389C24—612no—
390C24—1222no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8C2×D24412yes—
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—21no—
421C22×C4—21no—
422C22×C4—612no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—2448no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—614no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—2448no—
443C2×C8C2×D24412yes—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—4881no—
453C4⋊C4—2444no—
454C4⋊C4—2442no—
455C4⋊C4C2×D24412yes—
456C4⋊C4#1175822212yes—
457C22⋊C4—1224no—
458C22⋊C4—4882no—
459C22⋊C4—2444no—
460C22⋊C4—2444no—
461C22⋊C4—2444no—
462C42—42no—
463C42—42no—
464C42#1175822212yes—
465C42D4×D6112yes—
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—22no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
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—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4#1451975012yes—
519C2×C4#1451975012yes—
520C2×C4SD16×D6112yes—
521C2×C4#1451980012yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×D24411yescharacteristic
524C2×C4C2×#1175822211yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C2×C4#1451982311yescharacteristic
527C2×C4#1451982311yescharacteristic
528C8—22no—
529C8—41no—
530C8—1224no—
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×#1451975011yescharacteristic
546C22D4×D24411yescharacteristic
547C22#1684040811yescharacteristic
548C22C2×SD16×D6111yescharacteristic
549C22C2×#1451980011yescharacteristic
550C22#1684056411yescharacteristic
551C22#1684056911yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—81no—
557C4—4882no—
558C4—2442no—
559C4#1683999414yescomplemented
560C4#1684031211yescharacteristic
561C4#1684036711yescharacteristic
562C4D4×D24411yescharacteristic
563C4#1684043311yescharacteristic
564C2—9762no—
565C2—2444no—
566C2#2144940711yescharacteristic
567C2C2×#1683999411yescharacteristic
568C2#2145550211yescharacteristic
569C2SD16×D24411yescharacteristic
570C2#2145757811yescharacteristic
571C2#2145764511yescharacteristic
572C2#2145776011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radical#21450761C2
hypercenterC4×#103D61
hypocenterC61#25387
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. C2×#16839994C2semidirect
2. C2×#16839962C2semidirect
3. C2×C8×D244C2semidirect
5. #21450737C2semidirect
6. #21447011C2semidirect
7. #21446821C2semidirect
8. #21446625C2semidirect
9. #21446582C2semidirect
10. #21446246C2semidirect
11. C4×#103×C61C2semidirect
80. C4⋊#92418D4semidirect
99. #103×C61D4semidirect
139. C2×D244SD16semidirect
271. C4×C61#465semidirect
288. C4×#103D61semidirect
330. C61#25387semidirect
381. #103D244semidirect
559. C4#16839994semidirect

Maximal subgroups

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

subgroupindex
C2×#168399942
C2×#168399622
C2×C8×D2442
#214507612
#214507372
#214470112
#214468212
#214466252
#214465822
#214462462
C4×#103×C612
#2538761