Finite Group Databasebeta

← #29689390

Subgroups — #29689390

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840330C211yesmaximal, characteristic
2C2×#16839821C211yesmaximal, characteristic, complemented
3#21448429C211yesmaximal, characteristic, complemented
4#21446748C212yesmaximal, complemented
5#21446744C212yesmaximal, complemented
6#21446642C211yesmaximal, characteristic, complemented
7#21446636C211yesmaximal, characteristic, complemented
8#21446625C211yesmaximal, characteristic, complemented
9#21445758C212yesmaximal, complemented
10#21445748C212yesmaximal, complemented
11C61×#1958C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×D4×D61C2211yescharacteristic
14#16840330—24no—
15#16840330C418yescomplemented
16C2×#14519774C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19C2×#14519750—22no—
20C2×#14519747—22no—
21C2×#14519734C2211yescharacteristic
22#16839821—24no—
23C2×#14519706C2212yescomplemented
24C2×#14519701C2212yescomplemented
25C2×#14519699C2212yes—
26C2×#14519699C2212yescomplemented
27C2×#14519698C2212yes—
28C2×#14519697C2212yes—
29C2×#14519697C2212yescomplemented
30C2×#14519689C2212yes—
31C2×#14519688C2211yescharacteristic, complemented
32#16839654—22no—
33#16839649—22no—
34#16839632C2211yescharacteristic, complemented
35#16839619C2212yes—
36C4⋊C4×Dic61C2212yescomplemented
37C2×#124×C61C2211yescharacteristic
38C2×#103×C61C2212yescomplemented
39C2×#102×C61C2212yes—
40C2×C4×C8×C61C2211yescharacteristic, complemented
41C61×#382C2211yescharacteristic, complemented
42C24×D61—21no—
43C2×#11758226—22no—
44C2×#11758226—42no—
45C2×#11758226—42no—
46C2×#11758226C2×C414yes—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—22no—
53C2×D4×D61—22no—
54C2×D4×D61—42no—
55C2×D4×D61—42no—
56C2×D4×D61C2×C414yes—
57C22×C4×D61—21no—
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60#14519774—22no—
61#14519774C2×C414yes—
62#14519773—22no—
63#14519773C2×C414yes—
64#14519766—22no—
65#14519766C2×C414yes—
66#14519750—216no—
67#14519747—216no—
68#14519734C2×C414yes—
69#14519734D414yes—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196D412yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198D412yes—
76C2×C4×Dic61—22no—
77C2×C4×Dic61C2311yescharacteristic
78#14519706—24no—
79#14519701—42no—
80#14519699—42no—
81#14519699—42no—
82#14519698—42no—
83#14519697—42no—
84#14519697—42no—
85#14519689—24no—
86#14519688—22no—
87#14519684—22no—
88#14519684—22no—
89C22×Q8×C61C2311yescharacteristic
90C22×D4×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92#124×C61—22no—
93#124×C61C2×C414yes—
94C2×C4⋊C4×C61C2312yes—
95C2×C22⋊C4×C61—22no—
96C2×C42×C61C2311yescharacteristic
97#103×C61—42no—
98#102×C61—42no—
99C4×C8×C61—22no—
100#95×C61—22no—
101C24×C61—21no—
102C23×D61—21no—
103C23×D61—21no—
104C23×D61—24no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—82no—
108C23×D61—82no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C22×C412yes—
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61C22×C412yes—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—24no—
123C2×C22⋊D61—28no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—82no—
127C2×C22⋊D61—82no—
128C22×Dic61—22no—
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D412yes—
131#11758226—48no—
132#11758226—48no—
133D4×D61—48no—
134D4×D61—48no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—24no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—28no—
143C2×D244—42no—
144C2×D244—42no—
145C2×D244—82no—
146C2×D244—82no—
147C2×D244C22×C412yes—
148C2×D244C22×C412yes—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—24no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61C22×C412yes—
155C2×C4×D61C2×D412yes—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C22⋊C4×C61—28no—
162C42×C61C22×C412yes—
163C42×C61C2×D412yes—
164#11758198—28no—
165#11758198—82no—
166#11758198C22⋊C418yes—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—82no—
170#11758196C22⋊C418yes—
171C4×Dic61—82no—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C23×C61—22no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61C22×D411yescharacteristic
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—48no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—88no—
199C22×D61—84no—
200C22×D61—84no—
201C22×D61—88no—
202C22×D61C2×C22⋊C412yes—
203Q8×C61—24no—
204Q8×C61—42no—
205D4×C61—24no—
206D4×C61—42no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C22⋊C414yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61H(5,𝔽2)12yes—
219C2×C4×C61H(5,𝔽2)12yes—
220C22⋊D61—48no—
221C22⋊D61—88no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61C2×C22⋊C414yes—
230C2×Dic61C2×C22⋊C412yes—
231D244—24no—
232D244—24no—
233D244—48no—
234D244—42no—
235D244—42no—
236D244—88no—
237C4×D61—24no—
238C4×D61—24no—
239C4×D61—48no—
240C8×C61—42no—
241#92418—42no—
242#21813—611nomaximal, Sylow
243C22×C61—42no—
244C22×C61—44no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—82no—
248C22×C61C22×C22⋊C411yescharacteristic
249C22×C61#53511yescharacteristic
250C22×C61#53511yescharacteristic
251C22×C61#57612yes—
252C22×C61#57712yes—
253C2×D61—22no—
254C2×D61—24no—
255C2×D61—41no—
256C2×D61—41no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—44no—
261C2×D61—44no—
262C2×D61—816no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—82no—
268C2×D61—82no—
269C4×C61—24no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61#51812yes—
273C4×C61#53512yes—
274Dic61—22no—
275Dic61—24no—
276Dic61—82no—
277C2×#535—611no—
278C2×#433—611no—
279#2314—611no—
280#1983—612no—
281#1982—612no—
282#1961—611no—
283#1959—611no—
284#1958D6111yescharacteristic, complemented
285#1957—611no—
286#1788—612no—
287#1786—612no—
288C2×C61—82no—
289C2×C61#230911yescharacteristic
290C2×C61#289712yes—
291C2×C61#322011yescharacteristic
292C2×C61#322211yescharacteristic
293C2×C61#322412yes—
294D61—42no—
295D61—82no—
296D61—82no—
297C22×#41—611no—
298C23×D4—611no—
299#535—618no—
300#535—1224no—
301C2×#127—611no—
302C2×#127—611no—
303C2×#124—611no—
304C2×#124C2×D6111yescharacteristic
305C2×#121—1222no—
306C2×#120—1222no—
307C2×#117—611no—
308#433—1224no—
309C2×C4⋊C8—612no—
310C2×#103—612no—
311C2×#103C2×D6112yescomplemented
312C2×#102—612no—
313C2×#102—612no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102C2×D6112yes—
318C2×C22⋊C8—612no—
319C2×#97—611no—
320C2×C4×C8C2×D6111yescharacteristic, complemented
321#388—1222no—
322#387—1222no—
323#384—611no—
324#382C2×D6111yescharacteristic, complemented
325#380—612no—
326C4×C4⋊C4—612no—
327C61#2181311yescharacteristic, complemented, Sylow
328C25—1221no—
329C2×#41—614no—
330C2×#41—1222no—
331C2×#41—2442no—
332C2×#41—2442no—
333C22×Q8C22×D6111yescharacteristic
334C22×D4—614no—
335C22×D4—611no—
336C22×D4—611no—
337C22×D4—1221no—
338C22×D4—1222no—
339C22×D4—1222no—
340C22×D4—1221no—
341C22×D4—1222no—
342C22×D4—2442no—
343C22×D4—2442no—
344C22×D4C22×D6111yescharacteristic
345C23×C4—611no—
346C23×C4—1221no—
347C22×C8—612no—
348C22×C8C22×D6112yes—
349#127—614no—
350#127—614no—
351#127—1222no—
352#127—1222no—
353#124—22no—
354#124—614no—
355#124—1222no—
356#124C4×D6114yes—
357#121—12216no—
358#120—12216no—
359#117—614no—
360#117—614no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—612no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C42—611no—
370C2×C42—1222no—
371C2×C42C22×D6111yescharacteristic
372C4⋊C8—1224no—
373#103—42no—
374#103—2442no—
375#102—42no—
376#102—2442no—
377#102—2442no—
378#102—2442no—
379#102—2442no—
380#102—2442no—
381C22⋊C8—1224no—
382#97—1222no—
383C4×C8—22no—
384#95—22no—
385#95—1222no—
386#95—1222no—
387C24—21no—
388C24—611no—
389C24—1224no—
390C24—1221no—
391C24—1221no—
392C24—4882no—
393C24—4882no—
394C24—2442no—
395C24—2442no—
396#41—2448no—
397#41—2448no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8C2×C4×D6112yes—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—612no—
406C2×D4—612no—
407C2×D4—1224no—
408C2×D4—1228no—
409C2×D4—1224no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1228no—
417C2×D4—4882no—
418C2×D4—4882no—
419C2×D4—4882no—
420C2×D4—4882no—
421C2×D4—2442no—
422C2×D4—2442no—
423C2×D4—2448no—
424C2×D4—2448no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4C2×C4×D6112yes—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—1224no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4D4×D6112yes—
441C22×C4C23×D6111yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—618no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—4882no—
454C22⋊C4—28no—
455C22⋊C4—618no—
456C22⋊C4—1228no—
457C22⋊C4—4882no—
458C42—612no—
459C42—612no—
460C42—4882no—
461C42C2×C4×D6112yes—
462C42D4×D6112yes—
463C23—22no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—612no—
469C23—1222no—
470C23—1222no—
471C23—1221no—
472C23—1221no—
473C23—1222no—
474C23—1222no—
475C23—4888no—
476C23—4888no—
477C23—4884no—
478C23—4884no—
479C23—2448no—
480C23—2442no—
481C23—2442no—
482C23—2442no—
483C23—2442no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23—2441no—
488C23C2×D4×D6111yescharacteristic
489Q8—24no—
490Q8—42no—
491D4—24no—
492D4—42no—
493D4—1224no—
494D4—1224no—
495D4—4888no—
496D4—4888no—
497D4—2448no—
498D4—2448no—
499D4—2442no—
500D4—2442no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—614no—
510C2×C4—612no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1224no—
516C2×C4—1224no—
517C2×C4—2448no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4C22⋊C4×D6114yes—
522C2×C4C22×C4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4#1451982212yes—
525C2×C4#1451982312yes—
526C8—42no—
527C8—2442no—
528C22—44no—
529C22—42no—
530C22—82no—
531C22—81no—
532C22—81no—
533C22—1224no—
534C22—1222no—
535C22—48816no—
536C22—4882no—
537C22—4882no—
538C22—4881no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—2444no—
543C22—2444no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22—2441no—
548C22—2441no—
549C22C2×C22⋊C4×D6111yescharacteristic
550C22#1684032511yescharacteristic
551C22#1684033011yescharacteristic
552C22#1684056412yes—
553C22#1684056912yes—
554C4—24no—
555C4—42no—
556C4—82no—
557C4—1224no—
558C4—1222no—
559C4—4882no—
560C4#1684023412yes—
561C4#1684032812yes—
562C2—82no—
563C2—4882no—
564C2—4882no—
565C2—2442no—
566C2#2144839811yescharacteristic
567C2#2145434112yes—
568C2#2145687211yescharacteristic
569C2#2145689411yescharacteristic
570C2#2145690612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC2×#16840330C2
cosocleC22×C4×C61C24
hypercenter#1958D61
hypocenterC61#21813
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16839821C2semidirect
3. #21448429C2semidirect
4. #21446748C2semidirect
5. #21446744C2semidirect
6. #21446642C2semidirect
7. #21446636C2semidirect
8. #21446625C2semidirect
9. #21445758C2semidirect
10. #21445748C2semidirect
11. C61×#1958C2semidirect
15. #16840330C4semidirect
23. C2×#14519706C22semidirect
24. C2×#14519701C22semidirect
26. C2×#14519699C22semidirect
29. C2×#14519697C22semidirect
31. C2×#14519688C22semidirect
34. #16839632C22semidirect
36. C4⋊C4×Dic61C22semidirect
38. C2×#103×C61C22semidirect
40. C2×C4×C8×C61C22semidirect
41. C61×#382C22semidirect
284. #1958D61semidirect
311. C2×#103C2×D61semidirect
320. C2×C4×C8C2×D61semidirect
324. #382C2×D61semidirect
327. C61#21813semidirect

Maximal subgroups

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

subgroupindex
C2×#168403302
C2×#168398212
#214484292
#214467482
#214467442
#214466422
#214466362
#214466252
#214457582
#214457482
C61×#19582
#2181361