Finite Group Databasebeta

← #30610315

Subgroups — #30610315

Order 15616.

1346 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462680C212yesmaximal, complemented
2#21461096C214yesmaximal, complemented
3#21461090C212yesmaximal, complemented
4#21461078C211yesmaximal, characteristic, complemented
5#21461065C212yesmaximal, complemented
6#21461058C211yesmaximal, characteristic, complemented
7D61⋊#323C212yesmaximal, complemented
8C2×#16840403C211yesmaximal, characteristic
9#16840796—22no—
10#16840792—24no—
11#16840790—22no—
12#16840783C2212yescomplemented
13#16840779C2212yescomplemented
14#16840733C2214yescomplemented
15D61⋊(C22⋊C8)C2214yescomplemented
16#16840606C2214yescomplemented
17#16840606C2212yescomplemented
18#16840606C2214yescomplemented
19#16840606C2212yescomplemented
20#16840605C2211yescharacteristic
21#16840605C2212yes—
22#16840605C2211yescharacteristic
23C22×#11758226C411yescharacteristic
24C23×D244C411yescharacteristic
25#16840403—22no—
26#16840403—24no—
27#16840403—22no—
28C2×#14519766C2211yescharacteristic
29C2×#14519762C412yes—
30C2×#14519752C411yescharacteristic
31C2×#14519752C2212yes—
32C2×#14519747C412yes—
33C2×#14519739C2211yescharacteristic
34C2×C4×C22⋊D61C2211yescharacteristic
35C2×C22⋊C4×D61C2212yes—
36C2×C22⋊Q8×C61C411yescharacteristic
37C2×#11758213—24no—
38C2×D61⋊C8—24no—
39#14519844—22no—
40C4⋊#92420—22no—
41#14519834—24no—
42#14519834—24no—
43C22⋊#92420C2312yes—
44C22⋊#92420C2314yes—
45C22⋊#92420C2312yes—
46#14519832—22no—
47C4×#92420—22no—
48C24×D61C2×C412yes—
49C2×#11758226—21no—
50C2×#11758226—22no—
51C2×#11758226—21no—
52C2×#11758226—44no—
53C22×D244—21no—
54C22×D244—22no—
55C22×D244—21no—
56C22×D244—44no—
57C22×D244C2×C411yescharacteristic
58C22×D244C2×C412yes—
59C22×D244C2×C411yescharacteristic
60C22×D244C2×C412yes—
61C22×D244C2×C412yes—
62C22×C22⋊D61C2×C411yescharacteristic
63C22×C4×D61C2×C412yes—
64C22×C4×D61C2×C411yescharacteristic
65C22×C4×D61C2312yes—
66#14519766—22no—
67#14519766—22no—
68#14519762—24no—
69#14519762—24no—
70#14519752—24no—
71#14519752—24no—
72#14519752—22no—
73#14519752—22no—
74#14519747—24no—
75#14519747—24no—
76#14519739D412yes—
77#14519739D414yes—
78#14519739D412yes—
79C4×C22⋊D61—22no—
80C4×C22⋊D61—22no—
81C2×#11758196C2×C411yescharacteristic
82C2×C22⋊Dic61C2311yescharacteristic
83C2×#11758198C2×C412yes—
84C2×#11758198C2×C412yes—
85C2×#11758198C2×C412yes—
86C2×#11758198C2312yes—
87C2×#11758198C2311yescharacteristic
88C22⋊C4×D61—24no—
89C22⋊C4×D61—24no—
90C2×C4×Dic61C2311yescharacteristic
91C22×Q8×C61C2×C411yescharacteristic
92C23×C4×C61C2×C411yescharacteristic
93C22⋊Q8×C61—22no—
94C22⋊Q8×C61—22no—
95C2×C4⋊C4×C61C2×C412yes—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C22⋊C4×C61C2×C412yes—
98C24×C61C22×C411yescharacteristic
99C23×D61—24no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61—42no—
103C23×D61—44no—
104C23×D61—82no—
105C23×D61C22×C412yes—
106C23×D61C22×C411yescharacteristic
107C23×D61C22×C412yes—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—42no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—42no—
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C412yes—
118C2×C22⋊D61—22no—
119C2×C22⋊D61—22no—
120C2×C22⋊D61—41no—
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C2411yescharacteristic
124#11758226—48no—
125#11758226—44no—
126#11758226—44no—
127C2×D244—24no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—24no—
135C2×D244—48no—
136C2×D244—44no—
137C2×D244—44no—
138C2×D244—42no—
139C2×D244—44no—
140C2×D244—44no—
141C2×D244—82no—
142C2×D244C22⋊C412yes—
143C2×D244C22⋊C411yescharacteristic
144C2×D244C22⋊C411yescharacteristic
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—42no—
150C2×C4×D61—44no—
151C2×C4×D61—82no—
152C2×C4×D61C2×D414yes—
153C2×C4×D61C2×D414yes—
154C2×#92420—22no—
155C2×#92420—22no—
156C2×#92420—24no—
157#11758213—84no—
158D61⋊C8—84no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61C22⋊C412yes—
161C4⋊C4×C61C22⋊C412yes—
162C22⋊C4×C61—24no—
163C22⋊Dic61—22no—
164#11758198—24no—
165#11758198—24no—
166#11758198—24no—
167#11758198—22no—
168#11758198C22⋊C414yes—
169#11758198C22⋊C414yes—
170#11758196—22no—
171C4×Dic61C2×D412yes—
172C4×Dic61C2×D412yes—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61#13112yes—
178C23×C61C23×C411yescharacteristic
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—24no—
182C22×D61—22no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—22no—
186C22×D61—21no—
187C22×D61—22no—
188C22×D61—42no—
189C22×D61—44no—
190C22×D61—44no—
191C22×D61—44no—
192C22×D61—44no—
193C22×D61—41no—
194C22×D61—42no—
195C22×D61—41no—
196C22×D61—42no—
197C22×D61—82no—
198C22×D61—82no—
199C22×D61—88no—
200C22×D61—88no—
201C22×D61C2×C22⋊C412yes—
202C22×D61C2×C22⋊C412yes—
203Q8×C61—42no—
204Q8×C61—42no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61C2×C22⋊C414yes—
215C2×C4×C61C2×C22⋊C411yescharacteristic
216C22⋊D61—44no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61C22×D412yes—
224D244—24no—
225D244—22no—
226D244—42no—
227D244—44no—
228D244—44no—
229D244—44no—
230D244—44no—
231D244—42no—
232D244—84no—
233D244#10014yes—
234C4×D61—44no—
235C4×D61—44no—
236C4×D61—42no—
237C4×D61—44no—
238C4×D61—44no—
239C4×D61—42no—
240#92420—42no—
241#92420—44no—
242#92420—42no—
243#17421—611nomaximal, Sylow
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—22no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61C22×C22⋊C411yescharacteristic
252C22×C61C2×#13111yescharacteristic
253C22×C61#56811yescharacteristic
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—44no—
260C2×D61—44no—
261C2×D61—41no—
262C2×D61—42no—
263C2×D61—41no—
264C2×D61—42no—
265C2×D61—41no—
266C2×D61—42no—
267C2×D61—88no—
268C2×D61—88no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—82no—
272C2×D61—82no—
273C2×D61C2×#10014yes—
274C4×C61—24no—
275C4×C61—21no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61C2×#10012yes—
279Dic61—22no—
280Dic61—22no—
281Dic61—41no—
282C2×#546—611no—
283#2974—612no—
284#1471—614no—
285#1469—612no—
286#1466—611no—
287#1462—612no—
288#1460—611no—
289C2×#323—612no—
290C2×C61—21no—
291C2×C61—21no—
292C2×C61—42no—
293C2×C61#288011yescharacteristic
294C2×C61#288311yescharacteristic
295C2×C61C22×#10011yescharacteristic
296D61—24no—
297D61—42no—
298D61—81no—
299D61—82no—
300C22×#41—611no—
301C23×D4—611no—
302#546—1222no—
303#546—1224no—
304#546—1222no—
305C2×#124—611no—
306C2×#123—612no—
307C2×C22⋊Q8#692311yescharacteristic
308C2×#121—612no—
309C2×#121—611no—
310C2×#120—612no—
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C22×C22⋊C4—612no—
314#436—1222no—
315#435—1224no—
316C8×D4—1222no—
317#432—612no—
318#431—612no—
319#407—614no—
320C2×C22⋊C8—614no—
321#323—611no—
322#323—612no—
323#323—611no—
324#323—614no—
325#323—612no—
326#323—614no—
327#323—612no—
328C61#1742111yescharacteristic, complemented, Sylow
329C25—612no—
330C2×#41—1221no—
331C2×#41—1222no—
332C2×#41—1221no—
333C2×#41—2444no—
334C22×Q8C2×#692311yescharacteristic
335C22×D4—611no—
336C22×D4—611no—
337C22×D4—612no—
338C22×D4—612no—
339C22×D4—612no—
340C22×D4—611no—
341C22×D4—1221no—
342C22×D4—1222no—
343C22×D4—1221no—
344C22×D4—2444no—
345C23×C4—612no—
346C23×C4—612no—
347C23×C4—611no—
348C23×C4C2×#692311yescharacteristic
349C2×M16—1224no—
350C22×C8—1224no—
351#124—1222no—
352#124—1222no—
353#123—1224no—
354#123—1224no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357#121—1224no—
358#121—1224no—
359#121—1222no—
360#121—1222no—
361#120—1224no—
362#120—1224no—
363C4×D4—612no—
364C4×D4—614no—
365C4×D4—612no—
366C4×D4—1222no—
367C4×D4—1222no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4C2×#692312yes—
370C2×C4⋊C4C2×#692311yescharacteristic
371C2×C22⋊C4—612no—
372C2×C22⋊C4—611no—
373C2×C22⋊C4—612no—
374C2×C22⋊C4—612no—
375C2×C22⋊C4—611no—
376C2×C22⋊C4—612no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4C2×#692312yes—
380C2×C42—611no—
381C4⋊C8—1222no—
382C4⋊C8—1222no—
383C22⋊C8—612no—
384C22⋊C8—614no—
385C22⋊C8—612no—
386C22⋊C8—1224no—
387C22⋊C8—1224no—
388#97—1222no—
389C4×C8—1222no—
390C24—612no—
391C24—612no—
392C24—611no—
393C24—1222no—
394C24—1222no—
395C24—1224no—
396C24—4882no—
397C24—2442no—
398C24—2444no—
399C24C22×#692311yescharacteristic
400#41—2444no—
401#41—2448no—
402#41—2444no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—42no—
406C2×D4—611no—
407C2×D4—612no—
408C2×D4—611no—
409C2×D4—1224no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1224no—
417C2×D4—1221no—
418C2×D4—1221no—
419C2×D4—4882no—
420C2×D4—2442no—
421C2×D4—2444no—
422C2×D4—2448no—
423C2×D4—2444no—
424C2×D4—2444no—
425C2×D4—2444no—
426C2×D4—2441no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—614no—
433C22×C4—614no—
434C22×C4—611no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—4882no—
440C22×C4—2442no—
441C22×C4—2444no—
442C22×C4C22×#692311yescharacteristic
443C22×C4C22×#692312yes—
444C22×C4C22×#692311yescharacteristic
445C22×C4C22×#692312yes—
446M16—4884no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1224no—
450C2×C8—4884no—
451C4⋊C4—24no—
452C4⋊C4—1222no—
453C4⋊C4C22⋊#692312yes—
454C4⋊C4C22⋊#692312yes—
455C22⋊C4—24no—
456C22⋊C4—614no—
457C22⋊C4—614no—
458C22⋊C4—1224no—
459C22⋊C4—1222no—
460C22⋊C4—1224no—
461C22⋊C4—1222no—
462C22⋊C4—1224no—
463C42—612no—
464C42—612no—
465C23—21no—
466C23—21no—
467C23—41no—
468C23—41no—
469C23—612no—
470C23—612no—
471C23—1224no—
472C23—1222no—
473C23—1222no—
474C23—1221no—
475C23—1222no—
476C23—1221no—
477C23—1222no—
478C23—1222no—
479C23—1221no—
480C23—4882no—
481C23—4882no—
482C23—4888no—
483C23—4888no—
484C23—2442no—
485C23—2444no—
486C23—2444no—
487C23—2441no—
488C23—2441no—
489C23—2442no—
490C23—2442no—
491C23—2444no—
492C23—2444no—
493C23#1451986512yes—
494C23C23×#692311yescharacteristic
495Q8—42no—
496Q8—42no—
497D4—614no—
498D4—1224no—
499D4—1222no—
500D4—4884no—
501D4—2442no—
502D4—2444no—
503D4—2444no—
504D4—2444no—
505D4—2444no—
506D4—2444no—
507D4—2442no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—2444no—
522C2×C4—2444no—
523C2×C4—2442no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4—2441no—
527C2×C4—2442no—
528C2×C4—2441no—
529C2×C4C2×C22⋊#692311yescharacteristic
530C2×C4C2×C22⋊#692314yes—
531C2×C4C2×C22⋊#692311yescharacteristic
532C8—2442no—
533C8—2444no—
534C8—2442no—
535C22—21no—
536C22—21no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—614no—
543C22—1222no—
544C22—1222no—
545C22—1222no—
546C22—1222no—
547C22—1222no—
548C22—4882no—
549C22—4882no—
550C22—4881no—
551C22—4881no—
552C22—4888no—
553C22—4888no—
554C22—2444no—
555C22—2444no—
556C22—2441no—
557C22—2442no—
558C22—2441no—
559C22—2442no—
560C22—2441no—
561C22—2442no—
562C22C22×C22⋊#692311yescharacteristic
563C22C2×#1451986511yescharacteristic
564C22#1684084211yescharacteristic
565C4—24no—
566C4—21no—
567C4—41no—
568C4—42no—
569C4—1222no—
570C4—1222no—
571C4—2441no—
572C4D61⋊#10012yes—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—1224no—
577C2—4882no—
578C2—4881no—
579C2—2442no—
580C2#2146240811yescharacteristic
581C2#2146241911yescharacteristic
582C2C2×D61⋊#10011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840403C2
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#17421
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21462680C2semidirect
2. #21461096C2semidirect
3. #21461090C2semidirect
4. #21461078C2semidirect
5. #21461065C2semidirect
6. #21461058C2semidirect
7. D61⋊#323C2semidirect
12. #16840783C22semidirect
13. #16840779C22semidirect
14. #16840733C22semidirect
15. D61⋊(C22⋊C8)C22semidirect
16. #16840606C22semidirect
17. #16840606C22semidirect
18. #16840606C22semidirect
19. #16840606C22semidirect
328. C61#17421semidirect

Maximal subgroups

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

subgroupindex
#214626802
#214610962
#214610902
#214610782
#214610652
#214610582
D61⋊#3232
C2×#168404032
#1742161