Finite Group Databasebeta

← #30610004

Subgroups — #30610004

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462674C212yesmaximal, complemented
2#21461080C212yesmaximal, complemented
3#21461076C212yesmaximal, complemented
4#21461070C212yesmaximal, complemented
5#21461065C212yesmaximal, complemented
6#21461056C212yesmaximal, complemented
7D61⋊#323C212yesmaximal, complemented
8C2×#16840236C211yesmaximal, characteristic
9#16840792—28no—
10#16840782C2214yescomplemented
11#16840733C2214yescomplemented
12D61⋊(C22⋊C8)C2214yescomplemented
13#16840607C2214yescomplemented
14#16840606C2214yescomplemented
15#16840606C2214yescomplemented
16#16840605C2214yes—
17C23×D244C411yescharacteristic
18#16840236—28no—
19C2×C4×D244C411yescharacteristic
20C2×C4×D244C411yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519739C411yescharacteristic
24C2×#14519739C411yescharacteristic
25C2×#14519739C2211yescharacteristic
26C2×#14519739C2211yescharacteristic
27C2×#14519736C2211yescharacteristic
28C2×C22⋊C4×D61C411yescharacteristic
29C2×C22⋊C4×D61C411yescharacteristic
30C2×C22⋊C4×D61C2211yescharacteristic
31C2×C22⋊C4×D61C2211yescharacteristic
32C2×#117×C61C411yescharacteristic
33C2×#11758213—24no—
34C2×D61⋊C8—24no—
35#14519844—24no—
36#14519834—24no—
37#14519834—24no—
38C22⋊#92420C2318yes—
39#14519832—24no—
40C24×D61C2×C411yescharacteristic
41C24×D61C2×C411yescharacteristic
42C22×D244—24no—
43C22×D244—24no—
44C22×D244—24no—
45C22×D244C2×C411yescharacteristic
46C22×D244C2×C411yescharacteristic
47C22×D244C2×C411yescharacteristic
48C22×D244C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C4×D244—24no—
58C4×D244—24no—
59C4×D244—24no—
60C4×D244—24no—
61#14519739—24no—
62#14519739—24no—
63#14519739—24no—
64#14519739D418yes—
65#14519736—24no—
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C22⋊Dic61C2×C411yescharacteristic
69C2×C22⋊Dic61C2311yescharacteristic
70C2×#11758198C2×C411yescharacteristic
71C2×#11758198C2×C411yescharacteristic
72C2×#11758198C2×C411yescharacteristic
73C2×#11758198C2×C411yescharacteristic
74C2×#11758198C2×C411yescharacteristic
75C2×#11758198C2×C411yescharacteristic
76C2×#11758198C2311yescharacteristic
77C2×#11758198C2311yescharacteristic
78C22⋊C4×D61—24no—
79C22⋊C4×D61—24no—
80C22⋊C4×D61—24no—
81C22⋊C4×D61—24no—
82C2×C4×Dic61C2×C411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C23×C4×C61C2×C411yescharacteristic
85#117×C61—24no—
86C2×C4⋊C4×C61C2×C411yescharacteristic
87C2×C4⋊C4×C61C2×C411yescharacteristic
88C2×C22⋊C4×C61C2×C411yescharacteristic
89C2×C22⋊C4×C61C2×C411yescharacteristic
90C2×C42×C61C2×C411yescharacteristic
91C2×C42×C61C2×C411yescharacteristic
92C24×C61C22×C411yescharacteristic
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—81no—
99C23×D61—81no—
100C23×D61—81no—
101C23×D61—81no—
102C23×D61M1614yescomplemented
103C23×D61C22×C411yescharacteristic
104C23×D61C22×C411yescharacteristic
105C23×D61C22×C411yescharacteristic
106C23×D61C22×C411yescharacteristic
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61C22×C411yescharacteristic
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C22×C411yescharacteristic
116C22×Dic61C22×C411yescharacteristic
117C22×Dic61C22×C411yescharacteristic
118C22×Dic61C22×C411yescharacteristic
119C22×Dic61C2411yescharacteristic
120C2×D244—232no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244C22⋊C414yes—
132C2×D244M1614yescomplemented
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—81no—
140C2×C4×D61—81no—
141C2×C4×D61—81no—
142C2×C4×D61—81no—
143C2×C4×D61—81no—
144C2×C4×D61—81no—
145C2×C4×D61—81no—
146C2×C4×D61—81no—
147C2×C4×D61C2×D414yes—
148C2×C4×D61C2×D414yes—
149C2×#92420—28no—
150#11758213—84no—
151D61⋊C8—84no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C22⋊C414yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61—22no—
157C42×C61—22no—
158C22⋊Dic61—22no—
159C22⋊Dic61—22no—
160#11758198—22no—
161#11758198—22no—
162#11758198—22no—
163#11758198—22no—
164#11758198—22no—
165#11758198—22no—
166#11758198C22⋊C414yes—
167#11758198C22⋊C414yes—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4×Dic61—22no—
171C4×Dic61C2×D414yes—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61C2×M1612yes—
177C23×C61C23×C411yescharacteristic
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—28no—
185C22×D61—28no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—82no—
195C22×D61—82no—
196C22×D61—82no—
197C22×D61—82no—
198C22×D61—84no—
199C22×D61—84no—
200C22×D61—84no—
201C22×D61—84no—
202C22×D61C2×C22⋊C412yes—
203C22×D61C2×C22⋊C412yes—
204C22×D61C2×M1612yes—
205C22×D61C2×M1612yes—
206C22×D61C2×M1612yes—
207C22×D61C2×M1612yes—
208C2×C4×C61—28no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×M1612yes—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61C2×C22⋊C412yes—
230C2×Dic61C22×D412yes—
231D244—28no—
232D244—28no—
233D244—28no—
234D244—28no—
235C4×D61—44no—
236C4×D61—44no—
237C4×D61—82no—
238C4×D61—82no—
239C4×D61—82no—
240C4×D61—82no—
241C4×D61—82no—
242C4×D61—82no—
243#92420—48no—
244#17367—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61C22×C22⋊C411yescharacteristic
253C22×C61C22×M1611yescharacteristic
254C22×C61C22×M1611yescharacteristic
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—88no—
275C2×D61—88no—
276C2×D61#40714yes—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—41no—
283Dic61—22no—
284Dic61—22no—
285Dic61—41no—
286Dic61—41no—
287C2×#518—611no—
288#2973—612no—
289#1466—612no—
290#1465—612no—
291#1464—612no—
292#1462—612no—
293#1459—612no—
294C2×#323—612no—
295C2×C61—21no—
296C2×C61—21no—
297C2×C61—42no—
298C2×C61C2×#40711yescharacteristic
299C2×C61#288311yescharacteristic
300C2×C61#288811yescharacteristic
301D61—24no—
302D61—42no—
303D61—42no—
304D61—42no—
305C23×D4—611no—
306#518—1228no—
307C2×C4×D4—611no—
308C2×C4×D4—611no—
309C2×C4×D4—611no—
310C2×C4×D4—611no—
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C2×C4×D4—611no—
314C2×C4×D4—611no—
315C2×#117—611no—
316C2×#117#692311yescharacteristic
317C22×C22⋊C4—611no—
318C22×C22⋊C4—611no—
319C22×C22⋊C4—611no—
320C22×C22⋊C4—611no—
321#435—1228no—
322#432—614no—
323#407—614no—
324C2×C22⋊C8—614no—
325#324—614no—
326#323—614no—
327#323—614no—
328#323—614no—
329C61#1736711yescharacteristic, complemented, Sylow
330C25—611no—
331C25—611no—
332C22×D4—611no—
333C22×D4—611no—
334C22×D4—611no—
335C22×D4—611no—
336C22×D4—1224no—
337C22×D4—1224no—
338C22×D4—1224no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4C2×#692311yescharacteristic
348C2×M16—1224no—
349C22×C8—1224no—
350C4×D4—618no—
351C4×D4—1224no—
352C4×D4—1224no—
353C4×D4—1224no—
354C4×D4—1224no—
355C4×D4—1224no—
356C4×D4—1224no—
357C4×D4—1224no—
358#117—24no—
359#117—1224no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4C2×#692311yescharacteristic
363C2×C4⋊C4C2×#692311yescharacteristic
364C2×C22⋊C4—611no—
365C2×C22⋊C4—611no—
366C2×C22⋊C4—611no—
367C2×C22⋊C4—611no—
368C2×C22⋊C4—611no—
369C2×C22⋊C4—611no—
370C2×C22⋊C4—611no—
371C2×C22⋊C4—611no—
372C2×C22⋊C4—611no—
373C2×C22⋊C4—611no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4C2×#692311yescharacteristic
379C2×C22⋊C4C2×#692311yescharacteristic
380C2×C42—611no—
381C2×C42—611no—
382C2×C42C2×#692311yescharacteristic
383C2×C42C2×#692311yescharacteristic
384C4⋊C8—1224no—
385C22⋊C8—618no—
386C22⋊C8—1224no—
387C22⋊C8—1224no—
388#97—1224no—
389C24—611no—
390C24—614no—
391C24—611no—
392C24—611no—
393C24—611no—
394C24—1222no—
395C24—1222no—
396C24—1222no—
397C24—1222no—
398C24—1222no—
399C24—4881no—
400C24—4881no—
401C24—4881no—
402C24—4881no—
403C24C22×#692311yescharacteristic
404C2×D4—614no—
405C2×D4—614no—
406C2×D4—1222no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—12232no—
415C2×D4—1222no—
416C2×D4—1222no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—614no—
423C22×C4—614no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—4881no—
433C22×C4—4881no—
434C22×C4—4881no—
435C22×C4—4881no—
436C22×C4—4881no—
437C22×C4—4881no—
438C22×C4—4881no—
439C22×C4—4881no—
440C22×C4C22×#692311yescharacteristic
441C22×C4C22×#692311yescharacteristic
442C22×C4C22×#692311yescharacteristic
443C22×C4C22×#692311yescharacteristic
444C22×C4C22×#692311yescharacteristic
445C22×C4C22×#692311yescharacteristic
446M16—4884no—
447C2×C8—1228no—
448C2×C8—4884no—
449C4⋊C4—22no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4C22⋊#692314yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—614no—
456C22⋊C4—614no—
457C22⋊C4—1222no—
458C22⋊C4—1222no—
459C22⋊C4—1222no—
460C22⋊C4—1222no—
461C22⋊C4—1222no—
462C22⋊C4—1222no—
463C22⋊C4—1222no—
464C22⋊C4—1222no—
465C42—22no—
466C42—22no—
467C42—614no—
468C42—1222no—
469C23—21no—
470C23—21no—
471C23—41no—
472C23—41no—
473C23—612no—
474C23—612no—
475C23—612no—
476C23—612no—
477C23—612no—
478C23—612no—
479C23—1228no—
480C23—1228no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—1221no—
485C23—1221no—
486C23—1221no—
487C23—4882no—
488C23—4882no—
489C23—4882no—
490C23—4882no—
491C23—4884no—
492C23—4884no—
493C23—4884no—
494C23—4884no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23D61⋊M1612yes—
504C23C23×#692311yescharacteristic
505D4—1228no—
506D4—1228no—
507D4—1228no—
508D4—1228no—
509C2×C4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4882no—
525C2×C4—4882no—
526C2×C4—4882no—
527C2×C4—4882no—
528C2×C4—4882no—
529C2×C4—4882no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C2×C22⋊#692312yes—
537C2×C4C2×C22⋊#692312yes—
538C2×C4C2×C22⋊#692312yes—
539C2×C4D61⋊M1612yes—
540C8—2448no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—41no—
547C22—41no—
548C22—614no—
549C22—1222no—
550C22—1222no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22—1222no—
556C22—4888no—
557C22—4888no—
558C22—2442no—
559C22—2442no—
560C22—2442no—
561C22—2442no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2441no—
567C22—2441no—
568C22—2441no—
569C22—2441no—
570C22C22×C22⋊#692311yescharacteristic
571C22C2×D61⋊M1611yescharacteristic
572C22C2×D61⋊M1611yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—1222no—
580C4—1222no—
581C4—2441no—
582C4—2441no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—1224no—
587C2—2442no—
588C2—2442no—
589C2—2442no—
590C2D61⋊#40711yescharacteristic
591C2#2146241911yescharacteristic
592C2#2146243011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#117×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D244C4
cosocleC22×Dic61C24
hypercenterC2×#117#6923
hypocenterC61#17367
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21462674C2semidirect
2. #21461080C2semidirect
3. #21461076C2semidirect
4. #21461070C2semidirect
5. #21461065C2semidirect
6. #21461056C2semidirect
7. D61⋊#323C2semidirect
10. #16840782C22semidirect
11. #16840733C22semidirect
12. D61⋊(C22⋊C8)C22semidirect
13. #16840607C22semidirect
14. #16840606C22semidirect
15. #16840606C22semidirect
102. C23×D61M16semidirect
132. C2×D244M16semidirect
329. C61#17367semidirect

Maximal subgroups

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

subgroupindex
#214626742
#214610802
#214610762
#214610702
#214610652
#214610562
D61⋊#3232
C2×#168402362
#1736761