Finite Group Databasebeta

← #29683947

Subgroups — #29683947

Order 15616.

1566 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic
2C2×#16839805C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic
4C2×M16×Dic61C211yesmaximal, characteristic, complemented
5#21448307C211yesmaximal, characteristic
6C2×#16839669C211yesmaximal, characteristic, complemented
7#21446697C212yesmaximal, complemented
8#21446687C212yesmaximal, complemented
9#21446677C212yesmaximal, complemented
10#21446670C212yesmaximal, complemented
11C2×C61×#419C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519808—24no—
14C2×#14519808—24no—
15C2×#14519808C418yescomplemented
16C22×#11758202C2212yes—
17C22×#11758201C2212yes—
18C22×#11758194C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20C22×C4×Dic61C2211yescharacteristic
21#16839805—24no—
22C2×#14519710C2212yes—
23C2×#14519708C2212yes—
24#16839764—24no—
25C2×#14519702C2212yes—
26C2×#14519699C2212yes—
27M16×Dic61—24no—
28C2×#14519687C2212yes—
29C2×C8×Dic61C2212yes—
30#16839669—22no—
31#16839669C2212yes—
32#16839669C2212yes—
33#16839667C2211yescharacteristic
34#16839667C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839639—22no—
38C4×#11758198—22no—
39C2×#14519684—22no—
40C23×Q8×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#419—24no—
44C2×#103×C61C2212yes—
45C2×#103×C61C2212yes—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C23×Dic61D412yes—
57C22×C4×D61—21no—
58#14519808—232no—
59C2×#11758202—24no—
60C2×#11758202—24no—
61C2×#11758202C2×C418yescomplemented
62C2×#11758201—24no—
63C2×#11758201—24no—
64C2×#11758201C2×C418yescomplemented
65C2×#11758194C2×C414yes—
66C2×#11758194D414yes—
67C2×#11758194D414yes—
68C22×#92418C2312yes—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61D412yes—
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80#14519710—24no—
81#14519708—24no—
82#14519702—24no—
83#14519699—24no—
84#14519687—24no—
85C8×Dic61—24no—
86#14519684—24no—
87#14519684—24no—
88#14519684—22no—
89#14519684—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61—22no—
92C22×Q8×C61—22no—
93C22×Q8×C61—22no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61C2×C414yes—
97C22×Q8×C61C2312yes—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—22no—
101C2×M16×C61—22no—
102C2×M16×C61D414yes—
103C22×C8×C61C2312yes—
104#117×C61—22no—
105#117×C61—22no—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110#103×C61—24no—
111#103×C61—24no—
112C24×C61C2×D411yescharacteristic
113C23×D61—21no—
114C2×Q8×C61—24no—
115C2×Q8×C61—22no—
116C2×Q8×C61—28no—
117C2×Q8×C61—28no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—42no—
123C2×Q8×C61—44no—
124C2×Q8×C61—44no—
125C2×Q8×C61—42no—
126C2×Q8×C61C22×C414yes—
127C22×C4×C61—24no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C22×Dic61—24no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—22no—
148C22×Dic61—42no—
149C22×Dic61—44no—
150C22×Dic61—42no—
151C22×Dic61C2×D412yes—
152C22×Dic61C2×D412yes—
153#11758222—28no—
154#11758222—28no—
155#11758222—44no—
156#11758222—44no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×C4×D61—24no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122C22×C412yes—
172M16×C61—28no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61C2×D414yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C22⋊C4×C61—42no—
181C42×C61—42no—
182#11758202—216no—
183#11758201—216no—
184#11758198—44no—
185#11758198—44no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—42no—
189C4×Dic61—24no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194#11758194C22⋊C4116yescomplemented
195C2×#92418C22×C414yes—
196C2×#92418C2×D414yes—
197C2×#92418C2×D414yes—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61C4×D412yes—
201C23×C61#12112yes—
202C23×C61#12712yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—42no—
208C22×D61—42no—
209Q8×C61—24no—
210Q8×C61—24no—
211Q8×C61—42no—
212Q8×C61—42no—
213Q8×C61—48no—
214C2×C4×C61—24no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—48no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61C2×C22⋊C414yes—
223C2×C4×C61C4×D414yes—
224C2×C4×C61C4×D412yes—
225C2×C4×C61#12114yes—
226C2×C4×C61#12112yes—
227C2×C4×C61#12414yes—
228C2×C4×C61#12712yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—48no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—48no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—42no—
246C2×Dic61—44no—
247C2×Dic61—84no—
248C2×Dic61#13714yescomplemented
249D244—22no—
250D244—22no—
251D244—41no—
252D244—41no—
253C4×D61—48no—
254Dic122—22no—
255Dic122—22no—
256Dic122—41no—
257Dic122—41no—
258C8×C61—24no—
259#92418C2×C22⋊C418yes—
260#21237—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—24no—
265C22×C61#39014yes—
266C22×C61C2×C4×D411yescharacteristic
267C22×C61C2×#12111yescharacteristic
268C22×C61C2×#12711yescharacteristic
269C22×C61C2×#13712yes—
270C22×C61#57512yes—
271C2×D61—44no—
272C2×D61—42no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—82no—
276C4×C61—44no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61#39014yes—
280C4×C61C22×C22⋊C411yescharacteristic
281C4×C61C2×C4×D411yescharacteristic
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12411yescharacteristic
284Dic61—24no—
285Dic61—42no—
286Dic61—42no—
287Dic61—82no—
288C22×#137—611no—
289C2×#428—611no—
290C2×#417—611no—
291C2×#419D6111yescharacteristic, complemented
292C2×C4×M16—611no—
293#2297—611no—
294C2×#391—611no—
295#1973—612no—
296#1971—612no—
297#1969—612no—
298#1968—612no—
299C2×C61—24no—
300C2×C61C2×#39011yescharacteristic
301C2×C61C4×#13712yes—
302C2×C61#305612yes—
303C2×C61#315211yescharacteristic
304C2×C61#315311yescharacteristic
305D61—82no—
306C22×#41—611no—
307C23×Q8C2×D6111yescharacteristic
308C2×#137—618no—
309C2×#137—1224no—
310C2×#137—1224no—
311C22×Q16—612no—
312C22×SD16—612no—
313C22×M16—611no—
314C22×M16C2×D6111yescharacteristic
315C2×#117C2×D6111yescharacteristic
316C22×C4⋊C4—611no—
317C22×C42—611no—
318#428—1224no—
319C2×#107—612no—
320C2×#106—612no—
321#419—24no—
322#417—1224no—
323C2×#103—612no—
324C2×#103C2×D6112yes—
325C2×#103C2×D6112yes—
326C2×#102—612no—
327C4×M16—1224no—
328C2×#97—612no—
329C2×C4×C8—612no—
330#391—612no—
331#391—612no—
332#391—611no—
333#391—611no—
334#391—1222no—
335#390—611no—
336#390—611no—
337#386—1222no—
338C4×C22⋊C4—1222no—
339C2×#95—1222no—
340C61#2123711yescharacteristic, complemented, Sylow
341C2×#41—614no—
342C2×#41—1222no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C22×Q8—21no—
348C22×Q8—22no—
349C22×Q8—22no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—611no—
354C22×Q8C4×D6114yes—
355C22×Q8C22×D6112yes—
356C22×D4—611no—
357C22×D4—1221no—
358C23×C4—21no—
359C23×C4—612no—
360C23×C4—1221no—
361C23×C4—1221no—
362C23×C4C22×D6111yescharacteristic
363#137—12232no—
364C2×Q16—618no—
365C2×Q16—1224no—
366C2×Q16—1224no—
367C2×SD16—618no—
368C2×SD16—1224no—
369C2×SD16—1224no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—614no—
373C2×M16—614no—
374C2×M16—614no—
375C2×M16C22⋊D6114yes—
376C22×C8—612no—
377C22×C8C22×D6112yes—
378#117—22no—
379#117—22no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C42—21no—
390C2×C42—612no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—1222no—
394C2×C42—1222no—
395#107—1224no—
396#106—1224no—
397#103—24no—
398#103—24no—
399#103—1224no—
400#102—1224no—
401#97—1224no—
402C4×C8—1224no—
403#95—1224no—
404#95—1224no—
405#95—1222no—
406#95—1222no—
407C24—1221no—
408C24D4×D6111yescharacteristic
409#41—1228no—
410#41—1228no—
411#41—2444no—
412#41—2444no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—42no—
425C2×Q8—612no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8C2×C4×D6114yes—
432C2×D4—612no—
433C2×D4—1224no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—2442no—
440C2×D4—2442no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—1224no—
450C22×C4—1221no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—2442no—
455C22×C4—2444no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4C2×C4×D6112yes—
460C22×C4#1175822212yes—
461C22×C4D4×D6112yes—
462C22×C4D4×D6112yes—
463C22×C4D4×D6111yescharacteristic
464C22×C4#1175822412yes—
465C22×C4C2×C22⋊D6112yes—
466C22×C4C23×D6111yescharacteristic
467Q16—12216no—
468SD16—12216no—
469M16—28no—
470M16—6116no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—614no—
474C2×C8—614no—
475C2×C8—614no—
476C2×C8C2×C22⋊D6114yes—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—2442no—
484C22⋊C4—42no—
485C22⋊C4—2444no—
486C22⋊C4—2444no—
487C42—42no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—1224no—
493C23—22no—
494C23—22no—
495C23—1222no—
496C23—2441no—
497C23—2442no—
498C23—2441no—
499C23—2442no—
500C23#1451973712yes—
501C23#1451975012yes—
502C23#1451977312yes—
503C23C2×D4×D6111yescharacteristic
504Q8—24no—
505Q8—24no—
506Q8—42no—
507Q8—42no—
508Q8—48no—
509Q8—1222no—
510Q8—1222no—
511Q8—2441no—
512Q8—2441no—
513D4—1222no—
514D4—1222no—
515D4—2448no—
516D4—2441no—
517D4—2441no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—614no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—4884no—
534C2×C4—2448no—
535C2×C4—2442no—
536C2×C4—2444no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2448no—
540C2×C4C22⋊C4×D6114yes—
541C2×C4#1451973712yes—
542C2×C4C4×C22⋊D6114yes—
543C2×C4#1451974914yes—
544C2×C4#1451975012yes—
545C2×C4#1451976314yes—
546C2×C4#1451977312yes—
547C2×C4C22×C4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4C2×#1175822211yescharacteristic
553C8—24no—
554C8—618no—
555C22—24no—
556C22—24no—
557C22—22no—
558C22—22no—
559C22—4882no—
560C22—4881no—
561C22—4881no—
562C22—2444no—
563C22—2442no—
564C22#1683966214yes—
565C22C2×#1451973711yescharacteristic
566C22C2×#1451975011yescharacteristic
567C22C2×#1451977311yescharacteristic
568C22#137×D6112yes—
569C22#1684056112yes—
570C4—44no—
571C4—81no—
572C4—81no—
573C4—1224no—
574C4—4882no—
575C4—2442no—
576C4—2442no—
577C4#1683966214yes—
578C4C2×C22⋊C4×D6111yescharacteristic
579C4C2×C4×C22⋊D6111yescharacteristic
580C4C2×#1451974911yescharacteristic
581C4C2×#1451976311yescharacteristic
582C2—24no—
583C2—4882no—
584C2C2×#1683966211yescharacteristic
585C2#2145463412yes—
586C2#2145544712yes—
587C2#2145627411yescharacteristic
588C2#2145628611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#21237
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×#16839805C2semidirect
4. C2×M16×Dic61C2semidirect
6. C2×#16839669C2semidirect
7. #21446697C2semidirect
8. #21446687C2semidirect
9. #21446677C2semidirect
10. #21446670C2semidirect
11. C2×C61×#419C2semidirect
15. C2×#14519808C4semidirect
61. C2×#11758202C2×C4semidirect
64. C2×#11758201C2×C4semidirect
194. #11758194C22⋊C4semidirect
248. C2×Dic61#137semidirect
291. C2×#419D61semidirect
340. C61#21237semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168398052
C2×#168397642
C2×M16×Dic612
#214483072
C2×#168396692
#214466972
#214466872
#214466772
#214466702
C2×C61×#4192
#2123761