Finite Group Databasebeta

← #29734543

Subgroups — #29734543

Order 15616.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456372C211yesmaximal, characteristic, complemented
2#21456219C211yesmaximal, characteristic, complemented
3#21451015C211yesmaximal, characteristic, complemented
4#21447026C212yesmaximal, complemented
5#21447026C212yesmaximal, complemented
6#21446764C211yesmaximal, characteristic, complemented
7#21446739C211yesmaximal, characteristic, complemented
8#21446052C212yesmaximal, complemented
9#21446045C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C61×#3506C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840168—21no—
14#16840168—21no—
15#16840161—22no—
16#16840130—21no—
17#16840126—21no—
18#16840116—22no—
19#16839805C2212yes—
20#16839805C2212yes—
21C2×#14519704C2212yes—
22C2×#14519704C2212yes—
23#16839763C2212yes—
24#16839763C2212yes—
25C2×#14519694C2212yes—
26C4×#11758194C2211yescharacteristic
27C4×#11758194C2211yescharacteristic
28#16839685—22no—
29#16839665C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839635—22no—
32#16839631C2211yescharacteristic
33#16839627C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35#16839531C2212yes—
36#16839521C2212yes—
37#16839521C2212yes—
38C61×#559—22no—
39C61×#556—21no—
40C61×#555—21no—
41C61×#553—22no—
42C61×#550—21no—
43C61×#547—21no—
44C61×#538—22no—
45C61×#535C2212yes—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#525—22no—
49C61×#520C2212yes—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53#14519769—21no—
54#14519769—21no—
55#14519769—21no—
56#14519769—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758196—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C4×Dic61C2311yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67#14519710—22no—
68#14519710—22no—
69#14519708—22no—
70#14519708—22no—
71#14519704—24no—
72#14519704—24no—
73#14519701—22no—
74#14519701—22no—
75#14519698—22no—
76#14519698—22no—
77#14519694—24no—
78#14519688—21no—
79#14519688—21no—
80C4×#92418—21no—
81C4×#92418—21no—
82#14519684—22no—
83#14519684—21no—
84C2×#41×C61—22no—
85C2×#41×C61C2312yes—
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61D412yes—
88C4⋊Q8×C61D411yescharacteristic
89C4⋊Q8×C61D411yescharacteristic
90C4⋊Q8×C61D411yescharacteristic
91#127×C61—22no—
92#127×C61D411yescharacteristic
93#126×C61—24no—
94#125×C61D412yes—
95#124×C61—24no—
96#124×C61—22no—
97#124×C61D412yes—
98#123×C61—24no—
99#123×C61—24no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103#121×C61—24no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—21no—
107#121×C61—22no—
108#121×C61—22no—
109#120×C61—24no—
110C4×Q8×C61—22no—
111C4×Q8×C61—21no—
112C4×Q8×C61—21no—
113C4×D4×C61—24no—
114C4×D4×C61—22no—
115C4×D4×C61—24no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61—21no—
122#117×C61—22no—
123#117×C61—22no—
124#117×C61C2312yes—
125C2×C4⋊C4×C61—22no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61D412yes—
128C2×C42×C61C2311yescharacteristic
129C24×C61C2×D412yes—
130#41×C61—22no—
131#41×C61—22no—
132#41×C61—22no—
133#41×C61—22no—
134#41×C61—84no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61C2×D412yes—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—24no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61C2×D412yes—
148C22×C4×C61—24no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—82no—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61C2×D412yes—
171C4⋊C4×C61C2×D412yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—24no—
181C42×C61—22no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C22⋊Dic61—44no—
187C22⋊Dic61—82no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—42no—
193#11758196—41no—
194#11758196—41no—
195#11758196—41no—
196#11758196—41no—
197#11758196—82no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200#11758194—22no—
201#11758194—22no—
202#11758194—22no—
203#11758194#4114yescomplemented
204C2×#92418—22no—
205C2×#92418—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61C22×D411yescharacteristic
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—42no—
216Q8×C61—81no—
217D4×C61—22no—
218D4×C61—22no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—81no—
225D4×C61—84no—
226D4×C61—84no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—44no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—82no—
238C2×C4×C61—84no—
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4112yes—
246C2×C4×C612−1+412yes—
247C2×Dic61—22no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—41no—
252C2×Dic61—42no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257#92418—24no—
258#92418—42no—
259#25697—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—44no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—82no—
270C22×C61C2×#12011yescharacteristic
271C22×C61#53211yescharacteristic
272C22×C61#53611yescharacteristic
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61#54712yes—
282C4×C61#54812yes—
283Dic61—42no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287#3506D6111yescharacteristic
288#3166—611no—
289#3142—611no—
290#2566—611no—
291#2037—612no—
292#2037—612no—
293#1986—611no—
294#1980—611no—
295#1860—612no—
296#1859—612no—
297C4×#100—611no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—82no—
301C2×C61—82no—
302C2×C61#243211yescharacteristic
303C2×C61#301811yescharacteristic
304C2×C61#301911yescharacteristic
305#559—22no—
306#556—21no—
307#555—21no—
308#553—22no—
309#550—21no—
310#547—21no—
311#538—22no—
312#535C2×D6112yes—
313#533C2×D6111yescharacteristic
314#532C2×D6111yescharacteristic
315C2×#125—611no—
316#525—22no—
317#520C2×D6112yes—
318C2×C4×D4C2×D6111yescharacteristic
319#501—1221no—
320#501—1221no—
321#499—1222no—
322#491—1221no—
323#490—1221no—
324#488—1222no—
325#428—612no—
326#428—612no—
327C2×#104—612no—
328C2×#104—612no—
329#417—612no—
330#417—612no—
331C2×#100—612no—
332C4×M16—611no—
333C4×M16—611no—
334#396—1222no—
335#390—611no—
336#390—611no—
337#385—1222no—
338#384—611no—
339#382—611no—
340C4×C4⋊C4—611no—
341#343—612no—
342#339—612no—
343#339—612no—
344C61#2569711yescharacteristic, complemented, Sylow
345C2×#41—22no—
346C2×#41C22×D6112yes—
347C22×D4C22×D6111yescharacteristic
348C23×C4C22⋊D6112yes—
349C2×M16—612no—
350C2×M16—612no—
351C4⋊Q8C22⋊D6111yescharacteristic
352C4⋊Q8C22⋊D6111yescharacteristic
353C4⋊Q8C22⋊D6111yescharacteristic
354#127—22no—
355#127C22⋊D6111yescharacteristic
356#126—24no—
357#125—1221no—
358#125—1221no—
359#125—1221no—
360#125—1221no—
361#125C22⋊D6112yes—
362#124—24no—
363#124—22no—
364#124C22⋊D6112yes—
365#123—24no—
366#123—24no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370#121—22no—
371#121—24no—
372#121—21no—
373#121—21no—
374#121—22no—
375#121—22no—
376#120—24no—
377C4×Q8—22no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×D4—24no—
381C4×D4—22no—
382C4×D4—24no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389#117—22no—
390#117—22no—
391#117C22×D6112yes—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4C22⋊D6112yes—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42C22×D6111yescharacteristic
406#107—1222no—
407#107—1222no—
408#106—1222no—
409#106—1222no—
410#104—1224no—
411#104—1224no—
412#103—1222no—
413#103—1222no—
414#102—1222no—
415#102—1222no—
416#100—1224no—
417#97—1221no—
418#97—1221no—
419C4×C8—1221no—
420C4×C8—1221no—
421#95—1221no—
422#95—1222no—
423C24C2×C22⋊D6112yes—
424#41—22no—
425#41—22no—
426#41—22no—
427#41—22no—
428#41—84no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×C22⋊D6112yes—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C2×C22⋊D6112yes—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—82no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455M16—614no—
456M16—1222no—
457M16—1222no—
458M16—1222no—
459C2×C8—1222no—
460C2×C8—1222no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—4882no—
475C4⋊C4—2442no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4C2×C22⋊D6112yes—
481C4⋊C4C2×C22⋊D6112yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—4882no—
491C22⋊C4—2444no—
492C42—24no—
493C42—22no—
494C42—1222no—
495C42—1222no—
496C42C2×C22⋊D6111yescharacteristic
497C42C2×C22⋊D6111yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C2×C22⋊D6111yescharacteristic
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—42no—
506C23C22×C22⋊D6111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—42no—
510Q8—81no—
511D4—22no—
512D4—22no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—81no—
519D4—84no—
520D4—84no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—1222no—
534C2×C4—2441no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4#1451974611yescharacteristic
544C2×C4#1451974611yescharacteristic
545C2×C4#1451974611yescharacteristic
546C2×C4#1451974611yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4C22×C22⋊D6111yescharacteristic
549C2×C4#41×D6112yes—
550C2×C4#1451982412yes—
551C8—1224no—
552C8—2442no—
553C22—21no—
554C22—21no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—82no—
563C22C2×#1451974611yescharacteristic
564C22#1684030811yescharacteristic
565C22#1684033311yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—4881no—
575C4—4881no—
576C4—4881no—
577C4—2442no—
578C4#1684041112yes—
579C4#1684041912yes—
580C2—21no—
581C2—21no—
582C2—82no—
583C2—82no—
584C2#2144944311yescharacteristic
585C2#2145516811yescharacteristic
586C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#25697
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456372C2semidirect
2. #21456219C2semidirect
3. #21451015C2semidirect
4. #21447026C2semidirect
5. #21447026C2semidirect
6. #21446764C2semidirect
7. #21446739C2semidirect
8. #21446052C2semidirect
9. #21446045C2semidirect
10. C4×#14519694C2semidirect
203. #11758194#41semidirect
344. C61#25697semidirect

Maximal subgroups

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

subgroupindex
#214563722
#214562192
#214510152
#214470262
#214467642
#214467392
#214460522
#214460452
C4×#145196942
C61×#35062
#2569761