Finite Group Databasebeta

← #29657734

Subgroups — #29657734

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457135C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic, complemented
3#21455624C211yesmaximal, characteristic, complemented
4#21445082C212yesmaximal, complemented
5#21445019C212yesmaximal, complemented
6#21444984C212yesmaximal, complemented
7#21444946C212yesmaximal, complemented
8#21444496C211yesmaximal, characteristic, complemented
9#21444270C211yesmaximal, characteristic, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3497C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840158C2211yescharacteristic
14#16840134C2211yescharacteristic
15#16840114C2211yescharacteristic
16#16840108C2211yescharacteristic
17#16840069—22no—
18#16840057C2212yes—
19#16840052C2212yes—
20#16840047C2212yes—
21#16840044—21no—
22#16839989—22no—
23#16839989—22no—
24#16839967C2212yes—
25#16839959—21no—
26C4×#11758200—22no—
27#16839802—21no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839771—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839508C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C2×2−1+4×C61—21no—
39C61×#550—21no—
40C61×#550C2212yes—
41D4×Q8×C61—21no—
42D4×Q8×C61C2212yes—
43C61×#548—22no—
44C61×#548—22no—
45C61×#537—22no—
46C61×#537—21no—
47C61×#532—22no—
48C61×#532C2211yescharacteristic
49C61×#532C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54C22×#92418—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—22no—
65#14519708—22no—
66#14519708—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519701—22no—
73#14519701—22no—
74#14519700—22no—
75#14519700—22no—
76#14519698—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
812−1+4×C61—44no—
822−1+4×C61—44no—
83C2×#41×C61—24no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61D411yescharacteristic
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61—22no—
95#127×C61D411yescharacteristic
96#127×C61C2311yescharacteristic
97#124×C61—24no—
98#124×C61—21no—
99#124×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—24no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—24no—
104#121×C61—24no—
105#121×C61—22no—
106#121×C61—24no—
107#121×C61—24no—
108#121×C61—21no—
109#121×C61—21no—
110#121×C61—22no—
111#121×C61D412yes—
112#121×C61D412yes—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
115C4×Q8×C61D412yes—
116C4×Q8×C61C2312yes—
117C4×D4×C61—24no—
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—21no—
121C4×D4×C61—21no—
122C4×D4×C61D412yes—
123C4×D4×C61C2312yes—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#41×C61—416no—
129#41×C61—44no—
130#41×C61—48no—
131#41×C61—44no—
132#41×C61—42no—
133#41×C61—42no—
134#41×C61—42no—
135#41×C61—42no—
136C2×Q8×C61—24no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—22no—
140C2×Q8×C61—21no—
141C2×Q8×C61—44no—
142C2×Q8×C61—48no—
143C2×Q8×C61—44no—
144C2×Q8×C61SD1614yescomplemented
145C2×Q8×C61C2×D412yes—
146C2×D4×C61—24no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C2×D4×C61—24no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61C2×D412yes—
155C2×D4×C61C2×D412yes—
156C22×C4×C61—21no—
157C22×C4×C61—24no—
158C22×C4×C61—24no—
159C22×C4×C61—21no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D412yes—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D412yes—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C42×C61—22no—
183C42×C61—22no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758200—44no—
190#11758200—82no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—41no—
198#11758194—42no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—42no—
204C2×#92418—81no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61#12011yescharacteristic
208Q8×C61—22no—
209Q8×C61—24no—
210Q8×C61—48no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—42no—
214Q8×C61C2×SD1614yes—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61—42no—
218D4×C61—48no—
219D4×C61—44no—
220D4×C61—44no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—42no—
226D4×C61—81no—
227D4×C61—84no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—48no—
237C2×C4×C61—48no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—44no—
241C2×C4×C61—81no—
242C2×C4×C61#12011yescharacteristic
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61C2×SD1612yes—
246C2×C4×C61AGL(1,ℤ/8)12yes—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×Dic61—41no—
251C2×Dic61—41no—
252Dic122—42no—
253Dic122—41no—
254Dic122—41no—
255#92418—22no—
256#92418—41no—
257#92418—42no—
258#92418—42no—
259#18675—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61#45012yes—
269C22×C61C2×#12011yescharacteristic
270C4×C61—24no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61#45012yes—
278C4×C61C22×SD1611yescharacteristic
279C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C4×C61#57711yescharacteristic
281C4×C61#57811yescharacteristic
282Dic61—81no—
283Dic61—81no—
284#3497D6111yescharacteristic
285#3246—611no—
286#3167—611no—
287#3078—611no—
288#1651—612no—
289#1642—612no—
290#1637—612no—
291#1631—612no—
292#1547—611no—
293#1492—611no—
294C2×Q8⋊C8—611no—
295C2×C61—22no—
296C2×C61—82no—
297C2×C61—82no—
298C2×C61C2×#45011yescharacteristic
299C2×C61#301311yescharacteristic
300C2×C61#301811yescharacteristic
301C2×2−1+4—21no—
302#550—21no—
303#550C2×D6112yes—
304D4×Q8—21no—
305D4×Q8C2×D6112yes—
306#548—22no—
307#548—22no—
308#537—22no—
309#537—21no—
310#533—611no—
311#532—22no—
312#532C2×D6111yescharacteristic
313#532C2×D6111yescharacteristic
314C2×C4×Q8C2×D6111yescharacteristic
315#498—611no—
316#492—611no—
317#487—611no—
318#486—611no—
319#479—1222no—
320#477—612no—
321#476—612no—
322#475—612no—
323#475—1221no—
324#465—1222no—
325#465—1222no—
326#461—612no—
327#460—1221no—
328C4×SD16—1222no—
329#428—1221no—
330#423—611no—
331C2×C4⋊C8—611no—
332#419—1221no—
333C2×C4×C8—611no—
334#332—612no—
335#331—612no—
336Q8⋊C8—612no—
337Q8⋊C8—612no—
338D4⋊C8—612no—
339D4⋊C8—612no—
340C61#1867511yescharacteristic, complemented, Sylow
3412−1+4—44no—
3422−1+4—44no—
343C2×#41—24no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—21no—
347C2×#41—21no—
348C22×Q8—22no—
349C22×Q8—22no—
350C22×Q8C22⋊D6111yescharacteristic
351C2×SD16—1222no—
352C2×SD16—1222no—
353C2×M16—1221no—
354C22×C8—1221no—
355C22×C8—1221no—
356C4⋊Q8—22no—
357C4⋊Q8—611no—
358C4⋊Q8—611no—
359C4⋊Q8C22⋊D6111yescharacteristic
360C4⋊Q8C22×D6111yescharacteristic
361#127—22no—
362#127C22⋊D6111yescharacteristic
363#127C22×D6111yescharacteristic
364#125—1221no—
365#124—24no—
366#124—21no—
367#124—21no—
368C22⋊Q8—22no—
369C22⋊Q8—24no—
370C22⋊Q8—22no—
371C22⋊Q8—24no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374#121—24no—
375#121—22no—
376#121—24no—
377#121—24no—
378#121—21no—
379#121—21no—
380#121—22no—
381#121C22⋊D6112yes—
382#121C22⋊D6112yes—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—1221no—
386C4×Q8—1221no—
387C4×Q8C22⋊D6112yes—
388C4×Q8C22×D6112yes—
389C4×D4—24no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4C22⋊D6112yes—
395C4×D4C22×D6112yes—
396#117—1221no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4C22⋊D6111yescharacteristic
399C2×C42—21no—
400C2×C42C22×D6111yescharacteristic
401#107—1222no—
402#106—1222no—
403#106—1222no—
404#106—1222no—
405C4⋊C8—612no—
406C4⋊C8—612no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409#103—1222no—
410#103—1222no—
411#103—1222no—
412#103—1222no—
413#102—1222no—
414#102—1222no—
415C4×C8—611no—
416C4×C8—611no—
417C4×C8—1221no—
418#41—416no—
419#41—44no—
420#41—48no—
421#41—44no—
422#41—42no—
423#41—42no—
424#41—42no—
425#41—42no—
426C2×Q8—24no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—44no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8#1175820014yescomplemented
437C2×Q8C2×C22⋊D6112yes—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C22⋊D6112yes—
447C2×D4C2×C22⋊D6112yes—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457SD16—4882no—
458SD16—2444no—
459M16—2442no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—4881no—
465C2×C8—2442no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—2441no—
477C4⋊C4—2441no—
478C4⋊C4C2×C22⋊D6112yes—
479C4⋊C4C2×C22⋊D6112yes—
480C4⋊C4C2×C22⋊D6112yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—2441no—
488C42—22no—
489C42—22no—
490C42—2441no—
491C42C2×C22⋊D6111yescharacteristic
492C42C2×C22⋊D6111yescharacteristic
493C42C2×C22⋊D6111yescharacteristic
494C42C23×D6111yescharacteristic
495C23—22no—
496C23—22no—
497C23#1451974611yescharacteristic
498Q8—22no—
499Q8—24no—
500Q8—48no—
501Q8—41no—
502Q8—41no—
503Q8—42no—
504Q8—2442no—
505Q8—2441no—
506Q8—2441no—
507Q8C2×#1175820014yes—
508D4—22no—
509D4—22no—
510D4—42no—
511D4—48no—
512D4—44no—
513D4—44no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—42no—
519D4—81no—
520D4—84no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—81no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4#1451974611yescharacteristic
538C2×C4#1451974611yescharacteristic
539C2×C4#1451974611yescharacteristic
540C2×C4C2×#1175820012yes—
541C2×C4#1451980312yes—
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C8—1222no—
546C8—2441no—
547C8—2442no—
548C8—2442no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22#1683991312yes—
558C22C2×#1451974611yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—4881no—
567C4—4881no—
568C4#1683991312yes—
569C4C22×#1175820011yescharacteristic
570C4C2×#1451980311yescharacteristic
571C4#1684057011yescharacteristic
572C4#1684057711yescharacteristic
573C2—22no—
574C2—82no—
575C2—82no—
576C2C2×#1683991311yescharacteristic
577C2#2145513411yescharacteristic
578C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18675
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21457135C2semidirect
2. #21456374C2semidirect
3. #21455624C2semidirect
4. #21445082C2semidirect
5. #21445019C2semidirect
6. #21444984C2semidirect
7. #21444946C2semidirect
8. #21444496C2semidirect
9. #21444270C2semidirect
10. C2×Q8⋊#92418C2semidirect
144. C2×Q8×C61SD16semidirect
340. C61#18675semidirect
436. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214571352
#214563742
#214556242
#214450822
#214450192
#214449842
#214449462
#214444962
#214442702
C2×Q8⋊#924182
C61×#34972
#1867561