Finite Group Databasebeta

← #29661616

Subgroups — #29661616

Order 15616.

948 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic, complemented
2#21456366C211yesmaximal, characteristic, complemented
3C2×#16840157C211yesmaximal, characteristic, complemented
4#21445136C214yesmaximal, complemented
5#21445130C214yesmaximal, complemented
6#21444434C211yesmaximal, characteristic, complemented
7#21444417C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3561C211yesmaximal, characteristic
10C22×#11758202—21no—
11C22×#11758200—21no—
12C2×#14519777C2211yescharacteristic
13#16840172C2212yescomplemented
14#16840172C2214yescomplemented
15#16840165—21no—
16#16840158—21no—
17#16840157—22no—
18#16840157C2214yes—
19#16840121C2212yescomplemented
20#16840095—21no—
21#16840078—21no—
22#16840047—21no—
23#16840042—21no—
24#16839819C2211yescharacteristic
25#16839805—22no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2212yescomplemented
29#16839504C2214yescomplemented
30#16839502C2214yes—
31#16839502C2212yescomplemented
32C61×#563C2212yes—
33C61×#562C2212yes—
34C61×#556—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37C61×#548—21no—
38C61×#546—21no—
39C61×#543—21no—
40C61×#542—21no—
41C61×#542—22no—
42C61×#538—21no—
43C61×#536—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758202—24no—
49C2×#11758202—44no—
50C2×#11758200—24no—
51C2×#11758200—44no—
52C2×#11758194—21no—
53C22×#92418—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777C2312yes—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60#14519710—22no—
61#14519710—24no—
62#14519708—22no—
63#14519708—24no—
64C4⋊#92418—21no—
65#14519701—22no—
66#14519698—22no—
67C22⋊#92418—21no—
68#14519688—21no—
69#14519688C2312yes—
70#14519688C2312yes—
71#14519688C2312yes—
72C4×#92418—21no—
73C2×#41×C61—21no—
74C22×Q8×C61—21no—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—22no—
79C4⋊Q8×C61D411yescharacteristic
80C4⋊Q8×C61D411yescharacteristic
81#126×C61—22no—
82#126×C61—22no—
83#126×C61—22no—
84#126×C61—21no—
85#125×C61C2312yes—
86#125×C61C2311yescharacteristic
87#124×C61—21no—
88#124×C61—22no—
89#124×C61—22no—
90#124×C61—21no—
91#124×C61—21no—
92#124×C61D412yes—
93#124×C61C2311yescharacteristic
94#124×C61C2312yes—
95#123×C61—22no—
96#123×C61—22no—
97#123×C61—21no—
98#123×C61—22no—
99#123×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—21no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61D411yescharacteristic
111#121×C61—21no—
112#121×C61—22no—
113#121×C61—22no—
114#121×C61D411yescharacteristic
115#120×C61—22no—
116C4×Q8×C61—21no—
117C4×Q8×C61—21no—
118C4×Q8×C61—21no—
119C4×Q8×C61D411yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—22no—
123C4×D4×C61—21no—
124C4×D4×C61D411yescharacteristic
125#117×C61—22no—
126#117×C61—21no—
127#117×C61D412yes—
128C2×C22⋊C4×C61—22no—
129C2×C22⋊C4×C61—21no—
130C2×C22⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132C24×C61—21no—
133#41×C61—82no—
134C2×Q8×C61—22no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—42no—
139C2×Q8×C61—82no—
140C2×Q8×C61C2×D411yescharacteristic
141C2×Q8×C61C2×D412yes—
142C2×D4×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—42no—
147C2×D4×C61C2×D411yescharacteristic
148C2×D4×C61C2×D412yes—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×Dic61—41no—
158C2×Dic122—22no—
159C2×Dic122—42no—
160C2×Dic122—42no—
161C2×Dic122—41no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61C2×D412yes—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C42×C61—22no—
194C42×C61—22no—
195C42×C61—21no—
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2411yescharacteristic
200#11758202—48no—
201#11758200—48no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61—41no—
205C4⋊Dic61—41no—
206#11758194—24no—
207C2×#92418—24no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—22no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61—42no—
214C23×C61—81no—
215C23×C61—81no—
216C23×C61#12011yescharacteristic
217Q8×C61—42no—
218Q8×C61—41no—
219Q8×C61—42no—
220Q8×C61—42no—
221Q8×C61—81no—
222Q8×C61—82no—
223D4×C61—42no—
224D4×C61—41no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—82no—
228D4×C61—82no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—81no—
240C2×C4×C61—82no—
241C2×C4×C61—82no—
242C2×C4×C61—82no—
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×Dic61—42no—
250C2×Dic61—82no—
251Dic122—42no—
252Dic122—42no—
253Dic122—82no—
254#92418—24no—
255#92418—24no—
256#92418—42no—
257#19033—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—41no—
261C22×C61—42no—
262C22×C61—41no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—41no—
266C22×C61—162no—
267C22×C61#45612yes—
268C22×C61C2×#12011yescharacteristic
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—42no—
280C4×C61#45312yes—
281Dic61—82no—
282#3561D6111yescharacteristic
283#3239—611no—
284#3165—611no—
285C2×#497—611no—
286#1661—614no—
287#1660—614no—
288#1531—611no—
289#1527—611no—
290#1524—611no—
291C2×C61—22no—
292C2×C61—81no—
293C2×C61—82no—
294C2×C61C2×#45311yescharacteristic
295C2×C61C2×#45611yescharacteristic
296C2×C61#302211yescharacteristic
297C22×Q16—1221no—
298C22×SD16—1221no—
299#563C2×D6112yes—
300#562C2×D6112yes—
301#556—21no—
302#551—22no—
303D4×Q8—21no—
304#548—21no—
305#546—21no—
306#543—21no—
307#542—21no—
308#542—22no—
309#538—21no—
310#536—21no—
311#533C2×D6111yescharacteristic
312C2×C4⋊Q8—611no—
313#529C2×D6111yescharacteristic
314C2×#124C2×D6111yescharacteristic
315#501—614no—
316#501—612no—
317#500—1221no—
318#498—1221no—
319#497—614no—
320#497—1222no—
321#489—612no—
322#484—1221no—
323#481—1221no—
324#475—1221no—
325#474—1221no—
326#432—611no—
327#428—1222no—
328C4×M16—611no—
329C2×#97—611no—
330#330—612no—
331#330—614no—
332#329—614no—
333#329—612no—
334C61#1903311yescharacteristic, complemented, Sylow
335C2×#41—21no—
336C22×Q8—21no—
337C22×Q8—1221no—
338C22×Q8C22⋊D6111yescharacteristic
339C22×D4C22⋊D6111yescharacteristic
340C2×Q16—1224no—
341C2×Q16—2444no—
342C2×SD16—1224no—
343C2×SD16—2444no—
344C2×M16—1221no—
345C22×C8—1221no—
346C4⋊Q8—21no—
347C4⋊Q8—22no—
348C4⋊Q8—612no—
349C4⋊Q8—1221no—
350C4⋊Q8—1221no—
351C4⋊Q8—1221no—
352C4⋊Q8C22⋊D6111yescharacteristic
353C4⋊Q8C22⋊D6111yescharacteristic
354#126—22no—
355#126—22no—
356#126—22no—
357#126—21no—
358#125C22×D6111yescharacteristic
359#125C22×D6112yes—
360#124—21no—
361#124—22no—
362#124—22no—
363#124—21no—
364#124—21no—
365#124C22⋊D6112yes—
366#124C22×D6112yes—
367#124C22×D6111yescharacteristic
368#123—22no—
369#123—22no—
370#123—21no—
371#123—22no—
372#123—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8C22⋊D6111yescharacteristic
384#121—21no—
385#121—22no—
386#121—22no—
387#121C22⋊D6111yescharacteristic
388#120—22no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8C22⋊D6111yescharacteristic
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—22no—
396C4×D4—21no—
397C4×D4C22⋊D6111yescharacteristic
398#117—22no—
399#117—21no—
400#117C22⋊D6112yes—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C42C22×D6111yescharacteristic
407#107—1224no—
408#107—1222no—
409#106—1224no—
410#106—1222no—
411C4⋊C8—1221no—
412#103—1222no—
413#102—1222no—
414C22⋊C8—1221no—
415#97—612no—
416#97—612no—
417#97—612no—
418#97—1221no—
419C4×C8—1221no—
420C24—21no—
421#41—82no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—42no—
427C2×Q8—82no—
428C2×Q8—1222no—
429C2×Q8—2442no—
430C2×Q8—2442no—
431C2×Q8—2441no—
432C2×Q8C2×C22⋊D6111yescharacteristic
433C2×Q8C2×C22⋊D6112yes—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4C2×C22⋊D6111yescharacteristic
440C2×D4C2×C22⋊D6112yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—2441no—
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450Q16—2448no—
451SD16—2448no—
452M16—1224no—
453C2×C8—1224no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4C2×C22⋊D6112yes—
474C4⋊C4C2×C22⋊D6111yescharacteristic
475C4⋊C4C2×C22⋊D6112yes—
476C4⋊C4C2×C22⋊D6111yescharacteristic
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—21no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42C2×C22⋊D6111yescharacteristic
496C42C2×C22⋊D6111yescharacteristic
497C42C2×C22⋊D6111yescharacteristic
498C42C23×D6111yescharacteristic
499C23—22no—
500C23—21no—
501C23—42no—
502C23—81no—
503C23—81no—
504C23#1451974611yescharacteristic
505Q8—42no—
506Q8—41no—
507Q8—42no—
508Q8—42no—
509Q8—81no—
510Q8—82no—
511Q8—4882no—
512Q8—2442no—
513Q8—2442no—
514D4—42no—
515D4—41no—
516D4—42no—
517D4—42no—
518D4—82no—
519D4—82no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—4882no—
535C2×C4—2442no—
536C2×C4#1451974611yescharacteristic
537C2×C4#1451974611yescharacteristic
538C2×C4#1451974611yescharacteristic
539C2×C4C22×C22⋊D6111yescharacteristic
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C8—1224no—
543C8—1224no—
544C8—2442no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—42no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—162no—
554C22#1683994512yes—
555C22C2×#1451974611yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—4882no—
568C4#1683993012yes—
569C2—22no—
570C2—82no—
571C2—81no—
572C2C2×#1683993011yescharacteristic
573C2C2×#1683994511yescharacteristic
574C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3561C2
socleC22×C61C2×#120
hypercenter#3561D61
hypocenterC61#19033
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21457046C2semidirect
2. #21456366C2semidirect
3. C2×#16840157C2semidirect
4. #21445136C2semidirect
5. #21445130C2semidirect
6. #21444434C2semidirect
7. #21444417C2semidirect
8. #21444407C2semidirect
13. #16840172C22semidirect
14. #16840172C22semidirect
19. #16840121C22semidirect
28. #16839504C22semidirect
29. #16839504C22semidirect
31. #16839502C22semidirect
334. C61#19033semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214563662
C2×#168401572
#214451362
#214451302
#214444342
#214444172
#214444072
C61×#35612
#1903361