Finite Group Databasebeta

← #29663615

Subgroups — #29663615

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457163C211yesmaximal, characteristic
2#21456969C211yesmaximal, characteristic
3#21456362C211yesmaximal, characteristic
4#21454767C211yesmaximal, characteristic
5#21445346C211yesmaximal, characteristic, complemented
6#21445328C211yesmaximal, characteristic, complemented
7#21445325C211yesmaximal, characteristic, complemented
8#21445221C211yesmaximal, characteristic, complemented
9#21445220C211yesmaximal, characteristic, complemented
10#21445203C211yesmaximal, characteristic, complemented
11#21445202C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic
13#21444473C211yesmaximal, characteristic
14#333×D61C211yesmaximal, characteristic
15C61×#1696C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840316C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19#16840170—21no—
20#16840170—21no—
21#16840166—21no—
22#16840165C2211yescharacteristic
23#16840160C2211yescharacteristic
24#16840158C2211yescharacteristic
25#16840111—21no—
26#16840099—21no—
27#16840096—21no—
28#16840058C2211yescharacteristic
29#16840057C2211yescharacteristic
30#16840056—21no—
31#16840054C2211yescharacteristic
32#16840054C2211yescharacteristic
33#16840004—21no—
34#16840002—21no—
35#16839973C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839971—21no—
38#16839890—21no—
39#16839889—21no—
40#16839889C2211yescharacteristic
41Q8×#92418C2211yescharacteristic
42C8×Dic122C2211yescharacteristic
43#16839814C2211yescharacteristic
44#16839803—21no—
45#16839803—21no—
46#16839787—21no—
47C4⋊C8×D61C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839719—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839511C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839510C2211yescharacteristic
58#16839509C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#500C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#477C2211yescharacteristic
63C61×#446C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#333C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Dic244—21no—
69C2×#11758202—21no—
70C2×#11758189D411yescharacteristic
71C2×#11758189D411yescharacteristic
72C2×C8×D61—21no—
73C2×C8×D61—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519776—21no—
77#14519776—21no—
78#14519776D411yescharacteristic
79#14519775—21no—
80#14519775D411yescharacteristic
81#14519768—21no—
82#14519767—21no—
83#14519757—21no—
84#14519757D411yescharacteristic
85#14519755—21no—
86#14519753—21no—
87#14519753D411yescharacteristic
88Q8×Dic61—21no—
89C4×Dic122—21no—
90C4×Dic122C2311yescharacteristic
91#14519742—21no—
92#14519735—21no—
93#14519734—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C42×D61C2311yescharacteristic
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519709—21no—
104#14519709—21no—
105#14519708—21no—
106#14519708—21no—
107C4⋊#92418—21no—
108C4⋊#92418C2311yescharacteristic
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111#14519706—21no—
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519702—21no—
116#14519702—21no—
117#14519701—21no—
118#14519700—21no—
119#14519700—21no—
120#14519700—21no—
121#14519689—21no—
122#14519687—21no—
123#14519687D411yescharacteristic
124#14519687D411yescharacteristic
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C4⋊Q8×C61C2311yescharacteristic
130C4⋊Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147Q8×D61—82no—
148C2×C4×D61—41no—
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61#4111yescharacteristic
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—41no—
156C2×Dic122C2×D411yescharacteristic
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C2×D411yescharacteristic
168C42×C61—21no—
169C42×C61C2411yescharacteristic
170#11758202—81no—
171#11758198—21no—
172#11758198—21no—
173#11758198—41no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—41no—
180C4⋊Dic61C2×D411yescharacteristic
181#11758196—21no—
182#11758196—21no—
183#11758196—21no—
184#11758196—21no—
185#11758196—21no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189C4×Dic61—21no—
190C4×Dic61—41no—
191C4×Dic61C2×D411yescharacteristic
192C4×Dic61C2×D411yescharacteristic
193C4×Dic61#4111yescharacteristic
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418C2×D411yescharacteristic
198C2×#92418C2×D411yescharacteristic
199Dic244—81no—
200#11758189—22no—
201#11758189Q1614yescomplemented
202C8×D61—42no—
203C8×D61—81no—
204C22×D61#12111yescharacteristic
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—81no—
208Q8×C61—81no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61#12111yescharacteristic
221C2×Dic61#12111yescharacteristic
222C2×Dic61#12111yescharacteristic
223C4×D61—21no—
224C4×D61—21no—
225C4×D61—41no—
226C4×D61—82no—
227C4×D61C2×Q1612yes—
228C4×D61#13712yes—
229Dic122—21no—
230Dic122—21no—
231Dic122—41no—
232Dic122—41no—
233Dic122—81no—
234Dic122—82no—
235Dic122—81no—
236C8×C61—21no—
237C8×C61—41no—
238C8×C61—41no—
239C8×C61—41no—
240C8×C61C2×Q1612yes—
241#92418—21no—
242#92418—41no—
243#92418—41no—
244#92418—41no—
245#92418C2×Q1612yes—
246#19241—611nomaximal, Sylow
247C22×C61C2×#12111yescharacteristic
248C2×D61—21no—
249C2×D61—21no—
250C2×D61#46212yes—
251C4×C61—21no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—81no—
255C4×C61—81no—
256C4×C61C22×Q1611yescharacteristic
257C4×C61C2×#13711yescharacteristic
258C4×C61#57711yescharacteristic
259C4×C61#57811yescharacteristic
260Dic61—21no—
261Dic61—21no—
262Dic61—41no—
263Dic61—41no—
264Dic61—81no—
265Dic61—81no—
266Dic61#46212yes—
267#3249—611no—
268#3230—611no—
269#3165—611no—
270#2962—611no—
271#1696—611no—
272#1696D6111yescharacteristic, complemented
273#1692—611no—
274#1692—611no—
275#1676—611no—
276#1676—611no—
277#1672—611no—
278#1672—611no—
279#1565—611no—
280#1542—611no—
281C2×#333—611no—
282C2×C61C2×#46211yescharacteristic
283C2×C61#306311yescharacteristic
284C2×C61#306811yescharacteristic
285D61—22no—
286#533—611no—
287#533—611no—
288C2×C4⋊Q8—611no—
289#501—1221no—
290#501—1221no—
291#500—611no—
292#500—1221no—
293#500C2×D6111yescharacteristic
294#498—611no—
295#498—611no—
296#487—1221no—
297#484—1221no—
298#484—1221no—
299#477—611no—
300#477—611no—
301#477—611no—
302#477—611no—
303#477—1221no—
304#477C2×D6111yescharacteristic
305#477C2×D6111yescharacteristic
306#467—1221no—
307#467—1221no—
308#462—611no—
309#462—611no—
310#462—1221no—
311#446—611no—
312#446—1221no—
313#446—1221no—
314#446C2×D6111yescharacteristic
315C8×Q8—611no—
316C8×Q8—611no—
317#431—611no—
318#428—1221no—
319#428—1221no—
320#424—1221no—
321C2×C4⋊C8—611no—
322C2×C4⋊C8—611no—
323#405—1221no—
324C4×M16—611no—
325#335—611no—
326#335—611no—
327#334—611no—
328#334C2×D6111yescharacteristic
329#333—611no—
330#333—611no—
331#333—611no—
332#333—611no—
333#333C2×D6111yescharacteristic
334#333C2×D6111yescharacteristic
335#332—611no—
336#332—611no—
337C61#1924111yescharacteristic, complemented, Sylow
338C22×Q8—1221no—
339C2×Q16—1221no—
340C2×Q16—1221no—
341C2×M16—611no—
342C2×M16—611no—
343C22×C8—1221no—
344C22×C8—1221no—
345C4⋊Q8—611no—
346C4⋊Q8—611no—
347C4⋊Q8—611no—
348C4⋊Q8—611no—
349C4⋊Q8—1221no—
350C4⋊Q8—1221no—
351C4⋊Q8—1221no—
352C4⋊Q8C22×D6111yescharacteristic
353C4⋊Q8C22×D6111yescharacteristic
354#125—1221no—
355#125—1221no—
356C22⋊Q8—611no—
357C22⋊Q8—611no—
358C22⋊Q8—1221no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C4×Q8—611no—
362C4×Q8—1221no—
363C4×Q8—1221no—
364C4×Q8—1221no—
365C4×Q8C22×D6111yescharacteristic
366#117—1221no—
367#117—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C42—611no—
371#107—21no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107—1221no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#106—1221no—
382#106—1221no—
383#106—1221no—
384#106—1221no—
385C4⋊C8—21no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8C22×D6111yescharacteristic
395C4⋊C8C22×D6111yescharacteristic
396C4⋊C8C22×D6111yescharacteristic
397#103—21no—
398#103—21no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405C22⋊C8—1221no—
406#97—21no—
407#97—611no—
408#97—611no—
409#97—1221no—
410C4×C8—611no—
411C4×C8—1221no—
412C4×C8—1221no—
413C4×C8—1221no—
414C4×C8C22×D6111yescharacteristic
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—611no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—4882no—
422C2×Q8—2441no—
423C2×Q8D4×D6111yescharacteristic
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—2441no—
428Q16—4881no—
429Q16—4881no—
430M16—614no—
431M16—1222no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—4881no—
441C2×C8—2442no—
442C2×C8D4×D6111yescharacteristic
443C2×C8D4×D6111yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—611no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4D4×D6111yescharacteristic
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—2441no—
468C42—21no—
469C42—611no—
470C42—611no—
471C42—611no—
472C42—1221no—
473C42—2441no—
474C42C23×D6111yescharacteristic
475C23—611no—
476Q8—21no—
477Q8—21no—
478Q8—81no—
479Q8—81no—
480Q8—1221no—
481Q8—1221no—
482Q8—4881no—
483Q8—4881no—
484Q8—4882no—
485Q8—2441no—
486Q8—2441no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—611no—
492C2×C4—612no—
493C2×C4—612no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—4882no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C8—21no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—612no—
512C8—1221no—
513C8—2441no—
514C8—2441no—
515C8—2441no—
516C8Q16×D6112yes—
517C22—612no—
518C22—1221no—
519C22—1221no—
520C22#121×D6111yescharacteristic
521C4—21no—
522C4—41no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4—612no—
527C4—1221no—
528C4—1221no—
529C4—4881no—
530C4—4881no—
531C4—2441no—
532C4—2441no—
533C4C2×Q16×D6111yescharacteristic
534C4#137×D6111yescharacteristic
535C4#1684057111yescharacteristic
536C4#1684057811yescharacteristic
537C2—1222no—
538C2#462×D6111yescharacteristic
539C2#2145550711yescharacteristic
540C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1696C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1696D61
hypocenterC61#19241
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1696

Complements

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

NH ≅ G/Ntype
5. #21445346C2semidirect
6. #21445328C2semidirect
7. #21445325C2semidirect
8. #21445221C2semidirect
9. #21445220C2semidirect
10. #21445203C2semidirect
11. #21445202C2semidirect
15. C61×#1696C2semidirect
201. #11758189Q16semidirect
272. #1696D61semidirect
337. C61#19241semidirect

Maximal subgroups

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

subgroupindex
#214571632
#214569692
#214563622
#214547672
#214453462
#214453282
#214453252
#214452212
#214452202
#214452032
#214452022
#214445682
#214444732
#333×D612
C61×#16962
#1924161