Finite Group Databasebeta

← #29662458

Subgroups — #29662458

Order 15616.

618 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456404C211yesmaximal, characteristic, complemented
2#21455608C211yesmaximal, characteristic
3#21455600C211yesmaximal, characteristic
4C8×#11758224C211yesmaximal, characteristic
5#21445278C211yesmaximal, characteristic, complemented
6#21445274C211yesmaximal, characteristic, complemented
7#21445270C211yesmaximal, characteristic, complemented
8#21445222C211yesmaximal, characteristic, complemented
9#21445216C211yesmaximal, characteristic, complemented
10#21445188C211yesmaximal, characteristic, complemented
11#21445182C211yesmaximal, characteristic, complemented
12#335×D61C211yesmaximal, characteristic, complemented
13#21444475C211yesmaximal, characteristic, complemented
14#21444473C211yesmaximal, characteristic, complemented
15C61×#1683C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840317C2211yescharacteristic
18C4×#11758224C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840141C2211yescharacteristic
21#16840113C2211yescharacteristic
22#16840072—21no—
23#16840068—21no—
24#16840064—21no—
25#16840060—21no—
26#16840058C2211yescharacteristic
27#16840054C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840048C2211yescharacteristic
30#16840044C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16840002—21no—
33#16840002—21no—
34C8×Dic122C2211yescharacteristic
35#16839886C2211yescharacteristic
36#16839854—21no—
37#16839854—21no—
38D4×#92418C2211yescharacteristic
39C8×C22⋊D61—21no—
40#16839825—21no—
41#16839814C2211yescharacteristic
42#16839812—21no—
43#16839799—21no—
44#16839797—21no—
45C4⋊C8×D61C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839755—21no—
48#16839755—21no—
49C2×C8×Dic61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839510C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839509C2211yescharacteristic
58#16839508C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#476C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758224D411yescharacteristic
68C2×#11758202—21no—
69C2×#11758202—21no—
70C22×#92418—21no—
71C2×C8×D61—21no—
72C2×C8×D61—21no—
73C2×C8×D61D411yescharacteristic
74C2×C8×D61D411yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519768—21no—
78#14519768—21no—
79#14519757—21no—
80#14519757—21no—
81#14519753—21no—
82#14519753—21no—
83Q8×Dic61—21no—
84Q8×Dic61—21no—
85C4×Dic122C2311yescharacteristic
86#14519742—21no—
87#14519742—21no—
88#14519742D411yescharacteristic
89D4×Dic61D411yescharacteristic
90C4×C22⋊D61—21no—
91#14519737—21no—
92#14519735—21no—
93#14519735—21no—
94#14519735D411yescharacteristic
95#14519733—21no—
96C2×C4×Dic61—21no—
97C42×D61C2311yescharacteristic
98#14519711—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519709—21no—
104#14519708—21no—
105#14519708—21no—
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519706—21no—
113#14519702—21no—
114#14519702—21no—
115#14519702—21no—
116#14519702—21no—
117#14519700—21no—
118#14519700—21no—
119#14519698—21no—
120#14519698—21no—
121C22⋊#92418—21no—
122#14519689—21no—
123C4×#92418C2311yescharacteristic
124C8×Dic61—21no—
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61D411yescharacteristic
128C8×Dic61D411yescharacteristic
129C22×C8×C61—21no—
130C4⋊Q8×C61C2311yescharacteristic
131C4⋊Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61C2×D411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151#11758224—22no—
152#11758224Q1614yescomplemented
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122C2×D411yescharacteristic
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—42no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C2×D411yescharacteristic
170C22⋊C4×C61—21no—
171C42×C61C2411yescharacteristic
172C22⋊Dic61—21no—
173#11758202—42no—
174#11758202—42no—
175#11758198—21no—
176#11758198—41no—
177#11758198—41no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—41no—
183C4⋊Dic61—41no—
184C4⋊Dic61C2×D411yescharacteristic
185#11758196—21no—
186#11758196—21no—
187#11758196—41no—
188#11758196—41no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—41no—
193C4×Dic61—41no—
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C4×Dic61#4111yescharacteristic
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—42no—
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C8×D61—22no—
204C8×D61—41no—
205C8×D61—42no—
206C8×D61—42no—
207C23×C61—21no—
208C22×D61#12111yescharacteristic
209Q8×C61—41no—
210Q8×C61—41no—
211Q8×C61—81no—
212D4×C61—21no—
213D4×C61C2×Q1612yes—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C22⋊D61—24no—
223C2×Dic61—21no—
224C2×Dic61—24no—
225C2×Dic61—21no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61#12111yescharacteristic
230C2×Dic61#12111yescharacteristic
231C2×Dic61#12111yescharacteristic
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—41no—
235C4×D61C2×Q1612yes—
236C4×D61#13512yes—
237Dic122—21no—
238Dic122—41no—
239Dic122—41no—
240Dic122—81no—
241Dic122C2×Q1612yes—
242C8×C61—21no—
243C8×C61—21no—
244C8×C61—41no—
245C8×C61—41no—
246C8×C61—41no—
247#92418—21no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#92418#13512yes—
252#19126—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61C2×#12111yescharacteristic
257C2×D61—21no—
258C2×D61—21no—
259C2×D61#46712yes—
260C4×C61—21no—
261C4×C61—41no—
262C4×C61—81no—
263C4×C61—81no—
264C4×C61C22×Q1611yescharacteristic
265C4×C61C2×#13511yescharacteristic
266C4×C61C2×#13511yescharacteristic
267C4×C61#57511yescharacteristic
268Dic61—21no—
269Dic61—22no—
270Dic61—21no—
271Dic61—41no—
272Dic61—81no—
273Dic61—81no—
274Dic61#46712yes—
275#3170—611no—
276#3076—611no—
277#3076—611no—
278C8×#41—611no—
279#1684—611no—
280#1684—611no—
281#1683—611no—
282#1683D6111yescharacteristic, complemented
283#1676—611no—
284#1675—611no—
285#1670—611no—
286#1669—611no—
287C2×#335—611no—
288#1542—611no—
289#1542—611no—
290C2×C61—42no—
291C2×C61#303511yescharacteristic
292C2×C61C2×#46711yescharacteristic
293C2×C61#306711yescharacteristic
294D61—22no—
295#533—611no—
296#533—611no—
297C4×#41—611no—
298#500—611no—
299#494—611no—
300#487—611no—
301#487C2×D6111yescharacteristic
302#479—1221no—
303#479—1221no—
304#478—1221no—
305#478—1221no—
306#477—611no—
307#477—611no—
308#476—611no—
309#476C2×D6111yescharacteristic
310#475—611no—
311#475—611no—
312#474—611no—
313#474C2×D6111yescharacteristic
314#467—1221no—
315#467—1221no—
316C8×Q8—611no—
317C8×Q8—611no—
318C4×Q16—1221no—
319C4×Q16—1221no—
320C8×D4—611no—
321C8×D4—1221no—
322C8×D4—1221no—
323C8×D4C2×D6111yescharacteristic
324#431—611no—
325#431—1221no—
326#427—1221no—
327#427—1221no—
328C2×C4⋊C8—611no—
329C2×C4⋊C8—611no—
330#416—1221no—
331#416—1221no—
332C2×C4×C8—611no—
333C2×C4×C8—1221no—
334#335—611no—
335#335—611no—
336#335—611no—
337#335C2×D6111yescharacteristic
338#333—611no—
339#333—611no—
340#333—611no—
341#333C2×D6111yescharacteristic
342#332—611no—
343#332—611no—
344#332—611no—
345#332C2×D6111yescharacteristic
346C61#1912611yescharacteristic, complemented, Sylow
347C2×#41—611no—
348C2×Q16—1221no—
349C2×Q16—1221no—
350C22×C8—21no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8—1221no—
354C22×C8—1221no—
355C22×C8—1221no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8C22×D6111yescharacteristic
359C4⋊Q8C22×D6111yescharacteristic
360#125—1221no—
361#125—1221no—
362C22⋊Q8—1221no—
363C22⋊Q8—1221no—
364C22⋊Q8—1221no—
365C22⋊Q8—1221no—
366C4×Q8—611no—
367C4×Q8—611no—
368C4×Q8—1221no—
369C4×Q8—1221no—
370C4×Q8—1221no—
371C4×Q8—1221no—
372C4×D4—611no—
373C4×D4—1221no—
374C4×D4—1221no—
375C4×D4C22×D6111yescharacteristic
376#117—611no—
377#117—1221no—
378#117—1221no—
379#117—1221no—
380C2×C42—611no—
381C2×C42—1221no—
382#107—21no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#107—1221no—
387#107—1221no—
388#106—21no—
389#106—1221no—
390#106—1221no—
391#106—1221no—
392C4⋊C8—611no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—1221no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399C4⋊C8C22×D6111yescharacteristic
400C4⋊C8C22×D6111yescharacteristic
401C4⋊C8C22×D6111yescharacteristic
402#103—21no—
403#103—21no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#102—21no—
411#102—21no—
412#102—1221no—
413#102—1221no—
414C22⋊C8—21no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8—611no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8—1221no—
423C4×C8C22×D6111yescharacteristic
424#41—614no—
425#41—1222no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—611no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×D4—1221no—
432C2×D4D4×D6111yescharacteristic
433C22×C4—21no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1221no—
438C22×C4—1221no—
439Q16—2442no—
440Q16—2442no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—42no—
445C2×C8—611no—
446C2×C8—611no—
447C2×C8—1222no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—2441no—
452C2×C8—2442no—
453C2×C8—2442no—
454C2×C8—2442no—
455C2×C8D4×D6111yescharacteristic
456C2×C8D4×D6111yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—611no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4D4×D6111yescharacteristic
473C22⋊C4—21no—
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—2441no—
477C22⋊C4—2441no—
478C42—611no—
479C42—611no—
480C42—611no—
481C42—1221no—
482C42—1221no—
483C42—1221no—
484C42—2441no—
485C42—2441no—
486C42C23×D6111yescharacteristic
487C23—21no—
488C23—611no—
489Q8—41no—
490Q8—41no—
491Q8—81no—
492Q8—612no—
493Q8—1221no—
494Q8—4881no—
495Q8—2441no—
496Q8—2441no—
497D4—21no—
498D4—1224no—
499D4Q16×D6112yes—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—611no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—1224no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—612no—
528C8—1221no—
529C8—2441no—
530C8—2441no—
531C8—2441no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—612no—
536C22—1221no—
537C22—1221no—
538C22#121×D6111yescharacteristic
539C4—21no—
540C4—41no—
541C4—81no—
542C4—81no—
543C4—612no—
544C4—1222no—
545C4—1221no—
546C4—1221no—
547C4—4881no—
548C4—4881no—
549C4—2441no—
550C4C2×Q16×D6111yescharacteristic
551C4#135×D6111yescharacteristic
552C4#135×D6111yescharacteristic
553C4#1684055811yescharacteristic
554C2—42no—
555C2—1222no—
556C2#2145527111yescharacteristic
557C2#467×D6111yescharacteristic
558C2#2145553411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1683C2
socleC22×C61C2×#121
prime radicalC2×#11758224D4
hypercenter#1683D61
hypocenterC61#19126
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1683

Complements

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

NH ≅ G/Ntype
1. #21456404C2semidirect
5. #21445278C2semidirect
6. #21445274C2semidirect
7. #21445270C2semidirect
8. #21445222C2semidirect
9. #21445216C2semidirect
10. #21445188C2semidirect
11. #21445182C2semidirect
12. #335×D61C2semidirect
13. #21444475C2semidirect
14. #21444473C2semidirect
15. C61×#1683C2semidirect
152. #11758224Q16semidirect
282. #1683D61semidirect
346. C61#19126semidirect

Maximal subgroups

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

subgroupindex
#214564042
#214556082
#214556002
C8×#117582242
#214452782
#214452742
#214452702
#214452222
#214452162
#214451882
#214451822
#335×D612
#214444752
#214444732
C61×#16832
#1912661