Finite Group Databasebeta

← #29665126

Subgroups — #29665126

Order 15616.

572 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#21457192C211yesmaximal, characteristic
2#21456199C211yesmaximal, characteristic, complemented
3#21454824C211yesmaximal, characteristic
4#21445399C211yesmaximal, characteristic, complemented
5#21445392C211yesmaximal, characteristic, complemented
6#21445371C211yesmaximal, characteristic, complemented
7#21445364C211yesmaximal, characteristic, complemented
8#21445275C211yesmaximal, characteristic, complemented
9#21445268C211yesmaximal, characteristic, complemented
10#21445247C211yesmaximal, characteristic, complemented
11#21445240C211yesmaximal, characteristic, complemented
12#21444587C211yesmaximal, characteristic, complemented
13#21444530C211yesmaximal, characteristic, complemented
14#21444504C211yesmaximal, characteristic, complemented
15C61×#3239C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840130—21no—
18#16840130—21no—
19#16840112C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840112C2211yescharacteristic
22#16840112C2211yescharacteristic
23#16840066—21no—
24#16840057C2211yescharacteristic
25#16840052C2211yescharacteristic
26#16840018—21no—
27#16840006—21no—
28#16839981—21no—
29#16839972C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33#16839840—21no—
34#16839840C2211yescharacteristic
35#16839834—21no—
36D4×#92418C2211yescharacteristic
37#16839823—21no—
38#16839815C2211yescharacteristic
39#16839786C2211yescharacteristic
40#16839771—21no—
41#16839771—21no—
42#16839718—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839508C2211yescharacteristic
55#16839508C2211yescharacteristic
56C2×C4⋊Q8×C61C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#501—21no—
59C61×#489—21no—
60C61×#484—21no—
61C61×#481—21no—
62C61×#475C2211yescharacteristic
63C61×#475C2211yescharacteristic
64C61×#474C2211yescharacteristic
65C61×#474C2211yescharacteristic
66C61×#432C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758194—21no—
70C2×#11758194—21no—
71C2×#11758194D411yescharacteristic
72C2×#11758194D411yescharacteristic
73C22×#92418—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519769—21no—
77#14519756—21no—
78#14519756—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81#14519736—21no—
82#14519710—21no—
83#14519708—21no—
84C4⋊#92418—21no—
85C4⋊#92418—21no—
86C4⋊#92418—21no—
87C4⋊#92418C2311yescharacteristic
88C4⋊#92418C2311yescharacteristic
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91#14519701—21no—
92#14519701—21no—
93#14519701—21no—
94#14519701—21no—
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519701—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700—21no—
102#14519698—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106#14519688—21no—
107#14519688D411yescharacteristic
108#14519688D411yescharacteristic
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C22×Q8×C61—21no—
114C4⋊Q8×C61—21no—
115C4⋊Q8×C61—21no—
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61D411yescharacteristic
122#123×C61—21no—
123C22⋊Q8×C61D411yescharacteristic
124#121×C61D411yescharacteristic
125C4×Q8×C61C2311yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61C2311yescharacteristic
128#117×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
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—
143#97×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—42no—
147C2×Q8×C61—42no—
148C2×Q8×C61—41no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4111yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C42×C61—21no—
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61#4111yescharacteristic
182C42×C61C2411yescharacteristic
183C22⋊Dic61—41no—
184#11758202—81no—
185#11758200—81no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190#11758196—41no—
191#11758196—41no—
192C4×Dic61—41no—
193#11758194—41no—
194#11758194—41no—
195#11758194—42no—
196#11758194—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—42no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C23×C61—21no—
207C23×C61#12111yescharacteristic
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—82no—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—81no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—41no—
238C2×Dic61—41no—
239Dic122—41no—
240Dic122—41no—
241Dic122—81no—
242C8×C61—41no—
243C8×C61—41no—
244#92418—21no—
245#92418—21no—
246#92418—21no—
247#92418—21no—
248#92418—41no—
249#92418—41no—
250#92418—41no—
251#92418—41no—
252#19381—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61C2×#12111yescharacteristic
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—81no—
268C4×C61—81no—
269C4×C61#57511yescharacteristic
270C4×C61#57511yescharacteristic
271C4×C61#57711yescharacteristic
272C4×C61#57811yescharacteristic
273Dic61—81no—
274Dic61—81no—
275#3252—611no—
276#3239D6111yescharacteristic
277#3139—611no—
278#2969—611no—
279#1704—611no—
280#1703—611no—
281#1700—611no—
282#1699—611no—
283#1684—611no—
284#1683—611no—
285#1680—611no—
286#1679—611no—
287#1569—611no—
288#1554—611no—
289#1549—611no—
290C2×C61—41no—
291C2×C61—81no—
292C2×C61#304011yescharacteristic
293C2×C61#306311yescharacteristic
294C2×C61#306911yescharacteristic
295#533—611no—
296C2×C4⋊Q8C2×D6111yescharacteristic
297#529C2×D6111yescharacteristic
298#501—21no—
299#491—1221no—
300#491—1221no—
301#489—21no—
302#487—611no—
303#487—611no—
304#487—611no—
305#487—611no—
306#484—21no—
307#481—21no—
308#479—1221no—
309#477—611no—
310#476—611no—
311#475C2×D6111yescharacteristic
312#475C2×D6111yescharacteristic
313#474C2×D6111yescharacteristic
314#474C2×D6111yescharacteristic
315#470—1221no—
316#468—1221no—
317#464—1221no—
318#462—611no—
319#461—611no—
320#446—611no—
321C8×Q8—611no—
322#436—611no—
323#436—1221no—
324#435—1221no—
325C8×D4—611no—
326#433—1221no—
327#432C2×D6111yescharacteristic
328#431—611no—
329#423—611no—
330#419—1221no—
331#419—1221no—
332#405—1221no—
333C4×M16—611no—
334#335—611no—
335#335—611no—
336#334—611no—
337#334—611no—
338#333—611no—
339#333—611no—
340#332—611no—
341#332—611no—
342#332—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346C61#1938111yescharacteristic, complemented, Sylow
347C22×Q8—21no—
348C2×Q16—1221no—
349C2×SD16—1221no—
350C2×M16—611no—
351C2×M16—611no—
352C2×M16—1221no—
353C2×M16—1221no—
354C22×C8—1221no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—611no—
359C4⋊Q8—611no—
360C4⋊Q8C22×D6111yescharacteristic
361C4⋊Q8C22×D6111yescharacteristic
362#126—21no—
363#125—1221no—
364#125C22⋊D6111yescharacteristic
365#124C22⋊D6111yescharacteristic
366#123—21no—
367C22⋊Q8—1221no—
368C22⋊Q8—1221no—
369C22⋊Q8C22⋊D6111yescharacteristic
370#121C22⋊D6111yescharacteristic
371C4×Q8—1221no—
372C4×Q8—1221no—
373C4×Q8C22×D6111yescharacteristic
374C4×D4—21no—
375C4×D4C22×D6111yescharacteristic
376#117—21no—
377#117—1221no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C42C22×D6111yescharacteristic
381#107—21no—
382#107—21no—
383#107—1221no—
384#106—21no—
385#106—21no—
386#106—1221no—
387C4⋊C8—611no—
388C4⋊C8—611no—
389C4⋊C8—611no—
390C4⋊C8—611no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393C4⋊C8—1221no—
394C4⋊C8C22×D6111yescharacteristic
395C4⋊C8C22×D6111yescharacteristic
396#103—21no—
397#103—21no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#102—21no—
410#102—21no—
411#102—1221no—
412C22⋊C8—21no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416#97—21no—
417#97—611no—
418#97—611no—
419#97—1221no—
420C4×C8—611no—
421C4×C8—611no—
422C4×C8—1221no—
423C4×C8—1221no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—41no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8C2×C22⋊D6111yescharacteristic
432C2×D4—21no—
433C2×D4C2×C22⋊D6111yescharacteristic
434C22×C4—21no—
435C22×C4—41no—
436C22×C4D4×D6111yescharacteristic
437C22×C4#1175822411yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439Q16—4881no—
440SD16—4881no—
441M16—4881no—
442M16—2441no—
443M16—2441no—
444M16—2442no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—611no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—611no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—2442no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4C2×C22⋊D6111yescharacteristic
473C4⋊C4C2×C22⋊D6111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—2441no—
480C42—21no—
481C42—2441no—
482C42D4×D6111yescharacteristic
483C42#1175822411yescharacteristic
484C42C2×C22⋊D6111yescharacteristic
485C42C23×D6111yescharacteristic
486C23—21no—
487C23#1451974911yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—82no—
495Q8—4881no—
496Q8—2441no—
497Q8—2441no—
498D4—21no—
499D4—21no—
500D4—81no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4#1451974911yescharacteristic
517C2×C4#1451974911yescharacteristic
518C2×C4#1451974911yescharacteristic
519C2×C4C2×D4×D6111yescharacteristic
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4C2×#1175822411yescharacteristic
522C8—41no—
523C8—41no—
524C8—1221no—
525C8—1221no—
526C8—1221no—
527C8—1221no—
528C8—2441no—
529C8—2441no—
530C8—2441no—
531C8—2441no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22C2×#1451974911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—4881no—
549C4—4881no—
550C4#1684055811yescharacteristic
551C4#1684055811yescharacteristic
552C4#1684057011yescharacteristic
553C4#1684057711yescharacteristic
554C2—41no—
555C2—81no—
556C2#2145531111yescharacteristic
557C2#2145550611yescharacteristic
558C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3239C2
socleC22×C61C2×#121
prime radical#121×C61D4
hypercenter#3239D61
hypocenterC61#19381
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3239

Complements

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

NH ≅ G/Ntype
2. #21456199C2semidirect
4. #21445399C2semidirect
5. #21445392C2semidirect
6. #21445371C2semidirect
7. #21445364C2semidirect
8. #21445275C2semidirect
9. #21445268C2semidirect
10. #21445247C2semidirect
11. #21445240C2semidirect
12. #21444587C2semidirect
13. #21444530C2semidirect
14. #21444504C2semidirect
346. C61#19381semidirect

Maximal subgroups

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

subgroupindex
#214571922
#214561992
#214548242
#214453992
#214453922
#214453712
#214453642
#214452752
#214452682
#214452472
#214452402
#214445872
#214445302
#214445042
C61×#32392
#1938161