Finite Group Databasebeta

← #29662970

Subgroups — #29662970

Order 15616.

616 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#21456538C211yesmaximal, characteristic, complemented
2#21455880C211yesmaximal, characteristic
3#21455601C211yesmaximal, characteristic, complemented
4#21454831C211yesmaximal, characteristic, complemented
5#21445400C211yesmaximal, characteristic, complemented
6#21445362C211yesmaximal, characteristic, complemented
7#21445346C211yesmaximal, characteristic, complemented
8#21445294C211yesmaximal, characteristic, complemented
9#21445269C211yesmaximal, characteristic, complemented
10#21445231C211yesmaximal, characteristic, complemented
11#21445217C211yesmaximal, characteristic, complemented
12#21445163C211yesmaximal, characteristic, complemented
13#21444572C211yesmaximal, characteristic, complemented
14#21444515C211yesmaximal, characteristic, complemented
15C61×#1554C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840136C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840087—21no—
23#16840066—21no—
24#16840059—21no—
25#16840058C2211yescharacteristic
26#16840054C2211yescharacteristic
27#16840053C2211yescharacteristic
28#16840049C2211yescharacteristic
29#16840044C2211yescharacteristic
30#16840039C2211yescharacteristic
31#16840010—21no—
32#16840010—21no—
33#16839963C2211yescharacteristic
34#16839958C2211yescharacteristic
35#16839909—21no—
36#16839889C2211yescharacteristic
37#16839889C2211yescharacteristic
38#16839860—21no—
39#16839860—21no—
40C4×#11758190—21no—
41C4×D488—21no—
42#16839836—21no—
43#16839836—21no—
44#16839827C2211yescharacteristic
45#16839827C2211yescharacteristic
46#16839817—21no—
47#16839798—21no—
48C2×C4⋊#92418C2211yescharacteristic
49#16839756—21no—
50C4×#11758194C2211yescharacteristic
51#97×D61—21no—
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839512C2211yescharacteristic
55#16839512C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839510C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#533C2211yescharacteristic
61C61×#432C2211yescharacteristic
62C61×#431C2211yescharacteristic
63C61×#333C2211yescharacteristic
64C61×#333C2211yescharacteristic
65C61×#332C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222D411yescharacteristic
69C2×#11758190—21no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×D488—21no—
73C2×#11758194D411yescharacteristic
74C2×#11758194D411yescharacteristic
75C22×#92418—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78#14519777C2311yescharacteristic
79#14519774C2311yescharacteristic
80#14519765—21no—
81#14519751—21no—
82#14519751—21no—
83C4×Dic122D411yescharacteristic
84C4×Dic122C2311yescharacteristic
85C4×D244—21no—
86C4×D244—21no—
87C4×D244D411yescharacteristic
88C4×D244C2311yescharacteristic
89C4×C22⋊D61—21no—
90C4×C22⋊D61—21no—
91#14519736D411yescharacteristic
92#14519734—21no—
93C42×D61—21no—
94#14519711—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519709—21no—
100#14519708—21no—
101#14519708—21no—
102C4⋊#92418—21no—
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706—21no—
106#14519706—21no—
107#14519702—21no—
108#14519701—21no—
109#14519701—21no—
110#14519700—21no—
111#14519700—21no—
112#14519699—21no—
113#14519699—21no—
114#14519699—21no—
115#14519697—21no—
116#14519697—21no—
117#14519697—21no—
118#14519697—21no—
119#14519689—21no—
120#14519689—21no—
121#14519688D411yescharacteristic
122#14519688D411yescharacteristic
123#14519687—21no—
124#14519687—21no—
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C4⋊Q8×C61C2311yescharacteristic
128C4⋊Q8×C61C2311yescharacteristic
129#125×C61—21no—
130C22⋊Q8×C61—21no—
131C22⋊Q8×C61—21no—
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142#97×C61—21no—
143C4×C8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4111yescharacteristic
149C2×C22⋊D61—21no—
150C2×C22⋊D61—41no—
151#11758222—22no—
152#11758222—41no—
153#11758222—82no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—41no—
162C2×Dic122—21no—
163C2×Dic122C2×D411yescharacteristic
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C42×C61—41no—
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61#4111yescharacteristic
179C42×C61C2411yescharacteristic
180C22⋊Dic61—21no—
181#11758201—42no—
182#11758201—42no—
183#11758198—21no—
184#11758198—21no—
185#11758198—41no—
186#11758198—41no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C2×D411yescharacteristic
190C4⋊Dic61C2×D411yescharacteristic
191#11758196—21no—
192#11758196—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196#11758194—22no—
197#11758194—41no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—42no—
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205D488—42no—
206#11758190—42no—
207C8×D61—42no—
208C8×D61—42no—
209C23×C61#12111yescharacteristic
210C22×D61—21no—
211C22×D61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—42no—
229C22⋊D61—82no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—41no—
234D244—21no—
235D244—41no—
236D244—41no—
237D244—41no—
238D244—41no—
239D244—81no—
240D244#13712yes—
241C4×D61—42no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—81no—
245C4×D61—82no—
246Dic122—21no—
247Dic122—21no—
248Dic122—81no—
249C8×C61—41no—
250C8×C61—41no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—21no—
254#92418—21no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418#13512yes—
259#19179—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61C2×#12111yescharacteristic
264C2×D61—22no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61C2×#13511yescharacteristic
277C4×C61C2×#13711yescharacteristic
278C4×C61#57511yescharacteristic
279C4×C61#57511yescharacteristic
280Dic61—41no—
281Dic61—41no—
282Dic61—41no—
283Dic61—81no—
284#3182—611no—
285#3106—611no—
286#3076—611no—
287#2970—611no—
288#1704—611no—
289#1698—611no—
290#1696—611no—
291#1686—611no—
292#1683—611no—
293#1677—611no—
294#1675—611no—
295#1665—611no—
296#1565—611no—
297#1554D6111yescharacteristic, complemented
298#1551—611no—
299C2×C61—41no—
300C2×C61#303511yescharacteristic
301C2×C61#305611yescharacteristic
302C2×C61#306811yescharacteristic
303D61—42no—
304D61—161no—
305#533C2×D6111yescharacteristic
306#532—611no—
307C4×#41—611no—
308#500—611no—
309#492—611no—
310#487—611no—
311#486—611no—
312#482—1221no—
313#479—1221no—
314#478—1221no—
315#477—611no—
316#477—611no—
317#476—611no—
318#476—611no—
319#475—611no—
320#474—611no—
321#468—1221no—
322#468—1221no—
323#460—611no—
324#459—611no—
325#449—1221no—
326#446—611no—
327#446—611no—
328#440—1221no—
329#440—1221no—
330C4×SD16—1221no—
331C4×D8—1221no—
332#435—1221no—
333#435—1221no—
334C8×D4—611no—
335C8×D4—611no—
336#432—1221no—
337#432C2×D6111yescharacteristic
338#431C2×D6111yescharacteristic
339#427—1221no—
340C2×C4⋊C8—611no—
341#416—1221no—
342C4×M16—611no—
343C2×#97—1221no—
344#335—611no—
345#335—611no—
346#334—611no—
347#334—611no—
348#333—611no—
349#333—611no—
350#333C2×D6111yescharacteristic
351#333C2×D6111yescharacteristic
352#332C2×D6111yescharacteristic
353#332C2×D6111yescharacteristic
354#331—611no—
355#331—611no—
356C61#1917911yescharacteristic, complemented, Sylow
357C2×#41—611no—
358C2×#41—1221no—
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×D8—1221no—
363C2×M16—611no—
364C2×M16—611no—
365C22×C8—1221no—
366C22×C8—1221no—
367C22×C8—1221no—
368C4⋊Q8—611no—
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#127—611no—
372#125—21no—
373#124—1221no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—1221no—
377#121—1221no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—611no—
381C4×Q8—611no—
382C4×D4—611no—
383C4×D4—611no—
384C4×D4—1221no—
385C4×D4—1221no—
386C4×D4—1221no—
387C4×D4—1221no—
388#117—21no—
389#117—611no—
390#117—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1221no—
394#107—1221no—
395#107—1221no—
396#107—1221no—
397#107—1221no—
398#106—1221no—
399#106—1221no—
400#106—1221no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8C22×D6111yescharacteristic
407C4⋊C8C22×D6111yescharacteristic
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426#97—21no—
427#97—611no—
428#97—611no—
429#97—1221no—
430#97—1221no—
431C4×C8—21no—
432C4×C8—611no—
433C4×C8—611no—
434#41—1222no—
435#41—4882no—
436#41—2441no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—611no—
440C2×Q8—1221no—
441C2×D4—611no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—2441no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—2441no—
451C22×C4#1175822211yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454SD16—2442no—
455SD16—2442no—
456SD16—2442no—
457D8—2442no—
458M16—1222no—
459M16—2441no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—2442no—
471C2×C8—2442no—
472C2×C8—2442no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—611no—
480C4⋊C4—611no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C22⋊C4—41no—
486C22⋊C4—1221no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C42—41no—
492C42—1221no—
493C42—1221no—
494C42—1221no—
495C42#1175822211yescharacteristic
496C42D4×D6111yescharacteristic
497C42D4×D6111yescharacteristic
498C42C23×D6111yescharacteristic
499C23—1221no—
500C23—2441no—
501C23#1451975011yescharacteristic
502Q8—41no—
503Q8—41no—
504Q8—81no—
505Q8—1221no—
506Q8—1221no—
507Q8—4881no—
508D4—612no—
509D4—1221no—
510D4—4881no—
511D4—4882no—
512D4—2442no—
513D4—2441no—
514D4—2441no—
515D4—2441no—
516D4—2441no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—4882no—
528C2×C4—4881no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451975011yescharacteristic
534C2×C4#1451975011yescharacteristic
535C2×C4#1451975011yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4C2×#1175822211yescharacteristic
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—612no—
544C8—1221no—
545C8—1221no—
546C8—1221no—
547C8—2441no—
548C8—2441no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—1222no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—2441no—
557C22—2441no—
558C22C2×#1451975011yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—81no—
564C4—81no—
565C4—4881no—
566C4—2441no—
567C4—2441no—
568C4—2441no—
569C4#135×D6111yescharacteristic
570C4#137×D6111yescharacteristic
571C4#1684055911yescharacteristic
572C4#1684056111yescharacteristic
573C2—41no—
574C2—9761no—
575C2—2442no—
576C2#2145527211yescharacteristic
577C2#2145544711yescharacteristic
578C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1554C2
socleC22×C61C2×#121
prime radical#21455880C2
hypercenter#1554D61
hypocenterC61#19179
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1554

Complements

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

NH ≅ G/Ntype
1. #21456538C2semidirect
3. #21455601C2semidirect
4. #21454831C2semidirect
5. #21445400C2semidirect
6. #21445362C2semidirect
7. #21445346C2semidirect
8. #21445294C2semidirect
9. #21445269C2semidirect
10. #21445231C2semidirect
11. #21445217C2semidirect
12. #21445163C2semidirect
13. #21444572C2semidirect
14. #21444515C2semidirect
15. C61×#1554C2semidirect
297. #1554D61semidirect
356. C61#19179semidirect

Maximal subgroups

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

subgroupindex
#214565382
#214558802
#214556012
#214548312
#214454002
#214453622
#214453462
#214452942
#214452692
#214452312
#214452172
#214451632
#214445722
#214445152
C61×#15542
#1917961