Finite Group Databasebeta

← #29665538

Subgroups — #29665538

Order 15616.

616 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#21456951C211yesmaximal, characteristic, complemented
2#21455982C211yesmaximal, characteristic, complemented
3#21455873C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic, complemented
5#21445262C211yesmaximal, characteristic, complemented
6#21445255C211yesmaximal, characteristic, complemented
7#21445234C211yesmaximal, characteristic, complemented
8#21445227C211yesmaximal, characteristic, complemented
9#21445222C211yesmaximal, characteristic, complemented
10#21445217C211yesmaximal, characteristic, complemented
11#21445204C211yesmaximal, characteristic, complemented
12#21445197C211yesmaximal, characteristic, complemented
13#21444600C211yesmaximal, characteristic, complemented
14#21444477C211yesmaximal, characteristic, complemented
15C61×#1551C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840108C2211yescharacteristic
21#16840108C2211yescharacteristic
22#16840104—21no—
23#16840083—21no—
24#16840083—21no—
25#16840083—21no—
26#16840062—21no—
27#16840062—21no—
28#16840058C2211yescharacteristic
29#16840054C2211yescharacteristic
30#16840053C2211yescharacteristic
31#16840049C2211yescharacteristic
32#16839973C2211yescharacteristic
33#16839968C2211yescharacteristic
34#16839959C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839926—21no—
37#16839905—21no—
38#16839886C2211yescharacteristic
39#16839886C2211yescharacteristic
40#16839870—21no—
41#16839864—21no—
42C4×#11758200—21no—
43C4×#11758199—21no—
44#16839827C2211yescharacteristic
45#16839827C2211yescharacteristic
46C8×C22⋊D61—21no—
47C8×C22⋊D61—21no—
48#16839815C2211yescharacteristic
49#16839815C2211yescharacteristic
50#16839814—21no—
51C4⋊C8×D61—21no—
52#16839514C2211yescharacteristic
53#16839514C2211yescharacteristic
54#16839512C2211yescharacteristic
55#16839512C2211yescharacteristic
56#16839510C2211yescharacteristic
57#16839510C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#532C2211yescharacteristic
61C61×#432C2211yescharacteristic
62C2×C4⋊C8×C61C2211yescharacteristic
63C61×#333C2211yescharacteristic
64C61×#333C2211yescharacteristic
65C61×#331C2211yescharacteristic
66C61×#331C2211yescharacteristic
67C2×#11758222D411yescharacteristic
68C2×Dic244—21no—
69C2×#11758190—21no—
70C2×#11758200—21no—
71C2×#11758199—21no—
72C2×C8×D61—21no—
73C2×C8×D61—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519769—21no—
77#14519756—21no—
78#14519756—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×Dic122D411yescharacteristic
82C4×Dic122C2311yescharacteristic
83C4×D244D411yescharacteristic
84C4×D244C2311yescharacteristic
85C4×C22⋊D61—21no—
86C4×C22⋊D61—21no—
87#14519736—21no—
88#14519736D411yescharacteristic
89#14519734—21no—
90C42×D61—21no—
91#14519711—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—21no—
95#14519709—21no—
96#14519708—21no—
97#14519708—21no—
98#14519708—21no—
99C4⋊#92418D411yescharacteristic
100C4⋊#92418D411yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519702—21no—
106#14519702—21no—
107#14519702—21no—
108#14519700—21no—
109#14519699—21no—
110#14519698—21no—
111#14519698—21no—
112#14519698—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519697—21no—
117C22⋊#92418—21no—
118C22⋊#92418D411yescharacteristic
119C22⋊#92418D411yescharacteristic
120#14519689—21no—
121#14519689—21no—
122C4×#92418—21no—
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C8×Dic61—21no—
126C8×Dic61—21no—
127C2×#41×C61—21no—
128C22×C8×C61—21no—
129C4⋊Q8×C61C2311yescharacteristic
130#127×C61C2311yescharacteristic
131#124×C61—21no—
132#121×C61—21no—
133#121×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C22⋊C8×C61—21no—
143#97×C61—21no—
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—41no—
150C22×C4×C61—41no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61#4111yescharacteristic
154C2×C22⋊D61—21no—
155#11758222—22no—
156#11758222—22no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C2×D411yescharacteristic
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—81no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C22⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61#4111yescharacteristic
176C42×C61C2411yescharacteristic
177C22⋊Dic61—21no—
178C22⋊Dic61—41no—
179#11758200—42no—
180#11758199—42no—
181#11758198—21no—
182#11758198—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61C2×D411yescharacteristic
188C4⋊Dic61C2×D411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—41no—
192#11758196—41no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—41no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418C2×D411yescharacteristic
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203Dic244—42no—
204#11758190—42no—
205C8×D61—81no—
206C8×D61—81no—
207C23×C61—41no—
208C23×C61#12111yescharacteristic
209C22×D61—21no—
210Q8×C61—81no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—41no—
215D4×C61—81no—
216D4×C61—82no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—82no—
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—24no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236D244—21no—
237D244#13512yes—
238C4×D61—24no—
239C4×D61—41no—
240C4×D61—41no—
241C4×D61—81no—
242Dic122—21no—
243Dic122—41no—
244Dic122—41no—
245Dic122—81no—
246Dic122#13512yes—
247C8×C61—41no—
248C8×C61—41no—
249C8×C61—41no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—21no—
254#92418—21no—
255#92418—41no—
256#92418—41no—
257#19419—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61C2×#12111yescharacteristic
265C2×D61—22no—
266C2×D61—41no—
267C2×D61—41no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—81no—
273C4×C61C2×#13511yescharacteristic
274C4×C61C2×#13511yescharacteristic
275C4×C61#57711yescharacteristic
276C4×C61#57811yescharacteristic
277Dic61—22no—
278Dic61—41no—
279Dic61—41no—
280Dic61—81no—
281Dic61—81no—
282#3228—611no—
283#3116—611no—
284#3106—611no—
285#2988—611no—
286#1682—611no—
287#1681—611no—
288#1678—611no—
289#1677—611no—
290#1676—611no—
291#1675—611no—
292#1672—611no—
293#1671—611no—
294#1572—611no—
295#1551D6111yescharacteristic, complemented
296#1542—611no—
297C2×C61—41no—
298C2×C61—161no—
299C2×C61#304611yescharacteristic
300C2×C61#305011yescharacteristic
301C2×C61#306811yescharacteristic
302D61—42no—
303#533—611no—
304#532C2×D6111yescharacteristic
305C4×#41—611no—
306#500—611no—
307#498—611no—
308#486—611no—
309#486—611no—
310#485—1221no—
311#482—1221no—
312#482—1221no—
313#482—1221no—
314#478—1221no—
315#478—1221no—
316#477—611no—
317#477—611no—
318#476—611no—
319#476—611no—
320#462—611no—
321#461—611no—
322#460—611no—
323#459—611no—
324#452—1221no—
325#449—1221no—
326C8×Q8—611no—
327C8×Q8—611no—
328#441—1221no—
329#440—1221no—
330C4×SD16—1221no—
331C4×D8—1221no—
332C8×D4—611no—
333C8×D4—611no—
334C8×D4—1221no—
335C8×D4—1221no—
336#432C2×D6111yescharacteristic
337#431—611no—
338#431—611no—
339#431—1221no—
340C2×C4⋊C8—1221no—
341C2×C4⋊C8C2×D6111yescharacteristic
342#335—611no—
343#335—611no—
344#334—611no—
345#334—611no—
346#333—611no—
347#333—611no—
348#333C2×D6111yescharacteristic
349#333C2×D6111yescharacteristic
350#332—611no—
351#332—611no—
352#331C2×D6111yescharacteristic
353#331C2×D6111yescharacteristic
354C61#1941911yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C2×#41—611no—
357C2×Q16—1221no—
358C2×SD16—1221no—
359C2×SD16—1221no—
360C2×D8—1221no—
361C22×C8—21no—
362C22×C8—1221no—
363C22×C8—1221no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8C22×D6111yescharacteristic
367#127C22×D6111yescharacteristic
368#125—1221no—
369#124—21no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372#121—21no—
373#121—21no—
374C4×Q8—611no—
375C4×Q8—611no—
376C4×Q8—1221no—
377C4×Q8—1221no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—611no—
381C4×D4—611no—
382C4×D4—1221no—
383C4×D4—1221no—
384#117—611no—
385#117—1221no—
386#117—1221no—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396#106—1221no—
397C4⋊C8—21no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405C4⋊C8C22×D6111yescharacteristic
406C4⋊C8C22×D6111yescharacteristic
407C4⋊C8C22×D6111yescharacteristic
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#102—1221no—
413#102—1221no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420C22⋊C8—21no—
421C22⋊C8—611no—
422C22⋊C8—611no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426#97—21no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—1221no—
430C4×C8—1221no—
431C4×C8—1221no—
432#41—82no—
433#41—1222no—
434#41—1222no—
435C2×Q8—21no—
436C2×Q8—611no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—611no—
444C2×D4—1221no—
445C22×C4—41no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4#1175822211yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4D4×D6111yescharacteristic
452Q16—2442no—
453SD16—2442no—
454SD16—2442no—
455D8—2442no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—81no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—4881no—
468C2×C8—4881no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—611no—
472C4⋊C4—611no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—1221no—
484C22⋊C4—1221no—
485C22⋊C4—1221no—
486C22⋊C4—2441no—
487C42—1221no—
488C42—1221no—
489C42—1221no—
490C42—2441no—
491C42#1175822211yescharacteristic
492C42D4×D6111yescharacteristic
493C42D4×D6111yescharacteristic
494C42C23×D6111yescharacteristic
495C23—41no—
496C23—1221no—
497C23#1451975011yescharacteristic
498Q8—81no—
499Q8—612no—
500Q8—1221no—
501Q8—4881no—
502Q8—2441no—
503Q8—2441no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—81no—
509D4—82no—
510D4—612no—
511D4—1224no—
512D4—1221no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—1224no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4881no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4#1451975011yescharacteristic
530C2×C4#1451975011yescharacteristic
531C2×C4#1451975011yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×#1175822211yescharacteristic
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—1221no—
540C8—1221no—
541C8—1221no—
542C8—1221no—
543C8—2441no—
544C8—2441no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—1222no—
552C22—2441no—
553C22—2441no—
554C22C2×#1451975011yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—81no—
560C4—1222no—
561C4—4881no—
562C4—4881no—
563C4—2441no—
564C4—2441no—
565C4#135×D6111yescharacteristic
566C4#135×D6111yescharacteristic
567C4#1684057211yescharacteristic
568C4#1684057811yescharacteristic
569C2—41no—
570C2—161no—
571C2—2442no—
572C2#2145536811yescharacteristic
573C2#2145539911yescharacteristic
574C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1551C2
socleC22×C61C2×#121
prime radical#21455873C2
hypercenter#1551D61
hypocenterC61#19419
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1551

Complements

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

NH ≅ G/Ntype
1. #21456951C2semidirect
2. #21455982C2semidirect
4. #21454928C2semidirect
5. #21445262C2semidirect
6. #21445255C2semidirect
7. #21445234C2semidirect
8. #21445227C2semidirect
9. #21445222C2semidirect
10. #21445217C2semidirect
11. #21445204C2semidirect
12. #21445197C2semidirect
13. #21444600C2semidirect
14. #21444477C2semidirect
15. C61×#1551C2semidirect
295. #1551D61semidirect
354. C61#19419semidirect

Maximal subgroups

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

subgroupindex
#214569512
#214559822
#214558732
#214549282
#214452622
#214452552
#214452342
#214452272
#214452222
#214452172
#214452042
#214451972
#214446002
#214444772
C61×#15512
#1941961