Finite Group Databasebeta

← #29665534

Subgroups — #29665534

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#21455975C211yesmaximal, characteristic, complemented
2#21455873C211yesmaximal, characteristic
3#21454927C211yesmaximal, characteristic, complemented
4#21445260C211yesmaximal, characteristic, complemented
5#21445253C211yesmaximal, characteristic, complemented
6#21445232C211yesmaximal, characteristic, complemented
7#21445225C211yesmaximal, characteristic, complemented
8#21445220C211yesmaximal, characteristic, complemented
9#21445215C211yesmaximal, characteristic, complemented
10#21445198C211yesmaximal, characteristic, complemented
11#21445191C211yesmaximal, characteristic, complemented
12#21444599C211yesmaximal, characteristic, complemented
13#21444514C211yesmaximal, characteristic, complemented
14#21444476C211yesmaximal, characteristic, complemented
15C61×#3228C211yesmaximal, characteristic, complemented
16C4×#11758222C2211yescharacteristic
17#16840100—21no—
18#16840083—21no—
19#16840054C2211yescharacteristic
20#16840054C2211yescharacteristic
21#16840049C2211yescharacteristic
22#16840049C2211yescharacteristic
23#16839969C2211yescharacteristic
24#16839964C2211yescharacteristic
25#16839959C2211yescharacteristic
26#16839954C2211yescharacteristic
27#16839922—21no—
28#16839905—21no—
29C8×Dic122C2211yescharacteristic
30C8×Dic122C2211yescharacteristic
31#16839867—21no—
32#16839860—21no—
33C4×#11758200—21no—
34C4×#11758199—21no—
35C8×D244C2211yescharacteristic
36C8×D244C2211yescharacteristic
37#16839825—21no—
38#16839825—21no—
39#16839819C2211yescharacteristic
40#16839815C2211yescharacteristic
41#16839814—21no—
42C2×C4⋊#92418C2211yescharacteristic
43C4⋊C8×D61—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839511C2211yescharacteristic
51#16839511C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#532C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#498C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#482—21no—
63C61×#482—21no—
64C61×#478—21no—
65C61×#478—21no—
66C61×#431C2211yescharacteristic
67C2×#11758222D411yescharacteristic
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C22×#92418—21no—
73C2×C8×D61—21no—
74C2×C8×D61—21no—
75C4×Dic122D411yescharacteristic
76C4×Dic122C2311yescharacteristic
77C4×D244D411yescharacteristic
78C4×D244C2311yescharacteristic
79C4×C22⋊D61—21no—
80C4×C22⋊D61—21no—
81#14519736D411yescharacteristic
82#14519734—21no—
83C42×D61—21no—
84#14519710—21no—
85#14519710—21no—
86#14519708—21no—
87#14519708—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519706—21no—
96#14519706—21no—
97#14519701—21no—
98#14519701—21no—
99#14519700—21no—
100#14519700—21no—
101#14519698—21no—
102#14519698—21no—
103#14519697—21no—
104#14519697—21no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107#14519689—21no—
108#14519689—21no—
109#14519688—21no—
110C4×#92418—21no—
111C8×Dic61—21no—
112C8×Dic61—21no—
113C2×#41×C61—21no—
114C4⋊Q8×C61C2311yescharacteristic
115C4⋊Q8×C61C2311yescharacteristic
116C4⋊Q8×C61C2311yescharacteristic
117#127×C61C2311yescharacteristic
118#125×C61—21no—
119#124×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61—21no—
122#121×C61—21no—
123#121×C61—21no—
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128#117×C61—21no—
129C2×C42×C61C2311yescharacteristic
130#107×C61—21no—
131#107×C61—21no—
132#106×C61—21no—
133#106×C61—21no—
134C4⋊C8×C61D411yescharacteristic
135C4⋊C8×C61D411yescharacteristic
136#102×C61—21no—
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140C22⋊C8×C61D411yescharacteristic
141C22⋊C8×C61D411yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C22×C4×C61—41no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C2×C22⋊D61—21no—
157#11758222—22no—
158#11758222—22no—
159C2×D244C2×D411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×Dic122C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—41no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61#4111yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—21no—
185#11758202—42no—
186#11758201—42no—
187#11758200—42no—
188#11758199—42no—
189#11758198—21no—
190#11758198—21no—
191C4⋊Dic61C2×D411yescharacteristic
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—81no—
205C8×D61—81no—
206C8×D61—81no—
207C23×C61—41no—
208C23×C61#12111yescharacteristic
209C22×D61—21no—
210Q8×C61—41no—
211Q8×C61—41no—
212Q8×C61—81no—
213Q8×C61—81no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—81no—
219D4×C61—82no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—24no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239D244—21no—
240D244#13512yes—
241C4×D61—24no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—81no—
245Dic122—21no—
246Dic122#13512yes—
247C8×C61—21no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—21no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
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×C61—81no—
274C4×C61—81no—
275C4×C61C2×#13511yescharacteristic
276C4×C61C2×#13511yescharacteristic
277C4×C61#57711yescharacteristic
278C4×C61#57811yescharacteristic
279Dic61—22no—
280Dic61—41no—
281Dic61—41no—
282#3228D6111yescharacteristic, complemented
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#1551—611no—
296#1542—611no—
297C2×C61—41no—
298C2×C61—161no—
299C2×C61#304611yescharacteristic
300C2×C61#305011yescharacteristic
301C2×C61#306811yescharacteristic
302D61—42no—
303#533C2×D6111yescharacteristic
304#532C2×D6111yescharacteristic
305C4×#41—611no—
306#500C2×D6111yescharacteristic
307#498C2×D6111yescharacteristic
308#486C2×D6111yescharacteristic
309#486C2×D6111yescharacteristic
310#485—1221no—
311#482—21no—
312#482—21no—
313#482—1221no—
314#478—21no—
315#478—21no—
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#432—611no—
337#431—611no—
338#431—1221no—
339#431C2×D6111yescharacteristic
340C2×C4⋊C8—611no—
341C2×C4⋊C8—1221no—
342#335—611no—
343#335—611no—
344#334—611no—
345#334—611no—
346#333—611no—
347#333—611no—
348#333—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#331—611no—
353#331—611no—
354C61#1941911yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C2×#41—611no—
357C2×Q16—1221no—
358C2×SD16—1221no—
359C2×SD16—1221no—
360C2×D8—1221no—
361C22×C8—1221no—
362C22×C8—1221no—
363C22×C8—1221no—
364C4⋊Q8C22×D6111yescharacteristic
365C4⋊Q8C22×D6111yescharacteristic
366C4⋊Q8C22×D6111yescharacteristic
367#127C22×D6111yescharacteristic
368#125—21no—
369#124—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372#121—21no—
373#121—21no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—611no—
377C4×Q8—611no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—611no—
381C4×D4—611no—
382C4×D4—1221no—
383C4×D4—1221no—
384#117—21no—
385#117—611no—
386#117—1221no—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389#107—21no—
390#107—21no—
391#107—1221no—
392#107—1221no—
393#106—21no—
394#106—21no—
395#106—1221no—
396#106—1221no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8C22⋊D6111yescharacteristic
407C4⋊C8C22⋊D6111yescharacteristic
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#102—21no—
413#102—21no—
414#102—21no—
415#102—21no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8C22⋊D6111yescharacteristic
425C22⋊C8C22⋊D6111yescharacteristic
426#97—1221no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8—1221no—
430C4×C8C22×D6111yescharacteristic
431C4×C8C22×D6111yescharacteristic
432#41—82no—
433#41—1222no—
434#41—1222no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—611no—
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×C4D4×D6111yescharacteristic
450C22×C4#1175822411yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452Q16—2442no—
453SD16—2442no—
454SD16—2442no—
455D8—2442no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—4881no—
463C2×C8—4881no—
464C2×C8—4881no—
465C2×C8C2×C22⋊D6111yescharacteristic
466C2×C8C2×C22⋊D6111yescharacteristic
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—611no—
478C4⋊C4—611no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—1221no—
485C22⋊C4—1221no—
486C22⋊C4—1221no—
487C42—41no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42D4×D6111yescharacteristic
492C42#1175822411yescharacteristic
493C42C2×C22⋊D6111yescharacteristic
494C42C23×D6111yescharacteristic
495C23—41no—
496C23—1221no—
497C23#1451974811yescharacteristic
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—81no—
502Q8—612no—
503Q8—1221no—
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—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—1224no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—4881no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4#1451974811yescharacteristic
530C2×C4#1451974811yescharacteristic
531C2×C4#1451974811yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C2×#1175822411yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—2441no—
540C8—2441no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—1222no—
552C22—2441no—
553C22—2441no—
554C22C2×#1451974811yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—1222no—
563C4—2441no—
564C4—2441no—
565C4#135×D6111yescharacteristic
566C4#135×D6111yescharacteristic
567C4#1684057011yescharacteristic
568C4#1684057711yescharacteristic
569C2—41no—
570C2—161no—
571C2—2442no—
572C2#2145536511yescharacteristic
573C2#2145539611yescharacteristic
574C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3228C2
socleC22×C61C2×#121
prime radical#21455873C2
hypercenter#3228D61
hypocenterC61#19419
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3228

Complements

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

NH ≅ G/Ntype
1. #21455975C2semidirect
3. #21454927C2semidirect
4. #21445260C2semidirect
5. #21445253C2semidirect
6. #21445232C2semidirect
7. #21445225C2semidirect
8. #21445220C2semidirect
9. #21445215C2semidirect
10. #21445198C2semidirect
11. #21445191C2semidirect
12. #21444599C2semidirect
13. #21444514C2semidirect
14. #21444476C2semidirect
15. C61×#3228C2semidirect
282. #3228D61semidirect
354. C61#19419semidirect

Maximal subgroups

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

subgroupindex
#214559752
#214558732
#214549272
#214452602
#214452532
#214452322
#214452252
#214452202
#214452152
#214451982
#214451912
#214445992
#214445142
#214444762
C61×#32282
#1941961