Finite Group Databasebeta

← #29665544

Subgroups — #29665544

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#21456955C211yesmaximal, characteristic, complemented
2#21455879C211yesmaximal, characteristic
3#21454930C211yesmaximal, characteristic, complemented
4#21445265C211yesmaximal, characteristic, complemented
5#21445258C211yesmaximal, characteristic, complemented
6#21445237C211yesmaximal, characteristic, complemented
7#21445230C211yesmaximal, characteristic, complemented
8#21445221C211yesmaximal, characteristic, complemented
9#21445216C211yesmaximal, characteristic, complemented
10#21445199C211yesmaximal, characteristic, complemented
11#21445192C211yesmaximal, characteristic, complemented
12#21444600C211yesmaximal, characteristic, complemented
13#21444515C211yesmaximal, characteristic, complemented
14#21444476C211yesmaximal, characteristic, complemented
15C61×#3116C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840158C2211yescharacteristic
19#16840110C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840086—21no—
22#16840086—21no—
23#16840086—21no—
24#16840059—21no—
25#16840059—21no—
26#16840057C2211yescharacteristic
27#16840052C2211yescharacteristic
28#16839962C2211yescharacteristic
29#16839957C2211yescharacteristic
30#16839907—21no—
31Q8×#92418C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33C4×#11758201—21no—
34C4×#11758199—21no—
35D4×#92418—21no—
36D4×#92418—21no—
37D4×#92418C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839815—21no—
40#16839815C2211yescharacteristic
41#16839815C2211yescharacteristic
42C2×C4⋊#92418—21no—
43C2×C4⋊#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#485—21no—
59C61×#477C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#462C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#452—21no—
64C61×#441—21no—
65C61×#440—21no—
66C61×#432C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758199—21no—
70C22×#92418—21no—
71C22×#92418—21no—
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519774C2311yescharacteristic
75#14519765—21no—
76#14519751—21no—
77#14519751—21no—
78C4×D244—21no—
79C4×D244—21no—
80#14519710—21no—
81#14519710—21no—
82#14519710—21no—
83#14519708—21no—
84#14519708—21no—
85#14519708—21no—
86C4⋊#92418—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418D411yescharacteristic
90C4⋊#92418D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519698—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418D411yescharacteristic
107C22⋊#92418D411yescharacteristic
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61D411yescharacteristic
114C2×Q16×C61—21no—
115C2×SD16×C61—21no—
116C4⋊Q8×C61C2311yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118#125×C61—21no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61D411yescharacteristic
124C4×Q8×C61C2311yescharacteristic
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61D411yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131#117×C61D411yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143#97×C61—21no—
144#41×C61—22no—
145#41×C61—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4111yescharacteristic
157C2×C22⋊D61—41no—
158#11758222—82no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164Q16×C61—42no—
165SD16×C61—42no—
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—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—41no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61#4111yescharacteristic
189C42×C61C2411yescharacteristic
190#11758201—42no—
191#11758199—42no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—81no—
201C2×#92418—81no—
202C2×#92418—81no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61#12111yescharacteristic
209C22×D61—41no—
210Q8×C61—21no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—81no—
214Q8×C61#13512yes—
215D4×C61—24no—
216D4×C61—21no—
217D4×C61#13512yes—
218C2×C4×C61—24no—
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—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—81no—
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C22⋊D61—82no—
239C2×Dic61—41no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—41no—
244D244—81no—
245C4×D61—82no—
246Dic122—81no—
247C8×C61—41no—
248C8×C61—41no—
249#92418—21no—
250#92418—21no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19419—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—22no—
261C22×C61—21no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61C2×#12111yescharacteristic
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61C2×#13511yescharacteristic
278C4×C61C2×#13511yescharacteristic
279C4×C61#57711yescharacteristic
280C4×C61#57811yescharacteristic
281Dic61—81no—
282#3228—611no—
283#3116D6111yescharacteristic, complemented
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—42no—
299C2×C61#304611yescharacteristic
300C2×C61#305011yescharacteristic
301C2×C61#306811yescharacteristic
302D61—161no—
303#533C2×D6111yescharacteristic
304#532—611no—
305C4×#41C2×D6111yescharacteristic
306#500—611no—
307#498—611no—
308#486—611no—
309#486—611no—
310#485—21no—
311#482—1221no—
312#482—1221no—
313#482—1221no—
314#478—1221no—
315#478—1221no—
316#477—611no—
317#477C2×D6111yescharacteristic
318#476—611no—
319#476C2×D6111yescharacteristic
320#462C2×D6111yescharacteristic
321#461C2×D6111yescharacteristic
322#460—611no—
323#459—611no—
324#452—21no—
325#449—1221no—
326C8×Q8—611no—
327C8×Q8—611no—
328#441—21no—
329#440—21no—
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—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—1221no—
356C2×#41C22⋊D6111yescharacteristic
357C2×Q16—21no—
358C2×SD16—21no—
359C2×SD16—1221no—
360C2×D8—1221no—
361C22×C8—1221no—
362C22×C8—1221no—
363C22×C8—1221no—
364C4⋊Q8—611no—
365C4⋊Q8C22×D6111yescharacteristic
366C4⋊Q8C22×D6111yescharacteristic
367#127—611no—
368#125—21no—
369#124—1221no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372#121—1221no—
373#121—1221no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8C22⋊D6111yescharacteristic
377C4×Q8C22×D6111yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—1221no—
381C4×D4—1221no—
382C4×D4C22⋊D6111yescharacteristic
383C4×D4C22×D6111yescharacteristic
384#117—21no—
385#117—21no—
386#117C22⋊D6111yescharacteristic
387C2×C42—21no—
388C2×C42C22×D6111yescharacteristic
389#107—21no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#106—21no—
394#106—1221no—
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—21no—
409#103—21no—
410#103—21no—
411#103—1221no—
412#102—21no—
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—22no—
433#41—22no—
434#41—4882no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—1221no—
438C2×Q8C2×C22⋊D6111yescharacteristic
439C2×D4—21no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—2441no—
444C2×D4C2×C22⋊D6111yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—2441no—
449C22×C4D4×D6111yescharacteristic
450C22×C4#1175822411yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452Q16—42no—
453SD16—42no—
454SD16—2442no—
455D8—2442no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—4881no—
467C2×C8—4881no—
468C2×C8—4881no—
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—1221no—
478C4⋊C4—1221no—
479C4⋊C4C2×C22⋊D6111yescharacteristic
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42D4×D6111yescharacteristic
492C42#1175822411yescharacteristic
493C42C2×C22⋊D6111yescharacteristic
494C42C23×D6111yescharacteristic
495C23—21no—
496C23—2441no—
497C23#1451974911yescharacteristic
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—81no—
502Q8—4881no—
503Q8#1451979712yes—
504D4—24no—
505D4—21no—
506D4—4881no—
507D4—4882no—
508D4—2441no—
509D4—2441no—
510D4—2441no—
511D4—2441no—
512D4#1451979712yes—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—81no—
527C2×C4—4882no—
528C2×C4—2441no—
529C2×C4#1451974911yescharacteristic
530C2×C4#1451974911yescharacteristic
531C2×C4#1451974911yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C2×#1175822411yescharacteristic
535C8—41no—
536C8—41no—
537C8—1221no—
538C8—1221no—
539C8—1221no—
540C8—1221no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22C2×#1451974911yescharacteristic
555C4—21no—
556C4—21no—
557C4—22no—
558C4—21no—
559C4—21no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—4881no—
565C4C2×#1451979711yescharacteristic
566C4C2×#1451979711yescharacteristic
567C4#1684057111yescharacteristic
568C4#1684057811yescharacteristic
569C2—41no—
570C2—42no—
571C2—9761no—
572C2#2145537311yescharacteristic
573C2#2145539711yescharacteristic
574C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3116C2
socleC22×C61C2×#121
prime radical#21455879C2
hypercenter#3116D61
hypocenterC61#19419
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
1. #21456955C2semidirect
3. #21454930C2semidirect
4. #21445265C2semidirect
5. #21445258C2semidirect
6. #21445237C2semidirect
7. #21445230C2semidirect
8. #21445221C2semidirect
9. #21445216C2semidirect
10. #21445199C2semidirect
11. #21445192C2semidirect
12. #21444600C2semidirect
13. #21444515C2semidirect
14. #21444476C2semidirect
15. C61×#3116C2semidirect
283. #3116D61semidirect
354. C61#19419semidirect

Maximal subgroups

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

subgroupindex
#214569552
#214558792
#214549302
#214452652
#214452582
#214452372
#214452302
#214452212
#214452162
#214451992
#214451922
#214446002
#214445152
#214444762
C61×#31162
#1941961