Finite Group Databasebeta

← #29662740

Subgroups — #29662740

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456374C211yesmaximal, characteristic, complemented
2#21455890C211yesmaximal, characteristic, complemented
3#21454786C211yesmaximal, characteristic, complemented
4#21445403C211yesmaximal, characteristic, complemented
5#21445396C211yesmaximal, characteristic, complemented
6#21445385C211yesmaximal, characteristic, complemented
7#21445378C211yesmaximal, characteristic, complemented
8#21445327C211yesmaximal, characteristic, complemented
9#21445320C211yesmaximal, characteristic, complemented
10#21445289C211yesmaximal, characteristic, complemented
11#21445282C211yesmaximal, characteristic, complemented
12#21444565C211yesmaximal, characteristic, complemented
13#21444483C211yesmaximal, characteristic, complemented
14#21444481C211yesmaximal, characteristic, complemented
15C61×#3106C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840158C2211yescharacteristic
18#16840134C2211yescharacteristic
19#16840114C2211yescharacteristic
20#16840108C2211yescharacteristic
21#16840103—21no—
22#16840047C2211yescharacteristic
23#16840042C2211yescharacteristic
24#16839989—21no—
25#16839989—21no—
26#16839972C2211yescharacteristic
27#16839967C2211yescharacteristic
28#16839924—21no—
29#16839893C2211yescharacteristic
30#16839893C2211yescharacteristic
31C4×#11758202—21no—
32C4×#11758200—21no—
33C4×#11758200—21no—
34C4×#11758200—21no—
35#16839840C2211yescharacteristic
36#16839840C2211yescharacteristic
37#16839834—21no—
38#16839834—21no—
39#16839815—21no—
40#16839815C2211yescharacteristic
41#16839815C2211yescharacteristic
42C4×#11758194—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#482—21no—
59C61×#477C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C4×SD16×C61—21no—
65C4×D8×C61—21no—
66C61×#431C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758194—21no—
72C2×#11758194—21no—
73C22×#92418D411yescharacteristic
74C22×#92418D411yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519769—21no—
78#14519756—21no—
79#14519756—21no—
80C4×Dic122—21no—
81C4×Dic122—21no—
82#14519736—21no—
83#14519710—21no—
84#14519708—21no—
85#14519708—21no—
86#14519708—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519701—21no—
94#14519700—21no—
95#14519700—21no—
96#14519700—21no—
97#14519700—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688—21no—
107C4×#92418—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×#41×C61D411yescharacteristic
115C2×SD16×C61—21no—
116C2×D8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#124×C61—21no—
120#121×C61—21no—
121#121×C61—21no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61D411yescharacteristic
129C4×D4×C61C2311yescharacteristic
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#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61—21no—
144#41×C61—41no—
145#41×C61—41no—
146#41×C61—82no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—41no—
154C2×D4×C61C2×D411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61#4111yescharacteristic
162C2×Dic122—21no—
163C2×Dic122—21no—
164SD16×C61—42no—
165D8×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×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61#4111yescharacteristic
185C42×C61C2411yescharacteristic
186C22⋊Dic61—41no—
187#11758202—42no—
188#11758200—42no—
189#11758200—42no—
190#11758200—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—41no—
198#11758194—42no—
199#11758194—42no—
200C2×#92418—22no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—41no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C23×C61—21no—
211C23×C61—41no—
212C23×C61#12111yescharacteristic
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—81no—
216D4×C61—21no—
217D4×C61—21no—
218D4×C61—42no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—81no—
224D4×C61—82no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×Dic61—41no—
246C2×Dic61—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—41no—
250Dic122—41no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—21no—
254#92418—21no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#92418#13512yes—
261#19159—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61C2×#12111yescharacteristic
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61#47112yes—
280C4×C61C2×#13511yescharacteristic
281C4×C61#57511yescharacteristic
282C4×C61#57511yescharacteristic
283C4×C61#57511yescharacteristic
284Dic61—81no—
285Dic61—81no—
286#3167—611no—
287#3107—611no—
288#3106D6111yescharacteristic
289#2965—611no—
290#1704—611no—
291#1703—611no—
292#1702—611no—
293#1701—611no—
294#1692—611no—
295#1691—611no—
296#1686—611no—
297#1685—611no—
298#1564—611no—
299#1544—611no—
300#1543—611no—
301C2×C61—41no—
302C2×C61—81no—
303C2×C61—161no—
304C2×C61#303511yescharacteristic
305C2×C61#305411yescharacteristic
306C2×C61C2×#47111yescharacteristic
307#533—611no—
308#532C2×D6111yescharacteristic
309C4×#41C2×D6111yescharacteristic
310#498—611no—
311#492—611no—
312#487—611no—
313#486—611no—
314#485—1221no—
315#482—21no—
316#477C2×D6111yescharacteristic
317#476C2×D6111yescharacteristic
318#475—611no—
319#474—611no—
320#465—1221no—
321#465—1221no—
322#462—611no—
323#461—611no—
324#460C2×D6111yescharacteristic
325#459C2×D6111yescharacteristic
326#452—1221no—
327#449—21no—
328#446—611no—
329#446—611no—
330C4×Q16—1221no—
331C4×SD16—21no—
332C4×SD16—1221no—
333C4×SD16—1221no—
334C4×SD16—1221no—
335C4×D8—21no—
336#436—611no—
337#436—611no—
338#435—1221no—
339#435—1221no—
340#431—611no—
341#431—611no—
342#431—1221no—
343#431C2×D6111yescharacteristic
344C4×M16—1221no—
345C2×C4×C8—611no—
346#335—611no—
347#335—611no—
348#334—611no—
349#334—611no—
350#333—611no—
351#333—611no—
352#332—611no—
353#332—611no—
354#332—611no—
355#332—611no—
356#331—611no—
357#331—611no—
358C61#1915911yescharacteristic, complemented, Sylow
359C2×#41—21no—
360C2×#41C22⋊D6111yescharacteristic
361C2×Q16—1221no—
362C2×SD16—21no—
363C2×SD16—1221no—
364C2×SD16—1221no—
365C2×SD16—1221no—
366C2×D8—21no—
367C2×M16—1221no—
368C2×M16—1221no—
369C22×C8—611no—
370C22×C8—611no—
371C4⋊Q8—611no—
372C4⋊Q8—611no—
373C4⋊Q8C22×D6111yescharacteristic
374#127C22×D6111yescharacteristic
375#125—1221no—
376#124—21no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379#121—21no—
380#121—21no—
381C4×Q8—1221no—
382C4×Q8—1221no—
383C4×Q8C22⋊D6111yescharacteristic
384C4×Q8C22×D6111yescharacteristic
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4C22⋊D6111yescharacteristic
390C4×D4C22×D6111yescharacteristic
391#117—21no—
392#117—1221no—
393#117C22⋊D6111yescharacteristic
394C2×C42—21no—
395C2×C42C22×D6111yescharacteristic
396#107—21no—
397#107—1221no—
398#106—21no—
399#106—1221no—
400#106—1221no—
401#106—1221no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—21no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#102—21no—
417#102—21no—
418#102—21no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—21no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#97—1221no—
428#97—1221no—
429C4×C8—21no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—1221no—
435C4×C8—1221no—
436#41—41no—
437#41—41no—
438#41—82no—
439C2×Q8—21no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8C2×C22⋊D6111yescharacteristic
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—41no—
448C2×D4C2×C22⋊D6111yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4D4×D6111yescharacteristic
454C22×C4#1175822411yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456Q16—2442no—
457SD16—42no—
458SD16—2442no—
459SD16—2442no—
460SD16—2442no—
461D8—42no—
462M16—2442no—
463M16—2442no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—2441no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—2441no—
485C4⋊C4—2441no—
486C4⋊C4C2×C22⋊D6111yescharacteristic
487C4⋊C4C2×C22⋊D6111yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—2441no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—2441no—
498C42D4×D6111yescharacteristic
499C42#1175822411yescharacteristic
500C42C2×C22⋊D6111yescharacteristic
501C42C23×D6111yescharacteristic
502C23—21no—
503C23—41no—
504C23#1451974911yescharacteristic
505Q8—21no—
506Q8—21no—
507Q8—81no—
508Q8—2441no—
509Q8—2441no—
510Q8—2441no—
511Q8—2441no—
512D4—21no—
513D4—21no—
514D4—42no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—81no—
520D4—82no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4#1451974911yescharacteristic
538C2×C4#1451974911yescharacteristic
539C2×C4#1451974911yescharacteristic
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C2×#1175822411yescharacteristic
543C8—41no—
544C8—41no—
545C8—612no—
546C8—1221no—
547C8—1221no—
548C8—1221no—
549C8—2441no—
550C8—2441no—
551C8—2441no—
552C8—2441no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22C2×#1451974911yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—4881no—
571C4—4881no—
572C4#1684002312yes—
573C4#135×D6111yescharacteristic
574C4#1684055911yescharacteristic
575C4#1684056011yescharacteristic
576C4#1684056011yescharacteristic
577C2—41no—
578C2—81no—
579C2—161no—
580C2#2145527011yescharacteristic
581C2#2145542511yescharacteristic
582C2C2×#1684002311yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3106C2
socleC22×C61C2×#121
hypercenter#3106D61
hypocenterC61#19159
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #21456374C2semidirect
2. #21455890C2semidirect
3. #21454786C2semidirect
4. #21445403C2semidirect
5. #21445396C2semidirect
6. #21445385C2semidirect
7. #21445378C2semidirect
8. #21445327C2semidirect
9. #21445320C2semidirect
10. #21445289C2semidirect
11. #21445282C2semidirect
12. #21444565C2semidirect
13. #21444483C2semidirect
14. #21444481C2semidirect
358. C61#19159semidirect

Maximal subgroups

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

subgroupindex
#214563742
#214558902
#214547862
#214454032
#214453962
#214453852
#214453782
#214453272
#214453202
#214452892
#214452822
#214445652
#214444832
#214444812
C61×#31062
#1915961