Finite Group Databasebeta

← #30613515

Subgroups — #30613515

Order 15616.

640 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#21462665C211yesmaximal, characteristic
2#21462620C211yesmaximal, characteristic, complemented
3#21461460C211yesmaximal, characteristic, complemented
4#21461456C211yesmaximal, characteristic, complemented
5#21461449C211yesmaximal, characteristic, complemented
6#21461442C211yesmaximal, characteristic, complemented
7#21461430C211yesmaximal, characteristic, complemented
8#21461429C211yesmaximal, characteristic, complemented
9#21461413C211yesmaximal, characteristic, complemented
10#21461406C211yesmaximal, characteristic, complemented
11#21461383C211yesmaximal, characteristic, complemented
12#21461163C211yesmaximal, characteristic, complemented
13#21461154C211yesmaximal, characteristic
14#21461042C211yesmaximal, characteristic, complemented
15#21454643C211yesmaximal, characteristic
16#16840814—21no—
17#16840814—21no—
18#16840812C2211yescharacteristic
19#16840812C2211yescharacteristic
20#16840811C2211yescharacteristic
21#16840811C2211yescharacteristic
22Q8×#92420C2211yescharacteristic
23Q8×#92420C2211yescharacteristic
24#16840797—21no—
25#16840797—21no—
26#16840797C2211yescharacteristic
27#16840797C2211yescharacteristic
28#16840783—21no—
29#16840783C2211yescharacteristic
30#16840765—21no—
31#16840765C2211yescharacteristic
32C4×#11758217—21no—
33C4×#11758217—21no—
34C4×#11758217C2211yescharacteristic
35C4×#11758217C2211yescharacteristic
36#16840626C2211yescharacteristic
37#16840625C2211yescharacteristic
38#16840624C2211yescharacteristic
39#16840623C2211yescharacteristic
40#16840612C2211yescharacteristic
41#16840612C2211yescharacteristic
42#16840612C2211yescharacteristic
43#16840612C2211yescharacteristic
44Q8⋊#92420C2211yescharacteristic
45Q8⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic
49#92420⋊C8C2211yescharacteristic
50#16840603C2211yescharacteristic
51C8×#92420C2211yescharacteristic
52C2×#14519807C411yescharacteristic
53#41×Dic61C2211yescharacteristic
54C2×#14519742C2211yescharacteristic
55#16839866C411yescharacteristic
56#16839866C411yescharacteristic
57#16839859C411yescharacteristic
58#16839859C411yescharacteristic
59#16839854C2211yescharacteristic
60#16839854C2211yescharacteristic
61#16839853C2211yescharacteristic
62#16839853C2211yescharacteristic
63#16839802C411yescharacteristic
64#16839772C411yescharacteristic
65M16×Dic61C2211yescharacteristic
66C61×#417C411yescharacteristic
67C2×#11758217—21no—
68C2×#11758217—21no—
69C2×#11758217—21no—
70C2×#11758217—21no—
71C2×#11758217D411yescharacteristic
72C2×#11758217D411yescharacteristic
73#14519844—21no—
74#14519844—21no—
75#14519844—21no—
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420—21no—
79C4⋊#92420—21no—
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82C4⋊#92420C2311yescharacteristic
83C4⋊#92420C2311yescharacteristic
84C22⋊#92420—21no—
85C22⋊#92420—21no—
86C22⋊#92420—21no—
87#14519832—21no—
88#14519832—21no—
89#14519832—21no—
90#14519832—21no—
91#14519832D411yescharacteristic
92#14519832D411yescharacteristic
93C4×#92420—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C22×Dic122C2×C411yescharacteristic
102#14519807—24no—
103C2×#11758202C2×C411yescharacteristic
104C2×#11758202C2×C411yescharacteristic
105C2×#11758200C2×C411yescharacteristic
106C2×#11758200C2×C411yescharacteristic
107C2×#11758194C2×C411yescharacteristic
108Q8×Dic61C2×C411yescharacteristic
109Q8×Dic61C2311yescharacteristic
110#14519742—21no—
111#14519742—21no—
112#14519742C2×C411yescharacteristic
113#14519742C2×C411yescharacteristic
114#14519742C2311yescharacteristic
115#14519742C2311yescharacteristic
116D4×Dic61—21no—
117D4×Dic61—21no—
118D4×Dic61C2×C411yescharacteristic
119D4×Dic61C2311yescharacteristic
120#14519736—21no—
121#14519736C2×C411yescharacteristic
122C2×#11758196—21no—
123C2×C4×Dic61—21no—
124C2×C4×Dic61—21no—
125C2×C4×Dic61C2311yescharacteristic
126#14519710C2×C411yescharacteristic
127#14519710C2×C411yescharacteristic
128#14519708C2×C411yescharacteristic
129#14519708C2×C411yescharacteristic
130#14519702C2×C411yescharacteristic
131#14519702C2×C411yescharacteristic
132#14519702C2×C411yescharacteristic
133#14519702C2×C411yescharacteristic
134#14519687C2×C411yescharacteristic
135#14519687C2×C411yescharacteristic
136C8×Dic61C2311yescharacteristic
137C8×Dic61C2311yescharacteristic
138C2×#41×C61C2×C411yescharacteristic
139C2×M16×C61C2×C411yescharacteristic
140C2×C4⋊C4×C61C2×C411yescharacteristic
141#103×C61C2×C411yescharacteristic
142#103×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144#102×C61C2×C411yescharacteristic
145#41×C61—22no—
146#41×C61—41no—
147C2×Q8×C61C22×C411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61#4111yescharacteristic
158C2×Dic122—22no—
159C2×Dic122—21no—
160C2×Dic122—41no—
161C2×Dic122C22×C411yescharacteristic
162C2×Dic122C22×C411yescharacteristic
163#11758217—22no—
164#11758217—41no—
165#11758217—42no—
166#11758217—42no—
167#11758217—42no—
168#11758217—42no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420C2×D411yescharacteristic
176C2×#92420C2×D411yescharacteristic
177C2×#92420C2×D411yescharacteristic
178C2×#92420C2×D411yescharacteristic
179M16×C61—41no—
180C2×C8×C61C22×C411yescharacteristic
181C2×C8×C61C22×C411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C22⋊Dic61—21no—
186C22⋊Dic61—21no—
187C22⋊Dic61—21no—
188#11758202—22no—
189#11758202M1614yescomplemented
190#11758200—22no—
191#11758200—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61#4111yescharacteristic
210C4×Dic61C2411yescharacteristic
211#11758194—22no—
212C2×#92418C22×C411yescharacteristic
213C2×#92418C22×C411yescharacteristic
214C23×C61—21no—
215C23×C61C4×D411yescharacteristic
216Q8×C61—21no—
217Q8×C61C2×M1612yes—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—42no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—42no—
227C2×C4×C61—81no—
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61C23×C411yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—81no—
241C2×Dic61C4×D411yescharacteristic
242C2×Dic61C4×D411yescharacteristic
243C2×Dic61C22×D411yescharacteristic
244C2×Dic61C2×#4111yescharacteristic
245Dic122—21no—
246Dic122—21no—
247Dic122—21no—
248Dic122—42no—
249Dic122C2×M1612yes—
250#92420—22no—
251#92420—21no—
252#92420—21no—
253#92420—21no—
254#92420—41no—
255#92420—41no—
256#92420—41no—
257#92420—41no—
258#92420—41no—
259#92420#13712yescomplemented
260C8×C61—21no—
261C8×C61—21no—
262#92418—21no—
263#92418C2×M1612yes—
264#18049—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61C2×C4×D411yescharacteristic
272C4×C61—21no—
273C4×C61—22no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61C22×M1611yescharacteristic
277C4×C61#56811yescharacteristic
278Dic61—22no—
279Dic61—21no—
280Dic61—21no—
281Dic61—21no—
282Dic61—41no—
283Dic61—41no—
284Dic61—41no—
285Dic61C2×#13711yescharacteristic
286Dic61#57511yescharacteristic
287#2971—611no—
288Q8×M16—611no—
289C4×#137—611no—
290#1592—611no—
291#1590—611no—
292#1588—611no—
293#1587—611no—
294#1584—611no—
295#1583—611no—
296#1579—611no—
297C8×Q16—611no—
298#1569—611no—
299#1491—611no—
300#1489—611no—
301#1455—611no—
302C2×C61—41no—
303C2×C61—81no—
304C2×C61D4×M1611yescharacteristic
305C2×C61C4×#13711yescharacteristic
306C2×C61#295611yescharacteristic
307C2×#137—611no—
308C4×#41—611no—
309C2×C4×Q8—611no—
310#446—611no—
311#446—611no—
312#446—1221no—
313#446—1221no—
314C8×Q8—611no—
315C8×Q8—611no—
316C8×Q8—611no—
317C8×Q8—611no—
318#441—611no—
319#441—611no—
320#440—611no—
321#440—611no—
322C4×Q16—611no—
323C4×Q16—611no—
324C4×SD16—611no—
325C4×SD16—611no—
326#436—611no—
327#436—611no—
328#436—1221no—
329#436—1221no—
330#432—611no—
331#432—1221no—
332#428—611no—
333#423—611no—
334#423—1221no—
335#419—611no—
336#417#692311yescharacteristic
337C4×M16—611no—
338C4×M16—611no—
339C4×M16—611no—
340C4×M16—1221no—
341C4×M16—1221no—
342#335—611no—
343#335—611no—
344#334—611no—
345#334—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8—611no—
349Q8⋊C8—611no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352D4⋊C8—611no—
353D4⋊C8—611no—
354#322—611no—
355#322—611no—
356#322—611no—
357C82—611no—
358C61#1804911yescharacteristic, complemented, Sylow
359C2×#41C2×#692311yescharacteristic
360C22×Q8—611no—
361#137—1224no—
362C2×Q16—611no—
363C2×Q16—611no—
364C2×SD16—611no—
365C2×SD16—611no—
366C2×M16—611no—
367C2×M16—611no—
368C2×M16—611no—
369C2×M16—1221no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1221no—
373C2×M16C2×#692311yescharacteristic
374C4×Q8—611no—
375C4×Q8—611no—
376C4×Q8—611no—
377C4×Q8—611no—
378C4×Q8—611no—
379C4×Q8—611no—
380C4×Q8—1221no—
381C4×Q8—1221no—
382C4×D4—611no—
383C4×D4—611no—
384C4×D4—1221no—
385C4×D4—1221no—
386#117—611no—
387#117—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C2×#692311yescharacteristic
390C2×C42—611no—
391C2×C42—1221no—
392C2×C42—1221no—
393#107—611no—
394#107—611no—
395#106—611no—
396#106—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408#103—611no—
409#103—611no—
410#103—611no—
411#103—611no—
412#103C2×#692311yescharacteristic
413#103C2×#692311yescharacteristic
414#102C2×#692311yescharacteristic
415#102C2×#692311yescharacteristic
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419#97—611no—
420#97—611no—
421#97—611no—
422#97—611no—
423#97—1221no—
424#97—1221no—
425#97—1221no—
426#97—1221no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—1221no—
434C4×C8—1221no—
435C4×C8—1221no—
436C4×C8—1221no—
437#41—22no—
438#41—41no—
439C2×Q8—611no—
440C2×Q8—611no—
441C2×Q8—1222no—
442C2×Q8—1221no—
443C2×Q8—2441no—
444C2×Q8C22×#692311yescharacteristic
445C2×D4—21no—
446C2×D4C22×#692311yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4C22×#692311yescharacteristic
455Q16—614no—
456Q16—1222no—
457SD16—1222no—
458SD16—1222no—
459M16—41no—
460M16—1222no—
461M16—1222no—
462M16—2442no—
463M16—2441no—
464M16—2442no—
465M16—2442no—
466M16—2442no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—611no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8C22×#692311yescharacteristic
480C2×C8C22×#692311yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—611no—
483C4⋊C4—611no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4C22×#692311yescharacteristic
491C4⋊C4C22×#692311yescharacteristic
492C22⋊C4—1221no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—1221no—
500C42—1221no—
501C42—1221no—
502C42—1221no—
503C42—1221no—
504C42—1221no—
505C42—1221no—
506C23—21no—
507C23D4×#692311yescharacteristic
508Q8—21no—
509Q8—612no—
510Q8—1221no—
511Q8—1221no—
512Q8—1221no—
513Q8—2442no—
514Q8D61⋊M1612yes—
515D4—21no—
516D4—21no—
517D4—42no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—81no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—4881no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2442no—
540C2×C4D4×#692311yescharacteristic
541C2×C4C23×#692311yescharacteristic
542C8—21no—
543C8—21no—
544C8—612no—
545C8—612no—
546C8—1221no—
547C8—1221no—
548C8—1221no—
549C8—1222no—
550C8—1221no—
551C8—2441no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C8—2441no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22C2×D4×#692311yescharacteristic
563C4—21no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—611no—
568C4—611no—
569C4—1222no—
570C4—1221no—
571C4—1221no—
572C4—1221no—
573C4—2441no—
574C4—2441no—
575C4—2441no—
576C4C2×D61⋊M1611yescharacteristic
577C4#1684084011yescharacteristic
578C2—41no—
579C2—81no—
580C2#2146255711yescharacteristic
581C2#137×#692311yescharacteristic
582C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#417C4
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C2×C4
cosocleC4×Dic61C24
hypercenter#417#6923
hypocenterC61#18049
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #21462620C2semidirect
3. #21461460C2semidirect
4. #21461456C2semidirect
5. #21461449C2semidirect
6. #21461442C2semidirect
7. #21461430C2semidirect
8. #21461429C2semidirect
9. #21461413C2semidirect
10. #21461406C2semidirect
11. #21461383C2semidirect
12. #21461163C2semidirect
14. #21461042C2semidirect
189. #11758202M16semidirect
259. #92420#137semidirect
358. C61#18049semidirect

Maximal subgroups

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

subgroupindex
#214626652
#214626202
#214614602
#214614562
#214614492
#214614422
#214614302
#214614292
#214614132
#214614062
#214613832
#214611632
#214611542
#214610422
#214546432
#1804961