Finite Group Databasebeta

← #30615950

Subgroups — #30615950

Order 15616.

1178 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840752C211yesmaximal, characteristic
2C22×C8×#6923C211yesmaximal, characteristic, complemented
3#21462280C211yesmaximal, characteristic
4#21462279C211yesmaximal, characteristic, complemented
5#21461909C214yesmaximal, complemented
6#21461908C214yesmaximal, complemented
7#21461505C211yesmaximal, characteristic, complemented
8#21461504C211yesmaximal, characteristic, complemented
9#416×D61C211yesmaximal, characteristic
10C22×D61⋊C8C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13C22×C4×#6923C2211yescharacteristic
14#16840752—24no—
15C2×#14519840C2212yes—
16C2×#14519839C2212yes—
17C2×C8×#6923—24no—
18C2×C8×#6923C2214yes—
19(C2×#6923)⋊Q8C2212yes—
20#16840714C2212yes—
21#16840713C2214yes—
22#16840710—22no—
23#16840709—22no—
24C4×C4⋊#6923—22no—
25C4×C22⋊#6923—22no—
26#16840636C2214yes—
27#16840635C2214yes—
28C2×#41×D61C2211yescharacteristic
29C22×C8×D61C2211yescharacteristic
30#117×D61C2211yescharacteristic
31#16839759C411yescharacteristic, complemented
32#16839758C411yescharacteristic, complemented
33#16839757C412yescomplemented
34#16839756C411yescharacteristic, complemented
35#16839755C412yescomplemented
36#103×D61C2212yes—
37#102×D61C2212yes—
38C61×#416C411yescharacteristic, complemented
39C23×#6923D412yes—
40C2×D61⋊C8—21no—
41C2×D61⋊C8—22no—
42C2×D61⋊C8D414yes—
43C2×D61⋊C8C2312yes—
44C22×#92420—21no—
45#14519856—22no—
46#14519856—22no—
47#14519856—22no—
48#14519856—22no—
49C2×C4⋊#6923—24no—
50C2×C4⋊#6923C2312yes—
51C2×C4⋊#6923C2312yes—
52C2×C22⋊#6923—21no—
53C2×C22⋊#6923—21no—
54C2×C22⋊#6923—24no—
55C2×C4×#6923—22no—
56C2×C4×#6923—22no—
57C2×C4×#6923—21no—
58C2×C4×#6923—21no—
59C2×C4×#6923D412yes—
60C2×C4×#6923C2312yes—
61#14519840—24no—
62#14519839—24no—
63C8×#6923—28no—
64(C2×#6923)⋊C4—24no—
65(C2×#6923)⋊C4—24no—
66C2×#11758226C2×C411yescharacteristic
67C2×#11758222—21no—
68C2×#11758222C2×C411yescharacteristic
69C2×#11758224—21no—
70C2×#11758224C2×C411yescharacteristic
71#41×D61—22no—
72#41×D61—22no—
73#41×D61—22no—
74#41×D61—22no—
75C2×Q8×D61C2311yescharacteristic
76C2×D4×D61—21no—
77C2×D4×D61C2311yescharacteristic
78C22×C4×D61—21no—
79C22×C4×D61C2311yescharacteristic
80C22×#92418C2×C411yescharacteristic
81C2×C8×D61—22no—
82C2×C8×D61—22no—
83C2×C8×D61C2×C412yes—
84C2×C8×D61C2312yes—
85#14519736C2×C411yescharacteristic
86#14519735C2×C412yes—
87#14519734—21no—
88#14519733—21no—
89C4⋊C4×D61C2312yes—
90C22⋊C4×D61—21no—
91C42×D61—21no—
92#14519702C2×C412yescomplemented
93#14519701C2×C412yescomplemented
94#14519700C2×C412yescomplemented
95#14519699C2×C412yescomplemented
96#14519698C2×C412yescomplemented
97#14519697C2×C412yescomplemented
98C2×#41×C61C2×C411yescharacteristic
99C22×C8×C61C2×C411yescharacteristic
100#117×C61C2×C411yescharacteristic
101#103×C61C2×C412yescomplemented
102#102×C61C2×C412yescomplemented
103C23×D61—21no—
104C23×D61C2×D411yescharacteristic
105C22×#6923—24no—
106C22×#6923—28no—
107C22×#6923—22no—
108C22×#6923—22no—
109C22×#6923C2×D414yes—
110#41×C61—22no—
111#41×C61—22no—
112C2×Q8×C61C22×C411yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61C22×C411yescharacteristic
115C22×C4×C61—21no—
116C22×C4×C61C22×C411yescharacteristic
117C2×C22⋊D61—21no—
118C2×C22⋊D61—21no—
119C2×C22⋊D61—21no—
120C22×Dic61—21no—
121C22×Dic61#4111yescharacteristic
122#11758226—22no—
123#11758226—22no—
124Q8×D61—21no—
125Q8×D61—21no—
126Q8×D61—21no—
127Q8×D61—21no—
128#11758224—22no—
129#11758224—22no—
130#11758224—42no—
131#11758224—42no—
132D4×D61—21no—
133D4×D61—21no—
134D4×D61—21no—
135D4×D61—21no—
136D4×D61—42no—
137D4×D61—42no—
138#11758222—22no—
139#11758222—22no—
140#11758222—42no—
141#11758222—42no—
142C2×D244—21no—
143C2×D244C22×C411yescharacteristic
144C2×C4×D61—21no—
145C2×C4×D61—22no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61C22×C411yescharacteristic
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61#4112yes—
154C2×C4×D61#4112yes—
155C2×C4×D61#4112yes—
156C2×C4×D61#4111yescharacteristic
157C2×C4×D61C2411yescharacteristic
158C2×Dic122—21no—
159C2×Dic122C22×C411yescharacteristic
160C22⋊#6923—42no—
161C22⋊#6923—42no—
162C22⋊#6923—48no—
163C2×#92420—24no—
164C2×#92420—22no—
165C4⋊#6923—22no—
166C4⋊#6923—22no—
167C4⋊#6923—22no—
168C4⋊#6923—22no—
169C4⋊#6923—48no—
170C4×#6923—24no—
171C4×#6923—22no—
172C4×#6923—22no—
173C4×#6923—42no—
174C4×#6923—42no—
175D61⋊C8—24no—
176D61⋊C8—22no—
177D61⋊C8C2×D414yes—
178C2×C8×C61—22no—
179C2×C8×C61C22×C412yes—
180C4⋊C4×C61C22×C412yes—
181C22⋊C4×C61—21no—
182C42×C61—21no—
183C22⋊Dic61—21no—
184#11758198—22no—
185C4⋊Dic61C22×C412yes—
186#11758196—22no—
187C4×Dic61—21no—
188C4×Dic61—22no—
189C2×#92418—22no—
190C2×#92418C22×C412yes—
191C8×D61—24no—
192C8×D61—22no—
193C8×D61—22no—
194C23×C61—21no—
195C23×C61C4×D411yescharacteristic
196C22×D61—24no—
197C22×D61—21no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—44no—
201C22×D61C4×D411yescharacteristic
202C22×D61#12112yes—
203C22×D61#12412yes—
204C22×D61C22×D411yescharacteristic
205C2×#6923—216no—
206C2×#6923—24no—
207C2×#6923—24no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923—44no—
211C2×#6923—44no—
212C2×#6923#13518yes—
213Q8×C61—21no—
214Q8×C61—21no—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—42no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—42no—
221C2×C4×C61#11712yes—
222C2×C4×C61C4×D412yes—
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C23×C411yescharacteristic
225C22⋊D61—42no—
226C22⋊D61—42no—
227C22⋊D61—41no—
228C22⋊D61—41no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—42no—
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61#12312yes—
234C2×Dic61#12612yes—
235C2×Dic61C2×#4111yescharacteristic
236D244—21no—
237D244—21no—
238D244—42no—
239C4×D61—41no—
240C4×D61—41no—
241C4×D61—44no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61#11712yes—
245C4×D61C4×D412yes—
246C4×D61#12112yes—
247C4×D61#12312yes—
248C4×D61#12412yes—
249C4×D61#12612yes—
250C4×D61C22×D411yescharacteristic
251C4×D61C2×#4111yescharacteristic
252C4×D61C2×#4111yescharacteristic
253C4×D61C2×#4111yescharacteristic
254Dic122—21no—
255Dic122—21no—
256Dic122—41no—
257Dic122—41no—
258#92420—24no—
259C8×C61—22no—
260#92418—22no—
261#21083—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—42no—
264C22×C61—41no—
265C22×C61#38812yes—
266C22×C61C2×C4×D411yescharacteristic
267C2×D61—22no—
268C2×D61—24no—
269C2×D61—24no—
270C2×D61—42no—
271C2×D61—41no—
272C2×D61—82no—
273C2×D61—82no—
274C2×D61#38812yes—
275C2×D61C2×#12111yescharacteristic
276C2×D61C2×#12411yescharacteristic
277C2×D61C2×#13514yes—
278#6923—28no—
279#6923—82no—
280#6923—82no—
281C4×C61—42no—
282C4×C61—41no—
283C4×C61#38812yes—
284C4×C61C2×#11711yescharacteristic
285C4×C61C2×C4×D411yescharacteristic
286Dic61—42no—
287Dic61—41no—
288Dic61#38812yes—
289Dic61C2×#12311yescharacteristic
290Dic61C2×#12611yescharacteristic
291C2×#416—611no—
292C2×#416—611no—
293C22×C4×C8—611no—
294#2297—611no—
295#2296—611no—
296#1934—614no—
297#1933—614no—
298#1735—611no—
299#1735—611no—
300C2×C61—22no—
301C2×C61—81no—
302C2×C61C2×#38811yescharacteristic
303C2×C61C4×#13512yes—
304D61—22no—
305D61—81no—
306D61#305012yes—
307D61#313112yes—
308C22×#41—611no—
309C23×C8—611no—
310C23×C8—611no—
311C2×#117—611no—
312C2×#117—611no—
313C2×#117—611no—
314C22×C42—611no—
315#416—611no—
316#416—611no—
317#416—611no—
318#416—612no—
319#416—612no—
320#416—1224no—
321#416#692311yescharacteristic, complemented
322C2×#103—612no—
323C2×#103—612no—
324C2×#102—612no—
325C2×#102—612no—
326C2×C4×C8—614no—
327C2×C4×C8—1224no—
328#391—612no—
329#390—612no—
330#389—614no—
331#386—1222no—
332#385—1222no—
333C4×C4⋊C4—1222no—
334C4×C22⋊C4—1222no—
335#340—614no—
336#340—614no—
337C61#2108311yescharacteristic, complemented, Sylow
338C2×#41—611no—
339C2×#41—611no—
340C2×#41—611no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41C2×#692311yescharacteristic
348C22×Q8—611no—
349C22×D4—611no—
350C22×D4—1221no—
351C23×C4—611no—
352C23×C4—612no—
353C23×C4—1221no—
354C22×C8—614no—
355C22×C8—612no—
356C22×C8—611no—
357C22×C8—612no—
358C22×C8—612no—
359C22×C8—1222no—
360C22×C8—1222no—
361C22×C8—1221no—
362C22×C8—1222no—
363C22×C8—1221no—
364C22×C8C2×#692311yescharacteristic
365#117—611no—
366#117—612no—
367#117—1221no—
368#117—1221no—
369#117—1222no—
370#117—1222no—
371#117—1222no—
372#117—1222no—
373#117C2×#692311yescharacteristic
374C2×C4⋊C4—612no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1224no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1224no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C42—612no—
383C2×C42—612no—
384C2×C42—1221no—
385C2×C42—1222no—
386C2×C42—1222no—
387C2×C42—1221no—
388C2×C42—1221no—
389#103—612no—
390#103—612no—
391#103—612no—
392#103—1224no—
393#103C2×#692312yescomplemented
394#102—612no—
395#102—612no—
396#102—612no—
397#102—1224no—
398#102C2×#692312yescomplemented
399C4×C8—1228no—
400#95—1224no—
401#95—1224no—
402C24—611no—
403C24—1221no—
404#41—22no—
405#41—22no—
406#41—1222no—
407#41—1222no—
408#41—1222no—
409#41—1222no—
410#41—1222no—
411#41—1222no—
412#41—2442no—
413#41—2442no—
414#41—2442no—
415#41—2442no—
416C2×Q8—611no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8C22×#692311yescharacteristic
423C2×D4—21no—
424C2×D4—611no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—2442no—
434C2×D4—2442no—
435C2×D4C22×#692311yescharacteristic
436C22×C4—21no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—614no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1222no—
450C22×C4—1228no—
451C22×C4—1221no—
452C22×C4—1224no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4C22×#692311yescharacteristic
459C2×C8—22no—
460C2×C8—614no—
461C2×C8—612no—
462C2×C8—1222no—
463C2×C8—1224no—
464C2×C8—1222no—
465C2×C8—1224no—
466C2×C8—1222no—
467C2×C8—1224no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8C22×#692312yes—
471C4⋊C4—612no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—2448no—
478C4⋊C4C22×#692312yes—
479C22⋊C4—21no—
480C22⋊C4—1222no—
481C22⋊C4—1221no—
482C22⋊C4—2448no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C42—21no—
486C42—1222no—
487C42—1222no—
488C42—1224no—
489C42—1222no—
490C42—1221no—
491C42—2442no—
492C42—2442no—
493C23—21no—
494C23—611no—
495C23—612no—
496C23—612no—
497C23—611no—
498C23—1224no—
499C23—1221no—
500C23—2444no—
501C23—2441no—
502C23—2441no—
503C23D4×#692311yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—1221no—
507Q8—1221no—
508Q8—2441no—
509Q8—2441no—
510D4—21no—
511D4—21no—
512D4—42no—
513D4—1221no—
514D4—1221no—
515D4—2442no—
516D4—2441no—
517D4—2441no—
518D4—2442no—
519D4—2442no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—42no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—612no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—618no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—12216no—
541C2×C4—1222no—
542C2×C4—1221no—
543C2×C4—2442no—
544C2×C4—2444no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—2442no—
553C2×C4#1451985612yes—
554C2×C4D4×#692312yes—
555C2×C4D4×#692311yescharacteristic
556C2×C4C23×#692311yescharacteristic
557C8—22no—
558C8—1224no—
559C8—1222no—
560C22—22no—
561C22—41no—
562C22—42no—
563C22—611no—
564C22—611no—
565C22—614no—
566C22—612no—
567C22—1224no—
568C22—1224no—
569C22—1222no—
570C22—4882no—
571C22—4882no—
572C22—2441no—
573C22—2442no—
574C22#1684071212yes—
575C22C2×D4×#692311yescharacteristic
576C4—41no—
577C4—42no—
578C4—611no—
579C4—611no—
580C4—612no—
581C4—1228no—
582C4—4882no—
583C4—4882no—
584C4—2441no—
585C4—2442no—
586C4#1684071212yes—
587C4C2×#1451985611yescharacteristic
588C4C2×D4×#692311yescharacteristic
589C2—22no—
590C2—81no—
591C2—612no—
592C2—612no—
593C2—1222no—
594C2—4881no—
595C2C2×#1684071211yescharacteristic
596C2#135×#692312yes—
subgroupG/H
centerC2×C4D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#416C4
socleC22×C61C2×C4×D4
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#416#6923
hypocenterC61#21083
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×C8×#6923C2semidirect
4. #21462279C2semidirect
5. #21461909C2semidirect
6. #21461908C2semidirect
7. #21461505C2semidirect
8. #21461504C2semidirect
31. #16839759C4semidirect
32. #16839758C4semidirect
33. #16839757C4semidirect
34. #16839756C4semidirect
35. #16839755C4semidirect
38. C61×#416C4semidirect
92. #14519702C2×C4semidirect
93. #14519701C2×C4semidirect
94. #14519700C2×C4semidirect
95. #14519699C2×C4semidirect
96. #14519698C2×C4semidirect
97. #14519697C2×C4semidirect
101. #103×C61C2×C4semidirect
102. #102×C61C2×C4semidirect
321. #416#6923semidirect
337. C61#21083semidirect
393. #103C2×#6923semidirect
398. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407522
C22×C8×#69232
#214622802
#214622792
#214619092
#214619082
#214615052
#214615042
#416×D612
#2108361