Finite Group Databasebeta

← #30615919

Subgroups — #30615919

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519846C211yesmaximal, characteristic, complemented
2C2×#16840753C211yesmaximal, characteristic
3#21462294C211yesmaximal, characteristic
4#21462256C211yesmaximal, characteristic, complemented
5#21461899C214yesmaximal, complemented
6#21461896C214yesmaximal, complemented
7#21461508C211yesmaximal, characteristic, complemented
8#21461484C211yesmaximal, characteristic, complemented
9#416×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C22×C4⋊#6923C2211yescharacteristic
13C22×C4⋊#6923C2211yescharacteristic
14C2×#14519846—24no—
15C2×#14519846C2214yes—
16#16840753—24no—
17C2×#14519840C2212yes—
18C2×#14519839C2212yes—
19(C2×#6923)⋊Q8C2212yes—
20#16840714C2212yes—
21#16840710—22no—
22#16840704—22no—
23C4⋊C4×#6923C2214yes—
24C4×C4⋊#6923—22no—
25C22⋊C4×#6923—22no—
26#16840635C2214yes—
27(C2×#6923)⋊C8C2214yes—
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×#6923—21no—
40C23×#6923—21no—
41C2×#11758213—21no—
42D61⋊M16—22no—
43D61⋊M16—22no—
44C2×#11758217—21no—
45C2×D61⋊C8C2312yes—
46#14519856—22no—
47#14519856—22no—
48C2×C4⋊#6923—21no—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—22no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923D412yes—
54C2×C4⋊#6923D414yes—
55C2×C4⋊#6923C2312yes—
56C2×C4⋊#6923C2312yes—
57C2×C22⋊#6923—24no—
58C2×C22⋊#6923D412yes—
59C2×C4×#6923—24no—
60C2×C4×#6923C2312yes—
61#14519846—28no—
62#14519840—24no—
63#14519839—24no—
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—22no—
106C22×#6923—24no—
107C22×#6923—22no—
108C22×#6923—42no—
109C22×#6923—82no—
110C22×#6923—82no—
111C22×#6923C2×D414yes—
112#41×C61—22no—
113#41×C61—22no—
114C2×Q8×C61C22×C411yescharacteristic
115C2×D4×C61—21no—
116C2×D4×C61C22×C411yescharacteristic
117C22×C4×C61—21no—
118C22×C4×C61C22×C411yescharacteristic
119C2×C22⋊D61—21no—
120C2×C22⋊D61—21no—
121C2×C22⋊D61—21no—
122C22×Dic61—21no—
123C22×Dic61#4111yescharacteristic
124#11758226—22no—
125#11758226—22no—
126Q8×D61—21no—
127Q8×D61—21no—
128Q8×D61—21no—
129Q8×D61—21no—
130#11758224—22no—
131#11758224—22no—
132#11758224—42no—
133#11758224—42no—
134D4×D61—21no—
135D4×D61—21no—
136D4×D61—21no—
137D4×D61—21no—
138D4×D61—42no—
139D4×D61—42no—
140#11758222—22no—
141#11758222—22no—
142#11758222—42no—
143#11758222—42no—
144C2×D244—21no—
145C2×D244C22×C411yescharacteristic
146C2×C4×D61—21no—
147C2×C4×D61—22no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61C22×C411yescharacteristic
153C2×C4×D61C2×D412yes—
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61#4112yes—
156C2×C4×D61#4112yes—
157C2×C4×D61#4112yes—
158C2×C4×D61#4111yescharacteristic
159C2×C4×D61C2411yescharacteristic
160C2×Dic122—21no—
161C2×Dic122C22×C411yescharacteristic
162C22⋊#6923—24no—
163C22⋊#6923—84no—
164#11758217—42no—
165C2×#92420—22no—
166C4⋊#6923—24no—
167C4⋊#6923—24no—
168C4⋊#6923—22no—
169C4⋊#6923—22no—
170C4⋊#6923—22no—
171C4⋊#6923—42no—
172C4⋊#6923C2×D414yes—
173C4×#6923—22no—
174C4×#6923—22no—
175C4×#6923—84no—
176#11758213—42no—
177D61⋊C8—22no—
178D61⋊C8—22no—
179C2×C8×C61—22no—
180C2×C8×C61C22×C412yes—
181C4⋊C4×C61C22×C412yes—
182C22⋊C4×C61—21no—
183C42×C61—21no—
184C22⋊Dic61—21no—
185#11758198—22no—
186C4⋊Dic61C22×C412yes—
187#11758196—22no—
188C4×Dic61—21no—
189C4×Dic61—22no—
190C2×#92418—22no—
191C2×#92418C22×C412yes—
192C8×D61—24no—
193C8×D61—22no—
194C8×D61—22no—
195C23×C61—21no—
196C23×C61C4×D411yescharacteristic
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—21no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61C4×D411yescharacteristic
205C22×D61#12112yes—
206C22×D61#12312yes—
207C22×D61C22×D411yescharacteristic
208C2×#6923—24no—
209C2×#6923—24no—
210C2×#6923—24no—
211C2×#6923—24no—
212C2×#6923—44no—
213C2×#6923—42no—
214C2×#6923—44no—
215C2×#6923—84no—
216C2×#6923—84no—
217Q8×C61—21no—
218Q8×C61—21no—
219D4×C61—21no—
220D4×C61—21no—
221D4×C61—42no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—42no—
225C2×C4×C61#11712yes—
226C2×C4×C61C4×D412yes—
227C2×C4×C61C4×D411yescharacteristic
228C2×C4×C61C23×C411yescharacteristic
229C22⋊D61—42no—
230C22⋊D61—42no—
231C22⋊D61—41no—
232C22⋊D61—41no—
233C2×Dic61—22no—
234C2×Dic61—21no—
235C2×Dic61—42no—
236C2×Dic61C4×D411yescharacteristic
237C2×Dic61#12412yes—
238C2×Dic61#12612yes—
239C2×Dic61C2×#4111yescharacteristic
240D244—21no—
241D244—21no—
242D244—42no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61#11712yes—
250C4×D61C4×D412yes—
251C4×D61#12112yes—
252C4×D61#12312yes—
253C4×D61#12412yes—
254C4×D61#12612yes—
255C4×D61C22×D411yescharacteristic
256C4×D61C2×#4111yescharacteristic
257C4×D61C2×#4111yescharacteristic
258C4×D61C2×#4111yescharacteristic
259Dic122—21no—
260Dic122—21no—
261Dic122—41no—
262Dic122—41no—
263#92420—42no—
264C8×C61—22no—
265#92418—22no—
266#21067—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—42no—
269C22×C61—41no—
270C22×C61#38812yes—
271C22×C61C2×C4×D411yescharacteristic
272C2×D61—22no—
273C2×D61—24no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—42no—
277C2×D61—41no—
278C2×D61—82no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61#38812yes—
282C2×D61C2×#12111yescharacteristic
283C2×D61C2×#12311yescharacteristic
284C2×D61#57612yes—
285C2×D61#57812yes—
286#6923—44no—
287#6923—44no—
288#6923—82no—
289C4×C61—42no—
290C4×C61—41no—
291C4×C61#38812yes—
292C4×C61C2×#11711yescharacteristic
293C4×C61C2×C4×D411yescharacteristic
294Dic61—42no—
295Dic61—41no—
296Dic61#38812yes—
297Dic61C2×#12411yescharacteristic
298Dic61C2×#12611yescharacteristic
299C22×#107—611no—
300C2×#417—611no—
301C2×#416—611no—
302#2311—611no—
303#2274—611no—
304#1928—614no—
305#1925—614no—
306#1736—611no—
307#1729—611no—
308C2×C61—22no—
309C2×C61—81no—
310C2×C61C2×#38811yescharacteristic
311C2×C61#295112yes—
312D61—22no—
313D61—81no—
314D61#304111yescharacteristic
315D61#304411yescharacteristic
316D61#309812yes—
317C22×#41—611no—
318C22×M16—611no—
319C23×C8—611no—
320C2×#117—611no—
321C2×#117—611no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—611no—
324C2×#107—614no—
325C2×#107—1224no—
326#417—1224no—
327#416—611no—
328#416—611no—
329#416—611no—
330#416—612no—
331#416—612no—
332#416#692311yescharacteristic, complemented
333C2×#103—612no—
334C2×#103—612no—
335C2×#102—612no—
336C2×#102—612no—
337#391—612no—
338#390—612no—
339#386—1222no—
340#380—1222no—
341C4×C4⋊C4—614no—
342C4×C4⋊C4—1222no—
343C4×C22⋊C4—1222no—
344#340—614no—
345#336—614no—
346C61#2106711yescharacteristic, complemented, Sylow
347C2×#41—611no—
348C2×#41—611no—
349C2×#41—611no—
350C2×#41—1221no—
351C2×#41—1221no—
352C2×#41—1222no—
353C2×#41—1222no—
354C2×#41—1222no—
355C2×#41—1222no—
356C2×#41C2×#692311yescharacteristic
357C22×Q8—611no—
358C22×D4—611no—
359C22×D4—1221no—
360C23×C4—611no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4—1221no—
364C2×M16—1221no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—1221no—
368C22×C8—611no—
369C22×C8—612no—
370C22×C8—612no—
371C22×C8—612no—
372C22×C8—1222no—
373C22×C8—1222no—
374C22×C8C2×#692311yescharacteristic
375#117—611no—
376#117—612no—
377#117—1221no—
378#117—1221no—
379#117—1222no—
380#117—1222no—
381#117C2×#692311yescharacteristic
382C2×C4⋊C4—614no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C22⋊C4—612no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1224no—
395C2×C42—612no—
396C2×C42—1221no—
397C2×C42—1224no—
398#107—1228no—
399#103—612no—
400#103—612no—
401#103—612no—
402#103—1224no—
403#103C2×#692312yescomplemented
404#102—612no—
405#102—612no—
406#102—612no—
407#102—1224no—
408#102C2×#692312yescomplemented
409#95—1224no—
410#95—1224no—
411C24—611no—
412C24—1221no—
413#41—22no—
414#41—22no—
415#41—1222no—
416#41—1222no—
417#41—1222no—
418#41—1222no—
419#41—1222no—
420#41—1222no—
421#41—2442no—
422#41—2442no—
423#41—2442no—
424#41—2442no—
425C2×Q8—611no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8C22×#692311yescharacteristic
432C2×D4—21no—
433C2×D4—611no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—2442no—
443C2×D4—2442no—
444C2×D4C22×#692311yescharacteristic
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—611no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—614no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1221no—
460C22×C4—1224no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—4882no—
465C22×C4—4882no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4C22×#692311yescharacteristic
470M16—2442no—
471M16—2442no—
472C2×C8—22no—
473C2×C8—612no—
474C2×C8—1222no—
475C2×C8—1224no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8C22×#692312yes—
482C4⋊C4—614no—
483C4⋊C4—612no—
484C4⋊C4—1222no—
485C4⋊C4—1224no—
486C4⋊C4—1222no—
487C4⋊C4—1224no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—2442no—
491C4⋊C4C22×#692312yes—
492C22⋊C4—21no—
493C22⋊C4—1222no—
494C22⋊C4—1224no—
495C22⋊C4—1221no—
496C22⋊C4—4884no—
497C42—21no—
498C42—1222no—
499C42—1222no—
500C42—1222no—
501C42—1221no—
502C42—4884no—
503C23—21no—
504C23—611no—
505C23—612no—
506C23—612no—
507C23—611no—
508C23—1222no—
509C23—1222no—
510C23—1221no—
511C23—2442no—
512C23—2441no—
513C23—2441no—
514C23—2442no—
515C23D4×#692311yescharacteristic
516Q8—21no—
517Q8—21no—
518Q8—1221no—
519Q8—1221no—
520Q8—2441no—
521Q8—2441no—
522D4—21no—
523D4—21no—
524D4—42no—
525D4—1221no—
526D4—1221no—
527D4—2442no—
528D4—2441no—
529D4—2441no—
530D4—2442no—
531D4—2442no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—42no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—612no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1222no—
554C2×C4—1221no—
555C2×C4—4884no—
556C2×C4—4884no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2442no—
563C2×C4—2444no—
564C2×C4—2442no—
565C2×C4—2444no—
566C2×C4—2442no—
567C2×C4#1451985612yes—
568C2×C4D4×#692311yescharacteristic
569C2×C4D4×#692312yes—
570C2×C4C23×#692311yescharacteristic
571C8—22no—
572C8—1222no—
573C8—2442no—
574C22—22no—
575C22—41no—
576C22—42no—
577C22—611no—
578C22—611no—
579C22—612no—
580C22—612no—
581C22—612no—
582C22—1222no—
583C22—1224no—
584C22—1222no—
585C22—1222no—
586C22—4881no—
587C22—4882no—
588C22—4881no—
589C22—2441no—
590C22—2442no—
591C22#1684071212yes—
592C22C2×D4×#692311yescharacteristic
593C4—41no—
594C4—42no—
595C4—611no—
596C4—611no—
597C4—612no—
598C4—4882no—
599C4—2444no—
600C4—2444no—
601C4—2441no—
602C4—2442no—
603C4#1684071212yes—
604C4C2×#1451985611yescharacteristic
605C4C2×D4×#692311yescharacteristic
606C2—22no—
607C2—81no—
608C2—611no—
609C2—612no—
610C2—611no—
611C2—1222no—
612C2—4881no—
613C2C2×#1684071211yescharacteristic
614C2#2146258112yes—
subgroupG/H
centerC22C2×D4×#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#21067
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C22×#14519846C2semidirect
4. #21462256C2semidirect
5. #21461899C2semidirect
6. #21461896C2semidirect
7. #21461508C2semidirect
8. #21461484C2semidirect
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
332. #416#6923semidirect
346. C61#21067semidirect
403. #103C2×#6923semidirect
408. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C22×#145198462
C2×#168407532
#214622942
#214622562
#214618992
#214618962
#214615082
#214614842
#416×D612
#2106761