Finite Group Databasebeta

← #30615912

Subgroups — #30615912

Order 15616.

1342 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic, complemented
2C2×#16840754C211yesmaximal, characteristic
3C2×#16840714C211yesmaximal, characteristic
4#117×#6923C211yesmaximal, characteristic, complemented
5#21461897C214yesmaximal, complemented
6#21461896C214yesmaximal, complemented
7#21461508C211yesmaximal, characteristic, complemented
8#21461481C211yesmaximal, characteristic, complemented
9#418×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C22×C4⋊#6923C2211yescharacteristic
13C22×C22⋊#6923—22no—
14C22×C4×#6923C2211yescharacteristic
15#16840773—24no—
16C2×#14519846C2212yescomplemented
17C2×#14519845C2212yescomplemented
18#16840754—24no—
19C2×#14519839C2212yes—
20C2×#14519839C2212yes—
21#16840714—21no—
22#16840714—21no—
23#16840714C2214yes—
24C4⋊C4×#6923C2212yescomplemented
25C4⋊C4×#6923C2212yescomplemented
26C22⋊C4×#6923—22no—
27#16840698—21no—
28C42×#6923—21no—
29#16840635C2214yescomplemented
30(C2×#6923)⋊C8C2214yescomplemented
31C22×D4×D61C2211yescharacteristic
32C2×M16×D61C2211yescharacteristic
33#117×D61C2211yescharacteristic
34#16839768C411yescharacteristic, complemented
35#16839767C411yescharacteristic, complemented
36#16839766C411yescharacteristic, complemented
37#16839765C414yescomplemented
38#102×D61C2214yes—
39C61×#418C411yescharacteristic, complemented
40C23×#6923—21no—
41C23×#6923D412yes—
42C2×#11758213—21no—
43D61⋊M16—22no—
44D61⋊M16—22no—
45C2×#11758217—21no—
46C2×D61⋊C8C2312yes—
47#14519856—22no—
48#14519856D414yes—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923C2312yes—
53C2×C4⋊#6923C2311yescharacteristic
54C2×C4⋊#6923C2311yescharacteristic
55C2×C22⋊#6923—21no—
56C2×C22⋊#6923—24no—
57C2×C22⋊#6923—24no—
58C2×C22⋊#6923—48no—
59C2×C4×#6923—21no—
60C2×C4×#6923—22no—
61C2×C4×#6923—22no—
62C2×C4×#6923—22no—
63C2×C4×#6923—22no—
64C2×C4×#6923D411yescharacteristic
65C2×C4×#6923D411yescharacteristic
66C2×C4×#6923C2312yes—
67#14519846—24no—
68#14519845—24no—
69#14519839—24no—
70#14519839—24no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4—22no—
73C24×D61—21no—
74C22×D244C2×C411yescharacteristic
75C22×C22⋊D61—21no—
76C2×D4×D61—22no—
77C2×D4×D61—21no—
78C2×D4×D61—22no—
79C2×D4×D61—22no—
80C2×D4×D61—22no—
81C2×D4×D61—21no—
82C2×D4×D61C2×C412yes—
83C2×D4×D61C2312yes—
84C22×C4×D61C2311yescharacteristic
85C2×#11758194C2×C411yescharacteristic
86C2×#11758189C2×C412yes—
87M16×D61—22no—
88M16×D61—22no—
89C2×C8×D61C2312yes—
90#14519736C2×C411yescharacteristic
91#14519735C2×C412yes—
92#14519734—21no—
93#14519733—21no—
94C4⋊C4×D61C2312yes—
95C22⋊C4×D61—21no—
96C42×D61—21no—
97#14519699C2×C414yescomplemented
98#14519698C2×C414yescomplemented
99#14519697C2×C414yescomplemented
100C22×D4×C61C2×C411yescharacteristic
101C2×M16×C61C2×C411yescharacteristic
102#117×C61C2×C411yescharacteristic
103#102×C61C2×C414yescomplemented
104C24×C61—21no—
105C23×D61—21no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—44no—
109C23×D61—44no—
110C23×D61—42no—
111C23×D61—42no—
112C23×D61C2×D411yescharacteristic
113C22×#6923—24no—
114C22×#6923—21no—
115C22×#6923—21no—
116C22×#6923—24no—
117C22×#6923—22no—
118C22×#6923—42no—
119C22×#6923—44no—
120C22×#6923—42no—
121C22×#6923C2×D412yes—
122C22×#6923C2×D412yes—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61C22×C412yes—
127C22×C4×C61C22×C411yescharacteristic
128C2×C22⋊D61—22no—
129C2×C22⋊D61—22no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C22×Dic61#4111yescharacteristic
133D4×D61—24no—
134D4×D61—24no—
135D4×D61—22no—
136D4×D61—22no—
137D4×D61—22no—
138D4×D61—22no—
139D4×D61—42no—
140D4×D61—42no—
141D4×D61—42no—
142D4×D61—42no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—21no—
146C2×D244C22×C412yes—
147C2×C4×D61—22no—
148C2×C4×D61C22×C411yescharacteristic
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61#4112yes—
152C2×C4×D61#4112yes—
153C2×C4×D61#4112yes—
154C2×C4×D61#4111yescharacteristic
155C2×C4×D61C2411yescharacteristic
156C22⋊#6923—24no—
157C22⋊#6923—48no—
158C22⋊#6923—48no—
159#11758217—42no—
160C2×#92420—22no—
161C4⋊#6923—22no—
162C4⋊#6923—21no—
163C4⋊#6923—21no—
164C4⋊#6923—22no—
165C4⋊#6923—42no—
166C4⋊#6923C2×D412yes—
167C4⋊#6923C2×D412yes—
168C4×#6923—24no—
169C4×#6923—22no—
170C4×#6923—22no—
171C4×#6923—22no—
172C4×#6923—22no—
173C4×#6923—44no—
174C4×#6923—44no—
175#11758213—42no—
176D61⋊C8—22no—
177D61⋊C8—22no—
178M16×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—
189#11758194—22no—
190C2×#92418C22×C412yes—
191#11758189—24no—
192C8×D61—22no—
193C8×D61—22no—
194C23×C61—22no—
195C23×C61—44no—
196C23×C61—42no—
197C23×C61C4×D411yescharacteristic
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—48no—
202C22×D61—44no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—44no—
207C22×D61—44no—
208C22×D61—44no—
209C22×D61—88no—
210C22×D61—84no—
211C22×D61—84no—
212C22×D61C4×D411yescharacteristic
213C22×D61#12112yes—
214C22×D61#12312yes—
215C22×D61C22×D411yescharacteristic
216C2×#6923—22no—
217C2×#6923—22no—
218C2×#6923—22no—
219C2×#6923—22no—
220C2×#6923—22no—
221C2×#6923—22no—
222C2×#6923—22no—
223C2×#6923—48no—
224C2×#6923—42no—
225C2×#6923—42no—
226C2×#6923—42no—
227C2×#6923—44no—
228C2×#6923—84no—
229C2×#6923AGL(1,ℤ/8)14yes—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—42no—
233C2×C4×C61—22no—
234C2×C4×C61#11712yes—
235C2×C4×C61C4×D412yes—
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C22⋊D61—42no—
239C22⋊D61—42no—
240C22⋊D61—44no—
241C2×Dic61—22no—
242C2×Dic61C4×D411yescharacteristic
243C2×Dic61#12412yes—
244C2×Dic61#12612yes—
245C2×Dic61C2×#4111yescharacteristic
246D244—22no—
247D244—22no—
248D244—42no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61#11712yes—
252C4×D61C4×D412yes—
253C4×D61#12112yes—
254C4×D61#12312yes—
255C4×D61#12412yes—
256C4×D61#12612yes—
257C4×D61C22×D411yescharacteristic
258C4×D61C2×#4111yescharacteristic
259C4×D61C2×#4111yescharacteristic
260C4×D61C2×#4111yescharacteristic
261#92420—42no—
262C8×C61—22no—
263#92418—22no—
264#21060—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—44no—
267C22×C61—42no—
268C22×C61—84no—
269C22×C61#38812yes—
270C22×C61C2×C4×D411yescharacteristic
271C2×D61—22no—
272C2×D61—24no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—44no—
276C2×D61—42no—
277C2×D61—88no—
278C2×D61—84no—
279C2×D61—84no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61#38812yes—
283C2×D61C2×#12111yescharacteristic
284C2×D61C2×#12311yescharacteristic
285C2×D61C2×AGL(1,ℤ/8)12yes—
286C2×D61#57512yes—
287#6923—24no—
288#6923—42no—
289#6923—42no—
290#6923—42no—
291#6923—82no—
292C4×C61—42no—
293C4×C61#38812yes—
294C4×C61C2×#11711yescharacteristic
295C4×C61C2×C4×D411yescharacteristic
296Dic61—42no—
297Dic61#38812yes—
298Dic61C2×#12411yescharacteristic
299Dic61C2×#12611yescharacteristic
300C2×#428—611no—
301C2×#418—611no—
302C2×#418—611no—
303C2×#390—611no—
304C4×#117—611no—
305#1926—614no—
306#1925—614no—
307#1736—611no—
308#1728—611no—
309C2×C61—22no—
310C2×C61—82no—
311C2×C61C2×#38811yescharacteristic
312C2×C61C4×AGL(1,ℤ/8)12yes—
313D61—22no—
314D61—82no—
315D61#303811yescharacteristic
316D61#304011yescharacteristic
317D61#309412yes—
318C23×D4—611no—
319C22×M16—611no—
320C22×M16—611no—
321C2×#117—611no—
322C2×#117—611no—
323C22×C4⋊C4—611no—
324C22×C22⋊C4—1222no—
325C22×C42—611no—
326#428—1224no—
327C2×#107—612no—
328C2×#106—612no—
329#418—611no—
330#418—611no—
331#418—611no—
332#418—614no—
333#418—1224no—
334#418#692311yescharacteristic, complemented
335C2×#102—614no—
336C2×#102—612no—
337C2×#102—612no—
338#390—614no—
339#390—1221no—
340#390—1221no—
341C4×C4⋊C4—612no—
342C4×C4⋊C4—612no—
343C4×C22⋊C4—1222no—
344#376—1221no—
345C43—1221no—
346#340—614no—
347#336—614no—
348C61#2106011yescharacteristic, complemented, Sylow
349C25—1221no—
350C22×D4—612no—
351C22×D4—612no—
352C22×D4—611no—
353C22×D4—1222no—
354C22×D4—1222no—
355C22×D4—1222no—
356C22×D4—1222no—
357C22×D4—1221no—
358C22×D4—1221no—
359C22×D4—1221no—
360C22×D4C2×#692311yescharacteristic
361C23×C4—611no—
362C23×C4—612no—
363C23×C4—1221no—
364C2×M16—611no—
365C2×M16—612no—
366C2×M16—1222no—
367C2×M16—1222no—
368C2×M16—1221no—
369C2×M16—1222no—
370C2×M16—1222no—
371C2×M16—1221no—
372C2×M16C2×#692311yescharacteristic
373C22×C8—612no—
374C22×C8—612no—
375#117—614no—
376#117—611no—
377#117—612no—
378#117—1221no—
379#117—1221no—
380#117—1222no—
381#117C2×#692311yescharacteristic
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1224no—
393C2×C22⋊C4—2448no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—612no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1222no—
400C2×C42—1222no—
401C2×C42—1222no—
402C2×C42—1222no—
403#107—1224no—
404#106—1224no—
405#102—614no—
406#102—614no—
407#102—614no—
408#102—1224no—
409#102—1224no—
410#102C2×#692314yescomplemented
411#95—1222no—
412#95—1222no—
413C24—21no—
414C24—611no—
415C24—1222no—
416C24—1222no—
417C24—1221no—
418C24—2442no—
419C24—2442no—
420C24—2444no—
421C24—2444no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—612no—
426C2×D4—1222no—
427C2×D4—1224no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1224no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1221no—
437C2×D4—2442no—
438C2×D4—2442no—
439C2×D4—2442no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4—2442no—
443C2×D4C22×#692312yes—
444C22×C4—611no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—1222no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1224no—
460C22×C4—1222no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2444no—
464C22×C4C22×#692311yescharacteristic
465M16—22no—
466M16—1224no—
467M16—1222no—
468M16—2442no—
469M16—2442no—
470C2×C8—612no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8C22×#692312yes—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—612no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—2442no—
486C4⋊C4C22×#692312yes—
487C22⋊C4—21no—
488C22⋊C4—1222no—
489C22⋊C4—1224no—
490C22⋊C4—1221no—
491C22⋊C4—2448no—
492C22⋊C4—2448no—
493C42—21no—
494C42—1222no—
495C42—1222no—
496C42—1222no—
497C42—1222no—
498C42—1222no—
499C42—1224no—
500C42—1221no—
501C42—2444no—
502C42—2444no—
503C23—22no—
504C23—44no—
505C23—42no—
506C23—611no—
507C23—612no—
508C23—612no—
509C23—611no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—4884no—
514C23—4888no—
515C23—4884no—
516C23—2444no—
517C23—2444no—
518C23—2442no—
519C23—2442no—
520C23—2444no—
521C23—2444no—
522C23—2448no—
523C23—2442no—
524C23D4×#692311yescharacteristic
525D4—22no—
526D4—22no—
527D4—42no—
528D4—1222no—
529D4—1222no—
530D4—2442no—
531D4—2442no—
532D4—2444no—
533D4—2442no—
534C2×C4—22no—
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—614no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—4884no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2448no—
562C2×C4—2444no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4#1451985612yes—
567C2×C4D4×#692311yescharacteristic
568C2×C4D4×#692312yes—
569C2×C4C23×#692311yescharacteristic
570C8—22no—
571C8—1222no—
572C8—2442no—
573C22—22no—
574C22—42no—
575C22—44no—
576C22—84no—
577C22—611no—
578C22—611no—
579C22—612no—
580C22—612no—
581C22—612no—
582C22—1222no—
583C22—1224no—
584C22—1222no—
585C22—1222no—
586C22—4884no—
587C22—4884no—
588C22—4882no—
589C22—4882no—
590C22—4888no—
591C22—2442no—
592C22—2444no—
593C22#1684071212yes—
594C22C2×D4×#692311yescharacteristic
595C4—42no—
596C4—611no—
597C4—611no—
598C4—612no—
599C4—1224no—
600C4—4882no—
601C4—2442no—
602C4—2442no—
603C4—2442no—
604C4—2442no—
605C4#1684071212yes—
606C4C2×#1451985611yescharacteristic
607C4C2×D4×#692311yescharacteristic
608C2—22no—
609C2—82no—
610C2—612no—
611C2—611no—
612C2—611no—
613C2—1222no—
614C2—4882no—
615C2C2×#1684071211yescharacteristic
616C2AGL(1,ℤ/8)×#692312yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#418C4
socleC22×C61C2×C4×D4
prime radicalC22×D4×D61C22
cosocleC2×C4×D61C24
hypercenter#418#6923
hypocenterC61#21060
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16840773C2semidirect
4. #117×#6923C2semidirect
5. #21461897C2semidirect
6. #21461896C2semidirect
7. #21461508C2semidirect
8. #21461481C2semidirect
16. C2×#14519846C22semidirect
17. C2×#14519845C22semidirect
24. C4⋊C4×#6923C22semidirect
25. C4⋊C4×#6923C22semidirect
29. #16840635C22semidirect
30. (C2×#6923)⋊C8C22semidirect
34. #16839768C4semidirect
35. #16839767C4semidirect
36. #16839766C4semidirect
37. #16839765C4semidirect
39. C61×#418C4semidirect
97. #14519699C2×C4semidirect
98. #14519698C2×C4semidirect
99. #14519697C2×C4semidirect
103. #102×C61C2×C4semidirect
334. #418#6923semidirect
348. C61#21060semidirect
410. #102C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
C2×#168407542
C2×#168407142
#117×#69232
#214618972
#214618962
#214615082
#214614812
#418×D612
#2106061