Finite Group Databasebeta

← #29681597

Subgroups — #29681597

Order 15616.

1974 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840325C211yesmaximal, characteristic
2#518×D61C211yesmaximal, characteristic, complemented
3#21454333C212yesmaximal, complemented
4C2×#100×D61C211yesmaximal, characteristic, complemented
5#21446218C211yesmaximal, characteristic, complemented
6#21446214C212yesmaximal, complemented
7#21446207C211yesmaximal, characteristic, complemented
8#21446206C212yesmaximal, complemented
9#21446205C211yesmaximal, characteristic, complemented
10#21445527C212yesmaximal, complemented
11C61×#1887C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C22×D4×D61C2211yescharacteristic
14C2×M16×D61C2212yes—
15#16840325—24no—
16#16840325C418yescomplemented
17C2×#14519773C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×#14519749—22no—
21C2×#14519745—22no—
22#16840237C2211yescharacteristic
23#16840236C2212yes—
24#16840235—22no—
25#16840234—24no—
26C4×D4×D61—24no—
27C2×#14519734C2211yescharacteristic
28#117×D61C2212yes—
29C2×C22⋊C4×D61—22no—
30#16839772C2212yescomplemented
31#16839770C2212yescomplemented
32#16839767C2212yescomplemented
33#16839760—24no—
34#16839755—24no—
35C2×#14519694C2211yescharacteristic
36C2×#14519693C2212yes—
37#100×D61—24no—
38#16839523C2212yescomplemented
39#16839522C2212yescomplemented
40#16839521C2212yescomplemented
41C2×#127×C61C2211yescharacteristic
42C61×#518C2211yescharacteristic
43C61×#418C2212yescomplemented
44C2×#100×C61C2211yescharacteristic
45C61×#339C2212yescomplemented
46C24×D61D412yes—
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—24no—
51C2×#11758224C2×C414yescomplemented
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C22×C22⋊D61—22no—
56C22×C22⋊D61D412yes—
57C2×D4×D61—22no—
58C2×D4×D61—22no—
59C2×D4×D61—24no—
60C2×D4×D61C2×C414yes—
61C2×D4×D61D414yes—
62C23×Dic61—21no—
63C22×C4×D61—24no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758194C2312yes—
66C2×#11758189—22no—
67M16×D61—24no—
68M16×D61—24no—
69C2×C8×D61—22no—
70#14519773—22no—
71#14519773C2×C414yescomplemented
72#14519765—22no—
73#14519765C2×C414yescomplemented
74#14519764—22no—
75#14519764C2×C414yescomplemented
76#14519749—216no—
77#14519745—216no—
78C4×D244—24no—
79D4×Dic61—24no—
80#14519739—24no—
81C4×C22⋊D61—28no—
82#14519737—28no—
83#14519736C2312yes—
84#14519735D414yescomplemented
85#14519734—22no—
86#14519734—22no—
87#14519734C2×C414yescomplemented
88#14519733—22no—
89C2×#11758196—21no—
90C4⋊C4×D61D414yescomplemented
91C2×C22⋊Dic61—22no—
92C2×C22⋊Dic61—22no—
93C2×#11758198—24no—
94C2×#11758198—21no—
95C2×#11758198—22no—
96C22⋊C4×D61—24no—
97C22⋊C4×D61—24no—
98C22⋊C4×D61—22no—
99C22⋊C4×D61—24no—
100C2×C4×Dic61C2311yescharacteristic
101C42×D61—22no—
102#14519702—24no—
103#14519700—24no—
104#14519698—24no—
105#14519694—22no—
106#14519693—24no—
107C22×D4×C61—22no—
108C22×D4×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C2×M16×C61C2312yes—
111#127×C61—22no—
112#127×C61C2×C414yescomplemented
113C4×D4×C61—24no—
114#117×C61C2312yes—
115C2×C22⋊C4×C61—22no—
116C2×C42×C61C2311yescharacteristic
117#102×C61—24no—
118#100×C61—22no—
119C24×C61—21no—
120C24×C61C2×D412yes—
121C23×D61—22no—
122C23×D61—22no—
123C23×D61—24no—
124C23×D61—84no—
125C23×D61C22⋊C418yes—
126C23×D61C2×D412yes—
127C23×D61C2×D411yescharacteristic
128C2×D4×C61—28no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—21no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—84no—
135C2×D4×C61C22×C412yes—
136C2×D4×C61C22×C412yescomplemented
137C2×D4×C61C2×D412yes—
138C22×C4×C61—21no—
139C22×C4×C61—24no—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—28no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—24no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—84no—
149C2×C22⋊D61C22⋊C418yes—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—24no—
153C22×Dic61—24no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61C2×D411yescharacteristic
157#11758224—28no—
158#11758224—28no—
159#11758224—44no—
160#11758224—44no—
161D4×D61—28no—
162D4×D61—28no—
163D4×D61—28no—
164D4×D61—28no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244C22×C412yes—
168C2×D244C2×D412yes—
169C2×C4×D61—24no—
170C2×C4×D61—28no—
171C2×C4×D61—88no—
172C2×C4×D61C22×C412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61#4114yes—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—21no—
180C2×Dic122C22×C412yescomplemented
181M16×C61—22no—
182M16×C61—22no—
183C2×C8×C61—22no—
184C4⋊C4×C61C2×D414yes—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—22no—
187C42×C61—21no—
188C42×C61—22no—
189C42×C61C22×C412yescomplemented
190C22⋊Dic61—24no—
191C22⋊Dic61—28no—
192C22⋊Dic61—22no—
193#11758198—28no—
194#11758198—24no—
195#11758198—28no—
196#11758198—24no—
197C4⋊Dic61C2×D414yes—
198#11758196—24no—
199#11758196—24no—
200C4×Dic61—22no—
201C4×Dic61—21no—
202C4×Dic61—24no—
203C4×Dic61C22×C412yescomplemented
204#11758194—22no—
205#11758194—22no—
206C2×#92418—22no—
207#11758189—44no—
208C8×D61—44no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—44no—
213C23×C61—44no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61C2×C22⋊C414yes—
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—28no—
221C22×D61—28no—
222C22×D61—42no—
223C22×D61—48no—
224C22×D61—44no—
225C22×D61—84no—
226C22×D61—88no—
227C22×D61—88no—
228C22×D61C2×C22⋊C414yes—
229C22×D61C2×C22⋊C412yes—
230C22×D61#12011yescharacteristic
231C22×D61#12011yescharacteristic
232C22×D61#12312yes—
233D4×C61—24no—
234D4×C61—24no—
235D4×C61—24no—
236D4×C61—42no—
237D4×C61—88no—
238C2×C4×C61—22no—
239C2×C4×C61—24no—
240C2×C4×C61—42no—
241C2×C4×C61—84no—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D412yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4112yes—
246C22⋊D61—216no—
247C22⋊D61—44no—
248C22⋊D61—44no—
249C22⋊D61—88no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—42no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—48no—
257C2×Dic61—84no—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61#12012yes—
260C2×Dic61#12312yes—
261D244—24no—
262D244—24no—
263C4×D61—48no—
264C4×D61—88no—
265C4×D61C4×D414yes—
266C4×D61C4×D414yes—
267C4×D61#12114yes—
268C4×D61#12114yes—
269Dic122—24no—
270Dic122—42no—
271C8×C61—42no—
272#92418—42no—
273#20953—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—24no—
276C22×C61—44no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—42no—
280C22×C61—82no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C2×#12011yescharacteristic
285C22×C61C2×#12311yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—24no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—416no—
292C2×D61—44no—
293C2×D61—44no—
294C2×D61—88no—
295C2×D61—82no—
296C2×D61#38612yes—
297C2×D61#38612yes—
298C4×C61—42no—
299C4×C61—44no—
300C4×C61C2×C4×D412yes—
301C4×C61C2×#12112yes—
302Dic61—42no—
303Dic61—44no—
304Dic61—42no—
305Dic61#38614yes—
306C2×#535—611no—
307C2×#518—611no—
308#2896—612no—
309C22×#100—611no—
310#1887—612no—
311#1887—611no—
312#1887D6111yescharacteristic, complemented
313#1885—611no—
314#1885—612no—
315#1885—611no—
316#1733—612no—
317C2×C61—21no—
318C2×C61—21no—
319C2×C61—44no—
320C2×C61—82no—
321C2×C61C2×#38611yescharacteristic
322C2×C61#301912yes—
323D61—22no—
324D61—22no—
325D61—44no—
326C22×#41—611no—
327C23×D4—611no—
328C22×M16—612no—
329#535—618no—
330#535—1224no—
331C2×#127—611no—
332C2×#127C2×D6111yescharacteristic
333C2×#124—611no—
334C2×#124—611no—
335C2×#121—1222no—
336C2×#120—1222no—
337#518—611no—
338#518—612no—
339#518—1222no—
340#518—1224no—
341#518C2×D6111yescharacteristic
342C2×C4×D4—1224no—
343C2×#117—611no—
344C2×#117—612no—
345C22×C22⋊C4—1222no—
346#419—612no—
347#419—612no—
348#418—612no—
349#418C2×D6112yescomplemented
350#417—1224no—
351#416—1224no—
352C2×#100—611no—
353C2×#100—612no—
354C2×#100—1224no—
355C2×#100C2×D6111yescharacteristic
356#339—612no—
357#339—612no—
358#339—612no—
359#339C2×D6112yescomplemented
360C61#2095311yescharacteristic, complemented, Sylow
361C25—612no—
362C2×#41—614no—
363C2×#41—1222no—
364C2×#41—1224no—
365C2×#41—1222no—
366C2×#41—1222no—
367C22×Q8—611no—
368C22×D4—22no—
369C22×D4—612no—
370C22×D4—614no—
371C22×D4—614no—
372C22×D4—611no—
373C22×D4—1222no—
374C22×D4—1222no—
375C22×D4—1224no—
376C22×D4—1222no—
377C22×D4—1221no—
378C22×D4C22×D6111yescharacteristic
379C22×D4C22×D6111yescharacteristic
380C23×C4—611no—
381C23×C4—1224no—
382C23×C4—1221no—
383C2×M16—612no—
384C2×M16—1222no—
385C2×M16—1224no—
386C2×M16—1224no—
387C2×M16C22×D6112yes—
388C22×C8—1222no—
389#127—22no—
390#127—614no—
391#127—1222no—
392#127C4×D6114yescomplemented
393#124—614no—
394#124—614no—
395#124—1222no—
396#124—1222no—
397#121—12216no—
398#120—12216no—
399C4×D4—24no—
400C4×D4—1224no—
401C4×D4—1228no—
402C4×D4—1224no—
403C4×D4—1228no—
404C4×D4—1224no—
405#117—614no—
406#117—612no—
407#117—614no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117C22×D6112yes—
412C2×C4⋊C4—614no—
413C2×C4⋊C4—1221no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—1222no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4—1224no—
418C2×C22⋊C4—1224no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1224no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C42—611no—
425C2×C42—1222no—
426C2×C42C22×D6111yescharacteristic
427#103—1224no—
428#103—1224no—
429#102—24no—
430#102—1224no—
431#100—22no—
432#100—1224no—
433#100—1222no—
434C24—21no—
435C24—611no—
436C24—618no—
437C24—612no—
438C24—1222no—
439C24—1224no—
440C24—1222no—
441C24—4884no—
442C24D4×D6112yes—
443#41—1228no—
444#41—1228no—
445#41—2444no—
446#41—2444no—
447C2×Q8—612no—
448C2×Q8—1221no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×D4—28no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—84no—
458C2×D4—618no—
459C2×D4—612no—
460C2×D4—612no—
461C2×D4—1228no—
462C2×D4—1228no—
463C2×D4—1228no—
464C2×D4—1228no—
465C2×D4—1224no—
466C2×D4—1224no—
467C2×D4—1228no—
468C2×D4—1224no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—4884no—
472C2×D4—2441no—
473C2×D4—2441no—
474C2×D4—2442no—
475C2×D4C2×C4×D6112yes—
476C2×D4C2×C4×D6112yescomplemented
477C2×D4D4×D6112yes—
478C22×C4—21no—
479C22×C4—24no—
480C22×C4—611no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—612no—
484C22×C4—612no—
485C22×C4—614no—
486C22×C4—1228no—
487C22×C4—1224no—
488C22×C4—1224no—
489C22×C4—1224no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—4888no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4C23×D6111yescharacteristic
496M16—22no—
497M16—22no—
498M16—1222no—
499M16—1222no—
500M16—2444no—
501C2×C8—22no—
502C2×C8—1222no—
503C2×C8—2444no—
504C4⋊C4—614no—
505C4⋊C4—1224no—
506C4⋊C4—1224no—
507C4⋊C4D4×D6114yes—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—1224no—
511C22⋊C4—1224no—
512C22⋊C4—1228no—
513C22⋊C4—1228no—
514C22⋊C4—1224no—
515C22⋊C4—1222no—
516C22⋊C4—1228no—
517C42—22no—
518C42—21no—
519C42—612no—
520C42—1224no—
521C42—1221no—
522C42—1222no—
523C42C2×C4×D6112yescomplemented
524C23—22no—
525C23—22no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—41no—
530C23—41no—
531C23—611no—
532C23—611no—
533C23—612no—
534C23—612no—
535C23—614no—
536C23—1228no—
537C23—1228no—
538C23—1222no—
539C23—1222no—
540C23—4884no—
541C23—4888no—
542C23—4888no—
543C23—2442no—
544C23—2444no—
545C23—2448no—
546C23C22⋊C4×D6114yes—
547C23C2×D4×D6111yescharacteristic
548Q8—1224no—
549Q8—2442no—
550D4—24no—
551D4—24no—
552D4—24no—
553D4—42no—
554D4—88no—
555D4—12216no—
556D4—1224no—
557D4—1224no—
558D4—4888no—
559D4—2444no—
560D4—2444no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—614no—
569C2×C4—614no—
570C2×C4—614no—
571C2×C4—614no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—4888no—
576C2×C4—4884no—
577C2×C4—2448no—
578C2×C4—2448no—
579C2×C4—2442no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4C2×D4×D6112yes—
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4#41×D6112yes—
586C8—42no—
587C8—2442no—
588C22—22no—
589C22—24no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—612no—
598C22—612no—
599C22—1222no—
600C22—1222no—
601C22—1224no—
602C22—4882no—
603C22—4888no—
604C22—2442no—
605C22—2442no—
606C22—2444no—
607C22—24416no—
608C22—2444no—
609C22C2×C22⋊C4×D6111yescharacteristic
610C22#120×D6111yescharacteristic
611C22#123×D6111yescharacteristic
612C4—42no—
613C4—44no—
614C4—614no—
615C4—2442no—
616C4—2442no—
617C4—2444no—
618C4C4×D4×D6112yes—
619C4#121×D6112yes—
620C2—21no—
621C2—21no—
622C2—44no—
623C2—82no—
624C2—1222no—
625C2—1222no—
626C2—2444no—
627C2#386×D6111yescharacteristic
628C2#2145517312yes—
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1887C2
socleC22×C61C2×#120
prime radicalC2×#16840325C2
cosocleC22×C4×C61C24
hypercenter#1887D61
hypocenterC61#20953
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1887

Complements

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

NH ≅ G/Ntype
2. #518×D61C2semidirect
3. #21454333C2semidirect
4. C2×#100×D61C2semidirect
5. #21446218C2semidirect
6. #21446214C2semidirect
7. #21446207C2semidirect
8. #21446206C2semidirect
9. #21446205C2semidirect
10. #21445527C2semidirect
11. C61×#1887C2semidirect
16. #16840325C4semidirect
30. #16839772C22semidirect
31. #16839770C22semidirect
32. #16839767C22semidirect
38. #16839523C22semidirect
39. #16839522C22semidirect
40. #16839521C22semidirect
43. C61×#418C22semidirect
45. C61×#339C22semidirect
51. C2×#11758224C2×C4semidirect
71. #14519773C2×C4semidirect
73. #14519765C2×C4semidirect
75. #14519764C2×C4semidirect
84. #14519735D4semidirect
87. #14519734C2×C4semidirect
90. C4⋊C4×D61D4semidirect
112. #127×C61C2×C4semidirect
136. C2×D4×C61C22×C4semidirect
180. C2×Dic122C22×C4semidirect
189. C42×C61C22×C4semidirect
203. C4×Dic61C22×C4semidirect
312. #1887D61semidirect
349. #418C2×D61semidirect
359. #339C2×D61semidirect
360. C61#20953semidirect
392. #127C4×D61semidirect
476. C2×D4C2×C4×D61semidirect
523. C42C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403252
#518×D612
#214543332
C2×#100×D612
#214462182
#214462142
#214462072
#214462062
#214462052
#214455272
C61×#18872
#2095361