Finite Group Databasebeta

← #29713040

Subgroups — #29713040

Order 15616.

1484 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840408C211yesmaximal, characteristic
2C2×#16840097C211yesmaximal, characteristic, complemented
3C22×#14519697C211yesmaximal, characteristic, complemented
4#21447060C212yesmaximal, complemented
5#21447001C212yesmaximal, complemented
6#21446240C211yesmaximal, characteristic, complemented
7#21446112C211yesmaximal, characteristic, complemented
8#21446065C212yesmaximal, complemented
9#21445757C212yesmaximal, complemented
10#21445749C211yesmaximal, characteristic, complemented
11C61×#2487C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×D244C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840408—24no—
16#16840408—24no—
17C2×#14519765C2211yescharacteristic
18C2×#14519762—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×#14519750—21no—
22C2×#14519745—21no—
23C2×C4×D244C2211yescharacteristic
24C2×C22⋊C4×D61—21no—
25#16840097—24no—
26C2×#14519710C2212yescomplemented
27C2×#14519700C2212yescomplemented
28C2×#14519700C2212yescomplemented
29C2×#14519697—24no—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×C22⋊#92418C2211yescharacteristic
35C2×C22⋊#92418C2212yescomplemented
36#16839524C2212yescomplemented
37#16839517C2212yescomplemented
38C2×C4×D4×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#400—21no—
41C61×#399—21no—
42C61×#389C2212yescomplemented
43C61×#386—21no—
44C61×#385—21no—
45C61×#380C2211yescharacteristic
46C61×#380C2212yescomplemented
47C24×D61—21no—
48C2×#11758222—22no—
49C2×#11758222—24no—
50C2×#11758222—22no—
51C2×#11758222—42no—
52C22×Dic122C2311yescharacteristic
53C22×D244—24no—
54C22×D244—21no—
55C22×D244—22no—
56C22×D244—42no—
57C22×D244D414yes—
58C22×D244C2311yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C22⋊D61—21no—
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×#92418—22no—
65C22×#92418—42no—
66C22×#92418C2312yes—
67C22×#92418C2312yes—
68#14519765—22no—
69#14519765—22no—
70#14519762—28no—
71#14519756—22no—
72#14519756D414yes—
73#14519751—22no—
74#14519751—22no—
75#14519750—28no—
76#14519745—28no—
77C4×D244—22no—
78C4×D244—22no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C22⋊C4×D61—28no—
87#14519710—24no—
88#14519700—24no—
89#14519700—42no—
90#14519697—24no—
91#14519697—24no—
92#14519697—24no—
93#14519697—42no—
94C22⋊#92418—22no—
95C22⋊#92418—42no—
96C22×D4×C61D411yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—22no—
116#95×C61—22no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C23×D61—21no—
120C23×D61—21no—
121C23×D61—24no—
122C23×D61—44no—
123C23×D61—42no—
124C23×D61—42no—
125C23×D61—82no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—42no—
138C22×C4×C61—82no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—24no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61—82no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152#11758222—48no—
153#11758222—44no—
154#11758222—44no—
155C2×D244—21no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—28no—
160C2×D244—24no—
161C2×D244—21no—
162C2×D244—44no—
163C2×D244—48no—
164C2×D244—44no—
165C2×D244C2×D412yes—
166C2×D244C2×D412yes—
167C2×D244#4114yes—
168C2×C4×D61—24no—
169C2×C4×D61—24no—
170C2×C4×D61—44no—
171C2×C4×D61—82no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122C2×D412yes—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61—82no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—82no—
187C42×C61—21no—
188C42×C61—21no—
189C22⋊Dic61—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194C4⋊Dic61—21no—
195C4⋊Dic61C2×D412yes—
196#11758196—24no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—44no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61#12012yes—
212C23×C61#13512yes—
213C23×C61#13712yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—24no—
217C22×D61—22no—
218C22×D61—48no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—42no—
226C22×D61—48no—
227C22×D61—84no—
228C22×D61—88no—
229C22×D61—88no—
230D4×C61—44no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—84no—
252C2×C4×C61#12012yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4112yes—
256C2×C4×C61H(5,𝔽2)12yes—
257C22⋊D61—48no—
258C22⋊D61—84no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263D244—22no—
264D244—22no—
265D244—22no—
266D244—24no—
267D244—48no—
268D244—42no—
269D244#12314yes—
270C4×D61—48no—
271C4×D61—84no—
272Dic122—24no—
273Dic122—41no—
274Dic122—41no—
275#92418—42no—
276#92418—42no—
277#24039—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61#45214yes—
288C22×C61C2×#12011yescharacteristic
289C22×C61#54612yes—
290C22×C61C2×#13511yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61#57612yes—
293C2×D61—24no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—44no—
297C2×D61—44no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—816no—
302C2×D61—82no—
303C2×D61—82no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—82no—
309C4×C61—82no—
310C4×C61C2×#12311yescharacteristic
311C4×C61#53411yescharacteristic
312C4×C61#54612yes—
313Dic61—42no—
314Dic61—42no—
315C2×#546—611no—
316C2×#484—611no—
317C22×#102—611no—
318#2487D6111yescharacteristic, complemented
319#2042—612no—
320#2033—612no—
321#1892—611no—
322#1869—611no—
323#1862—612no—
324#1788—612no—
325#1786—611no—
326C2×C61—24no—
327C2×C61—81no—
328C2×C61#240511yescharacteristic
329C2×C61C2×#45211yescharacteristic
330C2×C61#300811yescharacteristic
331C2×C61#329712yes—
332C2×C61#331412yes—
333D61—44no—
334D61—82no—
335C22×#41—611no—
336C23×D4—611no—
337C23×C8—611no—
338#546—1224no—
339#546—1224no—
340C2×#124—611no—
341C2×#123—1221no—
342C2×C22⋊Q8—611no—
343C2×#121—611no—
344C2×#121—1221no—
345C2×#120—1221no—
346C2×C4×D4—611no—
347C2×C4×D4C2×D6111yescharacteristic
348C22×C4⋊C4C2×D6111yescharacteristic
349C22×C22⋊C4—1221no—
350#484—1224no—
351C2×#107—612no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#102—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102—612no—
358C2×#102—1224no—
359C2×C22⋊C8—611no—
360C2×C22⋊C8—612no—
361#400—21no—
362#399—21no—
363#389C2×D6112yescomplemented
364#386—21no—
365#385—21no—
366#380C2×D6111yescharacteristic
367#380C2×D6112yescomplemented
368#340—612no—
369#336—612no—
370C61#2403911yescharacteristic, complemented, Sylow
371C25—1221no—
372C2×#41—1222no—
373C2×#41—1222no—
374C2×#41—1224no—
375C2×#41—2442no—
376C22×Q8—611no—
377C22×D4—614no—
378C22×D4—611no—
379C22×D4—611no—
380C22×D4—1224no—
381C22×D4—1221no—
382C22×D4—1221no—
383C22×D4—1222no—
384C22×D4—1221no—
385C22×D4—2442no—
386C22×D4C22⋊D6111yescharacteristic
387C23×C4—21no—
388C23×C4—1221no—
389C23×C4—1221no—
390C23×C4C22×D6111yescharacteristic
391C23×C4C22×D6111yescharacteristic
392C22×C8—612no—
393C22×C8—612no—
394C22×C8—1222no—
395C22×C8—2442no—
396#124—1222no—
397#124—1222no—
398#123—1228no—
399C22⋊Q8—614no—
400C22⋊Q8—1222no—
401#121—1228no—
402#121—1222no—
403#121—1222no—
404#120—1228no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—1222no—
408C4×D4—1222no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22×D6112yes—
417C2×C4⋊C4C22×D6112yes—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—1228no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4C22⋊D6111yescharacteristic
427C2×C22⋊C4C22⋊D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429#107—1224no—
430#103—1224no—
431#103—2442no—
432#102—1224no—
433#102—1224no—
434#102—1224no—
435#102—2442no—
436C22⋊C8—1222no—
437C22⋊C8—2442no—
438#95—21no—
439#95—22no—
440#95—22no—
441C24—1221no—
442C24—1224no—
443C24—1221no—
444C24—4882no—
445C24—2444no—
446C24—2442no—
447C24—2442no—
448C24C2×C22⋊D6111yescharacteristic
449C24C2×C22⋊D6111yescharacteristic
450#41—2444no—
451#41—2448no—
452#41—2444no—
453C2×Q8—612no—
454C2×Q8—1221no—
455C2×Q8—1222no—
456C2×Q8—1222no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—612no—
461C2×D4—612no—
462C2×D4—614no—
463C2×D4—1228no—
464C2×D4—1224no—
465C2×D4—1224no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1224no—
472C2×D4—4882no—
473C2×D4—2444no—
474C2×D4—2448no—
475C2×D4—2444no—
476C2×D4—2444no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—82no—
487C22×C4—1224no—
488C22×C4—1224no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—4882no—
492C22×C4—2444no—
493C22×C4D4×D6112yes—
494C22×C4D4×D6112yes—
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—2444no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—82no—
514C4⋊C4—612no—
515C4⋊C4—1221no—
516C4⋊C4—1224no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—82no—
521C22⋊C4—1224no—
522C22⋊C4—1224no—
523C22⋊C4—1224no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C42—21no—
527C42—21no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—1222no—
536C23—1224no—
537C23—1222no—
538C23—4888no—
539C23—4888no—
540C23—4884no—
541C23—2448no—
542C23—2441no—
543C23—2441no—
544C23—2442no—
545C23—2442no—
546C23—2448no—
547C23—2441no—
548C23—2441no—
549C23—2442no—
550C23#1451974512yes—
551C23#1451979712yes—
552C23#1451980812yes—
553C23C22×C22⋊D6111yescharacteristic
554Q8—1224no—
555Q8—2441no—
556Q8—2441no—
557D4—44no—
558D4—614no—
559D4—1222no—
560D4—1222no—
561D4—1222no—
562D4—1224no—
563D4—4884no—
564D4—2448no—
565D4—2448no—
566D4—2442no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4884no—
591C2×C4—2448no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4#1451974512yes—
595C2×C4C2×D4×D6111yescharacteristic
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4#41×D6112yes—
598C2×C4#1451982312yes—
599C8—2442no—
600C8—2442no—
601C22—22no—
602C22—24no—
603C22—22no—
604C22—24no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—42no—
610C22—1224no—
611C22—48816no—
612C22—4882no—
613C22—4882no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22—2444no—
619C22—2442no—
620C22—2444no—
621C22#1683992214yes—
622C22C2×#1451974511yescharacteristic
623C22#1684040012yes—
624C22C2×#1451979711yescharacteristic
625C22C2×#1451980811yescharacteristic
626C22#1684056512yes—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4—82no—
633C4—2442no—
634C4—2442no—
635C4#123×D6111yescharacteristic
636C4#1684032311yescharacteristic
637C4#1684040412yes—
638C2—24no—
639C2—81no—
640C2—4882no—
641C2—2444no—
642C2#2144914311yescharacteristic
643C2C2×#1683992211yescharacteristic
644C2#2145508511yescharacteristic
645C2#2145766312yes—
646C2#2145789412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2487C2
socleC23×C61C22×D4
prime radicalC2×#16840408C2
hypercenter#2487D61
hypocenterC61#24039
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C2×#16840097C2semidirect
3. C22×#14519697C2semidirect
4. #21447060C2semidirect
5. #21447001C2semidirect
6. #21446240C2semidirect
7. #21446112C2semidirect
8. #21446065C2semidirect
9. #21445757C2semidirect
10. #21445749C2semidirect
11. C61×#2487C2semidirect
26. C2×#14519710C22semidirect
27. C2×#14519700C22semidirect
28. C2×#14519700C22semidirect
35. C2×C22⋊#92418C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
42. C61×#389C22semidirect
46. C61×#380C22semidirect
318. #2487D61semidirect
363. #389C2×D61semidirect
367. #380C2×D61semidirect
370. C61#24039semidirect

Maximal subgroups

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

subgroupindex
C2×#168404082
C2×#168400972
C22×#145196972
#214470602
#214470012
#214462402
#214461122
#214460652
#214457572
#214457492
C61×#24872
#2403961