Finite Group Databasebeta

← #29713029

Subgroups — #29713029

Order 15616.

1504 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840408C211yesmaximal, characteristic
2C2×#16840092C211yesmaximal, characteristic, complemented
3C22×#14519697C211yesmaximal, characteristic, complemented
4#21447060C212yesmaximal, complemented
5#21447008C212yesmaximal, complemented
6#21446240C211yesmaximal, characteristic, complemented
7#21446112C211yesmaximal, characteristic, complemented
8#21446079C212yesmaximal, complemented
9#21445757C212yesmaximal, complemented
10#21445749C211yesmaximal, characteristic, complemented
11C61×#2486C211yesmaximal, 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#16840092—24no—
26C2×#14519708C2212yescomplemented
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×#123×C61—21no—
39C2×C4×D4×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#398—21no—
42C61×#397—21no—
43C61×#389C2212yescomplemented
44C61×#381—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#14519708—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
100#123×C61—44no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#95×C61—21no—
116#95×C61—22no—
117#95×C61—22no—
118C24×C61C2×D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C23×D61—21no—
122C23×D61—24no—
123C23×D61—44no—
124C23×D61—42no—
125C23×D61—42no—
126C23×D61—82no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—82no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154#11758222—48no—
155#11758222—44no—
156#11758222—44no—
157C2×D244—21no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—28no—
162C2×D244—24no—
163C2×D244—21no—
164C2×D244—44no—
165C2×D244—48no—
166C2×D244—44no—
167C2×D244C2×D412yes—
168C2×D244C2×D412yes—
169C2×D244#4114yes—
170C2×C4×D61—24no—
171C2×C4×D61—24no—
172C2×C4×D61—44no—
173C2×C4×D61—82no—
174C2×Dic122—22no—
175C2×Dic122—22no—
176C2×Dic122—21no—
177C2×Dic122C2×D412yes—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—44no—
184C4⋊C4×C61—42no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—44no—
188C22⋊C4×C61—42no—
189C42×C61—21no—
190C42×C61—21no—
191C22⋊Dic61—24no—
192#11758198—24no—
193#11758198—24no—
194#11758198—24no—
195#11758198—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2×D412yes—
198#11758196—24no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—44no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—22no—
213C23×C61#12012yes—
214C23×C61#13512yes—
215C23×C61AGL(1,ℤ/8)12yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—22no—
218C22×D61—24no—
219C22×D61—22no—
220C22×D61—48no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—42no—
228C22×D61—48no—
229C22×D61—84no—
230C22×D61—88no—
231C22×D61—88no—
232D4×C61—44no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—24no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—84no—
254C2×C4×C61#12012yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4112yes—
258C2×C4×C61H(5,𝔽2)12yes—
259C22⋊D61—48no—
260C22⋊D61—84no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265D244—22no—
266D244—22no—
267D244—22no—
268D244—24no—
269D244—48no—
270D244—42no—
271D244#12314yescomplemented
272C4×D61—48no—
273C4×D61—84no—
274Dic122—24no—
275Dic122—41no—
276Dic122—41no—
277#92418—42no—
278#92418—42no—
279#24038—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—24no—
283C22×C61—24no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61#44914yes—
290C22×C61C2×#12011yescharacteristic
291C22×C61#54612yes—
292C22×C61C2×#13511yescharacteristic
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61#57712yes—
295C2×D61—24no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—816no—
304C2×D61—82no—
305C2×D61—82no—
306C4×C61—22no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—82no—
311C4×C61—82no—
312C4×C61C2×#12311yescharacteristic
313C4×C61#53411yescharacteristic
314C4×C61#54612yes—
315Dic61—42no—
316Dic61—42no—
317C2×#546—611no—
318C2×#483—611no—
319C22×#102—611no—
320#2486D6111yescharacteristic, complemented
321#2042—612no—
322#2034—612no—
323#1892—611no—
324#1869—611no—
325#1864—612no—
326#1788—612no—
327#1786—611no—
328C2×C61—24no—
329C2×C61—81no—
330C2×C61#240511yescharacteristic
331C2×C61C2×#44911yescharacteristic
332C2×C61#300911yescharacteristic
333C2×C61#329612yes—
334C2×C61#330612yes—
335D61—44no—
336D61—82no—
337C22×#41—611no—
338C23×D4—611no—
339C23×C8—611no—
340#546—1224no—
341#546—1224no—
342C2×#124—611no—
343C2×#123—21no—
344C2×#123—1221no—
345C2×C22⋊Q8—611no—
346C2×#121—611no—
347C2×#121—1221no—
348C2×#120—1221no—
349C2×C4×D4—611no—
350C2×C4×D4C2×D6111yescharacteristic
351C22×C4⋊C4C2×D6111yescharacteristic
352C22×C22⋊C4—1221no—
353#483—1224no—
354C2×#106—612no—
355C2×#103—612no—
356C2×#103—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×#102—612no—
360C2×#102—612no—
361C2×#102—1224no—
362C2×C22⋊C8—611no—
363C2×C22⋊C8—612no—
364#398—21no—
365#397—21no—
366#389C2×D6112yescomplemented
367#381—21no—
368#380C2×D6111yescharacteristic
369#380C2×D6112yescomplemented
370#340—612no—
371#336—612no—
372C61#2403811yescharacteristic, complemented, Sylow
373C25—1221no—
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41—1224no—
377C2×#41—2442no—
378C22×Q8—611no—
379C22×D4—614no—
380C22×D4—611no—
381C22×D4—611no—
382C22×D4—1224no—
383C22×D4—1221no—
384C22×D4—1221no—
385C22×D4—1222no—
386C22×D4—1221no—
387C22×D4—2442no—
388C22×D4C22⋊D6111yescharacteristic
389C23×C4—21no—
390C23×C4—1221no—
391C23×C4—1221no—
392C23×C4C22×D6111yescharacteristic
393C23×C4C22×D6111yescharacteristic
394C22×C8—612no—
395C22×C8—612no—
396C22×C8—1222no—
397C22×C8—2442no—
398#124—1222no—
399#124—1222no—
400#123—44no—
401#123—1228no—
402C22⋊Q8—614no—
403C22⋊Q8—1222no—
404#121—1228no—
405#121—1222no—
406#121—1222no—
407#120—1228no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—1222no—
411C4×D4—1222no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22⋊D6111yescharacteristic
419C2×C4⋊C4C22×D6112yes—
420C2×C4⋊C4C22×D6112yes—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—1228no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4C22⋊D6111yescharacteristic
430C2×C22⋊C4C22⋊D6111yescharacteristic
431C2×C42C22×D6111yescharacteristic
432#106—1224no—
433#103—1224no—
434#103—2442no—
435#102—1224no—
436#102—1224no—
437#102—1224no—
438#102—2442no—
439C22⋊C8—1222no—
440C22⋊C8—2442no—
441#95—21no—
442#95—22no—
443#95—22no—
444C24—1221no—
445C24—1224no—
446C24—1221no—
447C24—4882no—
448C24—2444no—
449C24—2442no—
450C24—2442no—
451C24C2×C22⋊D6111yescharacteristic
452C24C2×C22⋊D6111yescharacteristic
453#41—2444no—
454#41—2448no—
455#41—2444no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1222no—
459C2×Q8—1222no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—612no—
464C2×D4—612no—
465C2×D4—614no—
466C2×D4—1228no—
467C2×D4—1224no—
468C2×D4—1224no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1222no—
474C2×D4—1224no—
475C2×D4—4882no—
476C2×D4—2444no—
477C2×D4—2448no—
478C2×D4—2444no—
479C2×D4—2444no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—1224no—
492C22×C4—1224no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—4882no—
496C22×C4—2444no—
497C22×C4D4×D6112yes—
498C22×C4D4×D6112yes—
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2444no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—612no—
519C4⋊C4—1221no—
520C4⋊C4—1224no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C42—21no—
531C42—21no—
532C23—22no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—1222no—
540C23—1224no—
541C23—1222no—
542C23—4888no—
543C23—4888no—
544C23—4884no—
545C23—2448no—
546C23—2441no—
547C23—2441no—
548C23—2442no—
549C23—2442no—
550C23—2448no—
551C23—2441no—
552C23—2441no—
553C23—2442no—
554C23#1451974512yes—
555C23#1451979712yes—
556C23#1451980312yes—
557C23C22×C22⋊D6111yescharacteristic
558Q8—1224no—
559Q8—2441no—
560Q8—2441no—
561D4—44no—
562D4—614no—
563D4—1222no—
564D4—1222no—
565D4—1222no—
566D4—1224no—
567D4—4884no—
568D4—2448no—
569D4—2448no—
570D4—2442no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—4884no—
595C2×C4—2448no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4#1451974512yes—
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6112yes—
602C2×C4#1451982312yes—
603C8—2442no—
604C8—2442no—
605C22—24no—
606C22—22no—
607C22—24no—
608C22—22no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—1224no—
615C22—48816no—
616C22—4882no—
617C22—4882no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22—2444no—
622C22—2442no—
623C22—2444no—
624C22—2442no—
625C22#1683990514yes—
626C22C2×#1451974511yescharacteristic
627C22#1684040012yes—
628C22C2×#1451979711yescharacteristic
629C22C2×#1451980311yescharacteristic
630C22#1684057012yes—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—82no—
636C4—82no—
637C4—2442no—
638C4—2442no—
639C4#123×D6111yescharacteristic
640C4#1684032311yescharacteristic
641C4#1684040412yes—
642C2—24no—
643C2—81no—
644C2—4882no—
645C2—2444no—
646C2#2144914311yescharacteristic
647C2C2×#1683990511yescharacteristic
648C2#2145509411yescharacteristic
649C2#2145764812yes—
650C2#2145777412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2486C2
socleC23×C61C22×D4
prime radicalC2×#16840408C2
hypercenter#2486D61
hypocenterC61#24038
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. C2×#16840092C2semidirect
3. C22×#14519697C2semidirect
4. #21447060C2semidirect
5. #21447008C2semidirect
6. #21446240C2semidirect
7. #21446112C2semidirect
8. #21446079C2semidirect
9. #21445757C2semidirect
10. #21445749C2semidirect
11. C61×#2486C2semidirect
26. C2×#14519708C22semidirect
27. C2×#14519700C22semidirect
28. C2×#14519700C22semidirect
35. C2×C22⋊#92418C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
43. C61×#389C22semidirect
46. C61×#380C22semidirect
271. D244#123semidirect
320. #2486D61semidirect
366. #389C2×D61semidirect
369. #380C2×D61semidirect
372. C61#24038semidirect

Maximal subgroups

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

subgroupindex
C2×#168404082
C2×#168400922
C22×#145196972
#214470602
#214470082
#214462402
#214461122
#214460792
#214457572
#214457492
C61×#24862
#2403861