Finite Group Databasebeta

← #29712652

Subgroups — #29712652

Order 15616.

1500 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519753C211yesmaximal, characteristic
2C2×#16840091C212yesmaximal, complemented
3#380×D61C211yesmaximal, characteristic
4#21447014C212yesmaximal, complemented
5#21447013C212yesmaximal, complemented
6#21446123C212yesmaximal, complemented
7#21446014C211yesmaximal, characteristic, complemented
8#21446011C211yesmaximal, characteristic, complemented
9#21445754C211yesmaximal, characteristic, complemented
10#21445728C211yesmaximal, characteristic
11C61×#1787C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C2×#14519753—28no—
14C2×#14519753—21no—
15C2×#14519753—21no—
16C2×#14519753C2212yes—
17C2×#14519753C2212yes—
18C22×C4⋊Dic61C2211yescharacteristic
19C22×#11758196—21no—
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C22×#11758198—21no—
23C2×C22⋊C4×D61—21no—
24#16840091—28no—
25C2×#14519709C2212yescomplemented
26C2×#14519708C2212yescomplemented
27C2×#14519701C2212yescomplemented
28C2×#14519701C2212yescomplemented
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yes—
31C2×#14519689C2212yes—
32#16839622C2211yescharacteristic
33#16839621—21no—
34#16839620—21no—
35#16839619C2211yescharacteristic
36#16839619C2211yescharacteristic
37#95×D61—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40C23×Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yescomplemented
42C2×#103×C61C2212yescomplemented
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#380C2211yescharacteristic
45C24×D61D411yescharacteristic
46C23×Dic61—21no—
47C23×Dic61D411yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—24no—
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C22×C4×D61—42no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2312yes—
59#14519753—28no—
60#14519753—28no—
61#14519753—24no—
62#14519753—24no—
63#14519753—24no—
64#14519753D418yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—28no—
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C4⋊C4×D61—24no—
76C4⋊C4×D61—42no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—28no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C22⋊C4×D61—44no—
84#14519709—24no—
85#14519708—24no—
86#14519701—24no—
87#14519701—42no—
88C22⋊#92418—22no—
89#14519689—24no—
90#14519689—42no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61—24no—
95C22×Q8×C61—21no—
96C22×Q8×C61—22no—
97C22×Q8×C61—22no—
98C22×Q8×C61—42no—
99C22×Q8×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106#103×C61—24no—
107#103×C61—42no—
108C22⋊C8×C61—22no—
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—22no—
115C23×D61—41no—
116C23×D61—41no—
117C23×D61SD1614yes—
118C23×D61C2×D411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—24no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—48no—
127C2×Q8×C61—48no—
128C2×Q8×C61—42no—
129C2×Q8×C61—42no—
130C2×Q8×C61C2×D414yes—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—24no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—41no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—48no—
166C2×C4×D61—48no—
167C2×C4×D61—41no—
168C2×C4×D61—41no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61C2×D414yes—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C22⋊C4×C61—42no—
182C22⋊Dic61—42no—
183#11758198—28no—
184#11758198—28no—
185#11758198—42no—
186#11758198—42no—
187C4⋊Dic61—24no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C2×D414yes—
192#11758196—28no—
193#11758196—28no—
194#11758196—42no—
195#11758196—42no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61#12312yes—
204C23×C61C2×SD1612yes—
205C23×C61#13712yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—28no—
218C22×D61—28no—
219C22×D61—44no—
220C22×D61—44no—
221C22×D61—44no—
222C22×D61—44no—
223C22×D61—44no—
224C22×D61#12012yes—
225C22×D61C2×SD1612yes—
226C22×D61C2×SD1612yes—
227C22×D61#13712yes—
228Q8×C61—24no—
229Q8×C61—24no—
230Q8×C61—48no—
231Q8×C61—42no—
232Q8×C61—42no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—48no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61#12312yes—
248C2×C4×C61C22×D412yes—
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)12yes—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—24no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—44no—
259C2×Dic61—48no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61#12012yes—
269C4×D61—24no—
270C4×D61—24no—
271C4×D61—44no—
272C4×D61—44no—
273C4×D61—42no—
274C4×D61—42no—
275C4×D61—84no—
276C4×D61—84no—
277C8×C61—42no—
278#92418—42no—
279#24003—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61#48314yescomplemented
287C22×C61C2×#12011yescharacteristic
288C22×C61C2×#12311yescharacteristic
289C22×C61#53411yescharacteristic
290C22×C61C22×SD1611yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61C2×#13711yescharacteristic
293C22×C61#57511yescharacteristic
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—416no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61#44814yes—
309C4×C61—22no—
310C4×C61—44no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61#54612yes—
314C4×C61#54612yes—
315Dic61—22no—
316Dic61—41no—
317Dic61—44no—
318Dic61—81no—
319Dic61—81no—
320C22×C22⋊Q8—611no—
321C2×#483—612no—
322C2×#380—611no—
323#2035—612no—
324#2035—612no—
325#1872—612no—
326#1853—611no—
327#1853—611no—
328#1787—611no—
329#1787D6111yescharacteristic, complemented
330#1779—611no—
331C2×C61—24no—
332C2×C61#240511yescharacteristic
333C2×C61C2×#44811yescharacteristic
334C2×C61#300711yescharacteristic
335C2×C61C2×#48311yescharacteristic
336C2×C61#309511yescharacteristic
337C2×C61#318711yescharacteristic
338C2×C61#319511yescharacteristic
339D61—24no—
340D61—42no—
341C23×Q8C2×D6111yescharacteristic
342C24×C4—611no—
343C2×C22⋊Q8—612no—
344C2×C22⋊Q8—612no—
345C2×C22⋊Q8—1228no—
346C2×C22⋊Q8—1221no—
347C2×C22⋊Q8—1221no—
348C22×C4⋊C4—611no—
349C22×C4⋊C4—611no—
350C22×C4⋊C4—611no—
351C22×C4⋊C4—1221no—
352C22×C22⋊C4—1221no—
353C22×C22⋊C4—1221no—
354#483—1228no—
355C2×#106—612no—
356C2×#106—612no—
357C2×#103—612no—
358C2×#103—612no—
359C2×#103C2×D6112yescomplemented
360C2×#103C2×D6112yescomplemented
361C2×C22⋊C8—611no—
362C2×C22⋊C8—612no—
363C2×C22⋊C8—612no—
364C2×C22⋊C8C2×D6111yescharacteristic
365#380—611no—
366#380—611no—
367#380—611no—
368#380—1221no—
369#380—1221no—
370#380C2×D6111yescharacteristic
371C2×#95—1221no—
372#340—612no—
373#340—612no—
374C61#2400311yescharacteristic, complemented, Sylow
375C25—611no—
376C22×Q8—24no—
377C22×Q8—21no—
378C22×Q8—22no—
379C22×Q8—22no—
380C22×Q8—42no—
381C22×Q8C22×D6112yes—
382C23×C4—21no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—1224no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4—1221no—
392C23×C4—1222no—
393C23×C4—1222no—
394C23×C4—2442no—
395C23×C4C22×D6111yescharacteristic
396C22×C8—612no—
397C22×C8C22×D6112yes—
398C22⋊Q8—618no—
399C22⋊Q8—1224no—
400C22⋊Q8—1224no—
401C22⋊Q8—1228no—
402C22⋊Q8—1228no—
403C22⋊Q8—1224no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4—1224no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1228no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—2442no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1228no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—2444no—
426#106—1224no—
427#106—1224no—
428#103—24no—
429#103—42no—
430#103—1224no—
431#103—2442no—
432C22⋊C8—22no—
433C22⋊C8—1224no—
434C22⋊C8—1222no—
435C22⋊C8—2442no—
436#95—21no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440C24—611no—
441C24—611no—
442C24—614no—
443C24—1222no—
444C24—1222no—
445C24—1222no—
446C24—1222no—
447C24—2441no—
448C24—2441no—
449C24D4×D6111yescharacteristic
450C2×Q8—24no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—48no—
457C2×Q8—48no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8D4×D6114yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—611no—
470C22×C4—614no—
471C22×C4—611no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1222no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—2441no—
490C22×C4—2441no—
491C22×C4—2448no—
492C22×C4—2448no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4—2441no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4#1175822212yes—
503C22×C4D4×D6111yescharacteristic
504C22×C4#1175822412yes—
505C22×C4C23×D6111yescharacteristic
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—1222no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—614no—
515C4⋊C4—1228no—
516C4⋊C4—1228no—
517C4⋊C4—1224no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C22⋊C4—42no—
524C22⋊C4—1228no—
525C22⋊C4—1228no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—612no—
534C23—612no—
535C23—612no—
536C23—612no—
537C23—1228no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—1228no—
549C23—2444no—
550C23—2444no—
551C23—2444no—
552C23—2444no—
553C23—2444no—
554C23#1451975912yes—
555C23SD16×D6112yes—
556C23#1451980912yes—
557C23C2×D4×D6111yescharacteristic
558Q8—24no—
559Q8—24no—
560Q8—48no—
561Q8—42no—
562Q8—42no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—612no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—4884no—
587C2×C4—4884no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2448no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4#1451975912yes—
603C2×C4C2×D4×D6112yes—
604C2×C4C2×#1175822411yescharacteristic
605C2×C4C2×#1175822211yescharacteristic
606C2×C4#1451982212yes—
607C8—42no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—614no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—24416no—
626C22—2441no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22#1684009114yescomplemented
631C22#120×D6111yescharacteristic
632C22C2×#1451975911yescharacteristic
633C22#1684032211yescharacteristic
634C22C2×SD16×D6111yescharacteristic
635C22#137×D6111yescharacteristic
636C22C2×#1451980911yescharacteristic
637C22#1684056111yescharacteristic
638C4—22no—
639C4—44no—
640C4—81no—
641C4—81no—
642C4—1222no—
643C4—4881no—
644C4—4881no—
645C4—2441no—
646C4—2444no—
647C4#1684039912yes—
648C4#1684040212yes—
649C2—24no—
650C2—1224no—
651C2—2442no—
652C2#2144914411yescharacteristic
653C2#448×D6111yescharacteristic
654C2#2145507811yescharacteristic
655C2C2×#1684009111yescharacteristic
656C2#2145576511yescharacteristic
657C2#2145657411yescharacteristic
658C2#2145664111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC24×D61D4
hypercenter#1787D61
hypocenterC61#24003
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C2×#16840091C2semidirect
4. #21447014C2semidirect
5. #21447013C2semidirect
6. #21446123C2semidirect
7. #21446014C2semidirect
8. #21446011C2semidirect
9. #21445754C2semidirect
11. C61×#1787C2semidirect
25. C2×#14519709C22semidirect
26. C2×#14519708C22semidirect
27. C2×#14519701C22semidirect
28. C2×#14519701C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
41. C2×#103×C61C22semidirect
42. C2×#103×C61C22semidirect
286. C22×C61#483semidirect
329. #1787D61semidirect
359. C2×#103C2×D61semidirect
360. C2×#103C2×D61semidirect
374. C61#24003semidirect
630. C22#16840091semidirect

Maximal subgroups

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

subgroupindex
C22×#145197532
C2×#168400912
#380×D612
#214470142
#214470132
#214461232
#214460142
#214460112
#214457542
#214457282
C61×#17872
#2400361