Finite Group Databasebeta

← #29681706

Subgroups — #29681706

Order 15616.

1914 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D61C211yesmaximal, characteristic, complemented
2C2×#16840272C211yesmaximal, characteristic
3C2×#16839755C212yesmaximal, complemented
4#21448201C211yesmaximal, characteristic, complemented
5#21446258C211yesmaximal, characteristic, complemented
6#21446254C212yesmaximal, complemented
7#21446252C211yesmaximal, characteristic, complemented
8#21446251C212yesmaximal, complemented
9#21446250C211yesmaximal, characteristic, complemented
10#21445534C212yesmaximal, complemented
11C61×#1895C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15C22×#11758189C2212yes—
16C2×M16×D61—28no—
17C2×M16×D61—24no—
18C22×C8×D61C2212yes—
19#16840272—24no—
20#16840272C418yescomplemented
21C2×#14519759—21no—
22C2×#14519757C2211yescharacteristic
23C2×#14519756C2211yescharacteristic
24C2×#14519748C2211yescharacteristic
25C2×#14519745—21no—
26C2×D4×Dic61—21no—
27C2×#14519737—21no—
28C2×#14519735C2212yes—
29C2×#14519735C2211yescharacteristic
30#16839755—28no—
31C2×#14519702C2212yescomplemented
32C2×#14519700C2212yescomplemented
33C2×#14519698C2212yescomplemented
34#16839624—21no—
35#16839622C2211yescharacteristic
36#16839619C2212yes—
37#16839617—21no—
38C22⋊C4×Dic61—21no—
39#16839609—21no—
40#16839525C2212yescomplemented
41#16839524C2212yescomplemented
42#16839524C2212yescomplemented
43C22×M16×C61C2211yescharacteristic
44C2×#121×C61C2211yescharacteristic
45C2×#102×C61C2212yescomplemented
46C61×#380C2211yescharacteristic
47C61×#340C2212yescomplemented
48C24×D61D411yescharacteristic
49C2×#11758224—24no—
50C2×#11758224—22no—
51C2×#11758224—24no—
52C2×#11758224C2×C414yes—
53C22×Dic122C2311yescharacteristic
54C22×C22⋊D61—21no—
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C23×Dic61D411yescharacteristic
58C22×C4×D61—24no—
59C22×C4×D61—24no—
60C22×C4×D61C2×C414yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D414yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×#11758194—22no—
67C2×#11758194—24no—
68C2×#11758189—24no—
69C2×#11758189—24no—
70C2×#11758189—24no—
71M16×D61—232no—
72C22×#92418C2312yes—
73C2×C8×D61—24no—
74C2×C8×D61—24no—
75C2×C8×D61D418yescomplemented
76#14519759—28no—
77#14519757—22no—
78#14519757C2×C414yes—
79#14519756—22no—
80#14519756C2×C414yes—
81#14519748—22no—
82#14519748C2×C414yes—
83#14519745—28no—
84D4×Dic61—28no—
85#14519737—28no—
86#14519735—22no—
87#14519735—24no—
88#14519735—24no—
89#14519735C2×C414yes—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—22no—
100C2×#11758198—22no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—22no—
103#14519702—24no—
104#14519700—24no—
105#14519698—24no—
106#14519684—21no—
107#14519684—22no—
108C22×D4×C61—21no—
109C22×D4×C61C2311yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C2×M16×C61—22no—
112C2×M16×C61—24no—
113C22×C8×C61C2312yes—
114#121×C61—22no—
115#121×C61C2×C414yes—
116C2×C4⋊C4×C61C2312yes—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120#102×C61—24no—
121#95×C61—21no—
122C24×C61—21no—
123C24×C61C2×D411yescharacteristic
124C23×D61—24no—
125C23×D61—42no—
126C23×D61—41no—
127C23×D61—41no—
128C23×D61C22⋊C414yes—
129C23×D61C2×D411yescharacteristic
130C23×D61C2×D411yescharacteristic
131C2×D4×C61—24no—
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—44no—
136C2×D4×C61C22×C412yes—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—24no—
146C2×C22⋊D61—24no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—44no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—24no—
154C22×Dic61—22no—
155C22×Dic61—44no—
156C22×Dic61C22⋊C414yes—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159#11758224—216no—
160#11758224—44no—
161#11758224—44no—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—24no—
166C2×C4×D61—216no—
167C2×C4×D61—28no—
168C2×C4×D61—28no—
169C2×C4×D61C22⋊C414yes—
170C2×C4×D61C22⋊C414yes—
171C2×C4×D61C22×C412yes—
172C2×C4×D61C22×C412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D414yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61#4114yes—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—21no—
180C2×Dic122C22×C412yes—
181M16×C61—28no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61C2×D414yes—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—24no—
190C22⋊C4×C61—42no—
191C22⋊Dic61—24no—
192C22⋊Dic61—24no—
193C22⋊Dic61—42no—
194#11758198—24no—
195#11758198—24no—
196#11758198—44no—
197#11758198—44no—
198C4⋊Dic61—21no—
199C4⋊Dic61—24no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22×C412yes—
203#11758196—24no—
204C4×Dic61—24no—
205C4×Dic61—44no—
206#11758194—28no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418C2×D414yes—
210#11758189—216no—
211C8×D61—216no—
212C23×C61—22no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—42no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—42no—
221C23×C61C2×C22⋊C412yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—28no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61—48no—
234C22×D61—48no—
235C22×D61C2×C22⋊C412yes—
236C22×D61C2×C22⋊C412yes—
237C22×D61#12114yes—
238D4×C61—24no—
239D4×C61—48no—
240D4×C61—41no—
241D4×C61—41no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D412yes—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4112yes—
256C22⋊D61—48no—
257C22⋊D61—48no—
258C2×Dic61—22no—
259C2×Dic61—28no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—42no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—48no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61#12114yes—
272C4×D61—28no—
273C4×D61—24no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—28no—
277C4×D61C4×D418yes—
278C4×D61#12014yes—
279C4×D61#12314yes—
280Dic122—24no—
281Dic122—41no—
282Dic122—41no—
283C8×C61—24no—
284#92418—24no—
285#20970—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—82no—
295C22×C61—82no—
296C22×C61C22×C22⋊C411yescharacteristic
297C22×C61C2×#12112yes—
298C22×C61#57512yes—
299C22×C61#57511yescharacteristic
300C22×C61#57511yescharacteristic
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—24no—
304C2×D61—24no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—416no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61#38614yes—
312C4×C61—22no—
313C4×C61—42no—
314C4×C61—82no—
315C4×C61C2×C4×D412yes—
316C4×C61C2×#12011yescharacteristic
317C4×C61C2×#12311yescharacteristic
318Dic61—22no—
319Dic61—42no—
320Dic61—42no—
321Dic61—82no—
322Dic61#38614yes—
323C23×M16—611no—
324C2×#525—611no—
325C2×#416—612no—
326#2280—611no—
327#1895—612no—
328#1895—611no—
329#1895D6111yescharacteristic, complemented
330#1894—611no—
331#1894—612no—
332#1894—611no—
333#1735—612no—
334C2×C61—42no—
335C2×C61—82no—
336C2×C61C2×#38611yescharacteristic
337C2×C61#289611yescharacteristic
338C2×C61#289611yescharacteristic
339C2×C61#305412yes—
340C2×C61#305412yes—
341D61—24no—
342D61—42no—
343C22×#41—611no—
344C24×C4—611no—
345C22×M16—611no—
346C22×M16—612no—
347C22×M16—1228no—
348C22×M16—1224no—
349C22×M16C2×D6111yescharacteristic
350C23×C8—612no—
351#525—618no—
352#525—1224no—
353C2×#123—1221no—
354C2×C22⋊Q8—611no—
355C2×C22⋊Q8—611no—
356C2×#121—611no—
357C2×#121C2×D6111yescharacteristic
358C2×#120—1221no—
359C2×C4×D4—1221no—
360C2×C4×D4—1221no—
361C2×#117—611no—
362C2×#117—612no—
363#416—1228no—
364C2×#103—612no—
365C2×#103—612no—
366C2×#102—612no—
367C2×#102C2×D6112yescomplemented
368#381—1221no—
369#380—611no—
370#380—612no—
371#380C2×D6111yescharacteristic
372#379—1221no—
373C4×C22⋊C4—1221no—
374C4×C22⋊C4—1221no—
375#340—612no—
376#340—612no—
377#340—612no—
378#340C2×D6112yescomplemented
379C61#2097011yescharacteristic, complemented, Sylow
380C25—611no—
381C2×#41—614no—
382C2×#41—1222no—
383C2×#41—1224no—
384C2×#41—1224no—
385C22×Q8—611no—
386C22×D4—21no—
387C22×D4—1221no—
388C22×D4—1221no—
389C22×D4C22×D6111yescharacteristic
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—614no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—614no—
396C23×C4—611no—
397C23×C4—1224no—
398C23×C4—1224no—
399C23×C4—1221no—
400C23×C4C22×D6111yescharacteristic
401C2×M16—22no—
402C2×M16—24no—
403C2×M16—1224no—
404C2×M16—1224no—
405C2×M16—1224no—
406C2×M16—12232no—
407C2×M16—1222no—
408C2×M16—1224no—
409C22×C8—612no—
410C22×C8—618no—
411C22×C8—1224no—
412C22×C8—1224no—
413C22×C8C22×D6112yes—
414#123—1228no—
415C22⋊Q8—614no—
416C22⋊Q8—614no—
417C22⋊Q8—1222no—
418C22⋊Q8—1222no—
419#121—22no—
420#121—614no—
421#121—1222no—
422#121C4×D6114yes—
423#120—1228no—
424C4×D4—1228no—
425C4×D4—1228no—
426#117—614no—
427#117—1222no—
428#117—1224no—
429#117—1224no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6112yes—
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1221no—
445C2×C42—1221no—
446C2×C42—1222no—
447#103—1224no—
448#103—1224no—
449#102—24no—
450#102—1224no—
451#95—21no—
452#95—1222no—
453#95—1221no—
454C24—21no—
455C24—611no—
456C24—611no—
457C24—614no—
458C24—1224no—
459C24—2441no—
460C24—2442no—
461C24—2441no—
462C24D4×D6111yescharacteristic
463#41—12216no—
464#41—2444no—
465#41—2444no—
466C2×Q8—612no—
467C2×Q8—1221no—
468C2×Q8—1222no—
469C2×Q8—1222no—
470C2×D4—24no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—44no—
475C2×D4—1224no—
476C2×D4—1224no—
477C2×D4—2444no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4C2×C4×D6112yes—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—611no—
486C22×C4—612no—
487C22×C4—614no—
488C22×C4—612no—
489C22×C4—614no—
490C22×C4—611no—
491C22×C4—614no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—614no—
495C22×C4—614no—
496C22×C4—1228no—
497C22×C4—1224no—
498C22×C4—12216no—
499C22×C4—1228no—
500C22×C4—1224no—
501C22×C4—1224no—
502C22×C4—1224no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1222no—
508C22×C4—2444no—
509C22×C4C2×C4×D6112yes—
510C22×C4D4×D6112yes—
511C22×C4D4×D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513M16—28no—
514M16—12216no—
515M16—1228no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—614no—
519C2×C8—12216no—
520C2×C8—1222no—
521C2×C8—1222no—
522C2×C8D4×D6114yes—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—612no—
527C4⋊C4—1224no—
528C4⋊C4—1221no—
529C4⋊C4—1224no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4C2×C4×D6112yes—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—1224no—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—1224no—
539C22⋊C4—2444no—
540C22⋊C4—2444no—
541C22⋊C4—2442no—
542C42—1224no—
543C42—2444no—
544C23—22no—
545C23—21no—
546C23—22no—
547C23—21no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—612no—
554C23—614no—
555C23—612no—
556C23—1228no—
557C23—1222no—
558C23—1221no—
559C23—1221no—
560C23—1222no—
561C23—1222no—
562C23—1221no—
563C23—1221no—
564C23—2444no—
565C23—2444no—
566C23—2448no—
567C23—2448no—
568C23C22⋊C4×D6112yes—
569C23C2×D4×D6111yescharacteristic
570Q8—1224no—
571Q8—2441no—
572Q8—2441no—
573D4—24no—
574D4—48no—
575D4—41no—
576D4—41no—
577D4—2448no—
578D4—2448no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—612no—
589C2×C4—614no—
590C2×C4—618no—
591C2×C4—614no—
592C2×C4—614no—
593C2×C4—612no—
594C2×C4—1228no—
595C2×C4—1228no—
596C2×C4—1228no—
597C2×C4—1222no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—2448no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4C22⋊C4×D6112yes—
611C2×C4C22×C4×D6111yescharacteristic
612C2×C4C2×D4×D6112yes—
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4#41×D6112yes—
615C8—24no—
616C8—1224no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—82no—
626C22—82no—
627C22—614no—
628C22—1224no—
629C22—1224no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—24416no—
636C22—2442no—
637C22—2442no—
638C22C2×C22⋊C4×D6111yescharacteristic
639C22#121×D6112yes—
640C22#1684055812yes—
641C22#1684055811yescharacteristic
642C22#1684056111yescharacteristic
643C4—22no—
644C4—42no—
645C4—82no—
646C4—614no—
647C4—1222no—
648C4—4882no—
649C4—2442no—
650C4—2442no—
651C4C4×D4×D6112yes—
652C4#120×D6111yescharacteristic
653C4#123×D6111yescharacteristic
654C2—42no—
655C2—82no—
656C2—1224no—
657C2—2442no—
658C2#386×D6111yescharacteristic
659C2#2145433311yescharacteristic
660C2#2145433511yescharacteristic
661C2#2145542312yes—
662C2#2145542612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1895C2
socleC23×C61C22×D4
prime radicalC2×#16840272C2
cosocleC22×C4×C61C24
hypercenter#1895D61
hypocenterC61#20970
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
1. C22×M16×D61C2semidirect
3. C2×#16839755C2semidirect
4. #21448201C2semidirect
5. #21446258C2semidirect
6. #21446254C2semidirect
7. #21446252C2semidirect
8. #21446251C2semidirect
9. #21446250C2semidirect
10. #21445534C2semidirect
11. C61×#1895C2semidirect
20. #16840272C4semidirect
31. C2×#14519702C22semidirect
32. C2×#14519700C22semidirect
33. C2×#14519698C22semidirect
40. #16839525C22semidirect
41. #16839524C22semidirect
42. #16839524C22semidirect
45. C2×#102×C61C22semidirect
47. C61×#340C22semidirect
75. C2×C8×D61D4semidirect
329. #1895D61semidirect
367. C2×#102C2×D61semidirect
378. #340C2×D61semidirect
379. C61#20970semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D612
C2×#168402722
C2×#168397552
#214482012
#214462582
#214462542
#214462522
#214462512
#214462502
#214455342
C61×#18952
#2097061