Finite Group Databasebeta

← #29681719

Subgroups — #29681719

Order 15616.

1894 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758189C211yesmaximal, characteristic
2C2×C22⋊Q8×D61C211yesmaximal, characteristic
3C2×#16839769C212yesmaximal, complemented
4#21448201C211yesmaximal, characteristic
5#21446251C212yesmaximal, complemented
6#21446249C212yesmaximal, complemented
7#21446243C211yesmaximal, characteristic, complemented
8#21446242C211yesmaximal, characteristic, complemented
9#21446240C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1892C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758189—28no—
15C22×#11758189—24no—
16C22×#11758189C2212yes—
17C22×#11758189C2212yes—
18C23×#92418C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×#14519755—21no—
22C2×#14519754—21no—
23C2×#14519753C2211yescharacteristic
24C22⋊Q8×D61—24no—
25C22⋊Q8×D61C418yescomplemented
26C2×#14519735C2212yes—
27C2×C4⋊C4×D61—21no—
28C2×C4⋊C4×D61C2211yescharacteristic
29C2×C22⋊C4×D61—21no—
30#16839769—28no—
31C2×#14519702C2212yescomplemented
32C2×#14519701C2212yescomplemented
33C2×#14519700C2212yescomplemented
34#16839624—21no—
35#16839622C2211yescharacteristic
36#16839619C2212yes—
37#16839617—21no—
38C22⋊C4×Dic61—21no—
39#16839609—21no—
40#16839525C2212yescomplemented
41#16839524C2212yescomplemented
42#16839524C2212yescomplemented
43C23×C8×C61C2211yescharacteristic
44C2×C22⋊Q8×C61C2211yescharacteristic
45C2×#103×C61C2212yescomplemented
46C61×#380C2211yescharacteristic
47C61×#340C2212yescomplemented
48C24×D61D411yescharacteristic
49C22×Dic122—21no—
50C22×Dic122C2311yescharacteristic
51C2×Q8×D61—22no—
52C2×Q8×D61—24no—
53C2×Q8×D61—24no—
54C2×Q8×D61C2×C414yes—
55C23×Dic61D411yescharacteristic
56C22×C4×D61—24no—
57C22×C4×D61—24no—
58C22×C4×D61—21no—
59C22×C4×D61—21no—
60C22×C4×D61C2×C414yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D414yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×#11758189—232no—
67C2×#11758189—24no—
68C2×#11758189—24no—
69C2×#11758189—24no—
70C2×#11758189—24no—
71C2×#11758189—24no—
72C2×#11758189D418yes—
73C22×#92418—22no—
74C22×#92418—24no—
75C22×#92418C2312yes—
76#14519757—22no—
77#14519757C2×C414yes—
78#14519756—22no—
79#14519756C2×C414yes—
80#14519755—28no—
81#14519754—28no—
82#14519753—22no—
83#14519753C2×C414yes—
84#14519735—24no—
85#14519735—24no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61C2312yes—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C4⋊C4×D61—28no—
93C4⋊C4×D61—22no—
94C4⋊C4×D61C2×C414yes—
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—22no—
100C2×#11758198—22no—
101C22⋊C4×D61—28no—
102C2×C4×Dic61—22no—
103#14519702—24no—
104#14519701—24no—
105#14519700—24no—
106#14519684—21no—
107#14519684—22no—
108C22×Q8×C61C2311yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C22×C8×C61—22no—
111C22×C8×C61—24no—
112C22×C8×C61C2312yes—
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61C2×C414yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2312yes—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120#103×C61—24no—
121#95×C61—21no—
122C24×C61C2×D411yescharacteristic
123C23×D61—22no—
124C23×D61—22no—
125C23×D61—24no—
126C23×D61—42no—
127C23×D61C22⋊C414yes—
128C23×D61C2×D411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C2×Q8×C61—21no—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61C22×C412yes—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61C22⋊C414yes—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150Q8×D61—216no—
151Q8×D61—44no—
152Q8×D61—44no—
153C2×C4×D61—24no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—24no—
157C2×C4×D61—216no—
158C2×C4×D61—28no—
159C2×C4×D61—28no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D414yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61#4114yes—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122—24no—
176C2×Dic122—44no—
177C2×Dic122C22×C412yes—
178C2×C8×C61—28no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61C2×D414yes—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61C22×C412yes—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—42no—
189C22⋊Dic61—24no—
190C22⋊Dic61—42no—
191#11758198—24no—
192#11758198—24no—
193#11758198—44no—
194#11758198—44no—
195C4⋊Dic61—21no—
196C4⋊Dic61—24no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22×C412yes—
200#11758196—24no—
201#11758196—24no—
202#11758196—24no—
203C4×Dic61—44no—
204C2×#92418—28no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418C2×D414yes—
208#11758189—216no—
209#11758189—216no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61C2×C22⋊C412yes—
213C23×C61#13712yes—
214C23×C61#13712yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—22no—
223C22×D61—28no—
224C22×D61—28no—
225C22×D61—44no—
226C22×D61—48no—
227C22×D61—48no—
228C22×D61C2×C22⋊C412yes—
229C22×D61C2×C22⋊C412yes—
230C22×D61#12114yes—
231Q8×C61—24no—
232Q8×C61—41no—
233Q8×C61—41no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61C2×C22⋊C412yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D412yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4112yes—
250C2×Dic61—22no—
251C2×Dic61—28no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61#12114yes—
262C4×D61—28no—
263C4×D61—24no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—28no—
267C4×D61—48no—
268C4×D61—48no—
269C4×D61C4×D418yes—
270C4×D61#12014yes—
271C4×D61#12314yes—
272Dic122—24no—
273Dic122—41no—
274Dic122—41no—
275Dic122—48no—
276C8×C61—24no—
277#92418—24no—
278#20972—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—24no—
282C22×C61—24no—
283C22×C61#41914yescomplemented
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61C2×#12112yes—
286C22×C61C2×#13711yescharacteristic
287C22×C61C2×#13711yescharacteristic
288C22×C61#57512yes—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—416no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61#38614yes—
300C4×C61—22no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61—82no—
304C4×C61C2×C4×D412yes—
305C4×C61C2×#12011yescharacteristic
306C4×C61C2×#12311yescharacteristic
307Dic61—22no—
308Dic61—42no—
309Dic61—42no—
310Dic61—82no—
311Dic61#38614yes—
312C23×M16—611no—
313C22×C22⋊Q8—611no—
314C2×#419—612no—
315#2280—611no—
316#1894—612no—
317#1894—612no—
318#1892—611no—
319#1892—611no—
320#1892—611no—
321#1892D6111yescharacteristic, complemented
322#1736—612no—
323C2×C61—24no—
324C2×C61C2×#38611yescharacteristic
325C2×C61C2×#41911yescharacteristic
326C2×C61#289611yescharacteristic
327C2×C61#305612yes—
328C2×C61#305612yes—
329D61—24no—
330D61—42no—
331C23×Q8—611no—
332C24×C4—611no—
333C22×M16—612no—
334C22×M16—612no—
335C22×M16—1228no—
336C22×M16—1224no—
337C23×C8—611no—
338C23×C8C2×D6111yescharacteristic
339C2×C22⋊Q8—611no—
340C2×C22⋊Q8—611no—
341C2×C22⋊Q8—618no—
342C2×C22⋊Q8—611no—
343C2×C22⋊Q8—1221no—
344C2×C22⋊Q8—1221no—
345C2×C22⋊Q8—1224no—
346C2×C22⋊Q8C2×D6111yescharacteristic
347C2×#117—612no—
348C22×C4⋊C4—611no—
349C22×C4⋊C4—1221no—
350C22×C22⋊C4—1221no—
351#419—1228no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#103—612no—
355C2×#103C2×D6112yescomplemented
356#381—1221no—
357#380—611no—
358#380—612no—
359#380C2×D6111yescharacteristic
360#379—1221no—
361C4×C22⋊C4—1221no—
362C4×C22⋊C4—1221no—
363#340—612no—
364#340—612no—
365#340—612no—
366#340C2×D6112yescomplemented
367C61#2097211yescharacteristic, complemented, Sylow
368C25—611no—
369C22×Q8—614no—
370C22×Q8—611no—
371C22×Q8—1221no—
372C22×Q8—1222no—
373C22×Q8—1224no—
374C22×Q8—1224no—
375C22×Q8C22×D6111yescharacteristic
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—614no—
379C23×C4—611no—
380C23×C4—614no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—1224no—
384C23×C4—1224no—
385C23×C4—1221no—
386C23×C4—1221no—
387C23×C4C22×D6111yescharacteristic
388C2×M16—618no—
389C2×M16—1224no—
390C2×M16—1224no—
391C2×M16—1224no—
392C2×M16—1224no—
393C2×M16—12232no—
394C2×M16—1224no—
395C22×C8—22no—
396C22×C8—24no—
397C22×C8—612no—
398C22×C8—1224no—
399C22×C8—1222no—
400C22×C8C22×D6112yes—
401C22⋊Q8—22no—
402C22⋊Q8—614no—
403C22⋊Q8—614no—
404C22⋊Q8—614no—
405C22⋊Q8—1228no—
406C22⋊Q8—1228no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1222no—
410C22⋊Q8C4×D6114yes—
411#117—1224no—
412#117—1224no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—614no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1228no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6112yes—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1228no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1222no—
432C2×C22⋊C4—1222no—
433C2×C22⋊C4—1221no—
434C2×C42—1222no—
435#103—24no—
436#103—1224no—
437#103—1224no—
438#103—1224no—
439#95—21no—
440#95—1222no—
441#95—1221no—
442C24—611no—
443C24—611no—
444C24—614no—
445C24—1224no—
446C24—1222no—
447C24—1222no—
448C24—2442no—
449C24D4×D6111yescharacteristic
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—612no—
454C2×Q8—1224no—
455C2×Q8—12216no—
456C2×Q8—1221no—
457C2×Q8—1222no—
458C2×Q8—1222no—
459C2×Q8—2444no—
460C2×Q8—2444no—
461C2×Q8—2444no—
462C2×Q8C2×C4×D6112yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—611no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—611no—
474C22×C4—614no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—614no—
478C22×C4—614no—
479C22×C4—1228no—
480C22×C4—1224no—
481C22×C4—12216no—
482C22×C4—1228no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—2444no—
493C22×C4—2444no—
494C22×C4C2×C4×D6112yes—
495C22×C4D4×D6112yes—
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498M16—12216no—
499M16—12216no—
500C2×C8—28no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—614no—
504C2×C8—1228no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8D4×D6114yes—
508C4⋊C4—21no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—612no—
513C4⋊C4—1224no—
514C4⋊C4—1224no—
515C4⋊C4—1224no—
516C4⋊C4—1221no—
517C4⋊C4—1224no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4C2×C4×D6112yes—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—1224no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—2444no—
527C22⋊C4—2444no—
528C22⋊C4—2442no—
529C42—2444no—
530C23—22no—
531C23—22no—
532C23—612no—
533C23—614no—
534C23—612no—
535C23—1228no—
536C23—1221no—
537C23—1221no—
538C23—1222no—
539C23—1222no—
540C23—1228no—
541C23—1221no—
542C23—1221no—
543C23—1222no—
544C23—2444no—
545C23—2448no—
546C23—2448no—
547C23C22⋊C4×D6112yes—
548C23#1451980612yes—
549C23#1451980912yes—
550C23C2×D4×D6111yescharacteristic
551Q8—24no—
552Q8—41no—
553Q8—41no—
554Q8—1224no—
555Q8—2448no—
556Q8—2441no—
557Q8—2441no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—612no—
570C2×C4—614no—
571C2×C4—618no—
572C2×C4—614no—
573C2×C4—614no—
574C2×C4—612no—
575C2×C4—1228no—
576C2×C4—1228no—
577C2×C4—1228no—
578C2×C4—1222no—
579C2×C4—1224no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—2448no—
585C2×C4—2448no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4C22⋊C4×D6112yes—
592C2×C4C22×C4×D6111yescharacteristic
593C2×C4C2×D4×D6112yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6112yes—
596C8—24no—
597C8—1224no—
598C22—22no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—614no—
603C22—1224no—
604C22—1224no—
605C22—1222no—
606C22—1222no—
607C22—1222no—
608C22—1222no—
609C22—1222no—
610C22—24416no—
611C22—2442no—
612C22—2442no—
613C22#1683976914yescomplemented
614C22C2×C22⋊C4×D6111yescharacteristic
615C22#121×D6112yes—
616C22C2×#1451980611yescharacteristic
617C22C2×#1451980911yescharacteristic
618C22#1684055912yes—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—82no—
623C4—614no—
624C4—1222no—
625C4—4882no—
626C4—2442no—
627C4—2442no—
628C4C4×D4×D6112yes—
629C4#120×D6111yescharacteristic
630C4#123×D6111yescharacteristic
631C2—24no—
632C2—1224no—
633C2—2442no—
634C2#386×D6111yescharacteristic
635C2C2×#1683976911yescharacteristic
636C2#2145433411yescharacteristic
637C2#2145544212yes—
638C2#2145544612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61#20972
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. C2×#16839769C2semidirect
5. #21446251C2semidirect
6. #21446249C2semidirect
7. #21446243C2semidirect
8. #21446242C2semidirect
9. #21446240C2semidirect
10. #21445537C2semidirect
11. C61×#1892C2semidirect
25. C22⋊Q8×D61C4semidirect
31. C2×#14519702C22semidirect
32. C2×#14519701C22semidirect
33. C2×#14519700C22semidirect
40. #16839525C22semidirect
41. #16839524C22semidirect
42. #16839524C22semidirect
45. C2×#103×C61C22semidirect
47. C61×#340C22semidirect
283. C22×C61#419semidirect
321. #1892D61semidirect
355. C2×#103C2×D61semidirect
366. #340C2×D61semidirect
367. C61#20972semidirect
613. C22#16839769semidirect

Maximal subgroups

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

subgroupindex
C23×#117581892
C2×C22⋊Q8×D612
C2×#168397692
#214482012
#214462512
#214462492
#214462432
#214462422
#214462402
#214455372
C61×#18922
#2097261