Finite Group Databasebeta

← #29681710

Subgroups — #29681710

Order 15616.

1986 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758189C211yesmaximal, characteristic, complemented
2C2×#16840275C211yesmaximal, characteristic
3C2×#16839762C212yesmaximal, complemented
4#380×D61C211yesmaximal, characteristic, complemented
5#21446256C212yesmaximal, complemented
6#21446249C212yesmaximal, complemented
7#21446247C211yesmaximal, characteristic, complemented
8#21446245C211yesmaximal, characteristic, complemented
9#21446243C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1892C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758189—28no—
15C22×#11758189—24no—
16C22×#11758189C2212yes—
17C22×#11758189C2212yes—
18C23×#92418C2211yescharacteristic
19#16840275—24no—
20#16840275C418yescomplemented
21C2×#14519762—21no—
22C2×#14519753C2211yescharacteristic
23C2×#14519752C2211yescharacteristic
24C2×#14519751C2211yescharacteristic
25C2×#14519747—21no—
26C2×#14519739—21no—
27C2×#14519737—21no—
28C2×#14519735C2211yescharacteristic
29C2×C4⋊C4×D61C2212yes—
30C2×C22⋊C4×D61—21no—
31#16839762—28no—
32C2×#14519701C2212yescomplemented
33C2×#14519699C2212yescomplemented
34C2×#14519697C2212yescomplemented
35#16839622C2211yescharacteristic
36#16839621—21no—
37#16839620—21no—
38#16839619C2212yes—
39#95×D61—21no—
40#16839525C2212yescomplemented
41#16839524C2212yescomplemented
42#16839524C2212yescomplemented
43C23×C8×C61C2211yescharacteristic
44C2×C22⋊Q8×C61C2211yescharacteristic
45C2×#103×C61C2212yescomplemented
46C61×#380C2211yescharacteristic
47C61×#340C2212yescomplemented
48C24×D61D411yescharacteristic
49C2×#11758226—22no—
50C2×#11758226—24no—
51C2×#11758226—24no—
52C2×#11758226C2×C414yes—
53C22×D244—21no—
54C22×D244—21no—
55C22×D244C2311yescharacteristic
56C22×C22⋊D61—21no—
57C23×Dic61D411yescharacteristic
58C22×C4×D61—24no—
59C22×C4×D61—24no—
60C22×C4×D61—21no—
61C22×C4×D61—22no—
62C22×C4×D61C2×C414yes—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D414yes—
66C22×C4×D61C2311yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C2×#11758189—232no—
69C2×#11758189—24no—
70C2×#11758189—24no—
71C2×#11758189—24no—
72C2×#11758189—24no—
73C2×#11758189—24no—
74C2×#11758189D418yescomplemented
75C22×#92418—22no—
76C22×#92418—24no—
77C22×#92418C2312yes—
78#14519762—28no—
79#14519753—22no—
80#14519753C2×C414yescomplemented
81#14519752—22no—
82#14519752C2×C414yescomplemented
83#14519751—22no—
84#14519751C2×C414yescomplemented
85#14519747—28no—
86#14519739—28no—
87#14519737—28no—
88#14519735—22no—
89#14519735C2×C414yes—
90C2×C4⋊Dic61C2312yes—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—22no—
94C4⋊C4×D61—24no—
95C4⋊C4×D61—24no—
96C2×C22⋊Dic61—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—22no—
102C22⋊C4×D61—44no—
103C2×C4×Dic61—21no—
104#14519701—24no—
105#14519699—24no—
106#14519697—24no—
107#14519684—21no—
108#14519684—22no—
109C22×Q8×C61C2311yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C22×C8×C61—22no—
112C22×C8×C61—24no—
113C22×C8×C61C2312yes—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61C2×C414yescomplemented
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2312yes—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121#103×C61—24no—
122#95×C61—21no—
123C24×C61C2×D411yescharacteristic
124C23×D61—22no—
125C23×D61—22no—
126C23×D61—24no—
127C23×D61—21no—
128C23×D61—42no—
129C23×D61C22⋊C414yes—
130C23×D61C2×D411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61C22×C412yes—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
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—42no—
147C2×C22⋊D61—42no—
148C22×Dic61—24no—
149C22×Dic61—21no—
150C22×Dic61—22no—
151C22×Dic61C22⋊C414yes—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154#11758226—216no—
155#11758226—44no—
156#11758226—44no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—24no—
160C2×D244—24no—
161C2×D244—21no—
162C2×D244—44no—
163C2×D244—44no—
164C2×D244C22×C412yes—
165C2×C4×D61—24no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—24no—
169C2×C4×D61—216no—
170C2×C4×D61—28no—
171C2×C4×D61—28no—
172C2×C4×D61—24no—
173C2×C4×D61—44no—
174C2×C4×D61—44no—
175C2×C4×D61—44no—
176C2×C4×D61—44no—
177C2×C4×D61C22⋊C414yes—
178C2×C4×D61C22⋊C414yes—
179C2×C4×D61C22×C412yes—
180C2×C4×D61C22×C412yes—
181C2×C4×D61C2×D412yes—
182C2×C4×D61C2×D414yes—
183C2×C4×D61C2×D412yes—
184C2×C4×D61#4114yes—
185C2×C8×C61—28no—
186C2×C8×C61—22no—
187C2×C8×C61—22no—
188C2×C8×C61C2×D414yes—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61C22×C412yes—
194C22⋊C4×C61—24no—
195C22⋊C4×C61—42no—
196C22⋊Dic61—42no—
197#11758198—24no—
198#11758198—24no—
199#11758198—24no—
200#11758198—24no—
201#11758198—44no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C412yes—
206#11758196—24no—
207#11758196—44no—
208C4×Dic61—24no—
209C2×#92418—28no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418C2×D414yes—
213#11758189—216no—
214#11758189—216no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61C2×C22⋊C412yes—
218C23×C61AGL(1,ℤ/8)12yes—
219C23×C61#13712yes—
220C23×C61C22×D411yescharacteristic
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—22no—
228C22×D61—28no—
229C22×D61—28no—
230C22×D61—22no—
231C22×D61—44no—
232C22×D61—48no—
233C22×D61—48no—
234C22×D61—42no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—42no—
238C22×D61C2×C22⋊C412yes—
239C22×D61C2×C22⋊C412yes—
240C22×D61#12114yes—
241Q8×C61—24no—
242Q8×C61—41no—
243Q8×C61—41no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C23×C411yescharacteristic
257C2×C4×C61C22×D412yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4112yes—
260C22⋊D61—48no—
261C2×Dic61—22no—
262C2×Dic61—28no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61#12114yes—
272D244—24no—
273D244—48no—
274D244—48no—
275D244—41no—
276D244—41no—
277C4×D61—28no—
278C4×D61—24no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—28no—
282C4×D61—48no—
283C4×D61—88no—
284C4×D61C4×D418yes—
285C4×D61#12014yes—
286C4×D61#12314yes—
287C8×C61—24no—
288#92418—24no—
289#20971—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61#41714yescomplemented
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61C2×#12112yes—
297C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C61C2×#13711yescharacteristic
299C22×C61#57512yes—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—24no—
303C2×D61—24no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—416no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—82no—
314C2×D61—82no—
315C2×D61#38614yes—
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—82no—
320C4×C61C2×C4×D412yes—
321C4×C61C2×#12011yescharacteristic
322C4×C61C2×#12311yescharacteristic
323Dic61—22no—
324Dic61—42no—
325Dic61—82no—
326Dic61#38614yes—
327C23×M16—611no—
328C2×#525—611no—
329C2×#417—612no—
330C2×#380—611no—
331#1895—612no—
332#1894—612no—
333#1893—611no—
334#1893—611no—
335#1892—611no—
336#1892D6111yescharacteristic, complemented
337#1736—612no—
338C2×C61—24no—
339C2×C61C2×#38611yescharacteristic
340C2×C61C2×#41711yescharacteristic
341C2×C61#289611yescharacteristic
342C2×C61#305512yes—
343C2×C61#305612yes—
344D61—24no—
345D61—42no—
346D61—82no—
347C22×#41—611no—
348C24×C4—611no—
349C22×M16—612no—
350C22×M16—612no—
351C22×M16—1228no—
352C22×M16—1224no—
353C23×C8—611no—
354C23×C8C2×D6111yescharacteristic
355#525—618no—
356#525—1224no—
357C2×#123—1221no—
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8C2×D6111yescharacteristic
360C2×#121—611no—
361C2×#121—611no—
362C2×#120—1221no—
363C2×C4×D4—1221no—
364C2×C4×D4—1221no—
365C2×#117—611no—
366C22×C4⋊C4—612no—
367C22×C22⋊C4—1221no—
368#417—1228no—
369C2×#103—612no—
370C2×#103C2×D6112yescomplemented
371C2×#102—612no—
372C2×#102—612no—
373#380—611no—
374#380—612no—
375#380—1221no—
376#380—1221no—
377#380C2×D6111yescharacteristic
378C2×#95—1221no—
379#340—612no—
380#340—612no—
381#340—612no—
382#340C2×D6112yescomplemented
383C61#2097111yescharacteristic, complemented, Sylow
384C25—611no—
385C2×#41—614no—
386C2×#41—1222no—
387C2×#41—1224no—
388C2×#41—1224no—
389C22×Q8C22×D6111yescharacteristic
390C22×D4—611no—
391C22×D4—1221no—
392C22×D4—1221no—
393C22×D4—1221no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—614no—
397C23×C4—611no—
398C23×C4—611no—
399C23×C4—614no—
400C23×C4—611no—
401C23×C4—1224no—
402C23×C4—1224no—
403C23×C4—1221no—
404C23×C4—1222no—
405C23×C4C22×D6111yescharacteristic
406C2×M16—618no—
407C2×M16—1224no—
408C2×M16—1224no—
409C2×M16—1224no—
410C2×M16—1224no—
411C2×M16—12232no—
412C2×M16—1224no—
413C22×C8—22no—
414C22×C8—24no—
415C22×C8—612no—
416C22×C8—1224no—
417C22×C8—1222no—
418C22×C8C22×D6112yes—
419#123—1228no—
420C22⋊Q8—22no—
421C22⋊Q8—614no—
422C22⋊Q8—1222no—
423C22⋊Q8C4×D6114yescomplemented
424#121—614no—
425#121—614no—
426#121—1222no—
427#121—1222no—
428#120—1228no—
429C4×D4—1228no—
430C4×D4—1228no—
431#117—614no—
432#117—1222no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—1224no—
437C2×C4⋊C4—1224no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4C22×D6112yes—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—2444no—
451C2×C42—1221no—
452#103—24no—
453#103—1224no—
454#102—1224no—
455#102—1224no—
456#95—21no—
457#95—1222no—
458#95—1221no—
459C24—611no—
460C24—611no—
461C24—614no—
462C24—1224no—
463C24—1222no—
464C24—1222no—
465C24—1221no—
466C24—2442no—
467C24D4×D6111yescharacteristic
468#41—12216no—
469#41—2444no—
470#41—2444no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8C2×C4×D6112yes—
475C2×D4—612no—
476C2×D4—1224no—
477C2×D4—1224no—
478C2×D4—1224no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—1222no—
482C2×D4—2444no—
483C2×D4—2444no—
484C2×D4—2442no—
485C2×D4—2442no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—611no—
492C22×C4—612no—
493C22×C4—614no—
494C22×C4—612no—
495C22×C4—614no—
496C22×C4—611no—
497C22×C4—614no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—614no—
501C22×C4—614no—
502C22×C4—1228no—
503C22×C4—1224no—
504C22×C4—12216no—
505C22×C4—1228no—
506C22×C4—1224no—
507C22×C4—1224no—
508C22×C4—1224no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1221no—
512C22×C4—1222no—
513C22×C4—2444no—
514C22×C4—2444no—
515C22×C4—2444no—
516C22×C4—2444no—
517C22×C4C2×C4×D6112yes—
518C22×C4D4×D6112yes—
519C22×C4D4×D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521M16—12216no—
522M16—12216no—
523C2×C8—28no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—614no—
527C2×C8—1228no—
528C2×C8—1222no—
529C2×C8—1222no—
530C2×C8D4×D6114yes—
531C4⋊C4—21no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—612no—
536C4⋊C4—1224no—
537C4⋊C4—1221no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—2444no—
541C4⋊C4C2×C4×D6112yes—
542C22⋊C4—24no—
543C22⋊C4—42no—
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C22⋊C4—1224no—
547C22⋊C4—1224no—
548C22⋊C4—2444no—
549C22⋊C4—2442no—
550C42—1224no—
551C23—22no—
552C23—22no—
553C23—612no—
554C23—614no—
555C23—612no—
556C23—1228no—
557C23—1221no—
558C23—1221no—
559C23—1222no—
560C23—1222no—
561C23—1228no—
562C23—1221no—
563C23—1221no—
564C23—1222no—
565C23—1222no—
566C23—2444no—
567C23—2448no—
568C23—2448no—
569C23—2442no—
570C23—2441no—
571C23—2441no—
572C23—2442no—
573C23C22⋊C4×D6112yes—
574C23#1451980412yes—
575C23#1451980912yes—
576C23C2×D4×D6111yescharacteristic
577Q8—24no—
578Q8—41no—
579Q8—41no—
580D4—1224no—
581D4—2448no—
582D4—2448no—
583D4—2448no—
584D4—2441no—
585D4—2441no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—612no—
598C2×C4—614no—
599C2×C4—618no—
600C2×C4—614no—
601C2×C4—614no—
602C2×C4—612no—
603C2×C4—1228no—
604C2×C4—1228no—
605C2×C4—1228no—
606C2×C4—1222no—
607C2×C4—1224no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—4888no—
612C2×C4—2448no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4C22⋊C4×D6112yes—
619C2×C4C22×C4×D6111yescharacteristic
620C2×C4C2×D4×D6112yes—
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4#41×D6112yes—
623C8—24no—
624C8—1224no—
625C22—22no—
626C22—24no—
627C22—24no—
628C22—22no—
629C22—614no—
630C22—1224no—
631C22—1224no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—4882no—
638C22—4882no—
639C22—24416no—
640C22—2442no—
641C22—2442no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22#1683976214yescomplemented
646C22C2×C22⋊C4×D6111yescharacteristic
647C22#121×D6112yes—
648C22C2×#1451980411yescharacteristic
649C22C2×#1451980911yescharacteristic
650C22#1684055912yes—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—614no—
656C4—1222no—
657C4—4882no—
658C4—2442no—
659C4C4×D4×D6112yes—
660C4#120×D6111yescharacteristic
661C4#123×D6111yescharacteristic
662C2—24no—
663C2—1224no—
664C2—4882no—
665C2—2442no—
666C2#386×D6111yescharacteristic
667C2C2×#1683976211yescharacteristic
668C2#2145433411yescharacteristic
669C2#2145543112yes—
670C2#2145544612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC2×#16840275C2
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61#20971
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
1. C23×#11758189C2semidirect
3. C2×#16839762C2semidirect
4. #380×D61C2semidirect
5. #21446256C2semidirect
6. #21446249C2semidirect
7. #21446247C2semidirect
8. #21446245C2semidirect
9. #21446243C2semidirect
10. #21445537C2semidirect
11. C61×#1892C2semidirect
20. #16840275C4semidirect
32. C2×#14519701C22semidirect
33. C2×#14519699C22semidirect
34. C2×#14519697C22semidirect
40. #16839525C22semidirect
41. #16839524C22semidirect
42. #16839524C22semidirect
45. C2×#103×C61C22semidirect
47. C61×#340C22semidirect
74. C2×#11758189D4semidirect
80. #14519753C2×C4semidirect
82. #14519752C2×C4semidirect
84. #14519751C2×C4semidirect
115. C22⋊Q8×C61C2×C4semidirect
294. C22×C61#417semidirect
336. #1892D61semidirect
370. C2×#103C2×D61semidirect
382. #340C2×D61semidirect
383. C61#20971semidirect
423. C22⋊Q8C4×D61semidirect
645. C22#16839762semidirect

Maximal subgroups

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

subgroupindex
C23×#117581892
C2×#168402752
C2×#168397622
#380×D612
#214462562
#214462492
#214462472
#214462452
#214462432
#214455372
C61×#18922
#2097161