Finite Group Databasebeta

← #29681246

Subgroups — #29681246

Order 15616.

1954 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840272C211yesmaximal, characteristic
2C2×#117×D61C211yesmaximal, characteristic, complemented
3C2×#16839760C212yesmaximal, complemented
4C2×#16839724C211yesmaximal, characteristic, complemented
5#21446142C211yesmaximal, characteristic, complemented
6#21446138C212yesmaximal, complemented
7#21446131C211yesmaximal, characteristic, complemented
8#21446130C212yesmaximal, complemented
9#21446129C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1875C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758189C2212yes—
15#16840272—24no—
16#16840272C418yescomplemented
17C2×#14519759—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×#14519745—21no—
22C2×D4×Dic61—21no—
23C2×#14519737—21no—
24C2×#14519736C2211yescharacteristic
25C2×#14519735C2212yes—
26C2×#14519735C2211yescharacteristic
27C2×#14519734—21no—
28C2×#14519733—21no—
29#117×D61—28no—
30C2×C4⋊C4×D61C2212yes—
31C2×C22⋊C4×D61—21no—
32C2×C42×D61—21no—
33#16839760—28no—
34C2×#14519702C2212yescomplemented
35C2×#14519700C2212yescomplemented
36C2×#14519698C2212yescomplemented
37#16839724—24no—
38C2×C22⋊#92418C2211yescharacteristic
39C2×#14519689C2212yes—
40#16839525C2212yescomplemented
41#16839524C2212yescomplemented
42#16839524C2212yescomplemented
43C2×#121×C61C2211yescharacteristic
44C2×#117×C61C2211yescharacteristic
45C2×#102×C61C2212yescomplemented
46C2×C22⋊C8×C61C2211yescharacteristic
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×D61—22no—
61C22×C4×D61C2×C414yes—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D414yes—
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C2×#11758189—24no—
68C2×#11758189—24no—
69C2×#11758189—24no—
70C22×#92418C2312yes—
71#14519759—28no—
72#14519757—22no—
73#14519757C2×C414yescomplemented
74#14519756—22no—
75#14519756C2×C414yescomplemented
76#14519748—22no—
77#14519748C2×C414yescomplemented
78#14519745—28no—
79D4×Dic61—28no—
80#14519737—28no—
81#14519736—24no—
82#14519735—22no—
83#14519735—24no—
84#14519735—24no—
85#14519735C2×C414yes—
86#14519734—28no—
87#14519733—28no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2312yes—
91C2×#11758196—21no—
92C2×#11758196—22no—
93C4⋊C4×D61—24no—
94C4⋊C4×D61D418yescomplemented
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×#11758198—22no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C22⋊C4×D61—28no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61—21no—
105C42×D61—28no—
106#14519702—24no—
107#14519700—24no—
108#14519698—24no—
109C22⋊#92418—22no—
110#14519689—24no—
111C22×D4×C61—21no—
112C22×D4×C61C2311yescharacteristic
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2312yes—
115#121×C61—22no—
116#121×C61C2×C414yescomplemented
117#117×C61—24no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2312yes—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C42×C61—21no—
123#102×C61—24no—
124C22⋊C8×C61—22no—
125C24×C61—21no—
126C24×C61C2×D411yescharacteristic
127C23×D61—22no—
128C23×D61—22no—
129C23×D61—42no—
130C23×D61—42no—
131C23×D61C22⋊C414yes—
132C23×D61C2×D411yescharacteristic
133C23×D61C2×D411yescharacteristic
134C2×D4×C61—24no—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—44no—
139C2×D4×C61C22×C412yes—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D412yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—24no—
149C2×C22⋊D61—24no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—24no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61C22⋊C414yes—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164#11758224—216no—
165#11758224—48no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—216no—
173C2×C4×D61—216no—
174C2×C4×D61—28no—
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×Dic122—22no—
186C2×Dic122—22no—
187C2×Dic122—21no—
188C2×Dic122C22×C412yes—
189C2×C8×C61—22no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61C22×C412yes—
195C4⋊C4×C61C2×D414yes—
196C22⋊C4×C61—24no—
197C22⋊C4×C61—24no—
198C42×C61—24no—
199C22⋊Dic61—24no—
200C22⋊Dic61—24no—
201C22⋊Dic61—24no—
202#11758198—28no—
203#11758198—24no—
204#11758198—24no—
205C4⋊Dic61—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61—24no—
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C2×D414yes—
210#11758196—24no—
211#11758196—28no—
212C4×Dic61—28no—
213C4×Dic61—24no—
214C4×Dic61—24no—
215C2×#92418—22no—
216C2×#92418—22no—
217C2×#92418—22no—
218#11758189—48no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—42no—
227C23×C61—42no—
228C23×C61C2×C22⋊C412yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—28no—
237C22×D61—44no—
238C22×D61—44no—
239C22×D61—48no—
240C22×D61—48no—
241C22×D61C2×C22⋊C412yes—
242C22×D61C2×C22⋊C412yes—
243C22×D61#12012yes—
244C22×D61#12312yes—
245D4×C61—24no—
246D4×C61—48no—
247D4×C61—42no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—24no—
251C2×C4×C61—24no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D412yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4112yes—
262C22⋊D61—48no—
263C22⋊D61—48no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—28no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—24no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—48no—
276C2×Dic61C2×C22⋊C412yes—
277C2×Dic61C2×C22⋊C412yes—
278C2×Dic61#12012yes—
279C2×Dic61#12312yes—
280C4×D61—28no—
281C4×D61—24no—
282C4×D61—24no—
283C4×D61—28no—
284C4×D61—416no—
285C4×D61C4×D418yes—
286C4×D61#12118yes—
287Dic122—24no—
288Dic122—42no—
289C8×C61—42no—
290#92418—42no—
291#20910—611nomaximal, Sylow
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—82no—
301C22×C61—82no—
302C22×C61C22×C22⋊C411yescharacteristic
303C22×C61C2×#12011yescharacteristic
304C22×C61C2×#12311yescharacteristic
305C22×C61#57512yes—
306C22×C61#57512yes—
307C2×D61—24no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—24no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—416no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61#38614yes—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—44no—
321C4×C61C2×C4×D412yes—
322C4×C61C2×#12112yes—
323Dic61—22no—
324Dic61—42no—
325Dic61—44no—
326Dic61—42no—
327Dic61#38614yes—
328C2×#525—611no—
329C22×#117—611no—
330C2×#417—612no—
331C2×#407—611no—
332#1875—612no—
333#1875—611no—
334#1875D6111yescharacteristic, complemented
335#1873—611no—
336#1873—612no—
337#1873—611no—
338#1736—612no—
339C2×C61—42no—
340C2×C61—82no—
341C2×C61C2×#38611yescharacteristic
342C2×C61#289611yescharacteristic
343C2×C61#289611yescharacteristic
344C2×C61#300512yes—
345C2×C61#309312yes—
346D61—24no—
347D61—42no—
348C22×#41—611no—
349C24×C4—611no—
350C22×M16—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—611no—
363C2×#117—612no—
364C2×#117—1221no—
365C2×#117—1221no—
366C2×#117—1228no—
367C2×#117C2×D6111yescharacteristic
368C22×C4⋊C4—612no—
369C22×C22⋊C4—1221no—
370C22×C42—1221no—
371#417—1228no—
372C2×#103—612no—
373C2×#103—612no—
374C2×#102—612no—
375C2×#102C2×D6112yescomplemented
376#407—1224no—
377C2×C22⋊C8—611no—
378C2×C22⋊C8—612no—
379C2×C22⋊C8C2×D6111yescharacteristic
380#340—612no—
381#340—612no—
382#340—612no—
383#340C2×D6112yescomplemented
384C61#2091011yescharacteristic, complemented, Sylow
385C25—611no—
386C2×#41—614no—
387C2×#41—1222no—
388C2×#41—1224no—
389C2×#41—1224no—
390C22×Q8—611no—
391C22×D4—21no—
392C22×D4—1221no—
393C22×D4—1221no—
394C22×D4C22×D6111yescharacteristic
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—614no—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—614no—
401C23×C4—611no—
402C23×C4—1224no—
403C23×C4—1224no—
404C23×C4—1222no—
405C23×C4—1221no—
406C23×C4C22×D6111yescharacteristic
407C2×M16—1224no—
408C2×M16—1224no—
409C2×M16—1224no—
410C22×C8—612no—
411C22×C8C22×D6112yes—
412#123—1228no—
413C22⋊Q8—614no—
414C22⋊Q8—614no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417#121—22no—
418#121—614no—
419#121—1222no—
420#121C4×D6114yescomplemented
421#120—1228no—
422C4×D4—1228no—
423C4×D4—1228no—
424#117—24no—
425#117—614no—
426#117—1228no—
427#117—1228no—
428#117—1224no—
429#117—1222no—
430#117—1224no—
431#117—1224no—
432C2×C4⋊C4—612no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—618no—
435C2×C4⋊C4—1224no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6112yes—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1228no—
444C2×C22⋊C4—1222no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—21no—
451C2×C42—1228no—
452C2×C42—1222no—
453C2×C42—1221no—
454C2×C42—1221no—
455#103—1224no—
456#103—1224no—
457#102—24no—
458#102—1224no—
459C22⋊C8—22no—
460C22⋊C8—1224no—
461C22⋊C8—1222no—
462C24—21no—
463C24—611no—
464C24—611no—
465C24—614no—
466C24—1222no—
467C24—1222no—
468C24—2442no—
469C24—2442no—
470C24D4×D6111yescharacteristic
471#41—12216no—
472#41—2448no—
473C2×Q8—612no—
474C2×Q8—1221no—
475C2×Q8—1222no—
476C2×Q8—1222no—
477C2×D4—24no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—44no—
482C2×D4—1224no—
483C2×D4—1224no—
484C2×D4—2442no—
485C2×D4—2442no—
486C2×D4—2444no—
487C2×D4C2×C4×D6112yes—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—611no—
493C22×C4—612no—
494C22×C4—614no—
495C22×C4—612no—
496C22×C4—614no—
497C22×C4—611no—
498C22×C4—614no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—614no—
502C22×C4—614no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—12216no—
506C22×C4—12216no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1228no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1224no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—2444no—
519C22×C4—2444no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4C2×C4×D6112yes—
523C22×C4D4×D6112yes—
524C22×C4D4×D6111yescharacteristic
525C22×C4C23×D6111yescharacteristic
526M16—2448no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—612no—
536C4⋊C4—614no—
537C4⋊C4—1228no—
538C4⋊C4—1224no—
539C4⋊C4—1222no—
540C4⋊C4—1221no—
541C4⋊C4—1224no—
542C4⋊C4C2×C4×D6112yes—
543C4⋊C4D4×D6114yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—1224no—
547C22⋊C4—1224no—
548C22⋊C4—1228no—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C42—24no—
553C42—1224no—
554C42—1224no—
555C42—1228no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—612no—
566C23—612no—
567C23—612no—
568C23—612no—
569C23—1228no—
570C23—1222no—
571C23—1222no—
572C23—1221no—
573C23—1221no—
574C23—1222no—
575C23—1222no—
576C23—2444no—
577C23—2444no—
578C23—2448no—
579C23—2448no—
580C23C22⋊C4×D6112yes—
581C23C2×D4×D6111yescharacteristic
582Q8—1224no—
583Q8—2442no—
584D4—24no—
585D4—48no—
586D4—42no—
587D4—2448no—
588D4—2448no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—618no—
602C2×C4—618no—
603C2×C4—612no—
604C2×C4—1228no—
605C2×C4—1228no—
606C2×C4—1228no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1224no—
614C2×C4—24416no—
615C2×C4—2448no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4C22⋊C4×D6112yes—
622C2×C4C22×C4×D6111yescharacteristic
623C2×C4C2×D4×D6112yes—
624C2×C4C2×D4×D6111yescharacteristic
625C2×C4#41×D6112yes—
626C8—42no—
627C8—2442no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—82no—
637C22—82no—
638C22—614no—
639C22—1224no—
640C22—1224no—
641C22—1222no—
642C22—1222no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22—24416no—
647C22—2442no—
648C22—2442no—
649C22C2×C22⋊C4×D6111yescharacteristic
650C22#120×D6111yescharacteristic
651C22#123×D6111yescharacteristic
652C22#1684055812yes—
653C22#1684055912yes—
654C4—22no—
655C4—44no—
656C4—42no—
657C4—614no—
658C4—1222no—
659C4—2442no—
660C4—2444no—
661C4—2442no—
662C4C4×D4×D6112yes—
663C4#121×D6112yes—
664C2—42no—
665C2—82no—
666C2—1224no—
667C2—2442no—
668C2#386×D6111yescharacteristic
669C2#2145433311yescharacteristic
670C2#2145433411yescharacteristic
671C2#2145505712yes—
672C2#2145574712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1875C2
socleC23×C61C22×D4
prime radicalC2×#16840272C2
cosocleC22×C4×C61C24
hypercenter#1875D61
hypocenterC61#20910
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. C2×#117×D61C2semidirect
3. C2×#16839760C2semidirect
4. C2×#16839724C2semidirect
5. #21446142C2semidirect
6. #21446138C2semidirect
7. #21446131C2semidirect
8. #21446130C2semidirect
9. #21446129C2semidirect
10. #21445537C2semidirect
11. C61×#1875C2semidirect
16. #16840272C4semidirect
34. C2×#14519702C22semidirect
35. C2×#14519700C22semidirect
36. C2×#14519698C22semidirect
40. #16839525C22semidirect
41. #16839524C22semidirect
42. #16839524C22semidirect
45. C2×#102×C61C22semidirect
47. C61×#340C22semidirect
73. #14519757C2×C4semidirect
75. #14519756C2×C4semidirect
77. #14519748C2×C4semidirect
94. C4⋊C4×D61D4semidirect
116. #121×C61C2×C4semidirect
334. #1875D61semidirect
375. C2×#102C2×D61semidirect
383. #340C2×D61semidirect
384. C61#20910semidirect
420. #121C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402722
C2×#117×D612
C2×#168397602
C2×#168397242
#214461422
#214461382
#214461312
#214461302
#214461292
#214455372
C61×#18752
#2091061