Finite Group Databasebeta

← #29685667

Subgroups — #29685667

Order 15616.

1868 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839913C211yesmaximal, characteristic
2C2×#16839862C212yesmaximal, complemented
3C4×#14519756C211yesmaximal, characteristic, complemented
4#21446153C212yesmaximal, complemented
5#21446126C211yesmaximal, characteristic, complemented
6#21446066C212yesmaximal, complemented
7#21446007C211yesmaximal, characteristic, complemented
8#21445750C211yesmaximal, characteristic
9#21445644C212yesmaximal, complemented
10#21445581C211yesmaximal, characteristic, complemented
11C22×C4×D4×C61C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C2×#14519756C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic, complemented
15#16839913C418yescomplemented
16#16839862—28no—
17C2×#14519710C2212yescomplemented
18C2×#14519700C2212yescomplemented
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24C2×#14519688C2212yescomplemented
25#16839668C2211yescharacteristic, complemented
26#16839660C2211yescharacteristic, complemented
27#16839646—21no—
28#16839635—21no—
29#16839622C2211yescharacteristic
30C4×#11758196—21no—
31C4×C4⋊Dic61C2211yescharacteristic, complemented
32C4×C22⋊Dic61—21no—
33#16839524C2212yescomplemented
34#16839524C2212yescomplemented
35#16839517C2212yescomplemented
36C23×D4×C61C2211yescharacteristic
37C24×C4×C61—21no—
38C2×C4×D4×C61—28no—
39C2×C4×D4×C61—21no—
40C2×C4×D4×C61—21no—
41C2×C4×D4×C61C2212yes—
42C2×C4×D4×C61C2212yes—
43C22×C4⋊C4×C61C2211yescharacteristic
44C22×C22⋊C4×C61—21no—
45C22×C42×C61C2211yescharacteristic
46C22×Dic122C2311yescharacteristic
47C2×#11758200—24no—
48C2×#11758200—24no—
49C2×#11758200C2×C418yescomplemented
50C22×#92418C2312yes—
51C22×#92418C2312yes—
52#14519756—22no—
53#14519756C2×C414yescomplemented
54C4×Dic122—24no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65#14519710—24no—
66#14519700—24no—
67#14519698—24no—
68#14519698—24no—
69#14519698C2×C418yescomplemented
70C22⋊#92418—22no—
71C22⋊#92418C2×C414yescomplemented
72#14519688—24no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78C25×C61—21no—
79C22×D4×C61—24no—
80C22×D4×C61—24no—
81C22×D4×C61—21no—
82C22×D4×C61—22no—
83C22×D4×C61C2×C414yes—
84C22×D4×C61C2312yes—
85C23×C4×C61—28no—
86C23×C4×C61—21no—
87C23×C4×C61—22no—
88C23×C4×C61—22no—
89C23×C4×C61—44no—
90C23×C4×C61—44no—
91C23×C4×C61D411yescharacteristic
92C23×C4×C61D411yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C4×D4×C61—28no—
95C4×D4×C61—28no—
96C4×D4×C61—24no—
97C4×D4×C61—24no—
98C4×D4×C61—24no—
99C4×D4×C61D418yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—28no—
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61—22no—
106C2×C42×C61—24no—
107C2×C42×C61D411yescharacteristic
108C2×C42×C61D411yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C24×C61—24no—
112C24×C61—24no—
113C24×C61—24no—
114C24×C61—22no—
115C24×C61—42no—
116C24×C61—42no—
117C24×C61—42no—
118C24×C61—42no—
119C24×C61C2×D411yescharacteristic
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—216no—
123C2×D4×C61—216no—
124C2×D4×C61—24no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61C22×C414yes—
129C2×D4×C61C2×D414yes—
130C2×D4×C61#4114yes—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—24no—
137C22×C4×C61—28no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—28no—
142C22×C4×C61—41no—
143C22×C4×C61—432no—
144C22×C4×C61—44no—
145C22×C4×C61—44no—
146C22×C4×C61—44no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—44no—
150C22×C4×C61C22×C412yes—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4112yes—
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122C22×C412yescomplemented
166C4⋊C4×C61—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C2×D414yes—
169C22⋊C4×C61—28no—
170C22⋊C4×C61—28no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—22no—
174C42×C61C2×D414yes—
175C22⋊Dic61—24no—
176C22⋊Dic61—42no—
177#11758200—216no—
178C4⋊Dic61—21no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61C22×C412yescomplemented
182#11758196—24no—
183#11758196—42no—
184#11758196—42no—
185C4×Dic61—42no—
186C4×Dic61—42no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418C22×C414yescomplemented
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—216no—
198C23×C61—24no—
199C23×C61—24no—
200C23×C61—24no—
201C23×C61—48no—
202C23×C61—48no—
203C23×C61—48no—
204C23×C61—48no—
205C23×C61—48no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—48no—
211C23×C61C4×D412yes—
212C23×C61AGL(1,ℤ/8)12yes—
213C23×C61#13712yes—
214C23×C61C22×D411yescharacteristic
215D4×C61—216no—
216D4×C61—24no—
217D4×C61—24no—
218D4×C61—24no—
219D4×C61C4×D418yes—
220C2×C4×C61—28no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—21no—
232C2×C4×C61—24no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—44no—
236C2×C4×C61—416no—
237C2×C4×C61—416no—
238C2×C4×C61—48no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61C4×D412yes—
242C2×C4×C61#12012yes—
243C2×C4×C61#12012yes—
244C2×C4×C61AGL(1,ℤ/8)12yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D412yes—
247C2×C4×C61C2×#4112yes—
248C2×C4×C61C2×#4111yescharacteristic
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261Dic122—24no—
262Dic122—42no—
263#92418—24no—
264#92418—42no—
265#21453—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—28no—
273C22×C61—432no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—44no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61#44014yescomplemented
281C22×C61C2×C4×D411yescharacteristic
282C22×C61C2×#12011yescharacteristic
283C22×C61#52511yescharacteristic
284C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61C2×#13711yescharacteristic
287C22×C61#57511yescharacteristic
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—44no—
293C4×C61—41no—
294C4×C61C2×C4×D412yes—
295C4×C61#52512yes—
296Dic61—42no—
297Dic61—42no—
298Dic61—81no—
299Dic61—81no—
300C22×C4×D4D6111yescharacteristic
301C2×#450—611no—
302C2×#440—612no—
303C4×C22⋊Q8—611no—
304#1877—612no—
305#1872—611no—
306#1862—612no—
307#1852—611no—
308#1786—611no—
309#1762—612no—
310#1749—611no—
311C2×C61—24no—
312C2×C61—48no—
313C2×C61C4×#12011yescharacteristic
314C2×C61C2×#44011yescharacteristic
315C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C61#300311yescharacteristic
317C2×C61#300711yescharacteristic
318C2×C61#311011yescharacteristic
319C2×C61#311411yescharacteristic
320C23×D4C2×D6111yescharacteristic
321C24×C4—21no—
322C22×SD16—612no—
323C2×C22⋊Q8—611no—
324C2×C4×Q8—611no—
325C2×C4×D4—28no—
326C2×C4×D4—21no—
327C2×C4×D4—21no—
328C2×C4×D4C2×D6112yes—
329C2×C4×D4C2×D6112yes—
330C22×C4⋊C4C2×D6111yescharacteristic
331C22×C22⋊C4—21no—
332C22×C42C2×D6111yescharacteristic
333#450—618no—
334#440—1228no—
335C2×#107—612no—
336C2×#103—612no—
337C2×#102—612no—
338C2×#102—612no—
339C2×#102—612no—
340C2×C22⋊C8—611no—
341C2×C22⋊C8—611no—
342C2×#97—612no—
343#391—611no—
344#389—611no—
345#387—1221no—
346#385—1221no—
347#380—611no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—1221no—
350C4×C22⋊C4—1221no—
351#340—612no—
352#340—612no—
353#336—612no—
354C61#2145311yescharacteristic, complemented, Sylow
355C25—21no—
356C22×Q8—611no—
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—22no—
360C22×D4—21no—
361C22×D4C4×D6114yes—
362C22×D4C22×D6112yes—
363C23×C4—28no—
364C23×C4—21no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—44no—
368C23×C4—44no—
369C23×C4C22⋊D6111yescharacteristic
370C23×C4C22⋊D6111yescharacteristic
371C23×C4C22×D6111yescharacteristic
372C2×SD16—618no—
373C2×SD16—1224no—
374C2×SD16—1224no—
375C22×C8—612no—
376C22×C8—612no—
377C22⋊Q8—614no—
378C22⋊Q8—1222no—
379C4×Q8—1224no—
380C4×D4—28no—
381C4×D4—28no—
382C4×D4—24no—
383C4×D4—24no—
384C4×D4—24no—
385C4×D4C22⋊D6118yes—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—28no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—24no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42C22⋊D6111yescharacteristic
404C2×C42C22⋊D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406C2×C42C22×D6111yescharacteristic
407#107—1224no—
408#103—1224no—
409#102—618no—
410#102—1224no—
411#102—1224no—
412C22⋊C8—614no—
413C22⋊C8—1222no—
414#97—1224no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—24no—
421C24—24no—
422C24—22no—
423C24—24no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24C2×C22⋊D6111yescharacteristic
429C2×Q8—612no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—216no—
436C2×D4—216no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—24no—
441C2×D4C2×C4×D6114yes—
442C2×D4#1175822214yes—
443C2×D4C2×C22⋊D6114yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—28no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—41no—
456C22×C4—432no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4C2×C4×D6112yes—
468C22×C4#1175822212yes—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C23×D6111yescharacteristic
475SD16—12216no—
476C2×C8—614no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—612no—
485C4⋊C4—1221no—
486C4⋊C4—1224no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4C2×C22⋊D6114yes—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—1224no—
495C22⋊C4—2442no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—2442no—
500C42—2442no—
501C42C2×C22⋊D6114yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—216no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23C4×C22⋊D6112yes—
521C23#1451980312yes—
522C23#1451980712yes—
523C23C22×C22⋊D6111yescharacteristic
524Q8—1224no—
525Q8—2442no—
526D4—216no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4C4×C22⋊D6118yes—
531C2×C4—28no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—416no—
548C2×C4—416no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4C4×C22⋊D6112yes—
565C2×C4#1451974612yes—
566C2×C4#1451974612yes—
567C2×C4#1451980312yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C22×C22⋊D6112yes—
570C2×C4C2×#1175822212yes—
571C2×C4C2×#1175822211yescharacteristic
572C8—1224no—
573C8—2442no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—28no—
581C22—432no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22#1683986214yescomplemented
589C22C2×C4×C22⋊D6111yescharacteristic
590C22C2×#1451974611yescharacteristic
591C22#1684027311yescharacteristic
592C22C2×#1451980311yescharacteristic
593C22C2×#1451980311yescharacteristic
594C22C2×#1451980711yescharacteristic
595C22#1684056011yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—44no—
601C4—41no—
602C4—4881no—
603C4—4881no—
604C4—2442no—
605C4—2442no—
606C4C2×C4×C22⋊D6112yes—
607C4#1684027312yes—
608C2—24no—
609C2—48no—
610C2C4×#1451974611yescharacteristic
611C2C2×#1683986211yescharacteristic
612C2C4×#1451980311yescharacteristic
613C2#2145504611yescharacteristic
614C2#2145507911yescharacteristic
615C2#2145591811yescharacteristic
616C2#2145595711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×C4×D4×C61C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
cosocleC22×C4×C61C24
hypercenterC22×C4×D4D61
hypocenterC61#21453
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#16839862C2semidirect
3. C4×#14519756C2semidirect
4. #21446153C2semidirect
5. #21446126C2semidirect
6. #21446066C2semidirect
7. #21446007C2semidirect
9. #21445644C2semidirect
10. #21445581C2semidirect
14. C2×C4×Dic122C22semidirect
15. #16839913C4semidirect
17. C2×#14519710C22semidirect
18. C2×#14519700C22semidirect
24. C2×#14519688C22semidirect
25. #16839668C22semidirect
26. #16839660C22semidirect
31. C4×C4⋊Dic61C22semidirect
33. #16839524C22semidirect
34. #16839524C22semidirect
35. #16839517C22semidirect
49. C2×#11758200C2×C4semidirect
53. #14519756C2×C4semidirect
69. #14519698C2×C4semidirect
71. C22⋊#92418C2×C4semidirect
165. C2×Dic122C22×C4semidirect
181. C4⋊Dic61C22×C4semidirect
192. C2×#92418C22×C4semidirect
280. C22×C61#440semidirect
354. C61#21453semidirect
588. C22#16839862semidirect

Maximal subgroups

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

subgroupindex
C2×#168399132
C2×#168398622
C4×#145197562
#214461532
#214461262
#214460662
#214460072
#214457502
#214456442
#214455812
C22×C4×D4×C612
#2145361