Finite Group Databasebeta

← #29685657

Subgroups — #29685657

Order 15616.

1952 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic61C211yesmaximal, characteristic, complemented
2C2×#16839874C212yesmaximal, complemented
3#121×Dic61C211yesmaximal, characteristic, complemented
4#21446169C212yesmaximal, complemented
5#21446137C211yesmaximal, characteristic, complemented
6#21446082C212yesmaximal, complemented
7#21446014C211yesmaximal, characteristic, complemented
8#21445750C211yesmaximal, characteristic, complemented
9#21445646C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C2×C61×#447C211yesmaximal, characteristic
12C24×Dic61—21no—
13C2×D4×Dic61—28no—
14C2×D4×Dic61—21no—
15C2×D4×Dic61—21no—
16C2×D4×Dic61—21no—
17C2×D4×Dic61C2212yes—
18C2×D4×Dic61C2212yes—
19C2×D4×Dic61C2211yescharacteristic, complemented
20C22×C4⋊Dic61C2211yescharacteristic
21C22×C22⋊Dic61—21no—
22C22×C4×Dic61C2211yescharacteristic
23#16839874—28no—
24C2×#14519709C2212yescomplemented
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519698C2212yescomplemented
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519687C2212yescomplemented
30#16839665C2211yescharacteristic, complemented
31#16839661C2211yescharacteristic, complemented
32#16839655—21no—
33#16839642—21no—
34#16839622C2211yescharacteristic
35C4⋊C4×Dic61C2211yescharacteristic, complemented
36C22⋊C4×Dic61—21no—
37#16839525C2212yescomplemented
38#16839524C2212yescomplemented
39#16839516C2212yescomplemented
40C23×D4×C61C2211yescharacteristic
41C22×D8×C61C2212yes—
42C2×#121×C61C2211yescharacteristic
43C61×#447C418yescomplemented
44C2×#102×C61C2212yes—
45C2×C22⋊C8×C61C2211yescharacteristic
46C23×Dic61—28no—
47C23×Dic61—21no—
48C23×Dic61—22no—
49C23×Dic61—22no—
50C23×Dic61—21no—
51C23×Dic61—44no—
52C23×Dic61—44no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×#92418C2312yes—
56D4×Dic61—28no—
57D4×Dic61—28no—
58D4×Dic61—24no—
59D4×Dic61—24no—
60D4×Dic61—24no—
61D4×Dic61—24no—
62D4×Dic61—44no—
63D4×Dic61D418yescomplemented
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C22⋊Dic61—28no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—24no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83#14519709—24no—
84#14519698—24no—
85#14519698—24no—
86#14519698—24no—
87C22⋊#92418—22no—
88#14519687—24no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C25×C61—21no—
93C22×D4×C61—24no—
94C22×D4×C61—24no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61—22no—
98C22×D4×C61C2×C414yescomplemented
99C22×D4×C61C2311yescharacteristic
100C22×D4×C61C2312yes—
101C23×C4×C61C2311yescharacteristic
102C2×D8×C61—24no—
103C2×D8×C61—24no—
104C2×D8×C61C2×C418yescomplemented
105C22×C8×C61C2312yes—
106#121×C61—22no—
107#121×C61C2×C414yescomplemented
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110#102×C61C2×C418yescomplemented
111C22⋊C8×C61C2×C414yescomplemented
112C24×C61—24no—
113C24×C61—24no—
114C24×C61—24no—
115C24×C61—22no—
116C24×C61—21no—
117C24×C61—42no—
118C24×C61—42no—
119C24×C61—42no—
120C24×C61—42no—
121C24×C61C2×D411yescharacteristic
122C2×D4×C61—24no—
123C2×D4×C61—216no—
124C2×D4×C61—24no—
125C2×D4×C61—216no—
126C2×D4×C61—24no—
127C2×D4×C61—24no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61C22×C414yes—
137C2×D4×C61C22×C412yescomplemented
138C2×D4×C61C2×D414yes—
139C2×D4×C61#4114yes—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61#4112yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—24no—
151C22×Dic61—28no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—28no—
155C22×Dic61—22no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—41no—
159C22×Dic61—432no—
160C22×Dic61—44no—
161C22×Dic61—44no—
162C22×Dic61—44no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—44no—
168C22×Dic61—42no—
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173D8×C61—216no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61C22×C414yescomplemented
177C4⋊C4×C61—21no—
178C4⋊C4×C61C22×C412yescomplemented
179C22⋊C4×C61—24no—
180C22⋊Dic61—28no—
181C22⋊Dic61—28no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C4⋊Dic61—24no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—42no—
191C4⋊Dic61C2×D414yes—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—42no—
196C4×Dic61C2×D414yes—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—216no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—22no—
209C23×C61—48no—
210C23×C61—48no—
211C23×C61—48no—
212C23×C61—48no—
213C23×C61—48no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—48no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61C4×D412yes—
226C23×C61AGL(1,ℤ/8)12yes—
227C23×C61AGL(1,ℤ/8)12yes—
228C23×C61C22×D411yescharacteristic
229D4×C61—216no—
230D4×C61—24no—
231D4×C61—24no—
232D4×C61—24no—
233D4×C61—24no—
234D4×C61—48no—
235D4×C61—42no—
236D4×C61C4×D418yescomplemented
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61C4×D412yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D412yes—
247C2×C4×C61C2×#4112yes—
248C2×C4×C61C2×#4111yescharacteristic
249C2×Dic61—28no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—24no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—44no—
261C2×Dic61—416no—
262C2×Dic61—416no—
263C2×Dic61—48no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—84no—
273C2×Dic61#12012yes—
274C2×Dic61#12012yes—
275C2×Dic61AGL(1,ℤ/8)12yes—
276C8×C61—24no—
277#92418—42no—
278#21452—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—28no—
286C22×C61—432no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—44no—
291C22×C61—44no—
292C22×C61—44no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61#44214yescomplemented
301C22×C61C2×C4×D411yescharacteristic
302C22×C61C2×#12011yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C61#57511yescharacteristic
308C4×C61—22no—
309C4×C61—42no—
310C4×C61C2×C4×D412yes—
311C4×C61#52512yes—
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—44no—
316Dic61—41no—
317Dic61—81no—
318Dic61—81no—
319C22×C4×D4—611no—
320C2×#447D6111yescharacteristic
321C2×#442—612no—
322C4×#121—611no—
323#1879—612no—
324#1874—611no—
325#1864—612no—
326#1853—611no—
327#1786—611no—
328#1762—612no—
329#1749—611no—
330C2×C61—24no—
331C2×C61—48no—
332C2×C61—82no—
333C2×C61C4×#12011yescharacteristic
334C2×C61C2×#44211yescharacteristic
335C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
336C2×C61#300311yescharacteristic
337C2×C61#300611yescharacteristic
338C2×C61#310911yescharacteristic
339C2×C61#311411yescharacteristic
340C23×D4C2×D6111yescharacteristic
341C24×C4—1221no—
342C22×D8C2×D6112yes—
343C2×#121C2×D6111yescharacteristic
344C2×C4×D4—612no—
345C2×C4×D4—612no—
346C2×C4×D4—611no—
347C2×C4×D4—1221no—
348C2×C4×D4—1221no—
349C2×C4×D4—1228no—
350C2×C4×D4—1221no—
351C22×C4⋊C4—611no—
352C22×C22⋊C4—1221no—
353C22×C42—611no—
354#447Dic6118yescomplemented
355#442—1228no—
356C2×#106—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×#102—612no—
360C2×#102C2×D6112yes—
361C2×C22⋊C8—611no—
362C2×C22⋊C8C2×D6111yescharacteristic
363C2×#97—612no—
364#390—611no—
365#389—611no—
366#388—1221no—
367#386—1221no—
368#380—611no—
369C4×C4⋊C4—611no—
370C4×C22⋊C4—1221no—
371#340—612no—
372#340—612no—
373#336—612no—
374C61#2145211yescharacteristic, complemented, Sylow
375C25—21no—
376C22×D4—24no—
377C22×D4—24no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—22no—
381C22×D4C2×Dic6114yescomplemented
382C22×D4C22×D6111yescharacteristic
383C22×D4C22×D6112yes—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—1222no—
387C23×C4—1222no—
388C23×C4—1221no—
389C23×C4—1228no—
390C23×C4—1221no—
391C23×C4—2444no—
392C23×C4—2444no—
393C23×C4C22×D6111yescharacteristic
394C2×D8—24no—
395C2×D8—24no—
396C2×D8C2×Dic6118yescomplemented
397C22×C8—612no—
398C22×C8C22×D6112yes—
399#121—22no—
400#121C2×Dic6114yescomplemented
401C4×D4—618no—
402C4×D4—1224no—
403C4×D4—1224no—
404C4×D4—1224no—
405C4×D4—1228no—
406C4×D4—1228no—
407C4×D4—1224no—
408C4×D4—2444no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1224no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1228no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—1224no—
429C2×C42—1221no—
430#106—1224no—
431#102—1224no—
432#102—1224no—
433#102—1224no—
434#102C2×Dic6118yescomplemented
435C22⋊C8—1222no—
436C22⋊C8C2×Dic6114yescomplemented
437#97—1224no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—24no—
442C24—24no—
443C24—22no—
444C24—24no—
445C24—21no—
446C24—42no—
447C24—42no—
448C24—42no—
449C24—42no—
450C24D4×D6111yescharacteristic
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—216no—
454C2×D4—216no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—24no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4D4×D6114yes—
466C2×D4#1175822414yes—
467C2×D4C22×Dic6114yes—
468C2×D4C22×Dic6112yescomplemented
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1228no—
480C22×C4—1222no—
481C22×C4—1224no—
482C22×C4—1228no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—2442no—
488C22×C4—2444no—
489C22×C4—2441no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—24432no—
493C22×C4—2444no—
494C22×C4—2444no—
495C22×C4—2444no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4D4×D6111yescharacteristic
499C22×C4#1175822412yes—
500C22×C4C22×Dic6112yes—
501C22×C4C23×D6111yescharacteristic
502D8—216no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8C22×Dic6114yescomplemented
509C4⋊C4—21no—
510C4⋊C4—614no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—2442no—
516C4⋊C4C22×Dic6112yescomplemented
517C22⋊C4—24no—
518C22⋊C4—1228no—
519C22⋊C4—1228no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—614no—
525C42—1222no—
526C42—1222no—
527C42—1222no—
528C42—2442no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—22no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—48no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23D4×Dic6112yes—
555C23#1451980112yes—
556C23#1451980112yes—
557C23C2×D4×D6111yescharacteristic
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4—24no—
563D4—48no—
564D4—42no—
565D4D4×Dic6118yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—1228no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1224no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—4884no—
588C2×C4—2444no—
589C2×C4—24416no—
590C2×C4—24416no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2448no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4D4×Dic6112yes—
601C2×C4C23×Dic6111yescharacteristic
602C2×C4C2×D4×D6112yes—
603C2×C4C2×#1175822412yes—
604C2×C4C2×#1175822411yescharacteristic
605C8—24no—
606C8—2442no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22#1683987414yescomplemented
629C22C2×D4×Dic6111yescharacteristic
630C22#120×D6111yescharacteristic
631C22#1684026811yescharacteristic
632C22AGL(1,ℤ/8)×D6111yescharacteristic
633C22C2×#1451980111yescharacteristic
634C22C2×#1451980111yescharacteristic
635C22#1684055811yescharacteristic
636C4—22no—
637C4—42no—
638C4—1222no—
639C4—1222no—
640C4—1222no—
641C4—4881no—
642C4—4881no—
643C4—2444no—
644C4—2441no—
645C4C2×D4×Dic6112yes—
646C4#1684026812yes—
647C2—24no—
648C2—48no—
649C2—82no—
650C2#120×Dic6111yescharacteristic
651C2C2×#1683987411yescharacteristic
652C2AGL(1,ℤ/8)×Dic6111yescharacteristic
653C2#2145504311yescharacteristic
654C2#2145506611yescharacteristic
655C2#2145590511yescharacteristic
656C2#2145595211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#447C2
socleC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenterC2×#447D61
hypocenterC61#21452
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C22×D4×Dic61C2semidirect
2. C2×#16839874C2semidirect
3. #121×Dic61C2semidirect
4. #21446169C2semidirect
5. #21446137C2semidirect
6. #21446082C2semidirect
7. #21446014C2semidirect
8. #21445750C2semidirect
9. #21445646C2semidirect
10. #21445583C2semidirect
19. C2×D4×Dic61C22semidirect
24. C2×#14519709C22semidirect
27. C2×#14519698C22semidirect
29. C2×#14519687C22semidirect
30. #16839665C22semidirect
31. #16839661C22semidirect
35. C4⋊C4×Dic61C22semidirect
37. #16839525C22semidirect
38. #16839524C22semidirect
39. #16839516C22semidirect
43. C61×#447C4semidirect
63. D4×Dic61D4semidirect
98. C22×D4×C61C2×C4semidirect
104. C2×D8×C61C2×C4semidirect
107. #121×C61C2×C4semidirect
110. #102×C61C2×C4semidirect
111. C22⋊C8×C61C2×C4semidirect
137. C2×D4×C61C22×C4semidirect
176. C2×C8×C61C22×C4semidirect
178. C4⋊C4×C61C22×C4semidirect
236. D4×C61C4×D4semidirect
300. C22×C61#442semidirect
354. #447Dic61semidirect
374. C61#21452semidirect
381. C22×D4C2×Dic61semidirect
396. C2×D8C2×Dic61semidirect
400. #121C2×Dic61semidirect
434. #102C2×Dic61semidirect
436. C22⋊C8C2×Dic61semidirect
468. C2×D4C22×Dic61semidirect
508. C2×C8C22×Dic61semidirect
516. C4⋊C4C22×Dic61semidirect
565. D4D4×Dic61semidirect
628. C22#16839874semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic612
C2×#168398742
#121×Dic612
#214461692
#214461372
#214460822
#214460142
#214457502
#214456462
#214455832
C2×C61×#4472
#2145261