Finite Group Databasebeta

← #29685654

Subgroups — #29685654

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×C4×D244C211yesmaximal, characteristic, complemented
2C2×#16839895C211yesmaximal, characteristic
3C2×#16839872C212yesmaximal, complemented
4#21446170C212yesmaximal, complemented
5#21446137C211yesmaximal, characteristic, complemented
6#21446079C212yesmaximal, complemented
7#21446011C211yesmaximal, characteristic, complemented
8#21445749C211yesmaximal, characteristic, complemented
9#21445647C212yesmaximal, complemented
10#21445581C211yesmaximal, characteristic, complemented
11C4×#121×C61C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×C4×D61—21no—
14C22×#11758199C2212yes—
15C2×C4×D244—28no—
16C2×C4×D244—21no—
17C2×C4×D244—21no—
18C2×C4×D244C2212yes—
19C2×C4×D244C2212yes—
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758198—21no—
22#16839895C418yescomplemented
23#16839872—28no—
24C2×#14519708C2212yescomplemented
25C2×#14519698C2212yescomplemented
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519688C2212yescomplemented
32#16839524C2212yescomplemented
33#16839524C2212yescomplemented
34#16839517C2212yescomplemented
35C2×#121×C61C2211yescharacteristic
36C2×C4×D4×C61—21no—
37C2×C4×D4×C61C2211yescharacteristic, complemented
38C22×C42×C61C2211yescharacteristic
39C61×#390C2211yescharacteristic, complemented
40C61×#389C2211yescharacteristic, complemented
41C61×#388—21no—
42C61×#386—21no—
43C61×#380C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic, complemented
45C4×C22⋊C4×C61—21no—
46C24×D61—21no—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—24no—
51C22×D244C2×C414yescomplemented
52C22×D244C2312yes—
53C23×Dic61—21no—
54C22×C4×D61—28no—
55C22×C4×D61—22no—
56C22×C4×D61—22no—
57C22×C4×D61—44no—
58C22×C4×D61—44no—
59C2×#11758199—24no—
60C2×#11758199—24no—
61C2×#11758199C2×C418yescomplemented
62C22×#92418C2312yes—
63C22×#92418C2312yes—
64C4×D244—28no—
65C4×D244—28no—
66C4×D244—24no—
67C4×D244—24no—
68C4×D244—24no—
69C4×D244D418yescomplemented
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—24no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—28no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76#14519708—24no—
77#14519698—24no—
78#14519697—24no—
79#14519697—24no—
80#14519697C2×C418yescomplemented
81C22⋊#92418—22no—
82C22⋊#92418C2×C414yescomplemented
83#14519688—24no—
84C22×D4×C61—21no—
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90#121×C61—22no—
91#121×C61C2×C414yescomplemented
92C4×D4×C61—24no—
93C4×D4×C61—44no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—24no—
104C2×C42×C61—21no—
105C2×C42×C61D411yescharacteristic
106C2×C42×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109#95×C61—21no—
110#95×C61—21no—
111#95×C61—21no—
112C24×C61—21no—
113C24×C61C2×D411yescharacteristic
114C23×D61—24no—
115C23×D61—24no—
116C23×D61—24no—
117C23×D61—22no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—42no—
121C23×D61—42no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61C22×C412yescomplemented
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—41no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61#4112yes—
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—24no—
152C22×Dic61—22no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C2×D244—24no—
156C2×D244—216no—
157C2×D244—24no—
158C2×D244—24no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—216no—
163C2×D244C22×C414yes—
164C2×D244C2×D414yes—
165C2×D244#4114yes—
166C2×C4×D61—28no—
167C2×C4×D61—28no—
168C2×C4×D61—432no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61—44no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61C22×C412yescomplemented
178C22⋊C4×C61—24no—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C42×C61—22no—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61—42no—
187C42×C61C2×D414yes—
188#11758199—216no—
189#11758198—28no—
190#11758198—28no—
191C4⋊Dic61—24no—
192C4⋊Dic61—22no—
193C4⋊Dic61C2×D414yes—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418C22×C414yescomplemented
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C4×D412yes—
212C23×C61AGL(1,ℤ/8)12yes—
213C23×C61AGL(1,ℤ/8)12yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—216no—
216C22×D61—24no—
217C22×D61—24no—
218C22×D61—24no—
219C22×D61—48no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61—48no—
223C22×D61—48no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—48no—
229D4×C61—24no—
230D4×C61—48no—
231D4×C61—42no—
232C2×C4×C61—28no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—44no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—84no—
257C2×C4×C61C4×D412yes—
258C2×C4×C61#12012yes—
259C2×C4×C61#12012yes—
260C2×C4×C61AGL(1,ℤ/8)12yes—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D412yes—
263C2×C4×C61C2×#4112yes—
264C2×C4×C61C2×#4111yescharacteristic
265C2×Dic61—24no—
266C2×Dic61—48no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269D244—216no—
270D244—24no—
271D244—24no—
272D244—24no—
273D244C4×D418yescomplemented
274C4×D61—416no—
275C4×D61—416no—
276#92418—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—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61#44214yescomplemented
293C22×C61C2×C4×D411yescharacteristic
294C22×C61C2×#12011yescharacteristic
295C22×C61#52511yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C61#57511yescharacteristic
300C2×D61—28no—
301C2×D61—432no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—44no—
305C2×D61—44no—
306C2×D61—44no—
307C2×D61—44no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—42no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61C2×C4×D412yes—
317C4×C61#52512yes—
318Dic61—44no—
319C22×C4×D4—611no—
320C2×#447—611no—
321C2×#442—612no—
322C4×#121D6111yescharacteristic, complemented
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—82no—
332C2×C61C4×#12011yescharacteristic
333C2×C61C2×#44211yescharacteristic
334C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
335C2×C61#300311yescharacteristic
336C2×C61#300611yescharacteristic
337C2×C61#310911yescharacteristic
338C2×C61#311411yescharacteristic
339D61—48no—
340C23×D4—611no—
341C24×C4—1221no—
342C22×D8—612no—
343C2×#121C2×D6111yescharacteristic
344C2×C4×D4—21no—
345C2×C4×D4—612no—
346C2×C4×D4—612no—
347C2×C4×D4—1228no—
348C2×C4×D4—1221no—
349C2×C4×D4—1221no—
350C2×C4×D4C2×D6111yescharacteristic, complemented
351C22×C4⋊C4—611no—
352C22×C22⋊C4—1221no—
353C22×C42C2×D6111yescharacteristic
354#447—618no—
355#442—1228no—
356C2×#106—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×#102—612no—
360C2×#102—612no—
361C2×C22⋊C8—611no—
362C2×C22⋊C8—611no—
363C2×#97—612no—
364#390C2×D6111yescharacteristic, complemented
365#389C2×D6111yescharacteristic, complemented
366#388—21no—
367#386—21no—
368#380C2×D6111yescharacteristic
369C4×C4⋊C4C2×D6111yescharacteristic, complemented
370C4×C22⋊C4—21no—
371#340—612no—
372#340—612no—
373#336—612no—
374C61#2145211yescharacteristic, complemented, Sylow
375C25—1221no—
376C22×D4—21no—
377C22×D4—614no—
378C22×D4—612no—
379C22×D4—1224no—
380C22×D4—1224no—
381C22×D4—1222no—
382C22×D4—1221no—
383C22×D4C22×D6111yescharacteristic
384C23×C4—21no—
385C23×C4—1228no—
386C23×C4—1221no—
387C23×C4—1222no—
388C23×C4—1222no—
389C23×C4—2444no—
390C23×C4—2444no—
391C23×C4C22⋊D6111yescharacteristic
392C23×C4C22⋊D6111yescharacteristic
393C23×C4C22×D6111yescharacteristic
394C2×D8—618no—
395C2×D8—1224no—
396C2×D8—1224no—
397C22×C8—612no—
398C22×C8—612no—
399#121—22no—
400#121C4×D6114yescomplemented
401C4×D4—24no—
402C4×D4—44no—
403C4×D4—618no—
404C4×D4—1224no—
405C4×D4—1224no—
406C4×D4—1228no—
407C4×D4—1228no—
408C4×D4—1224no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1224no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1228no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C42—24no—
425C2×C42—21no—
426C2×C42C22⋊D6111yescharacteristic
427C2×C42C22⋊D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430#106—1224no—
431#102—618no—
432#102—1224no—
433#102—1224no—
434#102—1224no—
435C22⋊C8—614no—
436C22⋊C8—1222no—
437#97—1224no—
438#95—21no—
439#95—21no—
440#95—21no—
441C24—21no—
442C24—1224no—
443C24—1224no—
444C24—1222no—
445C24—1224no—
446C24—2442no—
447C24—2442no—
448C24—2442no—
449C24—2442no—
450C24C2×C22⋊D6111yescharacteristic
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—614no—
458C2×D4—614no—
459C2×D4—614no—
460C2×D4—12216no—
461C2×D4—1224no—
462C2×D4—1224no—
463C2×D4—12216no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1224no—
468C2×D4C2×C4×D6112yescomplemented
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—1228no—
484C22×C4—1224no—
485C22×C4—1222no—
486C22×C4—1228no—
487C22×C4—24432no—
488C22×C4—2444no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2444no—
494C22×C4C2×C4×D6112yes—
495C22×C4#1175822212yes—
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502D8—12216no—
503C2×C8—614no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—1222no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—614no—
514C4⋊C4—1224no—
515C4⋊C4—1222no—
516C4⋊C4C2×C4×D6112yescomplemented
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—1228no—
523C22⋊C4—1228no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—42no—
528C42D4×D6114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—12216no—
541C23—1224no—
542C23—1224no—
543C23—1224no—
544C23—2448no—
545C23—2448no—
546C23—2448no—
547C23—2448no—
548C23—2448no—
549C23—2442no—
550C23—2442no—
551C23—2442no—
552C23—2442no—
553C23—2448no—
554C23C4×C22⋊D6112yes—
555C23#1451980212yes—
556C23#1451980312yes—
557C23C22×C22⋊D6111yescharacteristic
558D4—24no—
559D4—48no—
560D4—42no—
561D4—618no—
562D4—12216no—
563D4—1224no—
564D4—1224no—
565D4—1224no—
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—1224no—
592C2×C4—24416no—
593C2×C4—24416no—
594C2×C4—2448no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4C4×C22⋊D6112yes—
598C2×C4#1451974512yes—
599C2×C4#1451974512yes—
600C2×C4#1451980212yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6112yes—
603C2×C4#41×D6112yes—
604C2×C4C2×#1175822211yescharacteristic
605C8—1224no—
606C8—2442no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—1228no—
621C22—24432no—
622C22—2444no—
623C22—2444no—
624C22—2444no—
625C22—2444no—
626C22—2444no—
627C22—2444no—
628C22#1683987214yescomplemented
629C22C2×C4×C22⋊D6111yescharacteristic
630C22C2×#1451974511yescharacteristic
631C22#1684026911yescharacteristic
632C22C2×#1451980211yescharacteristic
633C22C2×#1451980211yescharacteristic
634C22C2×#1451980311yescharacteristic
635C22#1684056011yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—2444no—
645C4C4×D4×D6112yes—
646C4#1684027112yes—
647C2—24no—
648C2—82no—
649C2—2448no—
650C2C4×#1451974511yescharacteristic
651C2C2×#1683987211yescharacteristic
652C2C4×#1451980211yescharacteristic
653C2#2145504511yescharacteristic
654C2#2145506911yescharacteristic
655C2#2145590611yescharacteristic
656C2#2145595311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#121×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839895C2
cosocleC22×C4×C61C24
hypercenterC4×#121D61
hypocenterC61#21452
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C22×C4×D244C2semidirect
3. C2×#16839872C2semidirect
4. #21446170C2semidirect
5. #21446137C2semidirect
6. #21446079C2semidirect
7. #21446011C2semidirect
8. #21445749C2semidirect
9. #21445647C2semidirect
10. #21445581C2semidirect
11. C4×#121×C61C2semidirect
22. #16839895C4semidirect
24. C2×#14519708C22semidirect
25. C2×#14519698C22semidirect
31. C2×#14519688C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
34. #16839517C22semidirect
37. C2×C4×D4×C61C22semidirect
39. C61×#390C22semidirect
40. C61×#389C22semidirect
44. C4×C4⋊C4×C61C22semidirect
51. C22×D244C2×C4semidirect
61. C2×#11758199C2×C4semidirect
69. C4×D244D4semidirect
80. #14519697C2×C4semidirect
82. C22⋊#92418C2×C4semidirect
91. #121×C61C2×C4semidirect
128. C2×D4×C61C22×C4semidirect
177. C4⋊C4×C61C22×C4semidirect
199. C2×#92418C22×C4semidirect
273. D244C4×D4semidirect
292. C22×C61#442semidirect
322. C4×#121D61semidirect
350. C2×C4×D4C2×D61semidirect
364. #390C2×D61semidirect
365. #389C2×D61semidirect
369. C4×C4⋊C4C2×D61semidirect
374. C61#21452semidirect
400. #121C4×D61semidirect
468. C2×D4C2×C4×D61semidirect
516. C4⋊C4C2×C4×D61semidirect
628. C22#16839872semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D2442
C2×#168398952
C2×#168398722
#214461702
#214461372
#214460792
#214460112
#214457492
#214456472
#214455812
C4×#121×C612
#2145261