Finite Group Databasebeta

← #29680743

Subgroups — #29680743

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C2×#16839803C212yesmaximal, complemented
3C2×#16839724C211yesmaximal, characteristic
4#21448201C211yesmaximal, characteristic
5#21446014C211yesmaximal, characteristic, complemented
6#21446013C212yesmaximal, complemented
7#21446012C211yesmaximal, characteristic, complemented
8#21446006C211yesmaximal, characteristic, complemented
9#21446005C212yesmaximal, complemented
10#21445537C212yesmaximal, complemented
11C61×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D61—21no—
13C23×C4×D61C2211yescharacteristic
14C22×#11758189C2212yes—
15C2×#14519735C2211yescharacteristic
16C2×#14519735C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758196—21no—
19C2×C4⋊C4×D61—28no—
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C4⋊C4×D61C2212yes—
24#16839803—28no—
25C2×#14519710C2212yescomplemented
26C2×#14519709C2212yescomplemented
27C2×#14519708C2212yescomplemented
28#16839724C418yescomplemented
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yes—
31#16839624—21no—
32#16839622C2211yescharacteristic
33#16839619C2211yescharacteristic
34#16839619C2211yescharacteristic
35#16839617—21no—
36C22⋊C4×Dic61—21no—
37#16839609—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×C4⋊C4×C61C2211yescharacteristic
42C2×#107×C61C2212yescomplemented
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C23×Dic61—21no—
48C23×Dic61D411yescharacteristic
49C22×C4×D61—24no—
50C22×C4×D61—24no—
51C22×C4×D61—28no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—44no—
55C22×C4×D61—44no—
56C22×C4×D61C2×C414yes—
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61Q814yes—
60C22×C4×D61C2311yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758189—24no—
63C2×#11758189—24no—
64C2×#11758189C2×C418yescomplemented
65C22×#92418C2312yes—
66#14519735—24no—
67#14519735—24no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—28no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C4⋊C4×D61—28no—
77C4⋊C4×D61—28no—
78C4⋊C4×D61—24no—
79C4⋊C4×D61—24no—
80C4⋊C4×D61—24no—
81C4⋊C4×D61D418yes—
82C2×C22⋊Dic61—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89#14519710—24no—
90#14519709—24no—
91#14519708—24no—
92C22⋊#92418C2×C414yescomplemented
93#14519689C2×C418yescomplemented
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106#107×C61—24no—
107C22⋊C8×C61C2×C414yescomplemented
108#95×C61—21no—
109C24×C61C2×D411yescharacteristic
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—22no—
115C23×D61—22no—
116C23×D61C4⋊C414yes—
117C23×D61C2×D411yescharacteristic
118C23×D61C2×D411yescharacteristic
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×Q812yes—
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—24no—
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61C4⋊C414yes—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—216no—
149C2×C4×D61—216no—
150C2×C4×D61—28no—
151C2×C4×D61—28no—
152C2×C4×D61—432no—
153C2×C4×D61—44no—
154C2×C4×D61—44no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61C4⋊C414yes—
158C2×C4×D61C4⋊C414yes—
159C2×C4×D61C22×C412yes—
160C2×C4×D61C22×C412yes—
161C2×C4×D61C2×D414yes—
162C2×C4×D61C2×Q812yes—
163C2×C4×D61C2×Q812yes—
164C2×C4×D61#4114yes—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61C22×C414yescomplemented
168C4⋊C4×C61—24no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61C2×D414yes—
173C22⋊C4×C61—42no—
174C22⋊Dic61—42no—
175#11758198—42no—
176#11758198—42no—
177#11758198—42no—
178#11758198—42no—
179C4⋊Dic61—24no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61C2×D414yes—
184#11758196—28no—
185#11758196—28no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418C22×C414yescomplemented
191#11758189—216no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61C2×C4⋊C412yes—
197C23×C61AGL(1,ℤ/8)12yes—
198C23×C61#13712yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—28no—
201C22×D61—28no—
202C22×D61—28no—
203C22×D61—28no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—28no—
213C22×D61—28no—
214C22×D61—28no—
215C22×D61C2×C4⋊C412yes—
216C22×D61C2×C4⋊C412yes—
217C22×D61#12012yes—
218C22×D61#12312yes—
219C22×D61AGL(1,ℤ/8)12yes—
220C22×D61#13712yes—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—48no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C2×C4⋊C412yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C22×D412yes—
238C2×C4×C61C22×Q811yescharacteristic
239C2×C4×C61C2×#4112yes—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—28no—
243C2×Dic61—24no—
244C2×Dic61—48no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61C2×C4⋊C412yes—
254C2×Dic61C2×C4⋊C412yes—
255C2×Dic61#12012yes—
256C2×Dic61#12312yes—
257C4×D61—28no—
258C4×D61—24no—
259C4×D61—24no—
260C4×D61—28no—
261C4×D61—416no—
262C4×D61—416no—
263C4×D61C4×D418yes—
264C4×D61C22⋊Q818yes—
265C8×C61—24no—
266#92418—24no—
267#20844—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61#42814yescomplemented
275C22×C61C22×C4⋊C411yescharacteristic
276C22×C61C2×#12011yescharacteristic
277C22×C61C2×#12311yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61C2×#13711yescharacteristic
281C22×C61C2×#13711yescharacteristic
282C2×D61—232no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61#38514yes—
296C2×D61#42814yescomplemented
297C4×C61—22no—
298C4×C61—44no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61C2×C4×D412yes—
302C4×C61C2×C22⋊Q812yes—
303Dic61—22no—
304Dic61—44no—
305Dic61—81no—
306Dic61—81no—
307Dic61#38514yes—
308C23×C4⋊C4—611no—
309C2×#428—612no—
310C2×#407—611no—
311#2280—611no—
312#1853—611no—
313#1853—612no—
314#1853—611no—
315#1852—612no—
316#1852—611no—
317#1852D6111yescharacteristic, complemented
318#1736—612no—
319C2×C61—24no—
320C2×C61C2×#38511yescharacteristic
321C2×C61C2×#42811yescharacteristic
322C2×C61C2×#42811yescharacteristic
323C2×C61#300311yescharacteristic
324C2×C61#300411yescharacteristic
325C2×C61#309211yescharacteristic
326C2×C61#309211yescharacteristic
327D61—24no—
328D61—24no—
329C24×C4—611no—
330C24×C4—1221no—
331C22×M16—612no—
332C2×#117—611no—
333C2×#117—611no—
334C22×C4⋊C4—612no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4—612no—
337C22×C4⋊C4—1228no—
338C22×C4⋊C4—1221no—
339C22×C4⋊C4—1221no—
340C22×C4⋊C4—1221no—
341C22×C4⋊C4C2×D6111yescharacteristic
342#428—1228no—
343C2×#107—612no—
344C2×#107C2×D6112yescomplemented
345C2×#106—612no—
346C2×#106—612no—
347#407—618no—
348C2×C22⋊C8—611no—
349C2×C22⋊C8—612no—
350C2×C22⋊C8C2×D6111yescharacteristic
351#381—1221no—
352#380—611no—
353#380—611no—
354#380—611no—
355#380C2×D6111yescharacteristic
356#379—1221no—
357C4×C22⋊C4—1221no—
358C4×C22⋊C4—1221no—
359#340—612no—
360#340—612no—
361#340—612no—
362#340C2×D6112yescomplemented
363C61#2084411yescharacteristic, complemented, Sylow
364C25—611no—
365C23×C4—21no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—614no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—614no—
372C23×C4—611no—
373C23×C4—1224no—
374C23×C4—1224no—
375C23×C4—1222no—
376C23×C4—1222no—
377C23×C4—1228no—
378C23×C4—1221no—
379C23×C4—2444no—
380C23×C4—2444no—
381C23×C4C22×D6111yescharacteristic
382C2×M16—618no—
383C2×M16—1224no—
384C2×M16—1224no—
385C22×C8—612no—
386C22×C8C22×D6112yes—
387#117—1224no—
388#117—1224no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—612no—
392C2×C4⋊C4—618no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—1224no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1228no—
398C2×C4⋊C4—1228no—
399C2×C4⋊C4—1224no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1228no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1224no—
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6112yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416#107—24no—
417#107—1224no—
418#106—1224no—
419#106—1224no—
420C22⋊C8—614no—
421C22⋊C8—618no—
422C22⋊C8C4×D6114yescomplemented
423#95—21no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—611no—
428C24—611no—
429C24—614no—
430C24—1222no—
431C24—1222no—
432C24—1222no—
433C24—1222no—
434C24—1222no—
435C24—1222no—
436C24D4×D6111yescharacteristic
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—614no—
448C22×C4—612no—
449C22×C4—614no—
450C22×C4—611no—
451C22×C4—614no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—614no—
455C22×C4—614no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—12216no—
459C22×C4—12216no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1228no—
463C22×C4—1228no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—2444no—
473C22×C4—24432no—
474C22×C4—2444no—
475C22×C4—2444no—
476C22×C4—2444no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4C2×C4×D6112yes—
480C22×C4D4×D6111yescharacteristic
481C22×C4Q8×D6112yes—
482C22×C4C23×D6111yescharacteristic
483M16—12216no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—614no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8C2×C4×D6114yescomplemented
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—614no—
495C4⋊C4—1228no—
496C4⋊C4—1228no—
497C4⋊C4—1224no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4D4×D6114yes—
502C22⋊C4—42no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C22⋊C4—2442no—
508C42—2442no—
509C42—2442no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—612no—
515C23—612no—
516C23—612no—
517C23—612no—
518C23—612no—
519C23—612no—
520C23—1228no—
521C23—1228no—
522C23—1228no—
523C23—1228no—
524C23—1228no—
525C23—1228no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23—1221no—
532C23—1221no—
533C23—1221no—
534C23—1228no—
535C23C4⋊C4×D6112yes—
536C23#1451980112yes—
537C23#1451980912yes—
538C23C2×D4×D6111yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—618no—
557C2×C4—618no—
558C2×C4—612no—
559C2×C4—1228no—
560C2×C4—1228no—
561C2×C4—1228no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—1224no—
567C2×C4—24416no—
568C2×C4—24416no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2448no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4C4⋊C4×D6112yes—
579C2×C4C22×C4×D6111yescharacteristic
580C2×C4C2×D4×D6112yes—
581C2×C4C2×Q8×D6111yescharacteristic
582C2×C4#41×D6112yes—
583C8—24no—
584C8—1224no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—614no—
592C22—614no—
593C22—12232no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22—1222no—
598C22—1222no—
599C22—1222no—
600C22—1222no—
601C22—1222no—
602C22—1222no—
603C22—1222no—
604C22—1222no—
605C22—1222no—
606C22#1683980314yescomplemented
607C22C2×C4⋊C4×D6111yescharacteristic
608C22#120×D6111yescharacteristic
609C22#123×D6111yescharacteristic
610C22AGL(1,ℤ/8)×D6111yescharacteristic
611C22C2×#1451980111yescharacteristic
612C22#137×D6111yescharacteristic
613C22C2×#1451980911yescharacteristic
614C4—22no—
615C4—44no—
616C4—81no—
617C4—81no—
618C4—614no—
619C4—1222no—
620C4—4881no—
621C4—4881no—
622C4—2444no—
623C4C4×D4×D6112yes—
624C4C22⋊Q8×D6112yes—
625C2—24no—
626C2—1224no—
627C2—1224no—
628C2#385×D6111yescharacteristic
629C2#428×D6111yescharacteristic
630C2C2×#1683980311yescharacteristic
631C2#2145504311yescharacteristic
632C2#2145505411yescharacteristic
633C2#2145573811yescharacteristic
634C2#2145574611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1852C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1852D61
hypocenterC61#20844
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#16839803C2semidirect
5. #21446014C2semidirect
6. #21446013C2semidirect
7. #21446012C2semidirect
8. #21446006C2semidirect
9. #21446005C2semidirect
10. #21445537C2semidirect
11. C61×#1852C2semidirect
25. C2×#14519710C22semidirect
26. C2×#14519709C22semidirect
27. C2×#14519708C22semidirect
28. #16839724C4semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
42. C2×#107×C61C22semidirect
45. C61×#340C22semidirect
64. C2×#11758189C2×C4semidirect
92. C22⋊#92418C2×C4semidirect
93. #14519689C2×C4semidirect
107. C22⋊C8×C61C2×C4semidirect
167. C2×C8×C61C22×C4semidirect
190. C2×#92418C22×C4semidirect
274. C22×C61#428semidirect
296. C2×D61#428semidirect
317. #1852D61semidirect
344. C2×#107C2×D61semidirect
362. #340C2×D61semidirect
363. C61#20844semidirect
422. C22⋊C8C4×D61semidirect
489. C2×C8C2×C4×D61semidirect
606. C22#16839803semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D612
C2×#168398032
C2×#168397242
#214482012
#214460142
#214460132
#214460122
#214460062
#214460052
#214455372
C61×#18522
#2084461