Finite Group Databasebeta

← #26221932

Subgroups — #26221932

Order 12032.

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×#15689274C211yesmaximal, characteristic
2C2×#15689223C212yesmaximal, complemented
3C4×#13094003C211yesmaximal, characteristic, complemented
4#20243722C212yesmaximal, complemented
5#20243695C211yesmaximal, characteristic, complemented
6#20243635C212yesmaximal, complemented
7#20243576C211yesmaximal, characteristic, complemented
8#20243319C211yesmaximal, characteristic
9#20243213C212yesmaximal, complemented
10#20243150C211yesmaximal, characteristic, complemented
11C22×C4×D4×C47C211yesmaximal, characteristic
12C22×#10625335C2212yes—
13C2×#13094003C2211yescharacteristic
14C2×C4×Dic94C2211yescharacteristic, complemented
15#15689274C418yescomplemented
16#15689223—28no—
17C2×#13093957C2212yescomplemented
18C2×#13093947C2212yescomplemented
19C2×#13093945C2212yes—
20C2×#13093945C2212yes—
21C2×#13093945C2212yes—
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24C2×#13093935C2212yescomplemented
25#15689029C2211yescharacteristic, complemented
26#15689021C2211yescharacteristic, complemented
27#15689007—21no—
28#15688996—21no—
29#15688983C2211yescharacteristic
30C4×Dic47⋊C4—21no—
31C4×C4⋊Dic47C2211yescharacteristic, complemented
32C4×C22⋊Dic47—21no—
33#15688885C2212yescomplemented
34#15688885C2212yescomplemented
35#15688878C2212yescomplemented
36C23×D4×C47C2211yescharacteristic
37C24×C4×C47—21no—
38C2×C4×D4×C47—28no—
39C2×C4×D4×C47—21no—
40C2×C4×D4×C47—21no—
41C2×C4×D4×C47C2212yes—
42C2×C4×D4×C47C2212yes—
43C22×C4⋊C4×C47C2211yescharacteristic
44C22×C22⋊C4×C47—21no—
45C22×C42×C47C2211yescharacteristic
46C22×Dic94C2311yescharacteristic
47C2×#10625335—24no—
48C2×#10625335—24no—
49C2×#10625335C2×C418yescomplemented
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2312yes—
52#13094003—22no—
53#13094003C2×C414yescomplemented
54C4×Dic94—24no—
55C2×C4⋊Dic47C2311yescharacteristic
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2311yescharacteristic
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4—21no—
60C2×Dic47⋊C4—21no—
61C2×C22⋊Dic47—21no—
62C2×C22⋊Dic47—21no—
63C2×C4×Dic47—21no—
64C2×C4×Dic47—21no—
65#13093957—24no—
66#13093947—24no—
67#13093945—24no—
68#13093945—24no—
69#13093945C2×C418yescomplemented
70C22⋊(C47⋊C8)—22no—
71C22⋊(C47⋊C8)C2×C414yescomplemented
72#13093935—24no—
73C47⋊#95—21no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C25×C47—21no—
79C22×D4×C47—24no—
80C22×D4×C47—24no—
81C22×D4×C47—21no—
82C22×D4×C47—22no—
83C22×D4×C47C2×C414yes—
84C22×D4×C47C2312yes—
85C23×C4×C47—28no—
86C23×C4×C47—21no—
87C23×C4×C47—22no—
88C23×C4×C47—22no—
89C23×C4×C47—44no—
90C23×C4×C47—44no—
91C23×C4×C47D411yescharacteristic
92C23×C4×C47D411yescharacteristic
93C23×C4×C47C2311yescharacteristic
94C4×D4×C47—28no—
95C4×D4×C47—28no—
96C4×D4×C47—24no—
97C4×D4×C47—24no—
98C4×D4×C47—24no—
99C4×D4×C47D418yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—24no—
102C2×C4⋊C4×C47C2312yes—
103C2×C22⋊C4×C47—28no—
104C2×C22⋊C4×C47—22no—
105C2×C22⋊C4×C47—22no—
106C2×C42×C47—24no—
107C2×C42×C47D411yescharacteristic
108C2×C42×C47D411yescharacteristic
109C2×C42×C47C2311yescharacteristic
110C2×C42×C47C2311yescharacteristic
111C24×C47—24no—
112C24×C47—24no—
113C24×C47—24no—
114C24×C47—22no—
115C24×C47—42no—
116C24×C47—42no—
117C24×C47—42no—
118C24×C47—42no—
119C24×C47C2×D411yescharacteristic
120C2×D4×C47—24no—
121C2×D4×C47—24no—
122C2×D4×C47—216no—
123C2×D4×C47—216no—
124C2×D4×C47—24no—
125C2×D4×C47—22no—
126C2×D4×C47—22no—
127C2×D4×C47—22no—
128C2×D4×C47C22×C414yes—
129C2×D4×C47C2×D414yes—
130C2×D4×C47#4114yes—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—24no—
137C22×C4×C47—28no—
138C22×C4×C47—22no—
139C22×C4×C47—22no—
140C22×C4×C47—22no—
141C22×C4×C47—28no—
142C22×C4×C47—41no—
143C22×C4×C47—432no—
144C22×C4×C47—44no—
145C22×C4×C47—44no—
146C22×C4×C47—44no—
147C22×C4×C47—42no—
148C22×C4×C47—42no—
149C22×C4×C47—44no—
150C22×C4×C47C22×C412yes—
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47#4112yes—
157C22×C4×C47C2411yescharacteristic
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47—21no—
161C22×Dic47—21no—
162C2×Dic94—22no—
163C2×Dic94—22no—
164C2×Dic94—21no—
165C2×Dic94C22×C412yescomplemented
166C4⋊C4×C47—24no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47C2×D414yes—
169C22⋊C4×C47—28no—
170C22⋊C4×C47—28no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—22no—
174C42×C47C2×D414yes—
175C22⋊Dic47—24no—
176C22⋊Dic47—42no—
177#10625335—216no—
178C4⋊Dic47—21no—
179C4⋊Dic47—22no—
180C4⋊Dic47—22no—
181C4⋊Dic47C22×C412yescomplemented
182Dic47⋊C4—24no—
183Dic47⋊C4—42no—
184Dic47⋊C4—42no—
185C4×Dic47—42no—
186C4×Dic47—42no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8—22no—
192C2×C47⋊C8C22×C414yescomplemented
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—216no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—24no—
201C23×C47—48no—
202C23×C47—48no—
203C23×C47—48no—
204C23×C47—48no—
205C23×C47—48no—
206C23×C47—42no—
207C23×C47—42no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47—48no—
211C23×C47C4×D412yes—
212C23×C47AGL(1,ℤ/8)12yes—
213C23×C47#13712yes—
214C23×C47C22×D411yescharacteristic
215D4×C47—216no—
216D4×C47—24no—
217D4×C47—24no—
218D4×C47—24no—
219D4×C47C4×D418yes—
220C2×C4×C47—28no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—21no—
232C2×C4×C47—24no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—44no—
236C2×C4×C47—416no—
237C2×C4×C47—416no—
238C2×C4×C47—48no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47C4×D412yes—
242C2×C4×C47#12012yes—
243C2×C4×C47#12012yes—
244C2×C4×C47AGL(1,ℤ/8)12yes—
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C22×D412yes—
247C2×C4×C47C2×#4112yes—
248C2×C4×C47C2×#4111yescharacteristic
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261Dic94—24no—
262Dic94—42no—
263C47⋊C8—24no—
264C47⋊C8—42no—
265#21453—471nomaximal, Sylow
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—28no—
273C22×C47—432no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—44no—
278C22×C47—44no—
279C22×C47—44no—
280C22×C47#44014yescomplemented
281C22×C47C2×C4×D411yescharacteristic
282C22×C47C2×#12011yescharacteristic
283C22×C47#52511yescharacteristic
284C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C47C2×#13711yescharacteristic
287C22×C47#57511yescharacteristic
288C4×C47—22no—
289C4×C47—22no—
290C4×C47—22no—
291C4×C47—22no—
292C4×C47—44no—
293C4×C47—41no—
294C4×C47C2×C4×D412yes—
295C4×C47#52512yes—
296Dic47—42no—
297Dic47—42no—
298Dic47—81no—
299Dic47—81no—
300C22×C4×D4D4711yescharacteristic
301C2×#450—471no—
302C2×#440—472no—
303C4×C22⋊Q8—471no—
304#1877—472no—
305#1872—471no—
306#1862—472no—
307#1852—471no—
308#1786—471no—
309#1762—472no—
310#1749—471no—
311C2×C47—24no—
312C2×C47—48no—
313C2×C47C4×#12011yescharacteristic
314C2×C47C2×#44011yescharacteristic
315C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C47#300311yescharacteristic
317C2×C47#300711yescharacteristic
318C2×C47#311011yescharacteristic
319C2×C47#311411yescharacteristic
320C23×D4C2×D4711yescharacteristic
321C24×C4—21no—
322C22×SD16—472no—
323C2×C22⋊Q8—471no—
324C2×C4×Q8—471no—
325C2×C4×D4—28no—
326C2×C4×D4—21no—
327C2×C4×D4—21no—
328C2×C4×D4C2×D4712yes—
329C2×C4×D4C2×D4712yes—
330C22×C4⋊C4C2×D4711yescharacteristic
331C22×C22⋊C4—21no—
332C22×C42C2×D4711yescharacteristic
333#450—478no—
334#440—948no—
335C2×#107—472no—
336C2×#103—472no—
337C2×#102—472no—
338C2×#102—472no—
339C2×#102—472no—
340C2×C22⋊C8—471no—
341C2×C22⋊C8—471no—
342C2×#97—472no—
343#391—471no—
344#389—471no—
345#387—941no—
346#385—941no—
347#380—471no—
348C4×C4⋊C4—471no—
349C4×C4⋊C4—941no—
350C4×C22⋊C4—941no—
351#340—472no—
352#340—472no—
353#336—472no—
354C47#2145311yescharacteristic, complemented, Sylow
355C25—21no—
356C22×Q8—471no—
357C22×D4—24no—
358C22×D4—24no—
359C22×D4—22no—
360C22×D4—21no—
361C22×D4C4×D4714yes—
362C22×D4C22×D4712yes—
363C23×C4—28no—
364C23×C4—21no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—44no—
368C23×C4—44no—
369C23×C4C22⋊D4711yescharacteristic
370C23×C4C22⋊D4711yescharacteristic
371C23×C4C22×D4711yescharacteristic
372C2×SD16—478no—
373C2×SD16—944no—
374C2×SD16—944no—
375C22×C8—472no—
376C22×C8—472no—
377C22⋊Q8—474no—
378C22⋊Q8—942no—
379C4×Q8—944no—
380C4×D4—28no—
381C4×D4—28no—
382C4×D4—24no—
383C4×D4—24no—
384C4×D4—24no—
385C4×D4C22⋊D4718yes—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—471no—
389C2×C4⋊C4—471no—
390C2×C4⋊C4—471no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4—941no—
393C2×C4⋊C4—941no—
394C2×C4⋊C4C22×D4712yes—
395C2×C22⋊C4—28no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C42—24no—
401C2×C42—941no—
402C2×C42—941no—
403C2×C42C22⋊D4711yescharacteristic
404C2×C42C22⋊D4711yescharacteristic
405C2×C42C22×D4711yescharacteristic
406C2×C42C22×D4711yescharacteristic
407#107—944no—
408#103—944no—
409#102—478no—
410#102—944no—
411#102—944no—
412C22⋊C8—474no—
413C22⋊C8—942no—
414#97—944no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95—941no—
419#95—941no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24C2×C22⋊D4711yescharacteristic
429C2×Q8—472no—
430C2×Q8—941no—
431C2×Q8—942no—
432C2×Q8—942no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—216no—
436C2×D4—216no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4C2×C4×D4714yes—
442C2×D4#1062535014yes—
443C2×D4C2×C22⋊D4714yes—
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—941no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4C2×C4×D4712yes—
468C22×C4#1062535012yes—
469C22×C4C2×C22⋊D4711yescharacteristic
470C22×C4C2×C22⋊D4711yescharacteristic
471C22×C4C2×C22⋊D4711yescharacteristic
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C2×C22⋊D4711yescharacteristic
474C22×C4C23×D4711yescharacteristic
475SD16—9416no—
476C2×C8—474no—
477C2×C8—942no—
478C2×C8—942no—
479C2×C8—942no—
480C2×C8—942no—
481C2×C8—942no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—472no—
485C4⋊C4—944no—
486C4⋊C4—941no—
487C4⋊C4—942no—
488C4⋊C4—942no—
489C4⋊C4—1882no—
490C4⋊C4—1882no—
491C4⋊C4C2×C22⋊D4714yes—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—944no—
495C22⋊C4—1882no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—1882no—
500C42—1882no—
501C42C2×C22⋊D4714yes—
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—48no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C4×C22⋊D4712yes—
521C23#1309405012yes—
522C23#1309405412yes—
523C23C22×C22⋊D4711yescharacteristic
524Q8—944no—
525Q8—1882no—
526D4—216no—
527D4—24no—
528D4—24no—
529D4—24no—
530D4C4×C22⋊D4718yes—
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—942no—
553C2×C4—942no—
554C2×C4—1881no—
555C2×C4—1881no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1881no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4—1881no—
562C2×C4—1881no—
563C2×C4—1881no—
564C2×C4C4×C22⋊D4712yes—
565C2×C4#1309399312yes—
566C2×C4#1309399312yes—
567C2×C4#1309405012yes—
568C2×C4C22×C4×D4711yescharacteristic
569C2×C4C22×C22⋊D4712yes—
570C2×C4C2×#1062535012yes—
571C2×C4C2×#1062535011yescharacteristic
572C8—944no—
573C8—1882no—
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#1568922314yescomplemented
589C22C2×C4×C22⋊D4711yescharacteristic
590C22C2×#1309399311yescharacteristic
591C22#1568963411yescharacteristic
592C22C2×#1309405011yescharacteristic
593C22C2×#1309405011yescharacteristic
594C22C2×#1309405411yescharacteristic
595C22#1568992111yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—44no—
601C4—41no—
602C4—3761no—
603C4—3761no—
604C4—1882no—
605C4—1882no—
606C4C2×C4×C22⋊D4712yes—
607C4#1568963412yes—
608C2—24no—
609C2—48no—
610C2C4×#1309399311yescharacteristic
611C2C2×#1568922311yescharacteristic
612C2C4×#1309405011yescharacteristic
613C2#2025261511yescharacteristic
614C2#2025264811yescharacteristic
615C2#2025348711yescharacteristic
616C2#2025352611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C4×D4×C47C2
socleC23×C47C22×D4
prime radicalC23×D4×C47C22
cosocleC22×C4×C47C24
hypercenterC22×C4×D4D47
hypocenterC47#21453
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#15689223C2semidirect
3. C4×#13094003C2semidirect
4. #20243722C2semidirect
5. #20243695C2semidirect
6. #20243635C2semidirect
7. #20243576C2semidirect
9. #20243213C2semidirect
10. #20243150C2semidirect
14. C2×C4×Dic94C22semidirect
15. #15689274C4semidirect
17. C2×#13093957C22semidirect
18. C2×#13093947C22semidirect
24. C2×#13093935C22semidirect
25. #15689029C22semidirect
26. #15689021C22semidirect
31. C4×C4⋊Dic47C22semidirect
33. #15688885C22semidirect
34. #15688885C22semidirect
35. #15688878C22semidirect
49. C2×#10625335C2×C4semidirect
53. #13094003C2×C4semidirect
69. #13093945C2×C4semidirect
71. C22⋊(C47⋊C8)C2×C4semidirect
165. C2×Dic94C22×C4semidirect
181. C4⋊Dic47C22×C4semidirect
192. C2×C47⋊C8C22×C4semidirect
280. C22×C47#440semidirect
354. C47#21453semidirect
588. C22#15689223semidirect

Maximal subgroups

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

subgroupindex
C2×#156892742
C2×#156892232
C4×#130940032
#202437222
#202436952
#202436352
#202435762
#202433192
#202432132
#202431502
C22×C4×D4×C472
#2145347