Finite Group Databasebeta

← #26182155

Subgroups — #26182155

Order 12032.

1738 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689087C212yesmaximal, complemented
2#20241726C214yesmaximal, complemented
3#20241718C212yesmaximal, complemented
4#20241715C212yesmaximal, complemented
5#20241706C214yesmaximal, complemented
6C22×C4×D4×C47C211yesmaximal, characteristic
7C22×#10625329—22no—
8#15689087—22no—
9#15689087—22no—
10#15689087C2214yes—
11#15689087C2214yescomplemented
12C2×C22⋊(C47⋊C8)—22no—
13C2×C22⋊(C47⋊C8)C2214yescomplemented
14#15688849C2218yescomplemented
15#15688847C2218yescomplemented
16C23×D4×C47C411yescharacteristic
17C24×C4×C47C411yescharacteristic
18C24×C4×C47C2211yescharacteristic
19C2×C4×D4×C47—22no—
20C2×C4×D4×C47—22no—
21C2×C4×D4×C47—22no—
22C2×C4×D4×C47—22no—
23C2×C4×D4×C47C412yes—
24C2×C4×D4×C47C412yes—
25C2×C4×D4×C47C2212yes—
26C2×C4×D4×C47C2212yes—
27C22×C4⋊C4×C47C2211yescharacteristic
28C22×C22⋊C4×C47C411yescharacteristic
29C22×C22⋊C4×C47C2211yescharacteristic
30C22×C42×C47C411yescharacteristic
31C2×#10625329—24no—
32C2×#10625329—24no—
33C2×#10625329—48no—
34C22×C47⋊C8—24no—
35C22⋊(C47⋊C8)—24no—
36C22⋊(C47⋊C8)—24no—
37C22⋊(C47⋊C8)—24no—
38C22⋊(C47⋊C8)C2318yes—
39C25×C47C2×C411yescharacteristic
40C25×C47C2×C411yescharacteristic
41C22×D4×C47—22no—
42C22×D4×C47—22no—
43C22×D4×C47—28no—
44C22×D4×C47C2×C412yes—
45C22×D4×C47C2×C412yes—
46C23×C4×C47—21no—
47C23×C4×C47—21no—
48C23×C4×C47—21no—
49C23×C4×C47—21no—
50C23×C4×C47—44no—
51C23×C4×C47—44no—
52C23×C4×C47C2×C411yescharacteristic
53C23×C4×C47C2×C411yescharacteristic
54C23×C4×C47C2×C411yescharacteristic
55C23×C4×C47C2×C411yescharacteristic
56C23×C4×C47C2×C412yes—
57C23×C4×C47C2×C412yes—
58C23×C4×C47C2×C411yescharacteristic
59C23×C4×C47C2×C411yescharacteristic
60C23×C4×C47D412yes—
61C23×C4×C47D412yes—
62C23×C4×C47D412yes—
63C23×C4×C47D412yes—
64C23×C4×C47C2311yescharacteristic
65C23×C4×C47C2311yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C4×D4×C47—24no—
68C4×D4×C47—24no—
69C4×D4×C47—24no—
70C4×D4×C47—24no—
71C4×D4×C47—24no—
72C4×D4×C47—24no—
73C4×D4×C47—24no—
74C4×D4×C47—24no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47—22no—
77C2×C4⋊C4×C47C2×C412yes—
78C2×C4⋊C4×C47C2312yes—
79C2×C22⋊C4×C47—22no—
80C2×C22⋊C4×C47—22no—
81C2×C22⋊C4×C47—22no—
82C2×C22⋊C4×C47—22no—
83C2×C22⋊C4×C47C2×C412yes—
84C2×C22⋊C4×C47C2×C412yes—
85C2×C22⋊C4×C47C2×C412yes—
86C2×C22⋊C4×C47C2312yes—
87C2×C42×C47—21no—
88C2×C42×C47—21no—
89C2×C42×C47—21no—
90C2×C42×C47—21no—
91C2×C42×C47C2×C412yes—
92C2×C42×C47C2×C412yes—
93C24×C47—21no—
94C24×C47—21no—
95C24×C47—24no—
96C24×C47—24no—
97C24×C47—42no—
98C24×C47—42no—
99C24×C47—42no—
100C24×C47—42no—
101C24×C47C22⋊C412yes—
102C24×C47C22⋊C412yes—
103C24×C47C22×C411yescharacteristic
104C24×C47C22×C411yescharacteristic
105C24×C47C22×C412yes—
106C24×C47C22×C411yescharacteristic
107C2×D4×C47—216no—
108C2×D4×C47—216no—
109C2×D4×C47—24no—
110C2×D4×C47—24no—
111C2×D4×C47—24no—
112C2×D4×C47—24no—
113C2×D4×C47—24no—
114C2×D4×C47M1618yescomplemented
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—24no—
118C22×C4×C47—28no—
119C22×C4×C47—24no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—44no—
131C22×C4×C47—44no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—44no—
135C22×C4×C47—42no—
136C22×C4×C47—44no—
137C22×C4×C47—44no—
138C22×C4×C47—44no—
139C22×C4×C47—44no—
140C22×C4×C47—44no—
141C22×C4×C47—44no—
142C22×C4×C47—42no—
143C22×C4×C47—42no—
144C22×C4×C47C22⋊C411yescharacteristic
145C22×C4×C47C22⋊C411yescharacteristic
146C22×C4×C47C22⋊C411yescharacteristic
147C22×C4×C47C22⋊C412yes—
148C22×C4×C47C22⋊C412yes—
149C22×C4×C47C22⋊C412yes—
150C22×C4×C47C22⋊C412yes—
151C22×C4×C47C22⋊C411yescharacteristic
152C22×C4×C47C22×C411yescharacteristic
153C22×C4×C47C22×C411yescharacteristic
154C22×C4×C47C22×C412yes—
155C22×C4×C47C22×C412yes—
156C22×C4×C47C22×C412yes—
157C22×C4×C47C22×C411yescharacteristic
158C22×C4×C47C2×D412yes—
159C22×C4×C47C2×D412yes—
160C22×C4×C47C2×D412yes—
161C22×C4×C47C2×D412yes—
162C22×C4×C47C2×D412yes—
163C22×C4×C47C2×D412yes—
164C22×C4×C47C2411yescharacteristic
165C4⋊C4×C47—24no—
166C4⋊C4×C47—24no—
167C22⋊C4×C47—24no—
168C22⋊C4×C47—24no—
169C22⋊C4×C47—24no—
170C22⋊C4×C47—24no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—22no—
174C42×C47—22no—
175#10625329—48no—
176#10625329—48no—
177C2×C47⋊C8—28no—
178C2×C47⋊C8—48no—
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—24no—
183C23×C47—24no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—24no—
188C23×C47—24no—
189C23×C47—24no—
190C23×C47—24no—
191C23×C47—42no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—42no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—416no—
201C23×C47—416no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47C2×C22⋊C412yes—
206C23×C47C2×C22⋊C412yes—
207C23×C47C2×C22⋊C411yescharacteristic
208C23×C47C2×C22⋊C411yescharacteristic
209C23×C47C2×M1614yes—
210C23×C47C2×M1614yescomplemented
211C23×C47C23×C411yescharacteristic
212D4×C47—216no—
213D4×C47—216no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—24no—
219C2×C4×C47—22no—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—22no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—41no—
233C2×C4×C47—42no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—41no—
237C2×C4×C47—44no—
238C2×C4×C47—44no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—44no—
244C2×C4×C47—44no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47#9914yes—
248C2×C4×C47C2×C22⋊C411yescharacteristic
249C2×C4×C47C2×C22⋊C411yescharacteristic
250C2×C4×C47C2×C22⋊C412yes—
251C2×C4×C47C2×C22⋊C411yescharacteristic
252C2×C4×C47C2×C22⋊C411yescharacteristic
253C2×C4×C47C2×M1614yescomplemented
254C2×C4×C47C22×D412yes—
255C47⋊C8—48no—
256#17362—471nomaximal, Sylow
257C22×C47—28no—
258C22×C47—21no—
259C22×C47—21no—
260C22×C47—21no—
261C22×C47—21no—
262C22×C47—22no—
263C22×C47—21no—
264C22×C47—21no—
265C22×C47—22no—
266C22×C47—22no—
267C22×C47—24no—
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—44no—
271C22×C47—48no—
272C22×C47—48no—
273C22×C47—44no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—48no—
277C22×C47—48no—
278C22×C47—42no—
279C22×C47—42no—
280C22×C47#40714yes—
281C22×C47C2×#9912yes—
282C22×C47#41312yes—
283C22×C47C22×C22⋊C411yescharacteristic
284C22×C47C22×M1612yes—
285C4×C47—24no—
286C4×C47—22no—
287C4×C47—21no—
288C4×C47—21no—
289C4×C47—22no—
290C4×C47—42no—
291C4×C47—42no—
292C4×C47C2×#9912yes—
293C4×C47C2×#9912yes—
294C22×C4×D4D4711yescharacteristic
295C2×#407—472no—
296#1465—474no—
297#1463—472no—
298#1462—472no—
299#1459—474no—
300C2×C47—22no—
301C2×C47—22no—
302C2×C47—22no—
303C2×C47—24no—
304C2×C47—42no—
305C2×C47—44no—
306C2×C47C2×#40711yescharacteristic
307C2×C47C22×#9911yescharacteristic
308C2×C47C2×#41311yescharacteristic
309C23×D4Dic4711yescharacteristic
310C24×C4Dic4711yescharacteristic
311C24×C4C2×D4711yescharacteristic
312C22×M16—942no—
313C2×C4×D4—22no—
314C2×C4×D4—22no—
315C2×C4×D4—22no—
316C2×C4×D4—22no—
317C2×C4×D4Dic4712yes—
318C2×C4×D4Dic4712yes—
319C2×C4×D4C2×D4712yes—
320C2×C4×D4C2×D4712yes—
321C22×C4⋊C4C2×D4711yescharacteristic
322C22×C22⋊C4Dic4711yescharacteristic
323C22×C22⋊C4C2×D4711yescharacteristic
324C22×C42Dic4711yescharacteristic
325#407—474no—
326#407—474no—
327#407—942no—
328#407—942no—
329C2×C22⋊C8—474no—
330C2×C22⋊C8—942no—
331#324—478no—
332#323—478no—
333C47#1736211yescharacteristic, complemented, Sylow
334C25C2×Dic4711yescharacteristic
335C25C2×Dic4711yescharacteristic
336C22×D4—22no—
337C22×D4—22no—
338C22×D4—28no—
339C22×D4C2×Dic4712yes—
340C22×D4C2×Dic4712yes—
341C23×C4—21no—
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—44no—
346C23×C4—44no—
347C23×C4C2×Dic4711yescharacteristic
348C23×C4C2×Dic4711yescharacteristic
349C23×C4C2×Dic4711yescharacteristic
350C23×C4C2×Dic4711yescharacteristic
351C23×C4C2×Dic4712yes—
352C23×C4C2×Dic4712yes—
353C23×C4C2×Dic4711yescharacteristic
354C23×C4C2×Dic4711yescharacteristic
355C23×C4C22⋊D4712yes—
356C23×C4C22⋊D4712yes—
357C23×C4C22⋊D4712yes—
358C23×C4C22⋊D4712yes—
359C23×C4C22×D4711yescharacteristic
360C23×C4C22×D4711yescharacteristic
361C23×C4C22×D4711yescharacteristic
362C2×M16—944no—
363C2×M16—944no—
364C2×M16—1888no—
365C22×C8—944no—
366C4×D4—24no—
367C4×D4—24no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—24no—
371C4×D4—24no—
372C4×D4—24no—
373C4×D4—24no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4C2×Dic4712yes—
377C2×C4⋊C4C22×D4712yes—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C2×Dic4712yes—
383C2×C22⋊C4C2×Dic4712yes—
384C2×C22⋊C4C2×Dic4712yes—
385C2×C22⋊C4C22×D4712yes—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42C2×Dic4712yes—
391C2×C42C2×Dic4712yes—
392C22⋊C8—478no—
393C22⋊C8—944no—
394C22⋊C8—944no—
395C22⋊C8—944no—
396C24—21no—
397C24—21no—
398C24—24no—
399C24—24no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22⋊Dic4712yes—
405C24C22⋊Dic4712yes—
406C24C22×Dic4711yescharacteristic
407C24C22×Dic4711yescharacteristic
408C24C22×Dic4712yes—
409C24C22×Dic4711yescharacteristic
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4#1062532918yescomplemented
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4C22⋊Dic4711yescharacteristic
448C22×C4C22⋊Dic4711yescharacteristic
449C22×C4C22⋊Dic4712yes—
450C22×C4C22⋊Dic4711yescharacteristic
451C22×C4C22⋊Dic4712yes—
452C22×C4C22⋊Dic4712yes—
453C22×C4C22⋊Dic4712yes—
454C22×C4C22⋊Dic4711yescharacteristic
455C22×C4C22×Dic4711yescharacteristic
456C22×C4C22×Dic4711yescharacteristic
457C22×C4C22×Dic4712yes—
458C22×C4C22×Dic4712yes—
459C22×C4C22×Dic4712yes—
460C22×C4C22×Dic4711yescharacteristic
461C22×C4C2×C22⋊D4712yes—
462C22×C4C2×C22⋊D4712yes—
463C22×C4C2×C22⋊D4712yes—
464C22×C4C2×C22⋊D4712yes—
465C22×C4C2×C22⋊D4712yes—
466C22×C4C2×C22⋊D4712yes—
467C22×C4C23×D4711yescharacteristic
468M16—1888no—
469M16—1888no—
470C2×C8—948no—
471C2×C8—1888no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—416no—
504C23—416no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C2×C22⋊Dic4712yes—
509C23C2×C22⋊Dic4712yes—
510C23C2×C22⋊Dic4711yescharacteristic
511C23C2×C22⋊Dic4711yescharacteristic
512C23C2×#1062532914yes—
513C23C2×#1062532914yescomplemented
514C23C23×Dic4711yescharacteristic
515D4—216no—
516D4—216no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4#1309393914yes—
551C2×C4C2×C22⋊Dic4711yescharacteristic
552C2×C4C2×C22⋊Dic4711yescharacteristic
553C2×C4C2×C22⋊Dic4712yes—
554C2×C4C2×C22⋊Dic4711yescharacteristic
555C2×C4C2×C22⋊Dic4711yescharacteristic
556C2×C4C2×#1062532914yescomplemented
557C2×C4C22×C22⋊D4712yes—
558C8—1888no—
559C22—28no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—48no—
574C22—48no—
575C22—44no—
576C22—48no—
577C22—48no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22#1568908714yes—
583C22C2×#1309393912yes—
584C22#1568910712yes—
585C22C22×C22⋊Dic4711yescharacteristic
586C22C22×#1062532912yes—
587C4—24no—
588C4—22no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4C2×#1309393912yes—
595C4C2×#1309393912yes—
596C2—22no—
597C2—22no—
598C2—22no—
599C2—24no—
600C2—42no—
601C2—44no—
602C2C2×#1568908711yescharacteristic
603C2C22×#1309393911yescharacteristic
604C2C2×#1568910711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×C4×D4×C47C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D4×C47C4
cosocleC22×C4×C47C24
hypercenterC22×C4×D4D47
hypocenterC47#17362
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15689087C2semidirect
2. #20241726C2semidirect
3. #20241718C2semidirect
4. #20241715C2semidirect
5. #20241706C2semidirect
11. #15689087C22semidirect
13. C2×C22⋊(C47⋊C8)C22semidirect
14. #15688849C22semidirect
15. #15688847C22semidirect
114. C2×D4×C47M16semidirect
210. C23×C47C2×M16semidirect
253. C2×C4×C47C2×M16semidirect
333. C47#17362semidirect
417. C2×D4#10625329semidirect
513. C23C2×#10625329semidirect
556. C2×C4C2×#10625329semidirect

Maximal subgroups

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

subgroupindex
C2×#156890872
#202417262
#202417182
#202417152
#202417062
C22×C4×D4×C472
#1736247