Finite Group Databasebeta

← #26222988

Subgroups — #26222988

Order 12032.

1002 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689379C211yesmaximal, characteristic
2C2×#15689228C211yesmaximal, characteristic, complemented
3C2×C4×#10625329C211yesmaximal, characteristic
4C4×#13094003C211yesmaximal, characteristic
5#20244248C211yesmaximal, characteristic, complemented
6#20244113C211yesmaximal, characteristic, complemented
7#20244067C211yesmaximal, characteristic, complemented
8#20244063C211yesmaximal, characteristic, complemented
9#20243822C211yesmaximal, characteristic
10#20243819C211yesmaximal, characteristic
11#20243419C211yesmaximal, characteristic
12#20243286C211yesmaximal, characteristic, complemented
13#20243221C211yesmaximal, characteristic, complemented
14#20243218C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C47C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C22×#10625329C2211yescharacteristic
18C22×#10625329C2211yescharacteristic
19C2×#13094003C2211yescharacteristic
20C2×C4×Dic94C2211yescharacteristic
21#15689379C418yescomplemented
22#15689228—24no—
23C2×#13093957C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×#13093948C2211yescharacteristic
27C2×#13093948C2211yescharacteristic
28C2×#13093948C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C4×#10625329—24no—
33C2×#13093935C2211yescharacteristic
34C2×#13093935C2211yescharacteristic
35C2×C4×C47⋊C8C2211yescharacteristic
36C2×C4×C47⋊C8C2211yescharacteristic
37#15689029C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689007—21no—
40#15688996—21no—
41#15688983C2211yescharacteristic
42C4×Dic47⋊C4—21no—
43C4×C4⋊Dic47C2211yescharacteristic
44C4×C22⋊Dic47—21no—
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51C2×C22⋊Q8×C47C2211yescharacteristic
52C2×C4×Q8×C47C2211yescharacteristic
53C22×C42×C47C2211yescharacteristic
54C47×#391C2211yescharacteristic
55C47×#389C2211yescharacteristic
56C47×#387—21no—
57C47×#385—21no—
58C47×#380C2211yescharacteristic
59C4×C4⋊C4×C47—21no—
60C4×C4⋊C4×C47C2211yescharacteristic
61C4×C22⋊C4×C47—21no—
62C22×Dic94C2311yescharacteristic
63C2×#10625337—22no—
64C2×#10625337—22no—
65C2×#10625337C2×C414yescomplemented
66C2×#10625329—22no—
67C2×#10625329—22no—
68C2×#10625329—22no—
69C2×#10625329—22no—
70C2×#10625329—22no—
71C2×#10625329C2×C414yes—
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74C22×C47⋊C8C2311yescharacteristic
75C22×C47⋊C8C2311yescharacteristic
76#13094003—22no—
77#13094003C2×C414yes—
78C4×Dic94—22no—
79C4×Dic94—22no—
80C2×C4⋊Dic47C2311yescharacteristic
81C2×C4⋊Dic47C2311yescharacteristic
82C2×C4⋊Dic47C2311yescharacteristic
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×Dic47⋊C4—21no—
86C2×C22⋊Dic47—21no—
87C2×C22⋊Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—21no—
90#13093957—22no—
91#13093955—22no—
92#13093955C2×C414yescomplemented
93#13093948—22no—
94#13093948—22no—
95#13093948C2×C414yescomplemented
96#13093947—22no—
97#13093947—22no—
98#13093947C2×C414yescomplemented
99#13093935—22no—
100#13093935D414yes—
101C4×C47⋊C8—22no—
102C4×C47⋊C8—22no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×Q8×C47C2311yescharacteristic
109C23×C4×C47D411yescharacteristic
110C23×C4×C47D411yescharacteristic
111C23×C4×C47C2311yescharacteristic
112C22⋊Q8×C47—22no—
113C22⋊Q8×C47C2×C414yes—
114C4×Q8×C47—22no—
115C4×Q8×C47—22no—
116C2×C4⋊C4×C47—21no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C4⋊C4×C47C2311yescharacteristic
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×C47—21no—
124C2×C42×C47—22no—
125C2×C42×C47—22no—
126C2×C42×C47—21no—
127C2×C42×C47—21no—
128C2×C42×C47D411yescharacteristic
129C2×C42×C47D411yescharacteristic
130C2×C42×C47C2311yescharacteristic
131C2×C42×C47C2311yescharacteristic
132#95×C47—21no—
133#95×C47—21no—
134#95×C47—21no—
135#95×C47—21no—
136#95×C47—21no—
137C24×C47C2×D411yescharacteristic
138C2×Q8×C47—21no—
139C2×Q8×C47—21no—
140C2×Q8×C47—21no—
141C2×Q8×C47—21no—
142C2×Q8×C47—21no—
143C2×Q8×C47C22×C412yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—21no—
148C22×C4×C47—21no—
149C22×C4×C47—21no—
150C22×C4×C47—22no—
151C22×C4×C47—22no—
152C22×C4×C47—22no—
153C22×C4×C47—22no—
154C22×C4×C47—21no—
155C22×C4×C47—21no—
156C22×C4×C47—41no—
157C22×C4×C47—41no—
158C22×C4×C47C22×C412yes—
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2×D411yescharacteristic
163C22×C4×C47C2×D411yescharacteristic
164C22×C4×C47#4112yes—
165C22×C4×C47C2411yescharacteristic
166C22×Dic47—21no—
167C22×Dic47—21no—
168C22×Dic47—21no—
169C22×Dic47—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94C22×C412yes—
176C4⋊C4×C47—24no—
177C4⋊C4×C47—21no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47—42no—
183C4⋊C4×C47—42no—
184C4⋊C4×C47C22×C412yes—
185C22⋊C4×C47—24no—
186C22⋊C4×C47—42no—
187C42×C47—22no—
188C42×C47—22no—
189C42×C47—21no—
190C42×C47—21no—
191C42×C47—42no—
192C42×C47—42no—
193C42×C47C2×D412yes—
194C42×C47#4112yes—
195C22⋊Dic47—24no—
196C22⋊Dic47—42no—
197#10625337—28no—
198C4⋊Dic47—21no—
199C4⋊Dic47—21no—
200C4⋊Dic47—21no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47C22×C412yes—
204Dic47⋊C4—24no—
205Dic47⋊C4—42no—
206Dic47⋊C4—42no—
207C4×Dic47—42no—
208C4×Dic47—42no—
209#10625329—28no—
210#10625329—28no—
211C2×C47⋊C8—21no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—21no—
215C2×C47⋊C8—21no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8C22×C412yes—
218C2×C47⋊C8C22×C412yes—
219C2×C47⋊C8C2×D412yes—
220C2×C47⋊C8C2×D412yes—
221C2×C47⋊C8#4112yes—
222C2×C47⋊C8#4112yes—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—21no—
226C23×C47—21no—
227C23×C47—21no—
228C23×C47—21no—
229C23×C47C4×D412yes—
230C23×C47C22×D411yescharacteristic
231Q8×C47—22no—
232Q8×C47—22no—
233Q8×C47—41no—
234Q8×C47—41no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—21no—
238C2×C4×C47—21no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—21no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—22no—
250C2×C4×C47—22no—
251C2×C4×C47—44no—
252C2×C4×C47—44no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—41no—
257C2×C4×C47—41no—
258C2×C4×C47—41no—
259C2×C4×C47—41no—
260C2×C4×C47—41no—
261C2×C4×C47—41no—
262C2×C4×C47—41no—
263C2×C4×C47C4×D412yes—
264C2×C4×C47#12112yes—
265C2×C4×C47#12112yes—
266C2×C4×C47#13712yes—
267C2×C4×C47#13712yes—
268C2×C4×C47C23×C411yescharacteristic
269C2×C4×C47C22×D411yescharacteristic
270C2×C4×C47C2×#4111yescharacteristic
271C2×C4×C47C2×#4111yescharacteristic
272C2×C4×C47C2×#4111yescharacteristic
273C2×C4×C47C2×#4111yescharacteristic
274C2×Dic47—22no—
275C2×Dic47—22no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—41no—
284C2×Dic47—41no—
285C2×Dic47—41no—
286Dic94—22no—
287Dic94—22no—
288Dic94—41no—
289Dic94—41no—
290C47⋊C8—22no—
291C47⋊C8—22no—
292C47⋊C8—22no—
293C47⋊C8C4×D414yes—
294#21539—471nomaximal, Sylow
295C22×C47—22no—
296C22×C47—22no—
297C22×C47—22no—
298C22×C47—41no—
299C22×C47—41no—
300C22×C47—41no—
301C22×C47—41no—
302C22×C47C2×C4×D411yescharacteristic
303C22×C47C2×#12111yescharacteristic
304C22×C47#53211yescharacteristic
305C22×C47C2×#13711yescharacteristic
306C22×C47C2×#13711yescharacteristic
307C22×C47#57511yescharacteristic
308C22×C47#57511yescharacteristic
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—22no—
313C4×C47—22no—
314C4×C47—42no—
315C4×C47—42no—
316C4×C47—81no—
317C4×C47—81no—
318C4×C47C2×C4×D411yescharacteristic
319C4×C47C4×#4111yescharacteristic
320C4×C47#52511yescharacteristic
321C4×C47#52911yescharacteristic
322Dic47—42no—
323Dic47—42no—
324Dic47—81no—
325Dic47—81no—
326C2×#470—471no—
327C2×#441—471no—
328C2×C4×M16—471no—
329C4×C22⋊Q8—471no—
330C4×C22⋊Q8D4711yescharacteristic
331#1969—471no—
332#1941—471no—
333#1933—471no—
334#1933—471no—
335#1894—471no—
336#1894—471no—
337#1809—471no—
338#1776—471no—
339#1763—471no—
340#1763—471no—
341C2×C47—42no—
342C2×C47C4×#12111yescharacteristic
343C2×C47C4×#13711yescharacteristic
344C2×C47C4×#13711yescharacteristic
345C2×C47#305611yescharacteristic
346C2×C47#305611yescharacteristic
347C2×C47#318311yescharacteristic
348C2×C47#318511yescharacteristic
349C22×Q16—471no—
350C22×M16—471no—
351C22×M16—471no—
352C2×C22⋊Q8—471no—
353C2×C22⋊Q8C2×D4711yescharacteristic
354C2×C4×Q8—471no—
355C2×C4×Q8C2×D4711yescharacteristic
356C22×C42C2×D4711yescharacteristic
357#470—478no—
358#441—944no—
359C2×#107—471no—
360C2×#106—471no—
361C2×#106—471no—
362C2×#103—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#103—471no—
366C2×#103—471no—
367C2×#103—471no—
368C4×M16—944no—
369C2×#97—471no—
370C2×#97—471no—
371C2×C4×C8—471no—
372C2×C4×C8—471no—
373#391—471no—
374#391C2×D4711yescharacteristic
375#389—471no—
376#389C2×D4711yescharacteristic
377#387—21no—
378#387—941no—
379#385—21no—
380#385—941no—
381#380—471no—
382#380C2×D4711yescharacteristic
383C4×C4⋊C4—21no—
384C4×C4⋊C4—471no—
385C4×C4⋊C4—941no—
386C4×C4⋊C4C2×D4711yescharacteristic
387C4×C22⋊C4—21no—
388C4×C22⋊C4—941no—
389#340—471no—
390#340—471no—
391#340—471no—
392#340—471no—
393#340—471no—
394#340—471no—
395C47#2153911yescharacteristic, complemented, Sylow
396C22×Q8—471no—
397C22×Q8C22×D4711yescharacteristic
398C23×C4C22⋊D4711yescharacteristic
399C23×C4C22⋊D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401C2×Q16—474no—
402C2×Q16—942no—
403C2×Q16—942no—
404C2×M16—474no—
405C2×M16—942no—
406C2×M16—942no—
407C2×M16—942no—
408C2×M16—942no—
409C2×M16—942no—
410C22×C8—471no—
411C22×C8—471no—
412C22×C8—471no—
413C22×C8—471no—
414C22⋊Q8—22no—
415C22⋊Q8—474no—
416C22⋊Q8—942no—
417C22⋊Q8C4×D4714yes—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—942no—
421C4×Q8—942no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—471no—
426C2×C4⋊C4—471no—
427C2×C4⋊C4—471no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4—941no—
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C4⋊C4C22×D4711yescharacteristic
433C2×C4⋊C4C22×D4711yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—941no—
437C2×C22⋊C4—941no—
438C2×C42—22no—
439C2×C42—22no—
440C2×C42—21no—
441C2×C42—21no—
442C2×C42—941no—
443C2×C42—941no—
444C2×C42C22⋊D4711yescharacteristic
445C2×C42C22⋊D4711yescharacteristic
446C2×C42C22×D4711yescharacteristic
447C2×C42C22×D4711yescharacteristic
448#107—942no—
449#106—474no—
450#106—942no—
451#103—474no—
452#103—474no—
453#103—942no—
454#103—942no—
455#103—942no—
456#103—942no—
457#97—474no—
458#97—942no—
459C4×C8—942no—
460C4×C8—942no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—941no—
467#95—941no—
468#95—941no—
469#95—941no—
470#95—941no—
471C24C2×C22⋊D4711yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—472no—
478C2×Q8—941no—
479C2×Q8—941no—
480C2×Q8—941no—
481C2×Q8—941no—
482C2×Q8—941no—
483C2×Q8C2×C4×D4712yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4C2×C4×D4712yes—
503C22×C4#1062535012yes—
504C22×C4C2×C22⋊D4711yescharacteristic
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C2×C22⋊D4711yescharacteristic
507C22×C4C2×C22⋊D4711yescharacteristic
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C23×D4711yescharacteristic
510Q16—948no—
511M16—948no—
512M16—948no—
513C2×C8—472no—
514C2×C8—472no—
515C2×C8—472no—
516C2×C8—472no—
517C2×C8—472no—
518C2×C8—472no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8—941no—
522C2×C8—941no—
523C2×C8—941no—
524C2×C8—941no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—472no—
534C4⋊C4—944no—
535C4⋊C4—941no—
536C4⋊C4—941no—
537C4⋊C4—941no—
538C4⋊C4—941no—
539C4⋊C4—941no—
540C4⋊C4—1882no—
541C4⋊C4—1882no—
542C4⋊C4C2×C4×D4712yes—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—944no—
546C22⋊C4—1882no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—42no—
552C42—42no—
553C42—1882no—
554C42—1882no—
555C42D4×D4712yes—
556C42#1062535212yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23C4×C22⋊D4712yes—
564C23C22×C22⋊D4711yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—942no—
570Q8—942no—
571Q8—1881no—
572Q8—1881no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—1881no—
604C2×C4—1881no—
605C2×C4—1881no—
606C2×C4—1881no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4C4×C22⋊D4712yes—
614C2×C4#1309399612yes—
615C2×C4#1309399612yes—
616C2×C4#1309405412yes—
617C2×C4#1309405512yes—
618C2×C4C22×C4×D4711yescharacteristic
619C2×C4C2×D4×D4711yescharacteristic
620C2×C4#41×D4711yescharacteristic
621C2×C4#41×D4711yescharacteristic
622C2×C4C2×#1062535211yescharacteristic
623C2×C4C2×#1062535011yescharacteristic
624C8—474no—
625C8—942no—
626C8—942no—
627C8—942no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×C4×C22⋊D4711yescharacteristic
636C22C2×#1309399611yescharacteristic
637C22#1568966911yescharacteristic
638C22C2×#1309405411yescharacteristic
639C22C2×#1309405511yescharacteristic
640C22#1568992111yescharacteristic
641C22#1568992111yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—3761no—
652C4—3761no—
653C4—1882no—
654C4—1882no—
655C4C4×D4×D4711yescharacteristic
656C4C4×#1062535211yescharacteristic
657C4#1568963211yescharacteristic
658C4#1568965311yescharacteristic
659C2—42no—
660C2C4×#1309399611yescharacteristic
661C2C4×#1309405411yescharacteristic
662C2C4×#1309405511yescharacteristic
663C2#2025301411yescharacteristic
664C2#2025302011yescharacteristic
665C2#2025411111yescharacteristic
666C2#2025412911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×C22⋊Q8×C47C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC4×C22⋊Q8D47
hypocenterC47#21539
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15689228C2semidirect
5. #20244248C2semidirect
6. #20244113C2semidirect
7. #20244067C2semidirect
8. #20244063C2semidirect
12. #20243286C2semidirect
13. #20243221C2semidirect
14. #20243218C2semidirect
21. #15689379C4semidirect
65. C2×#10625337C2×C4semidirect
92. #13093955C2×C4semidirect
95. #13093948C2×C4semidirect
98. #13093947C2×C4semidirect
395. C47#21539semidirect

Maximal subgroups

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

subgroupindex
C2×#156893792
C2×#156892282
C2×C4×#106253292
C4×#130940032
#202442482
#202441132
#202440672
#202440632
#202438222
#202438192
#202434192
#202432862
#202432212
#202432182
C4×C22⋊Q8×C472
#2153947