Finite Group Databasebeta

← #26222677

Subgroups — #26222677

Order 12032.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689328C211yesmaximal, characteristic
2C2×#15689223C211yesmaximal, characteristic
3C2×C4×#10625335C211yesmaximal, characteristic
4#20246107C211yesmaximal, characteristic, complemented
5#20244311C211yesmaximal, characteristic, complemented
6#20244156C211yesmaximal, characteristic, complemented
7#20244029C211yesmaximal, characteristic, complemented
8#20244022C211yesmaximal, characteristic, complemented
9#20243722C211yesmaximal, characteristic
10#20243722C211yesmaximal, characteristic
11#20243398C211yesmaximal, characteristic
12#20243245C211yesmaximal, characteristic, complemented
13#20243221C211yesmaximal, characteristic, complemented
14C4×#13093947C211yesmaximal, characteristic, complemented
15C47×#2275C211yesmaximal, characteristic
16C22×#10625335C2211yescharacteristic
17C22×#10625335C2211yescharacteristic
18C2×#13094024C2211yescharacteristic
19C2×C4×Dic94C2211yescharacteristic
20C2×C4×Dic94C2211yescharacteristic
21#15689328C418yescomplemented
22#15689223—24no—
23C4×#10625335—24no—
24C2×#13093957C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30C2×#13093947C2211yescharacteristic
31C2×#13093947C2211yescharacteristic
32C2×#13093947C2211yescharacteristic
33C2×#13093947C2211yescharacteristic
34C2×#13093945C2211yescharacteristic
35C2×#13093945C2211yescharacteristic
36C2×#13093935C2211yescharacteristic
37C2×C4×C47⋊C8C2211yescharacteristic
38#15689029C2211yescharacteristic
39#15689029C2211yescharacteristic
40#15689008—21no—
41#15689008—21no—
42#15689008—21no—
43#15689008—21no—
44#15688989C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49#15688878C2211yescharacteristic
50#15688878C2211yescharacteristic
51C2×C4×D4×C47C2211yescharacteristic
52C2×C4×D4×C47C2211yescharacteristic
53C2×C4×D4×C47C2211yescharacteristic
54C4×C4⋊C4×C47C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C22⋊C4×C47—21no—
58C4×C22⋊C4×C47—21no—
59C4×C22⋊C4×C47—21no—
60C47×#376C2211yescharacteristic
61C2×#95×C47—21no—
62C22×Dic94C2311yescharacteristic
63C22×Dic94C2311yescharacteristic
64C2×#10625335—22no—
65C2×#10625335—22no—
66C2×#10625335—22no—
67C2×#10625335—22no—
68C2×#10625335C2×C414yes—
69C2×#10625335C2×C414yes—
70C22×C47⋊C8C2311yescharacteristic
71C22×C47⋊C8C2311yescharacteristic
72C22×C47⋊C8C2311yescharacteristic
73C22×C47⋊C8C2311yescharacteristic
74#13094024—22no—
75#13094024C2×C414yescomplemented
76C4×Dic94—22no—
77C4×Dic94—22no—
78C4×Dic94—22no—
79C4×Dic94—22no—
80C2×C4⋊Dic47—21no—
81C2×C4⋊Dic47—21no—
82C2×C4⋊Dic47C2311yescharacteristic
83C2×C4⋊Dic47C2311yescharacteristic
84C2×Dic47⋊C4—21no—
85C2×Dic47⋊C4—21no—
86C2×C4×Dic47—21no—
87C2×C4×Dic47—21no—
88#13093957—22no—
89#13093955—22no—
90C4⋊(C47⋊C8)—22no—
91C4⋊(C47⋊C8)C2×C414yescomplemented
92#13093947—22no—
93#13093947—22no—
94#13093947—22no—
95#13093947—22no—
96#13093947C2×C414yescomplemented
97#13093947C2×C414yescomplemented
98#13093945—22no—
99#13093945—22no—
100#13093935—22no—
101C4×C47⋊C8—22no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C47⋊#95—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C22×D4×C47C2311yescharacteristic
109C23×C4×C47—21no—
110C23×C4×C47—21no—
111C23×C4×C47—21no—
112C4×D4×C47—22no—
113C4×D4×C47—22no—
114C4×D4×C47—22no—
115C4×D4×C47—22no—
116C4×D4×C47C2×C414yes—
117C4×D4×C47D414yes—
118C2×C4⋊C4×C47C2311yescharacteristic
119C2×C4⋊C4×C47C2311yescharacteristic
120C2×C4⋊C4×C47C2311yescharacteristic
121C2×C22⋊C4×C47—21no—
122C2×C22⋊C4×C47—21no—
123C2×C22⋊C4×C47—21no—
124C2×C42×C47—21no—
125C2×C42×C47—21no—
126C2×C42×C47—21no—
127C2×C42×C47C2311yescharacteristic
128C2×C42×C47C2311yescharacteristic
129C2×C42×C47C2311yescharacteristic
130#95×C47—21no—
131#95×C47—21no—
132#95×C47—21no—
133#95×C47D411yescharacteristic
134#95×C47D411yescharacteristic
135#95×C47D411yescharacteristic
136#95×C47D411yescharacteristic
137C24×C47—21no—
138C2×D4×C47—21no—
139C2×D4×C47—21no—
140C2×D4×C47—21no—
141C2×D4×C47C22×C412yes—
142C2×D4×C47C2×D412yes—
143C2×D4×C47#4112yes—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—21no—
147C22×C4×C47—24no—
148C22×C4×C47—24no—
149C22×C4×C47—24no—
150C22×C4×C47—81no—
151C22×C4×C47—81no—
152C22×C4×C47—81no—
153C22×C4×C47—81no—
154C22×C4×C47—81no—
155C22×C4×C47—81no—
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D411yescharacteristic
158C22×C4×C47C2×D411yescharacteristic
159C22×C4×C47C2×D411yescharacteristic
160C22×C4×C47C2×D411yescharacteristic
161C22×C4×C47C2×D411yescharacteristic
162C22×C4×C47C2411yescharacteristic
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C2×Dic94—21no—
168C2×Dic94—21no—
169C2×Dic94—21no—
170C2×Dic94—21no—
171C2×Dic94—21no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—21no—
175C2×Dic94—21no—
176C2×Dic94—21no—
177C2×Dic94C22×C412yes—
178C2×Dic94C22×C412yes—
179C4⋊C4×C47—21no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47C22×C412yes—
183C4⋊C4×C47C2×D412yes—
184C4⋊C4×C47#4112yes—
185C22⋊C4×C47—24no—
186C22⋊C4×C47—24no—
187C22⋊C4×C47—24no—
188C42×C47—21no—
189C42×C47—21no—
190C42×C47—81no—
191C42×C47—81no—
192C42×C47—81no—
193C42×C47C22×C412yes—
194C42×C47C2×D412yes—
195C42×C47#4112yes—
196C42×C47#4112yes—
197#10625335—28no—
198#10625335—28no—
199C4⋊Dic47—24no—
200C4⋊Dic47—24no—
201C4⋊Dic47—21no—
202C4⋊Dic47—21no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205Dic47⋊C4—42no—
206Dic47⋊C4—42no—
207C4×Dic47—42no—
208C4×Dic47—42no—
209C2×C47⋊C8—21no—
210C2×C47⋊C8—21no—
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⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8C22×C412yes—
220C2×C47⋊C8C22×C412yes—
221C23×C47—22no—
222C23×C47—22no—
223C23×C47—22no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47C22×D411yescharacteristic
229D4×C47—22no—
230D4×C47—22no—
231D4×C47—22no—
232D4×C47C4×D414yes—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
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—22no—
242C2×C4×C47—22no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—22no—
246C2×C4×C47—21no—
247C2×C4×C47—21no—
248C2×C4×C47—21no—
249C2×C4×C47—21no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—84no—
257C2×C4×C47—84no—
258C2×C4×C47—84no—
259C2×C4×C47C4×D412yes—
260C2×C4×C47C4×D412yes—
261C2×C4×C47#12112yes—
262C2×C4×C47#12112yes—
263C2×C4×C47C23×C411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C2×C4×C47C2×#4111yescharacteristic
266C2×C4×C47C2×#4111yescharacteristic
267C2×C4×C47C2×#4111yescharacteristic
268C2×C4×C47C2×#4111yescharacteristic
269C2×Dic47—22no—
270C2×Dic47—22no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281Dic94—22no—
282Dic94—22no—
283Dic94—22no—
284Dic94—22no—
285Dic94—41no—
286Dic94—41no—
287Dic94—41no—
288Dic94—41no—
289C47⋊C8—22no—
290C47⋊C8—22no—
291C47⋊C8—41no—
292C47⋊C8—41no—
293#21514—471nomaximal, Sylow
294C22×C47—24no—
295C22×C47—42no—
296C22×C47—42no—
297C22×C47—42no—
298C22×C47—42no—
299C22×C47—42no—
300C22×C47—42no—
301C22×C47C2×C4×D411yescharacteristic
302C22×C47C2×#12111yescharacteristic
303C22×C47#52511yescharacteristic
304C22×C47#57711yescharacteristic
305C22×C47#57711yescharacteristic
306C22×C47#57711yescharacteristic
307C22×C47#57811yescharacteristic
308C4×C47—22no—
309C4×C47—22no—
310C4×C47—22no—
311C4×C47—22no—
312C4×C47—41no—
313C4×C47—41no—
314C4×C47—42no—
315C4×C47—42no—
316C4×C47—42no—
317C4×C47C2×C4×D411yescharacteristic
318C4×C47C4×#4111yescharacteristic
319C4×C47#52911yescharacteristic
320C4×C47#53211yescharacteristic
321Dic47—42no—
322Dic47—42no—
323Dic47—81no—
324Dic47—81no—
325C2×#461—471no—
326C2×#440—471no—
327C2×C4×SD16—471no—
328#2327—471no—
329#2275D4711yescharacteristic
330#1981—471no—
331#1949—471no—
332#1928—471no—
333#1927—471no—
334#1877—471no—
335#1877—471no—
336#1804—471no—
337#1768—471no—
338#1763—471no—
339C4×#103—471no—
340C2×C47—44no—
341C2×C47C4×#12111yescharacteristic
342C2×C47#294711yescharacteristic
343C2×C47#294911yescharacteristic
344C2×C47#304211yescharacteristic
345C2×C47#304611yescharacteristic
346C2×C47#312011yescharacteristic
347C2×C47#312311yescharacteristic
348C22×SD16—471no—
349C22×SD16—471no—
350C2×C4⋊Q8—471no—
351C2×C4×Q8—471no—
352C2×C4×Q8—471no—
353C2×C4×D4C2×D4711yescharacteristic
354C2×C4×D4C2×D4711yescharacteristic
355C2×C4×D4C2×D4711yescharacteristic
356#461—478no—
357#440—944no—
358C4×SD16—944no—
359C2×#107—471no—
360C2×#106—471no—
361C2×C4⋊C8—471no—
362C2×C4⋊C8—471no—
363C2×#103—471no—
364C2×#103—471no—
365C2×#103—471no—
366C2×#103—471no—
367C2×#103—471no—
368C2×#103—471no—
369C2×#102—471no—
370C2×#102—471no—
371C2×#97—471no—
372C2×C4×C8—471no—
373#391—471no—
374#391—471no—
375#387—941no—
376#387—941no—
377#387—941no—
378#387—941no—
379#382—471no—
380C4×C4⋊C4C2×D4711yescharacteristic
381C4×C4⋊C4C2×D4711yescharacteristic
382C4×C4⋊C4C2×D4711yescharacteristic
383C4×C22⋊C4—21no—
384C4×C22⋊C4—21no—
385C4×C22⋊C4—21no—
386#376C2×D4711yescharacteristic
387C2×#95—21no—
388#340—471no—
389#340—471no—
390#340—471no—
391#340—471no—
392#336—471no—
393#336—471no—
394C47#2151411yescharacteristic, complemented, Sylow
395C22×Q8—471no—
396C22×Q8—471no—
397C22×D4C22×D4711yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4—21no—
401C2×SD16—474no—
402C2×SD16—474no—
403C2×SD16—942no—
404C2×SD16—942no—
405C2×SD16—942no—
406C2×SD16—942no—
407C22×C8—471no—
408C22×C8—471no—
409C22×C8—471no—
410C22×C8—471no—
411C4⋊Q8—474no—
412C4⋊Q8—942no—
413C4×Q8—942no—
414C4×Q8—942no—
415C4×Q8—942no—
416C4×Q8—942no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4C4×D4714yes—
422C4×D4D18814yes—
423C2×C4⋊C4—471no—
424C2×C4⋊C4—471no—
425C2×C4⋊C4—941no—
426C2×C4⋊C4—941no—
427C2×C4⋊C4—941no—
428C2×C4⋊C4—941no—
429C2×C4⋊C4C22×D4711yescharacteristic
430C2×C4⋊C4C22×D4711yescharacteristic
431C2×C4⋊C4C22×D4711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—941no—
439C2×C42—941no—
440C2×C42C22×D4711yescharacteristic
441C2×C42C22×D4711yescharacteristic
442C2×C42C22×D4711yescharacteristic
443#107—942no—
444#106—942no—
445C4⋊C8—474no—
446C4⋊C8—942no—
447#103—474no—
448#103—474no—
449#103—942no—
450#103—942no—
451#103—942no—
452#103—942no—
453#102—942no—
454#102—942no—
455#97—942no—
456C4×C8—942no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—941no—
461#95—941no—
462#95—941no—
463#95—941no—
464#95—941no—
465#95—941no—
466#95C22⋊D4711yescharacteristic
467#95C22⋊D4711yescharacteristic
468#95C22⋊D4711yescharacteristic
469#95C22⋊D4711yescharacteristic
470C24—21no—
471C2×Q8—472no—
472C2×Q8—472no—
473C2×Q8—941no—
474C2×Q8—941no—
475C2×Q8—941no—
476C2×Q8—941no—
477C2×Q8—941no—
478C2×Q8—941no—
479C2×Q8—941no—
480C2×Q8—941no—
481C2×Q8—941no—
482C2×Q8—941no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D4712yes—
487C2×D4C2×D18812yes—
488C2×D4#1062535012yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—941no—
505C22×C4C2×C22⋊D4711yescharacteristic
506C22×C4C2×C22⋊D4711yescharacteristic
507C22×C4C2×C22⋊D4711yescharacteristic
508C22×C4C2×C22⋊D4711yescharacteristic
509C22×C4C2×C22⋊D4711yescharacteristic
510C22×C4C2×C22⋊D4711yescharacteristic
511C22×C4C23×D4711yescharacteristic
512SD16—948no—
513SD16—948no—
514C2×C8—472no—
515C2×C8—472no—
516C2×C8—941no—
517C2×C8—941no—
518C2×C8—941no—
519C2×C8—941no—
520C2×C8—941no—
521C2×C8—941no—
522C2×C8—941no—
523C2×C8—941no—
524C2×C8—941no—
525C2×C8—941no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—944no—
530C4⋊C4—944no—
531C4⋊C4—941no—
532C4⋊C4—941no—
533C4⋊C4—941no—
534C4⋊C4—941no—
535C4⋊C4—1882no—
536C4⋊C4—1882no—
537C4⋊C4C2×C4×D4712yes—
538C4⋊C4C2×D18812yes—
539C4⋊C4#1062535012yes—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C42—21no—
544C42—21no—
545C42—81no—
546C42—81no—
547C42—81no—
548C42—1882no—
549C42—1882no—
550C42C2×C4×D4712yes—
551C42C2×D18812yes—
552C42#1062535012yes—
553C42#1062535012yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×C22⋊D4711yescharacteristic
562Q8—942no—
563Q8—942no—
564Q8—942no—
565Q8—942no—
566Q8—1881no—
567Q8—1881no—
568Q8—1881no—
569Q8—1881no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4C4×D18814yes—
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—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—1881no—
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×C4C4×C22⋊D4712yes—
613C2×C4C4×C22⋊D4712yes—
614C2×C4#1309399812yes—
615C2×C4#1309399812yes—
616C2×C4C22×C4×D4711yescharacteristic
617C2×C4C22×D18811yescharacteristic
618C2×C4C2×#1062535011yescharacteristic
619C2×C4C2×#1062535011yescharacteristic
620C2×C4C2×#1062535011yescharacteristic
621C2×C4C2×#1062535011yescharacteristic
622C8—942no—
623C8—942no—
624C8—1881no—
625C8—1881no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22C2×C4×C22⋊D4711yescharacteristic
634C22C2×#1309399811yescharacteristic
635C22#1568963411yescharacteristic
636C22#1568993411yescharacteristic
637C22#1568993411yescharacteristic
638C22#1568993411yescharacteristic
639C22#1568993811yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—42no—
648C4—42no—
649C4—3761no—
650C4—3761no—
651C4—1882no—
652C4—1882no—
653C4C2×C4×D18811yescharacteristic
654C4C4×#1062535011yescharacteristic
655C4#1568965711yescharacteristic
656C4#1568967311yescharacteristic
657C2—44no—
658C2C4×#1309399811yescharacteristic
659C2#2025222711yescharacteristic
660C2#2025224511yescharacteristic
661C2#2025290011yescharacteristic
662C2#2025293811yescharacteristic
663C2#2025360211yescharacteristic
664C2#2025364011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2275C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
cosocleC22×C4×C47C24
hypercenter#2275D47
hypocenterC47#21514
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
4. #20246107C2semidirect
5. #20244311C2semidirect
6. #20244156C2semidirect
7. #20244029C2semidirect
8. #20244022C2semidirect
12. #20243245C2semidirect
13. #20243221C2semidirect
14. C4×#13093947C2semidirect
21. #15689328C4semidirect
75. #13094024C2×C4semidirect
91. C4⋊(C47⋊C8)C2×C4semidirect
96. #13093947C2×C4semidirect
97. #13093947C2×C4semidirect
394. C47#21514semidirect

Maximal subgroups

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

subgroupindex
C2×#156893282
C2×#156892232
C2×C4×#106253352
#202461072
#202443112
#202441562
#202440292
#202440222
#202437222
#202433982
#202432452
#202432212
C4×#130939472
C47×#22752
#2151447