Finite Group Databasebeta

← #26139101

Subgroups — #26139101

Order 12032.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688995C211yesmaximal, characteristic
2C2×#15688986C211yesmaximal, characteristic
3C2×C22⋊C4×Dic47C211yesmaximal, characteristic
4C22×C47⋊#95C211yesmaximal, characteristic
5#20241649C212yesmaximal, complemented
6#20241643C212yesmaximal, complemented
7#20241619C212yesmaximal, complemented
8#20241612C212yesmaximal, complemented
9#20241610C211yesmaximal, characteristic
10#20241610C211yesmaximal, characteristic
11C2×C47×#381C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C24×Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2211yescharacteristic
15C22×Dic47⋊C4C2211yescharacteristic
16C22×C22⋊Dic47C2211yescharacteristic
17C22×C22⋊Dic47C2211yescharacteristic
18C22×C4×Dic47C2212yes—
19#15688995—24no—
20#15688986C418yescomplemented
21C22⋊C4×Dic47—24no—
22C2×C47⋊#95—24no—
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C24×C4×C47C2211yescharacteristic
35C22×C22⋊C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2211yescharacteristic
37C47×#381—24no—
38C2×#95×C47C2212yes—
39C2×#95×C47C2212yes—
40C23×Dic47—22no—
41C23×Dic47—22no—
42C23×Dic47—22no—
43C23×Dic47—24no—
44C23×Dic47—24no—
45C23×Dic47—22no—
46C23×Dic47—22no—
47C23×Dic47—42no—
48C23×Dic47—42no—
49C23×Dic47D414yes—
50C23×Dic47C2312yes—
51C23×Dic47C2311yescharacteristic
52C23×Dic47C2311yescharacteristic
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2312yes—
55C23×Dic47C2311yescharacteristic
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—22no—
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4—42no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C22⋊Dic47C2×C414yes—
65C2×C22⋊Dic47C2×C414yes—
66C2×C4×Dic47—24no—
67C2×C4×Dic47—24no—
68C2×C4×Dic47—42no—
69C47⋊#95—24no—
70C47⋊#95—24no—
71C47⋊#95—24no—
72C47⋊#95—22no—
73C47⋊#95—22no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—42no—
77C47⋊#95—42no—
78C47⋊#95C2×C418yescomplemented
79C47⋊#95C2×C418yescomplemented
80C25×C47C2311yescharacteristic
81C23×C4×C47—22no—
82C23×C4×C47—22no—
83C23×C4×C47—22no—
84C23×C4×C47—22no—
85C23×C4×C47—22no—
86C23×C4×C47—22no—
87C23×C4×C47C2×C414yes—
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2311yescharacteristic
90C23×C4×C47C2312yes—
91C23×C4×C47C2312yes—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—22no—
94C2×C22⋊C4×C47—22no—
95C2×C22⋊C4×C47—22no—
96C2×C22⋊C4×C47—22no—
97C2×C22⋊C4×C47D414yes—
98#95×C47—24no—
99#95×C47—24no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47C22×C412yes—
105C24×C47C2×D412yes—
106C24×C47C2×D412yes—
107C24×C47#4112yes—
108C24×C47#4112yes—
109C24×C47#4112yes—
110C24×C47#4112yes—
111C24×C47C2411yescharacteristic
112C22×C4×C47—28no—
113C22×C4×C47—28no—
114C22×C4×C47—28no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—21no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—44no—
138C22×C4×C47—44no—
139C22×C4×C47—44no—
140C22×C4×C47—44no—
141C22×C4×C47C22×C414yes—
142C22×C4×C47C2×D414yes—
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—28no—
147C22×Dic47—28no—
148C22×Dic47—21no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—21no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—22no—
172C22×Dic47—28no—
173C22×Dic47—44no—
174C22×Dic47—48no—
175C22×Dic47—48no—
176C22×Dic47—48no—
177C22×Dic47—48no—
178C22×Dic47—44no—
179C22×Dic47—44no—
180C22×Dic47C22×C412yes—
181C22×Dic47C22×C414yes—
182C22×Dic47C22×C412yes—
183C22×Dic47C2×D412yes—
184C22×Dic47C2×D412yes—
185C22×Dic47#4114yes—
186C22×Dic47#4114yes—
187C22⋊C4×C47—28no—
188C22⋊C4×C47—44no—
189C22⋊Dic47—28no—
190C22⋊Dic47—28no—
191Dic47⋊C4—44no—
192Dic47⋊C4—44no—
193C4×Dic47—48no—
194C23×C47—24no—
195C23×C47—24no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—24no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—24no—
202C23×C47—24no—
203C23×C47—24no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—24no—
209C23×C47—24no—
210C23×C47#11714yes—
211C23×C47C4×D414yes—
212C23×C47C4×D414yes—
213C23×C47#12114yes—
214C23×C47#12314yes—
215C23×C47#12414yes—
216C23×C47#12614yes—
217C23×C47C23×C411yescharacteristic
218C23×C47C22×D411yescharacteristic
219C23×C47C22×D411yescharacteristic
220C23×C47C2×#4112yes—
221C23×C47C2×#4112yes—
222C23×C47C2×#4111yescharacteristic
223C23×C47C2×#4111yescharacteristic
224C23×C47C2×#4111yescharacteristic
225C23×C47C2×#4111yescharacteristic
226C23×C47H(5,𝔽2)12yes—
227C23×C472−1+412yes—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—42no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—416no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×Dic47—22no—
254C2×Dic47—22no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—22no—
258C2×Dic47—22no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—24no—
267C2×Dic47—22no—
268C2×Dic47—24no—
269C2×Dic47—24no—
270C2×Dic47—416no—
271C2×Dic47—416no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—41no—
277C2×Dic47—42no—
278C2×Dic47—41no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C2×Dic47—42no—
284C2×Dic47—42no—
285C2×Dic47—42no—
286C2×Dic47—42no—
287C2×Dic47—42no—
288C2×Dic47#12314yes—
289#13972—471nomaximal, Sylow
290C22×C47—24no—
291C22×C47—24no—
292C22×C47—28no—
293C22×C47—28no—
294C22×C47—24no—
295C22×C47—24no—
296C22×C47—28no—
297C22×C47—28no—
298C22×C47—24no—
299C22×C47—24no—
300C22×C47#38818yes—
301C22×C47C2×#11711yescharacteristic
302C22×C47C2×C4×D411yescharacteristic
303C22×C47C2×C4×D411yescharacteristic
304C22×C47C4×#4112yes—
305C22×C47#52012yes—
306C22×C47C2×#12111yescharacteristic
307C22×C47C2×#12311yescharacteristic
308C22×C47C2×#12311yescharacteristic
309C22×C47#52511yescharacteristic
310C22×C47C2×#12411yescharacteristic
311C22×C47C2×#12611yescharacteristic
312C22×C47#52912yes—
313C22×C47#53411yescharacteristic
314C22×C47#53611yescharacteristic
315C22×C47#53912yes—
316C22×C47#54612yes—
317C22×C47#54712yes—
318C22×C47#55112yes—
319C22×C47#55112yes—
320C22×C47#55112yes—
321C22×C47#55212yes—
322C22×C47#55212yes—
323C22×C47#55212yes—
324C4×C47—44no—
325C4×C47—44no—
326C4×C47—81no—
327C4×C47—81no—
328Dic47—24no—
329Dic47—44no—
330Dic47—42no—
331Dic47—44no—
332Dic47—42no—
333Dic47—42no—
334C2×#385—471no—
335C2×#381—471no—
336C2×#381D4711yescharacteristic
337C2×C4×C22⋊C4—471no—
338C22×#95—471no—
339#1443—472no—
340#1442—472no—
341#1436—472no—
342#1434—472no—
343C22⋊#95—471no—
344C22⋊#95—471no—
345C2×C47—28no—
346C2×C47C2×#38811yescharacteristic
347C2×C47C4×#12311yescharacteristic
348C2×C47#232211yescharacteristic
349C2×C47#234312yes—
350C2×C47#235912yes—
351C2×C47#241211yescharacteristic
352C2×C47#241311yescharacteristic
353C2×C47#241912yes—
354C2×C47#249312yes—
355C2×C47#250311yescharacteristic
356C2×C47#256111yescharacteristic
357C24×C4—471no—
358C24×C4—471no—
359C24×C4C2×D4711yescharacteristic
360C22×C4⋊C4—471no—
361C22×C4⋊C4—471no—
362C22×C22⋊C4—471no—
363C22×C22⋊C4—471no—
364C22×C22⋊C4C2×D4711yescharacteristic
365C22×C22⋊C4C2×D4711yescharacteristic
366C22×C42—472no—
367#385—944no—
368#381—24no—
369#381—478no—
370C4×C22⋊C4—944no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95—472no—
374C2×#95—472no—
375C2×#95—471no—
376C2×#95—472no—
377C2×#95—472no—
378C2×#95—472no—
379C2×#95—471no—
380C2×#95—472no—
381C2×#95—472no—
382C2×#95—944no—
383C2×#95C2×D4712yes—
384C2×#95C2×D4712yes—
385C47#1397211yescharacteristic, complemented, Sylow
386C25C22×D4711yescharacteristic
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—471no—
394C23×C4—471no—
395C23×C4—471no—
396C23×C4—472no—
397C23×C4—471no—
398C23×C4—472no—
399C23×C4—474no—
400C23×C4—944no—
401C23×C4—944no—
402C23×C4—942no—
403C23×C4—942no—
404C23×C4—942no—
405C23×C4—942no—
406C23×C4—942no—
407C23×C4—1882no—
408C23×C4—1882no—
409C23×C4C4×D4714yes—
410C23×C4C22×D4711yescharacteristic
411C23×C4C22×D4711yescharacteristic
412C23×C4C22×D4712yes—
413C23×C4C22×D4712yes—
414C2×C4⋊C4—944no—
415C2×C4⋊C4—942no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4—1882no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—474no—
425C2×C22⋊C4—942no—
426C2×C22⋊C4—942no—
427C2×C22⋊C4—942no—
428C2×C22⋊C4—942no—
429C2×C22⋊C4C22⋊D4714yes—
430C2×C42—944no—
431C2×C42—944no—
432C2×C42—1882no—
433#95—24no—
434#95—24no—
435#95—478no—
436#95—478no—
437#95—944no—
438#95—944no—
439#95—944no—
440#95—944no—
441#95—942no—
442#95—942no—
443#95—944no—
444#95—1882no—
445#95—1882no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×C4×D4712yes—
451C24#1062535012yes—
452C24D4×D4712yes—
453C24#1062535212yes—
454C24#1062535212yes—
455C24#1062535212yes—
456C24C2×C22⋊D4712yes—
457C24C23×D4711yescharacteristic
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—472no—
488C22×C4—474no—
489C22×C4—472no—
490C22×C4—472no—
491C22×C4—474no—
492C22×C4—472no—
493C22×C4—474no—
494C22×C4—948no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—942no—
498C22×C4—948no—
499C22×C4—948no—
500C22×C4—942no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—941no—
512C22×C4—941no—
513C22×C4—942no—
514C22×C4—941no—
515C22×C4—942no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—942no—
521C22×C4—942no—
522C22×C4—942no—
523C22×C4—942no—
524C22×C4—1884no—
525C22×C4—1888no—
526C22×C4—1888no—
527C22×C4—1888no—
528C22×C4—1888no—
529C22×C4—1884no—
530C22×C4—1884no—
531C22×C4C2×C4×D4714yes—
532C22×C4C2×C22⋊D4714yes—
533C4⋊C4—1884no—
534C4⋊C4—1884no—
535C22⋊C4—28no—
536C22⋊C4—44no—
537C22⋊C4—948no—
538C22⋊C4—948no—
539C42—1888no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23#1309398014yes—
557C23(C22⋊D47)⋊C414yes—
558C23C4×C22⋊D4714yes—
559C23#1309399514yes—
560C23#1309400514yes—
561C23#1309401014yes—
562C23#1309401714yes—
563C23C22×C4×D4711yescharacteristic
564C23C2×D4×D4711yescharacteristic
565C23C22×C22⋊D4711yescharacteristic
566C23#41×D4712yes—
567C23#41×D4712yes—
568C23C2×#1062535211yescharacteristic
569C23C2×#1062535211yescharacteristic
570C23C2×#1062535211yescharacteristic
571C23C2×#1062535011yescharacteristic
572C23#1309407012yes—
573C23#1309407112yes—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—474no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—944no—
605C2×C4—944no—
606C2×C4—942no—
607C2×C4—942no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—944no—
612C2×C4—944no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—944no—
616C2×C4—942no—
617C2×C4—18816no—
618C2×C4—18816no—
619C2×C4—1882no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1882no—
623C2×C4—1882no—
624C2×C4—1882no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4—1881no—
628C2×C4—1881no—
629C2×C4—1882no—
630C2×C4—1882no—
631C2×C4—1881no—
632C2×C4—1882no—
633C2×C4—1881no—
634C2×C4—1882no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—28no—
639C22—28no—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22#1568901118yes—
646C22C2×#1309398011yescharacteristic
647C22C2×(C22⋊D47)⋊C411yescharacteristic
648C22C2×C4×C22⋊D4711yescharacteristic
649C22#1568959112yes—
650C22#1568960312yes—
651C22C2×#1309399511yescharacteristic
652C22#123×D4711yescharacteristic
653C22C2×#1309400511yescharacteristic
654C22#1568963111yescharacteristic
655C22C2×#1309401011yescharacteristic
656C22C2×#1309401711yescharacteristic
657C22#1568965012yes—
658C22#1568968411yescharacteristic
659C22#1568969511yescharacteristic
660C22#1568971712yes—
661C22#1568976112yes—
662C22#1568977212yes—
663C22#1568979612yes—
664C22#1568979612yes—
665C22#1568979712yes—
666C22#1568980612yes—
667C22#1568981012yes—
668C22#1568981012yes—
669C4—44no—
670C4—44no—
671C4—81no—
672C4—81no—
673C4—944no—
674C4—1882no—
675C4—1884no—
676C4—1882no—
677C4—1884no—
678C4—1882no—
679C2—28no—
680C2C2×#1568901111yescharacteristic
681C2#2024590511yescharacteristic
682C2#2024605411yescharacteristic
683C2#2024625012yes—
684C2#2024638012yes—
685C2#2024678011yescharacteristic
686C2#2024679111yescharacteristic
687C2#2024686112yes—
688C2#2024777712yes—
689C2#2024787111yescharacteristic
690C2#2024854511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#381C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×#381D47
hypocenterC47#13972
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
5. #20241649C2semidirect
6. #20241643C2semidirect
7. #20241619C2semidirect
8. #20241612C2semidirect
20. #15688986C4semidirect
78. C47⋊#95C2×C4semidirect
79. C47⋊#95C2×C4semidirect
385. C47#13972semidirect

Maximal subgroups

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

subgroupindex
C2×#156889952
C2×#156889862
C2×C22⋊C4×Dic472
C22×C47⋊#952
#202416492
#202416432
#202416192
#202416122
#202416102
C2×C47×#3812
#1397247