Finite Group Databasebeta

← #26122791

Subgroups — #26122791

Order 12032.

1578 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093948C212yesmaximal, complemented
2C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic
3C2×#15688983C211yesmaximal, characteristic
4C2×#15688885C212yesmaximal, complemented
5C22×C22⋊Q8×C47C211yesmaximal, characteristic
6C23×C47⋊C8C2212yes—
7C22×C4⋊Dic47C2212yes—
8C22×C22⋊Dic47—21no—
9C2×#13093948—24no—
10C2×#13093948—24no—
11C2×#13093948—24no—
12C2×C22⋊(C47⋊C8)—24no—
13C2×C22⋊(C47⋊C8)—22no—
14#15688983—24no—
15C2×C47⋊#95—21no—
16#15688885—24no—
17C23×Q8×C47C2211yescharacteristic
18C24×C4×C47C2211yescharacteristic
19C2×C22⋊Q8×C47—24no—
20C2×C22⋊Q8×C47—22no—
21C2×C22⋊Q8×C47—24no—
22C2×C22⋊Q8×C47C414yescomplemented
23C22×C4⋊C4×C47—21no—
24C22×C4⋊C4×C47C2211yescharacteristic
25C22×C22⋊C4×C47—21no—
26C23×Dic47—22no—
27C22×C47⋊C8—22no—
28C22×C47⋊C8—22no—
29C22×C47⋊C8—22no—
30C22×C47⋊C8—22no—
31C22×C47⋊C8—22no—
32C22×C47⋊C8—22no—
33C22×C47⋊C8D414yes—
34C2×C4⋊Dic47—22no—
35C2×C4⋊Dic47—22no—
36C2×C4⋊Dic47—22no—
37C2×C4⋊Dic47—22no—
38C2×C4⋊Dic47—22no—
39C2×C4⋊Dic47D414yes—
40C2×C22⋊Dic47—24no—
41C2×C22⋊Dic47—42no—
42C2×C22⋊Dic47—42no—
43#13093948—216no—
44C22⋊(C47⋊C8)—28no—
45C47⋊#95—24no—
46C25×C47D411yescharacteristic
47C22×Q8×C47—21no—
48C22×Q8×C47—22no—
49C22×Q8×C47—22no—
50C22×Q8×C47—22no—
51C22×Q8×C47—22no—
52C22×Q8×C47—22no—
53C22×Q8×C47C2×C412yes—
54C22×Q8×C47D414yes—
55C23×C4×C47—21no—
56C23×C4×C47—21no—
57C23×C4×C47—22no—
58C23×C4×C47—22no—
59C23×C4×C47—22no—
60C23×C4×C47—22no—
61C23×C4×C47—22no—
62C23×C4×C47—22no—
63C23×C4×C47C2×C412yes—
64C23×C4×C47D411yescharacteristic
65C23×C4×C47D412yes—
66C23×C4×C47C2311yescharacteristic
67C22⋊Q8×C47—216no—
68C22⋊Q8×C47—28no—
69C22⋊Q8×C47—28no—
70C2×C4⋊C4×C47—28no—
71C2×C4⋊C4×C47—24no—
72C2×C4⋊C4×C47—22no—
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47C2×C412yes—
75C2×C4⋊C4×C47D414yes—
76C2×C22⋊C4×C47—28no—
77C2×C22⋊C4×C47—24no—
78C24×C47—22no—
79C24×C47—21no—
80C24×C47—22no—
81C24×C47—21no—
82C24×C47—22no—
83C24×C47—21no—
84C24×C47—21no—
85C24×C47—22no—
86C24×C47C22⋊C412yes—
87C24×C47SD1612yes—
88C24×C47Q1612yes—
89C24×C47C2×D411yescharacteristic
90C2×Q8×C47—22no—
91C2×Q8×C47—24no—
92C2×Q8×C47—24no—
93C2×Q8×C47—24no—
94C2×Q8×C47—24no—
95C2×Q8×C47—24no—
96C2×Q8×C47—24no—
97C2×Q8×C47—24no—
98C2×Q8×C47—41no—
99C2×Q8×C47—42no—
100C2×Q8×C47—42no—
101C2×Q8×C47—42no—
102C2×Q8×C47—41no—
103C2×Q8×C47—42no—
104C2×Q8×C47—42no—
105C2×Q8×C47C22⋊C414yescomplemented
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—24no—
109C22×C4×C47—24no—
110C22×C4×C47—22no—
111C22×C4×C47—21no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—24no—
117C22×C4×C47—24no—
118C22×C4×C47—24no—
119C22×C4×C47—22no—
120C22×C4×C47—24no—
121C22×C4×C47—22no—
122C22×C4×C47—41no—
123C22×C4×C47—42no—
124C22×C4×C47—41no—
125C22×C4×C47—42no—
126C22×C4×C47—41no—
127C22×C4×C47—41no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47—42no—
134C22×C4×C47—42no—
135C22×C4×C47—42no—
136C22×C4×C47C22⋊C412yes—
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D411yescharacteristic
143C22×C4×C47#4112yes—
144C22×C4×C47#4112yes—
145C22×Dic47—24no—
146C22×Dic47—42no—
147C22×Dic47—42no—
148C22×Dic47—42no—
149C22×Dic47—42no—
150C22×Dic47—42no—
151C22×Dic47—42no—
152C4⋊C4×C47—216no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—24no—
155C4⋊C4×C47C22⋊C414yescomplemented
156C22⋊C4×C47—216no—
157C22⋊Dic47—48no—
158C4⋊Dic47—24no—
159C4⋊Dic47—24no—
160C4⋊Dic47—42no—
161C4⋊Dic47—42no—
162C2×C47⋊C8—24no—
163C2×C47⋊C8—24no—
164C2×C47⋊C8—24no—
165C2×C47⋊C8—42no—
166C2×C47⋊C8—42no—
167C2×C47⋊C8—42no—
168C2×C47⋊C8—42no—
169C23×C47—22no—
170C23×C47—22no—
171C23×C47—22no—
172C23×C47—22no—
173C23×C47—22no—
174C23×C47—22no—
175C23×C47—24no—
176C23×C47—22no—
177C23×C47—22no—
178C23×C47—24no—
179C23×C47—24no—
180C23×C47—24no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47#10314yescomplemented
192C23×C47C2×C22⋊C411yescharacteristic
193C23×C47#12011yescharacteristic
194C23×C47#12012yes—
195C23×C47#12112yes—
196C23×C47#12311yescharacteristic
197C23×C47C2×SD1611yescharacteristic
198C23×C47C2×Q1611yescharacteristic
199C23×C47#13512yes—
200C23×C47AGL(1,ℤ/8)11yescharacteristic
201C23×C47#13711yescharacteristic
202C23×C47#13712yes—
203Q8×C47—28no—
204Q8×C47—44no—
205Q8×C47—44no—
206Q8×C47—44no—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—22no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—44no—
216C2×C4×C47—42no—
217C2×C4×C47—44no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—44no—
221C2×C4×C47—44no—
222C2×C4×C47—44no—
223C2×C4×C47—44no—
224C2×C4×C47—44no—
225C2×C4×C47—44no—
226C2×C4×C47C2×C22⋊C411yescharacteristic
227C2×C4×C47C2×C22⋊C411yescharacteristic
228C2×C4×C47C4×D412yes—
229C2×C4×C47C4×D412yes—
230C2×C4×C47#12012yes—
231C2×C4×C47#12011yescharacteristic
232C2×C4×C47#12012yes—
233C2×C4×C47#12112yes—
234C2×C4×C47#12112yes—
235C2×C4×C47#12112yes—
236C2×C4×C47#12112yes—
237C2×C4×C47#12112yes—
238C2×C4×C47#12312yes—
239C2×C4×C47#12311yescharacteristic
240C2×C4×C47#12412yes—
241C2×C4×C47#12712yes—
242C2×Dic47—44no—
243C2×Dic47—44no—
244C2×Dic47—44no—
245C2×Dic47—82no—
246C2×Dic47—82no—
247C2×Dic47—82no—
248C2×Dic47—82no—
249C47⋊C8—44no—
250#10970—471nomaximal, Sylow
251C22×C47—24no—
252C22×C47—24no—
253C22×C47—24no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—44no—
258C22×C47—44no—
259C22×C47—44no—
260C22×C47—44no—
261C22×C47—42no—
262C22×C47#37911yescharacteristic
263C22×C47#38611yescharacteristic
264C22×C47#38611yescharacteristic
265C22×C47#39212yes—
266C22×C47#39412yes—
267C22×C47C2×#10311yescharacteristic
268C22×C47#41611yescharacteristic
269C22×C47#41711yescharacteristic
270C22×C47#41911yescharacteristic
271C22×C47#44812yes—
272C22×C47#44912yes—
273C22×C47#45011yescharacteristic
274C22×C47#45111yescharacteristic
275C22×C47#45112yes—
276C22×C47#45212yes—
277C22×C47#45212yes—
278C22×C47#46512yes—
279C22×C47#46712yes—
280C22×C47#46812yes—
281C22×C47#47012yes—
282C22×C47#48311yescharacteristic
283C22×C47#48411yescharacteristic
284C22×C47#48512yes—
285C4×C47—42no—
286C4×C47—44no—
287C4×C47—44no—
288C4×C47#38612yes—
289C4×C47#39012yes—
290C4×C47#39212yes—
291C4×C47#39412yes—
292Dic47—84no—
293C22×C22⋊Q8D4711yescharacteristic
294C22×#103—472no—
295C22×C22⋊C8—471no—
296C2×#380—471no—
297C2×#340—472no—
298C2×C47—44no—
299C2×C47#143811yescharacteristic
300C2×C47#178711yescharacteristic
301C2×C47#178811yescharacteristic
302C2×C47#187211yescharacteristic
303C2×C47#187311yescharacteristic
304C2×C47#189211yescharacteristic
305C2×C47#189411yescharacteristic
306C2×C47#200012yes—
307C2×C47#200112yes—
308C2×C47#203512yes—
309C2×C47#203612yes—
310C23×Q8C2×D4711yescharacteristic
311C24×C4C2×D4711yescharacteristic
312C23×C8—472no—
313C2×C22⋊Q8—24no—
314C2×C22⋊Q8—22no—
315C2×C22⋊Q8—24no—
316C2×C22⋊Q8Dic4714yescomplemented
317C22×C4⋊C4—21no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4C2×D4711yescharacteristic
320C22×C22⋊C4—21no—
321C22×C22⋊C4—941no—
322C2×#103—944no—
323C2×#103—944no—
324C2×#103—944no—
325C2×C22⋊C8—944no—
326C2×C22⋊C8—942no—
327#380—944no—
328C2×#95—941no—
329#340—944no—
330C47#1097011yescharacteristic, complemented, Sylow
331C25C22⋊D4711yescharacteristic
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8C2×Dic4712yes—
339C22×Q8C22⋊D4714yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—22no—
347C23×C4—22no—
348C23×C4—942no—
349C23×C4C2×Dic4712yes—
350C23×C4C22⋊D4711yescharacteristic
351C23×C4C22⋊D4712yes—
352C23×C4C22×D4711yescharacteristic
353C22×C8—474no—
354C22×C8—942no—
355C22×C8—942no—
356C22×C8—942no—
357C22×C8—942no—
358C22×C8—942no—
359C22×C8—942no—
360C22⋊Q8—216no—
361C22⋊Q8—28no—
362C22⋊Q8—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—474no—
368C2×C4⋊C4—942no—
369C2×C4⋊C4—942no—
370C2×C4⋊C4—942no—
371C2×C4⋊C4—942no—
372C2×C4⋊C4—942no—
373C2×C4⋊C4C2×Dic4712yes—
374C2×C4⋊C4C22⋊D4714yes—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—944no—
378C2×C22⋊C4—1882no—
379C2×C22⋊C4—1882no—
380#103—9416no—
381C22⋊C8—948no—
382#95—944no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24#1062533612yes—
392C24#1062533712yes—
393C24C22⋊Dic4712yes—
394C24C2×C22⋊D4711yescharacteristic
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—41no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—41no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C22⋊Dic4714yescomplemented
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—944no—
442C22×C4—1882no—
443C22×C4—1882no—
444C22×C4—1882no—
445C22×C4—1882no—
446C22×C4—1882no—
447C22×C4—1882no—
448C22×C4C22⋊Dic4712yes—
449C22×C4D4×D4712yes—
450C22×C4D4×D4712yes—
451C22×C4#1062535212yes—
452C22×C4#1062535212yes—
453C22×C4C22×Dic4711yescharacteristic
454C22×C4C2×C22⋊D4712yes—
455C22×C4C2×C22⋊D4712yes—
456C22×C4C2×C22⋊D4711yescharacteristic
457C2×C8—944no—
458C2×C8—944no—
459C2×C8—944no—
460C2×C8—1882no—
461C2×C8—1882no—
462C2×C8—1882no—
463C2×C8—1882no—
464C4⋊C4—216no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—944no—
468C4⋊C4—944no—
469C4⋊C4—1882no—
470C4⋊C4—1882no—
471C4⋊C4C22⋊Dic4714yescomplemented
472C22⋊C4—216no—
473C22⋊C4—1888no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#1309394814yescomplemented
497C23C2×C22⋊Dic4711yescharacteristic
498C23#1309399211yescharacteristic
499C23#1309399312yes—
500C23#1309399612yes—
501C23#1309400511yescharacteristic
502C23C2×#1062533611yescharacteristic
503C23C2×#1062533711yescharacteristic
504C23#1309404412yes—
505C23#1309404911yescharacteristic
506C23#1309405411yescharacteristic
507C23#1309405512yes—
508Q8—28no—
509Q8—44no—
510Q8—44no—
511Q8—44no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—3762no—
532C2×C4—3762no—
533C2×C4—3762no—
534C2×C4—3762no—
535C2×C4—1884no—
536C2×C4—1884no—
537C2×C4—1884no—
538C2×C4C2×C22⋊Dic4711yescharacteristic
539C2×C4C2×C22⋊Dic4711yescharacteristic
540C2×C4D4×Dic4712yes—
541C2×C4D4×Dic4712yes—
542C2×C4#1309399212yes—
543C2×C4#1309399211yescharacteristic
544C2×C4#1309399312yes—
545C2×C4#1309399512yes—
546C2×C4#1309399512yes—
547C2×C4#1309399612yes—
548C2×C4#1309399612yes—
549C2×C4#1309399612yes—
550C2×C4#1309400512yes—
551C2×C4#1309400511yescharacteristic
552C2×C4#1309401112yes—
553C2×C4#1309402012yes—
554C8—1884no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22#1568897911yescharacteristic
567C22#1568900311yescharacteristic
568C22#1568900311yescharacteristic
569C22#1568903112yes—
570C22#1568903712yes—
571C22C2×#1309394811yescharacteristic
572C22#1568911911yescharacteristic
573C22#1568912411yescharacteristic
574C22#1568913211yescharacteristic
575C22#1568926212yes—
576C22#1568926812yes—
577C22#1568927311yescharacteristic
578C22#1568927811yescharacteristic
579C22#1568927912yes—
580C22#1568928312yes—
581C22#1568928512yes—
582C22#1568935412yes—
583C22#1568936212yes—
584C22#1568937112yes—
585C22#1568937912yes—
586C22#1568945111yescharacteristic
587C22#1568945611yescharacteristic
588C22#1568946112yes—
589C4—44no—
590C4—42no—
591C4—44no—
592C4—3764no—
593C4#1568900312yes—
594C4#1568902612yes—
595C4#1568903112yes—
596C4#1568903712yes—
597C2—44no—
598C2#2024162911yescharacteristic
599C2#2024332311yescharacteristic
600C2#2024332811yescharacteristic
601C2#2024369611yescharacteristic
602C2#2024370111yescharacteristic
603C2#2024381211yescharacteristic
604C2#2024382211yescharacteristic
605C2#2024438912yes—
606C2#2024439612yes—
607C2#2024458112yes—
608C2#2024458812yes—
subgroupG/H
centerC24C2×C22⋊D47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC22×C22⋊Q8×C47C2
socleC24×C47C2×D4
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC22×C22⋊Q8D47
hypocenterC47#10970
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13093948C2semidirect
4. C2×#15688885C2semidirect
22. C2×C22⋊Q8×C47C4semidirect
105. C2×Q8×C47C22⋊C4semidirect
155. C4⋊C4×C47C22⋊C4semidirect
191. C23×C47#103semidirect
316. C2×C22⋊Q8Dic47semidirect
330. C47#10970semidirect
410. C2×Q8C22⋊Dic47semidirect
471. C4⋊C4C22⋊Dic47semidirect
496. C23#13093948semidirect

Maximal subgroups

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

subgroupindex
C22×#130939482
C22×C22⋊(C47⋊C8)2
C2×#156889832
C2×#156888852
C22×C22⋊Q8×C472
#1097047