Finite Group Databasebeta

← #26111650

Subgroups — #26111650

Order 12032.

1136 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
2#20243298C211yesmaximal, characteristic, complemented
3C2×#15688847C212yesmaximal, complemented
4#20241231C212yesmaximal, complemented
5C2×C47×#386C211yesmaximal, characteristic
6C23×C47⋊C8—21no—
7C2×C22⋊(C47⋊C8)—24no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)—21no—
10C2×C22⋊(C47⋊C8)—21no—
11C2×C22⋊(C47⋊C8)C2212yes—
12C2×C22⋊(C47⋊C8)C2212yes—
13#15688847—24no—
14C23×D4×C47C411yescharacteristic
15C24×C4×C47C2211yescharacteristic
16C22×C22⋊C4×C47C411yescharacteristic
17C22×C22⋊C4×C47C411yescharacteristic
18C22×C22⋊C4×C47C2211yescharacteristic
19C47×#386—21no—
20C47×#386—21no—
21C47×#386—21no—
22C47×#386—21no—
23C2×#95×C47C411yescharacteristic
24C2×#95×C47C2211yescharacteristic
25C22×C47⋊C8—24no—
26C22×C47⋊C8—22no—
27C22×C47⋊C8—22no—
28C22×C47⋊C8—42no—
29C22×C47⋊C8—42no—
30C22⋊(C47⋊C8)—24no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—42no—
35C22⋊(C47⋊C8)—42no—
36C22⋊(C47⋊C8)D414yescomplemented
37C25×C47C2×C411yescharacteristic
38C25×C47C2×C411yescharacteristic
39C22×D4×C47—21no—
40C22×D4×C47—21no—
41C22×D4×C47—22no—
42C22×D4×C47—42no—
43C22×D4×C47—42no—
44C22×D4×C47C814yescomplemented
45C23×C4×C47—22no—
46C23×C4×C47—22no—
47C23×C4×C47—42no—
48C23×C4×C47—42no—
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47C2×C411yescharacteristic
52C23×C4×C47C2×C411yescharacteristic
53C23×C4×C47D411yescharacteristic
54C23×C4×C47D411yescharacteristic
55C23×C4×C47D411yescharacteristic
56C23×C4×C47D411yescharacteristic
57C23×C4×C47C2311yescharacteristic
58C2×C22⋊C4×C47—21no—
59C2×C22⋊C4×C47—21no—
60C2×C22⋊C4×C47—21no—
61C2×C22⋊C4×C47—21no—
62C2×C22⋊C4×C47—22no—
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—22no—
66C2×C22⋊C4×C47—42no—
67C2×C22⋊C4×C47—42no—
68#95×C47—21no—
69#95×C47—21no—
70#95×C47—21no—
71#95×C47—21no—
72C24×C47—22no—
73C24×C47—22no—
74C24×C47—22no—
75C24×C47—22no—
76C24×C47—21no—
77C24×C47—21no—
78C24×C47—22no—
79C24×C47—22no—
80C24×C47—41no—
81C24×C47—41no—
82C24×C47—41no—
83C24×C47—41no—
84C24×C47—41no—
85C24×C47—41no—
86C24×C47C22⋊C411yescharacteristic
87C24×C47C22⋊C411yescharacteristic
88C24×C47C2×C812yes—
89C24×C47C2×C812yescomplemented
90C24×C47M1612yes—
91C24×C47C22×C411yescharacteristic
92C2×D4×C47—24no—
93C2×D4×C47—24no—
94C2×D4×C47—48no—
95C2×D4×C47—42no—
96C2×D4×C47—42no—
97C2×D4×C47—42no—
98C2×D4×C47—42no—
99C2×D4×C47—48no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47—22no—
107C22×C4×C47—21no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—41no—
119C22×C4×C47—41no—
120C22×C4×C47—41no—
121C22×C4×C47—41no—
122C22×C4×C47—41no—
123C22×C4×C47—41no—
124C22×C4×C47—41no—
125C22×C4×C47—48no—
126C22×C4×C47—48no—
127C22×C4×C47—41no—
128C22×C4×C47—41no—
129C22×C4×C47—41no—
130C22×C4×C47—41no—
131C22×C4×C47—41no—
132C22×C4×C47—41no—
133C22×C4×C47C22⋊C411yescharacteristic
134C22×C4×C47C22⋊C411yescharacteristic
135C22×C4×C47C2×C812yescomplemented
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×C4×C47#4112yes—
140C22⋊C4×C47—42no—
141C22⋊C4×C47—42no—
142C22⋊C4×C47—42no—
143C22⋊C4×C47—42no—
144C22⋊C4×C47—41no—
145C22⋊C4×C47—42no—
146C22⋊C4×C47—41no—
147C2×C47⋊C8—24no—
148C2×C47⋊C8—48no—
149C2×C47⋊C8—42no—
150C2×C47⋊C8—42no—
151C2×C47⋊C8—42no—
152C2×C47⋊C8—42no—
153C2×C47⋊C8—42no—
154C23×C47—21no—
155C23×C47—21no—
156C23×C47—21no—
157C23×C47—21no—
158C23×C47—21no—
159C23×C47—21no—
160C23×C47—21no—
161C23×C47—21no—
162C23×C47—24no—
163C23×C47—24no—
164C23×C47—21no—
165C23×C47—21no—
166C23×C47—21no—
167C23×C47—21no—
168C23×C47—21no—
169C23×C47—21no—
170C23×C47—24no—
171C23×C47—24no—
172C23×C47—22no—
173C23×C47—24no—
174C23×C47—42no—
175C23×C47—42no—
176C23×C47—44no—
177C23×C47—44no—
178C23×C47—44no—
179C23×C47—44no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—41no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47—41no—
188C23×C47—41no—
189C23×C47—41no—
190C23×C47—42no—
191C23×C47—82no—
192C23×C47—82no—
193C23×C47—82no—
194C23×C47—82no—
195C23×C47C22⋊C812yes—
196C23×C47C22⋊C812yes—
197C23×C47#9912yes—
198C23×C47#10012yes—
199C23×C47C2×C22⋊C411yescharacteristic
200C23×C47C4×D412yes—
201C23×C47C22×C811yescharacteristic
202C23×C47C2×M1611yescharacteristic
203C23×C47#13112yes—
204D4×C47—48no—
205D4×C47—48no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—22no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—44no—
221C2×C4×C47—44no—
222C2×C4×C47—44no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—42no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—82no—
236C2×C4×C47—82no—
237C2×C4×C47#9912yes—
238C2×C4×C47#12011yescharacteristic
239C2×C4×C47#12112yes—
240C2×C4×C47#12311yescharacteristic
241C47⋊C8—44no—
242C47⋊C8—82no—
243#9079—471nomaximal, Sylow
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47—22no—
248C22×C47—22no—
249C22×C47—21no—
250C22×C47—21no—
251C22×C47—21no—
252C22×C47—21no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—48no—
262C22×C47—48no—
263C22×C47—41no—
264C22×C47—41no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—42no—
268C22×C47—42no—
269C22×C47—42no—
270C22×C47—84no—
271C22×C47—84no—
272C22×C47—81no—
273C22×C47—81no—
274C22×C47—81no—
275C22×C47—81no—
276C22×C47#32314yescomplemented
277C22×C47#38611yescharacteristic
278C22×C47C2×C22⋊C811yescharacteristic
279C22×C47#40811yescharacteristic
280C22×C47C2×#9911yescharacteristic
281C22×C47C2×#9911yescharacteristic
282C22×C47C2×#10011yescharacteristic
283C22×C47#41311yescharacteristic
284C22×C47C8×D412yes—
285C22×C47#43512yes—
286C4×C47—21no—
287C4×C47—22no—
288C4×C47—22no—
289C4×C47—21no—
290C4×C47—41no—
291C4×C47—42no—
292C4×C47—81no—
293C4×C47—81no—
294C22×C22⋊C8—471no—
295C2×#386D4711yescharacteristic
296#1779—471no—
297C2×#323—472no—
298#1280—472no—
299C2×C47—22no—
300C2×C47—22no—
301C2×C47—24no—
302C2×C47—42no—
303C2×C47—82no—
304C2×C47C2×#32311yescharacteristic
305C2×C47#145911yescharacteristic
306C2×C47#146811yescharacteristic
307C2×C47#147111yescharacteristic
308C2×C47#186911yescharacteristic
309C2×C47#189811yescharacteristic
310C2×C47#190011yescharacteristic
311C23×D4Dic4711yescharacteristic
312C24×C4C2×D4711yescharacteristic
313C23×C8—941no—
314C22×C22⋊C4Dic4711yescharacteristic
315C22×C22⋊C4Dic4711yescharacteristic
316C22×C22⋊C4C2×D4711yescharacteristic
317C2×C22⋊C8—472no—
318C2×C22⋊C8—472no—
319C2×C22⋊C8—944no—
320C2×C22⋊C8—941no—
321C2×C22⋊C8—941no—
322C2×C22⋊C8—941no—
323#386—21no—
324#386—21no—
325#386—21no—
326#386—21no—
327C2×#95Dic4711yescharacteristic
328C2×#95C2×D4711yescharacteristic
329#323—944no—
330C47#907911yescharacteristic, complemented, Sylow
331C25C2×Dic4711yescharacteristic
332C25C2×Dic4711yescharacteristic
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—42no—
337C22×D4—42no—
338C22×D4C47⋊C814yescomplemented
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4C2×Dic4711yescharacteristic
344C23×C4C2×Dic4711yescharacteristic
345C23×C4C2×Dic4711yescharacteristic
346C23×C4C2×Dic4711yescharacteristic
347C23×C4C22⋊D4711yescharacteristic
348C23×C4C22⋊D4711yescharacteristic
349C23×C4C22⋊D4711yescharacteristic
350C23×C4C22⋊D4711yescharacteristic
351C23×C4C22×D4711yescharacteristic
352C22×C8—944no—
353C22×C8—942no—
354C22×C8—942no—
355C22×C8—1882no—
356C22×C8—1882no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C22⋊C8—474no—
368C22⋊C8—944no—
369C22⋊C8—942no—
370C22⋊C8—942no—
371C22⋊C8—942no—
372C22⋊C8—1882no—
373C22⋊C8—1882no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24C2×C47⋊C812yes—
393C24C2×C47⋊C812yescomplemented
394C24#1062532912yes—
395C24C22⋊Dic4711yescharacteristic
396C24C22⋊Dic4711yescharacteristic
397C24C22×Dic4711yescharacteristic
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C2×C47⋊C812yescomplemented
440C22×C4C22⋊Dic4711yescharacteristic
441C22×C4C22⋊Dic4711yescharacteristic
442C22×C4D4×D4712yes—
443C22×C4#1062535212yes—
444C22×C4C2×C22⋊D4711yescharacteristic
445C22×C4C2×C22⋊D4711yescharacteristic
446C2×C8—944no—
447C2×C8—1882no—
448C2×C8—1882no—
449C2×C8—1882no—
450C2×C8—1888no—
451C2×C8—1882no—
452C2×C8—1882no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—24no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23C22⋊(C47⋊C8)12yes—
502C23C22⋊(C47⋊C8)12yes—
503C23#1309393912yes—
504C23#1309394112yes—
505C23C2×C22⋊Dic4711yescharacteristic
506C23D4×Dic4712yes—
507C23C22×C47⋊C811yescharacteristic
508C23C2×#1062532911yescharacteristic
509C23#1309403112yes—
510D4—48no—
511D4—48no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#1309393912yes—
544C2×C4#1309399211yescharacteristic
545C2×C4#1309399612yes—
546C2×C4#1309400511yescharacteristic
547C8—3762no—
548C8—1884no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—48no—
567C22—48no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—84no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22#1568884714yescomplemented
582C22#1568900311yescharacteristic
583C22C2×C22⋊(C47⋊C8)11yescharacteristic
584C22#1568909011yescharacteristic
585C22C2×#1309393911yescharacteristic
586C22C2×#1309393911yescharacteristic
587C22C2×#1309394111yescharacteristic
588C22#1568910711yescharacteristic
589C22D4×C47⋊C812yes—
590C22#1568919512yes—
591C4—22no—
592C4—21no—
593C4—22no—
594C4—21no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—42no—
603C2—82no—
604C2C2×#1568884711yescharacteristic
605C2#2024170611yescharacteristic
606C2#2024173611yescharacteristic
607C2#2024174811yescharacteristic
608C2#2024368111yescharacteristic
609C2#2024384211yescharacteristic
610C2#2024385211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC2×C47×#386C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D4×C47C4
cosocleC23×C4×C47C23
hypercenterC2×#386D47
hypocenterC47#9079
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C47⋊C8)C2semidirect
2. #20243298C2semidirect
3. C2×#15688847C2semidirect
4. #20241231C2semidirect
36. C22⋊(C47⋊C8)D4semidirect
44. C22×D4×C47C8semidirect
89. C24×C47C2×C8semidirect
135. C22×C4×C47C2×C8semidirect
276. C22×C47#323semidirect
330. C47#9079semidirect
338. C22×D4C47⋊C8semidirect
393. C24C2×C47⋊C8semidirect
439. C22×C4C2×C47⋊C8semidirect
581. C22#15688847semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C47⋊C8)2
#202432982
C2×#156888472
#202412312
C2×C47×#3862
#907947