Finite Group Databasebeta

← #26139810

Subgroups — #26139810

Order 12032.

1326 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689021C211yesmaximal, characteristic
2C2×#15688988C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic
4#20241648C212yesmaximal
5#20241643C212yesmaximal
6#20241619C212yesmaximal
7#20241617C211yesmaximal, characteristic
8#20241617C211yesmaximal, characteristic
9#20241613C212yesmaximal
10C4×C47⋊#95C211yesmaximal, characteristic
11C47×C4⋊#95C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×C4×Dic47C2212yes—
16C22×C4×Dic47C2211yescharacteristic
17C22×C4×Dic47C2211yescharacteristic
18#15689021—24no—
19#15688988C418yescomplemented
20C4⋊C4×Dic47—24no—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C22×C4⋊C4×C47C2211yescharacteristic
34C22×C4⋊C4×C47C2211yescharacteristic
35C22×C42×C47C2211yescharacteristic
36C2×#95×C47C2212yes—
37C2×#95×C47C2212yes—
38C23×Dic47C2311yescharacteristic
39C23×Dic47C2311yescharacteristic
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47C2×C414yes—
49C2×C4⋊Dic47D414yes—
50C2×Dic47⋊C4—24no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—42no—
53C2×C4×Dic47—22no—
54C2×C4×Dic47—22no—
55C2×C4×Dic47—22no—
56C2×C4×Dic47—22no—
57C2×C4×Dic47—24no—
58C2×C4×Dic47—24no—
59C2×C4×Dic47—42no—
60C2×C4×Dic47C2×C414yes—
61C2×C4×Dic47Q814yes—
62C47⋊#95—22no—
63C47⋊#95—22no—
64C47⋊#95—24no—
65C47⋊#95—24no—
66C47⋊#95—22no—
67C47⋊#95—24no—
68C47⋊#95—22no—
69C47⋊#95—42no—
70C47⋊#95—42no—
71C47⋊#95—42no—
72C47⋊#95C2×C418yes—
73C47⋊#95C2×C418yes—
74C23×C4×C47C2312yes—
75C23×C4×C47C2312yes—
76C23×C4×C47C2311yescharacteristic
77C23×C4×C47C2311yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47—22no—
84C2×C4⋊C4×C47C2×C414yes—
85C2×C42×C47—22no—
86C2×C42×C47—22no—
87C2×C42×C47—22no—
88#95×C47—24no—
89#95×C47—42no—
90C24×C47C2411yescharacteristic
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—21no—
105C22×C4×C47—21no—
106C22×C4×C47—22no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47C22×C414yes—
117C22×C4×C47C22×C412yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2×Q812yes—
120C22×C4×C47#4112yes—
121C22×C4×C47#4112yes—
122C22×C4×C47#4112yes—
123C22×C4×C47#4112yes—
124C22×C4×C47#4112yes—
125C22×C4×C47#4112yes—
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—21no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—21no—
160C22×Dic47C22×C412yes—
161C22×Dic47C22×C412yes—
162C22×Dic47C22×C414yes—
163C22×Dic47C2×D414yes—
164C22×Dic47C2×Q812yes—
165C22×Dic47C2×Q812yes—
166C22×Dic47#4112yes—
167C22×Dic47#4112yes—
168C4⋊C4×C47—28no—
169C4⋊C4×C47—44no—
170C42×C47—44no—
171C4⋊Dic47—28no—
172C4⋊Dic47—28no—
173Dic47⋊C4—48no—
174C4×Dic47—28no—
175C4×Dic47—28no—
176C4×Dic47—48no—
177C23×C47C23×C411yescharacteristic
178C23×C47C22×D411yescharacteristic
179C23×C47C22×Q811yescharacteristic
180C23×C47C2×#4111yescharacteristic
181C23×C47C2×#4111yescharacteristic
182C23×C47C2×#4111yescharacteristic
183C23×C47C2×#4111yescharacteristic
184C23×C47C2×#4111yescharacteristic
185C23×C47C2×#4111yescharacteristic
186C23×C47C2×#4111yescharacteristic
187C23×C47C2×#4111yescharacteristic
188C23×C47H(5,𝔽2)11yescharacteristic
189C23×C47H(5,𝔽2)11yescharacteristic
190C23×C472−1+411yescharacteristic
191C23×C472−1+411yescharacteristic
192C2×C4×C47—24no—
193C2×C4×C47—24no—
194C2×C4×C47—24no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—41no—
204C2×C4×C47—41no—
205C2×C4×C47—42no—
206C2×C4×C47—41no—
207C2×C4×C47—42no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—42no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47#11714yes—
221C2×C4×C47C4×D414yes—
222C2×C4×C47C4×Q814yes—
223C2×C4×C47C22⋊Q814yes—
224C2×C4×C47#12314yes—
225C2×C4×C47#12514yes—
226C2×C4×C47#12614yes—
227C2×Dic47—22no—
228C2×Dic47—22no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—24no—
239C2×Dic47—22no—
240C2×Dic47—22no—
241C2×Dic47—24no—
242C2×Dic47—22no—
243C2×Dic47—24no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—42no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C2×Dic47—41no—
257C2×Dic47—42no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—42no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47#12514yes—
269#14030—471nomaximal, Sylow
270C22×C47C2×#11711yescharacteristic
271C22×C47C2×C4×D411yescharacteristic
272C22×C47C2×C4×Q811yescharacteristic
273C22×C47C4×#4111yescharacteristic
274C22×C47C4×#4111yescharacteristic
275C22×C47#52011yescharacteristic
276C22×C47#52011yescharacteristic
277C22×C47C2×C22⋊Q811yescharacteristic
278C22×C47C2×#12311yescharacteristic
279C22×C47C2×#12511yescharacteristic
280C22×C47C2×#12511yescharacteristic
281C22×C47C2×#12611yescharacteristic
282C22×C47#52911yescharacteristic
283C22×C47#52911yescharacteristic
284C22×C47#53311yescharacteristic
285C22×C47#54011yescharacteristic
286C22×C47#54111yescharacteristic
287C22×C47#54411yescharacteristic
288C22×C47#54411yescharacteristic
289C22×C47#54611yescharacteristic
290C22×C47#54611yescharacteristic
291C22×C47#54811yescharacteristic
292C22×C47#54811yescharacteristic
293C22×C47#55211yescharacteristic
294C22×C47#55211yescharacteristic
295C22×C47#55311yescharacteristic
296C22×C47#55311yescharacteristic
297C22×C47#55511yescharacteristic
298C22×C47#55511yescharacteristic
299C22×C47#55611yescharacteristic
300C22×C47#55611yescharacteristic
301C22×C47#55611yescharacteristic
302C22×C47#55611yescharacteristic
303C22×C47#55911yescharacteristic
304C22×C47#55911yescharacteristic
305C4×C47—44no—
306C4×C47—42no—
307C4×C47—42no—
308C4×C47—82no—
309C4×C47#38718yes—
310Dic47—24no—
311Dic47—42no—
312Dic47—42no—
313Dic47—44no—
314Dic47—44no—
315Dic47—42no—
316C2×#389—471no—
317C2×#382—471no—
318C2×C4×C4⋊C4—471no—
319#1443—472no—
320#1442—472no—
321#1436—472no—
322C4⋊#95—471no—
323C4⋊#95—471no—
324C4⋊#95D4711yescharacteristic
325#1434—472no—
326C4×#95—471no—
327C2×C47C2×#38711yescharacteristic
328C2×C47C4×#12511yescharacteristic
329C2×C47#233011yescharacteristic
330C2×C47#234611yescharacteristic
331C2×C47#234711yescharacteristic
332C2×C47#236711yescharacteristic
333C2×C47#236711yescharacteristic
334C2×C47#242011yescharacteristic
335C2×C47#242111yescharacteristic
336C2×C47#243911yescharacteristic
337C2×C47#244211yescharacteristic
338C2×C47#251311yescharacteristic
339C2×C47#251411yescharacteristic
340C2×C47#257011yescharacteristic
341C2×C47#257111yescharacteristic
342C22×C4⋊C4—471no—
343C22×C4⋊C4—471no—
344C22×C4⋊C4—472no—
345C22×C4⋊C4C2×D4711yescharacteristic
346C22×C4⋊C4C2×D4711yescharacteristic
347C22×C42—471no—
348C22×C42—471no—
349C22×C42—472no—
350C22×C42C2×D4711yescharacteristic
351#389—944no—
352#382—478no—
353C4×C4⋊C4—944no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—471no—
357C2×#95—471no—
358C2×#95—471no—
359C2×#95—471no—
360C2×#95—472no—
361C2×#95—472no—
362C2×#95—472no—
363C2×#95—472no—
364C2×#95—472no—
365C2×#95—472no—
366C2×#95C2×D4712yes—
367C2×#95C2×D4712yes—
368C47#1403011yescharacteristic, complemented, Sylow
369C23×C4—471no—
370C23×C4—472no—
371C23×C4—472no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—471no—
375C23×C4C22×D4712yes—
376C23×C4C22×D4712yes—
377C23×C4C22×D4711yescharacteristic
378C23×C4C22×D4711yescharacteristic
379C23×C4C22×D4711yescharacteristic
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—474no—
386C2×C4⋊C4—474no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—942no—
389C2×C4⋊C4—942no—
390C2×C4⋊C4—942no—
391C2×C4⋊C4—944no—
392C2×C4⋊C4—944no—
393C2×C4⋊C4—1882no—
394C2×C4⋊C4C4×D4714yes—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—474no—
399C2×C42—474no—
400C2×C42—942no—
401C2×C42—942no—
402C2×C42—942no—
403C2×C42—942no—
404C2×C42—944no—
405C2×C42—944no—
406C2×C42—1882no—
407#95—24no—
408#95—42no—
409#95—478no—
410#95—478no—
411#95—942no—
412#95—942no—
413#95—944no—
414#95—944no—
415#95—944no—
416#95—942no—
417#95—942no—
418#95—1882no—
419#95—1882no—
420#95—1882no—
421C24C23×D4711yescharacteristic
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—472no—
448C22×C4—472no—
449C22×C4—472no—
450C22×C4—472no—
451C22×C4—472no—
452C22×C4—474no—
453C22×C4—474no—
454C22×C4—472no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—942no—
461C22×C4—941no—
462C22×C4—942no—
463C22×C4—941no—
464C22×C4—942no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—941no—
470C22×C4—941no—
471C22×C4—941no—
472C22×C4—941no—
473C22×C4—941no—
474C22×C4—942no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—941no—
480C22×C4—942no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—942no—
488C22×C4—941no—
489C22×C4C2×C4×D4714yes—
490C22×C4C2×C4×D4712yes—
491C22×C4#1062535012yes—
492C22×C4#1062535012yes—
493C22×C4D4×D4712yes—
494C22×C4#1062535212yes—
495C22×C4#1062535212yes—
496C22×C4#1062535212yes—
497C22×C4Q8×D4712yes—
498C22×C4#1062535412yes—
499C4⋊C4—28no—
500C4⋊C4—44no—
501C4⋊C4—948no—
502C4⋊C4—948no—
503C4⋊C4—1888no—
504C42—44no—
505C42—948no—
506C42—948no—
507C42—1888no—
508C23C22×C4×D4711yescharacteristic
509C23C2×D4×D4711yescharacteristic
510C23C2×Q8×D4711yescharacteristic
511C23#41×D4711yescharacteristic
512C23#41×D4711yescharacteristic
513C23C2×#1062535211yescharacteristic
514C23C2×#1062535211yescharacteristic
515C23C2×#1062535211yescharacteristic
516C23C2×#1062535011yescharacteristic
517C23C2×#1062535011yescharacteristic
518C23C2×#1062535411yescharacteristic
519C23#1309406911yescharacteristic
520C23#1309407011yescharacteristic
521C23#1309407111yescharacteristic
522C23#1309407211yescharacteristic
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—474no—
552C2×C4—942no—
553C2×C4—942no—
554C2×C4—942no—
555C2×C4—942no—
556C2×C4—942no—
557C2×C4—942no—
558C2×C4—942no—
559C2×C4—944no—
560C2×C4—944no—
561C2×C4—942no—
562C2×C4—942no—
563C2×C4—944no—
564C2×C4—942no—
565C2×C4—942no—
566C2×C4—944no—
567C2×C4—942no—
568C2×C4—944no—
569C2×C4—942no—
570C2×C4—942no—
571C2×C4—944no—
572C2×C4—942no—
573C2×C4—1882no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1882no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1882no—
581C2×C4—1881no—
582C2×C4—1882no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4#1309398114yes—
594C2×C4(C22⋊D47)⋊C414yes—
595C2×C4#1309398914yes—
596C2×C4#1309400214yes—
597C2×C4#1309400614yes—
598C2×C4#1309401414yes—
599C2×C4#1309401714yes—
600C22C2×#1309398111yescharacteristic
601C22C2×(C22⋊D47)⋊C411yescharacteristic
602C22C2×#1309398911yescharacteristic
603C22C4×#1062535211yescharacteristic
604C22C4×#1062535411yescharacteristic
605C22#1568960211yescharacteristic
606C22#1568960511yescharacteristic
607C22C2×#1309400211yescharacteristic
608C22C2×#1309400611yescharacteristic
609C22#125×D4711yescharacteristic
610C22C2×#1309401411yescharacteristic
611C22C2×#1309401711yescharacteristic
612C22#1568965411yescharacteristic
613C22#1568965911yescharacteristic
614C22#1568967811yescharacteristic
615C22#1568972611yescharacteristic
616C22#1568973311yescharacteristic
617C22#1568975011yescharacteristic
618C22#1568975311yescharacteristic
619C22#1568976411yescharacteristic
620C22#1568977011yescharacteristic
621C22#1568977911yescharacteristic
622C22#1568978311yescharacteristic
623C22#1568980811yescharacteristic
624C22#1568981411yescharacteristic
625C22#1568982111yescharacteristic
626C22#1568982311yescharacteristic
627C22#1568983611yescharacteristic
628C22#1568983811yescharacteristic
629C22#1568984111yescharacteristic
630C22#1568984211yescharacteristic
631C22#1568984611yescharacteristic
632C22#1568984711yescharacteristic
633C22#1568985511yescharacteristic
634C22#1568985911yescharacteristic
635C4—42no—
636C4—42no—
637C4—44no—
638C4—82no—
639C4—944no—
640C4—1882no—
641C4—1882no—
642C4—1884no—
643C4—1884no—
644C4—1882no—
645C4#1568900518yes—
646C2C2×#1568900511yescharacteristic
647C2#2024594711yescharacteristic
648C2#2024612811yescharacteristic
649C2#2024626911yescharacteristic
650C2#2024627711yescharacteristic
651C2#2024645811yescharacteristic
652C2#2024646411yescharacteristic
653C2#2024688211yescharacteristic
654C2#2024689311yescharacteristic
655C2#2024708111yescharacteristic
656C2#2024711411yescharacteristic
657C2#2024798311yescharacteristic
658C2#2024799411yescharacteristic
659C2#2024861811yescharacteristic
660C2#2024862811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C4⋊#95C2
socle, prime radical, cosocleC24×C47C24
hypercenterC4⋊#95D47
hypocenterC47#14030
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
19. #15688988C4semidirect
368. C47#14030semidirect

Maximal subgroups

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

subgroupindex
C2×#156890212
C2×#156889882
C2×C4⋊C4×Dic472
#202416482
#202416432
#202416192
#202416172
#202416132
C4×C47⋊#952
C47×C4⋊#952
#1403047