Finite Group Databasebeta

← #26138372

Subgroups — #26138372

Order 12032.

1746 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689014C212yesmaximal
2C2×#15688982C211yesmaximal, characteristic
3C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
4#20241651C212yesmaximal, complemented
5#20241630C212yesmaximal
6#20241621C212yesmaximal, complemented
7Dic47⋊#95C211yesmaximal, characteristic, complemented
8#20241615C211yesmaximal, characteristic, complemented
9#20241613C211yesmaximal, characteristic, complemented
10(C2×D47)⋊#95C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×Dic47⋊C4C2212yes—
14C22×Dic47⋊C4C2212yes—
15C22×(C2×D47)⋊C4C2212yes—
16C22×(C2×D47)⋊C4C2212yes—
17C22×(C2×D47)⋊C4C2211yescharacteristic
18C22×(C2×D47)⋊C4C2211yescharacteristic
19C22×C4×Dic47C2211yescharacteristic
20#15689014—28no—
21#15688982—24no—
22C4×(C2×D47)⋊C4C418yescomplemented
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2211yescharacteristic
33C22×C4⋊C4×C47C2212yes—
34C22×C42×C47C2212yes—
35C22×C42×C47C2211yescharacteristic
36C4×C4⋊C4×C47—24no—
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2211yescharacteristic
39C24×D47C2311yescharacteristic
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2311yescharacteristic
45C22×C4×D47—22no—
46C22×C4×D47—22no—
47C22×C4×D47—24no—
48C22×C4×D47—42no—
49C22×C4×D47C2×C414yes—
50C2×Dic47⋊C4—24no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—24no—
54C2×Dic47⋊C4—42no—
55C2×Dic47⋊C4—42no—
56C2×(C2×D47)⋊C4—24no—
57C2×(C2×D47)⋊C4—24no—
58C2×(C2×D47)⋊C4—24no—
59C2×(C2×D47)⋊C4—24no—
60C2×(C2×D47)⋊C4—24no—
61C2×(C2×D47)⋊C4—42no—
62C2×(C2×D47)⋊C4—42no—
63C2×(C2×D47)⋊C4C2×C414yes—
64C2×(C2×D47)⋊C4C2×C414yes—
65C2×(C2×D47)⋊C4D414yes—
66C2×(C2×D47)⋊C4Q814yes—
67C2×C4×Dic47—42no—
68C2×C4×Dic47C2×C414yes—
69C47⋊#95—22no—
70C47⋊#95—24no—
71C47⋊#95—24no—
72C47⋊#95—22no—
73C47⋊#95—24no—
74C47⋊#95—24no—
75C47⋊#95—42no—
76C47⋊#95—42no—
77C47⋊#95C2×C414yes—
78C47⋊#95C2×C414yes—
79C23×C4×C47C2312yes—
80C23×C4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82C23×C4×C47C2311yescharacteristic
83C23×C4×C47C2311yescharacteristic
84C2×C4⋊C4×C47—24no—
85C2×C4⋊C4×C47—24no—
86C2×C4⋊C4×C47—24no—
87C2×C42×C47—22no—
88C2×C42×C47—22no—
89C2×C42×C47—24no—
90C2×C42×C47—24no—
91C2×C42×C47—42no—
92C2×C42×C47C2×C414yes—
93#95×C47—22no—
94#95×C47D414yes—
95C24×C47C2411yescharacteristic
96C23×D47—21no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47—22no—
100C23×D47—21no—
101C23×D47—21no—
102C23×D47—22no—
103C23×D47—21no—
104C23×D47C22×C412yes—
105C23×D47C2×D412yes—
106C23×D47C2×Q812yes—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47—21no—
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—22no—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C22×C412yes—
131C22×C4×C47C2×D414yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×Q812yes—
135C22×C4×C47#4114yes—
136C22×C4×C47#4114yes—
137C22×C4×C47#4112yes—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—21no—
145C22×Dic47—22no—
146C22×Dic47—21no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47C22×C412yes—
162C22×Dic47C22×C412yes—
163C22×Dic47C22×C412yes—
164C22×Dic47C2×D412yes—
165C22×Dic47C2×Q812yes—
166C22×Dic47#4114yes—
167C2×C4×D47—28no—
168C2×C4×D47—28no—
169C2×C4×D47—44no—
170C2×C4×D47—48no—
171C2×C4×D47—48no—
172C4⋊C4×C47—216no—
173C42×C47—28no—
174C42×C47—48no—
175(C2×D47)⋊C4—28no—
176(C2×D47)⋊C4—48no—
177(C2×D47)⋊C4—48no—
178(C2×D47)⋊C4C4⋊C4116yescomplemented
179Dic47⋊C4—48no—
180Dic47⋊C4—48no—
181C4×Dic47—44no—
182C23×C47C23×C411yescharacteristic
183C23×C47C22×D411yescharacteristic
184C23×C47C22×D411yescharacteristic
185C23×C47C22×Q811yescharacteristic
186C23×C47C2×#4112yes—
187C23×C47C2×#4112yes—
188C23×C47C2×#4111yescharacteristic
189C23×C47C2×#4112yes—
190C23×C47H(5,𝔽2)11yescharacteristic
191C23×C47H(5,𝔽2)12yes—
192C23×C472−1+411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—24no—
200C22×D47—24no—
201C22×D47—22no—
202C22×D47—24no—
203C22×D47—22no—
204C22×D47—22no—
205C22×D47—22no—
206C22×D47—24no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—42no—
212C22×D47—42no—
213C22×D47—42no—
214C22×D47—42no—
215C22×D47C2×C4⋊C414yes—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—22no—
225C2×C4×C47—24no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—24no—
229C2×C4×C47—22no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
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—42no—
243C2×C4×C47—42no—
244C2×C4×C47C2×C4⋊C414yes—
245C2×C4×C47#11714yes—
246C2×C4×C47C4×D414yes—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—24no—
252C2×Dic47—24no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47C2×C4⋊C414yes—
275C2×Dic47#11714yes—
276C4×D47—416no—
277#13891—471nomaximal, Sylow
278C22×C47C22×C4⋊C411yescharacteristic
279C22×C47C2×#11711yescharacteristic
280C22×C47C2×#11711yescharacteristic
281C22×C47C2×C4×D411yescharacteristic
282C22×C47C4×#4111yescharacteristic
283C22×C47#51811yescharacteristic
284C22×C47#52011yescharacteristic
285C22×C47#52512yes—
286C22×C47#52912yes—
287C22×C47#53212yes—
288C22×C47#53312yes—
289C22×C47#54012yes—
290C22×C47#54212yes—
291C22×C47D4211yescharacteristic
292C22×C47#54612yes—
293C22×C47#54612yes—
294C22×C47#54711yescharacteristic
295C22×C47D4×Q811yescharacteristic
296C22×C47#55112yes—
297C22×C47#55312yes—
298C22×C47#55411yescharacteristic
299C22×C47#55412yes—
300C22×C47#55612yes—
301C2×D47—24no—
302C2×D47—24no—
303C2×D47—24no—
304C2×D47—42no—
305C2×D47—42no—
306C2×D47—42no—
307C2×D47—44no—
308C2×D47—44no—
309C2×D47—44no—
310C2×D47—44no—
311C2×D47—42no—
312C4×C47—24no—
313C4×C47—24no—
314C4×C47—24no—
315C4×C47—44no—
316C4×C47—44no—
317Dic47—24no—
318Dic47—24no—
319Dic47—42no—
320Dic47—82no—
321Dic47—82no—
322C2×#388—472no—
323C2×#380—471no—
324C2×C4×C4⋊C4D4711yescharacteristic, complemented
325C2×C4×C22⋊C4—471no—
326#1444—472no—
327#1439—472no—
328#1436—472no—
329C4⋊#95—471no—
330#1434—471no—
331#1434—471no—
332C22⋊#95—471no—
333C2×C47#228511yescharacteristic
334C2×C47#229011yescharacteristic
335C2×C47#231911yescharacteristic
336C2×C47#234211yescharacteristic
337C2×C47#234711yescharacteristic
338C2×C47D4×C4⋊C411yescharacteristic
339C2×C47#234911yescharacteristic
340C2×C47#246412yes—
341C2×C47#246512yes—
342C2×C47#252112yes—
343C2×C47#252212yes—
344D47—44no—
345C24×C4—471no—
346C22×C4⋊C4—472no—
347C22×C4⋊C4—472no—
348C22×C4⋊C4C2×D4712yes—
349C22×C22⋊C4—471no—
350C22×C22⋊C4—471no—
351C22×C22⋊C4—472no—
352C22×C22⋊C4—472no—
353C22×C42—471no—
354C22×C42C2×D4712yes—
355C22×C42C2×D4711yescharacteristic
356#388—948no—
357#380—944no—
358C4×C4⋊C4—24no—
359C4×C22⋊C4—478no—
360C2×#95—471no—
361C2×#95—471no—
362C2×#95—471no—
363C2×#95—472no—
364C2×#95—471no—
365C2×#95—472no—
366C2×#95—472no—
367C2×#95—472no—
368C2×#95—472no—
369C2×#95—472no—
370C2×#95C2×D4711yescharacteristic
371C2×#95C2×D4711yescharacteristic
372C47#1389111yescharacteristic, complemented, Sylow
373C25—471no—
374C23×C4—471no—
375C23×C4—471no—
376C23×C4—471no—
377C23×C4—474no—
378C23×C4—472no—
379C23×C4—472no—
380C23×C4—944no—
381C23×C4—942no—
382C23×C4—942no—
383C23×C4—1882no—
384C23×C4C22×D4712yes—
385C23×C4C22×D4712yes—
386C23×C4C22×D4711yescharacteristic
387C23×C4C22×D4711yescharacteristic
388C23×C4C22×D4711yescharacteristic
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—944no—
393C2×C4⋊C4—944no—
394C2×C4⋊C4—944no—
395C2×C4⋊C4—944no—
396C2×C4⋊C4—1882no—
397C2×C4⋊C4—1882no—
398C2×C22⋊C4—474no—
399C2×C22⋊C4—474no—
400C2×C22⋊C4—474no—
401C2×C22⋊C4—474no—
402C2×C22⋊C4—944no—
403C2×C22⋊C4—944no—
404C2×C22⋊C4—944no—
405C2×C22⋊C4—944no—
406C2×C22⋊C4—944no—
407C2×C22⋊C4—1882no—
408C2×C22⋊C4—1882no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—24no—
412C2×C42—24no—
413C2×C42—42no—
414C2×C42—474no—
415C2×C42—1882no—
416C2×C42C4×D4714yes—
417#95—22no—
418#95—474no—
419#95—474no—
420#95—944no—
421#95—944no—
422#95—944no—
423#95—942no—
424#95—944no—
425#95—942no—
426#95—1882no—
427#95—1882no—
428#95C22⋊D4714yes—
429C24—472no—
430C24—472no—
431C24—472no—
432C24—941no—
433C24—942no—
434C24—942no—
435C24—942no—
436C24—941no—
437C24—941no—
438C24—942no—
439C24—941no—
440C24C23×D4711yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—472no—
463C22×C4—472no—
464C22×C4—472no—
465C22×C4—472no—
466C22×C4—474no—
467C22×C4—472no—
468C22×C4—948no—
469C22×C4—941no—
470C22×C4—942no—
471C22×C4—941no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—941no—
476C22×C4—942no—
477C22×C4—941no—
478C22×C4—948no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—1884no—
494C22×C4—1888no—
495C22×C4—1888no—
496C22×C4C2×C4×D4712yes—
497C22×C4C2×C4×D4712yes—
498C22×C4C2×C4×D4712yes—
499C22×C4#1062535014yes—
500C22×C4#1062535014yes—
501C22×C4#1062535012yes—
502C22×C4D4×D4712yes—
503C22×C4Q8×D4712yes—
504C22×C4C2×C22⋊D4714yes—
505C22×C4C2×C22⋊D4712yes—
506C4⋊C4—216no—
507C4⋊C4—1888no—
508C4⋊C4—1888no—
509C22⋊C4—4716no—
510C22⋊C4—948no—
511C22⋊C4—1888no—
512C22⋊C4—1888no—
513C42—28no—
514C42—48no—
515C42—1884no—
516C23—474no—
517C23—942no—
518C23—944no—
519C23—942no—
520C23—942no—
521C23—942no—
522C23—942no—
523C23—942no—
524C23—944no—
525C23—942no—
526C23—942no—
527C23—942no—
528C23—942no—
529C23—944no—
530C23—944no—
531C23—1882no—
532C23—1882no—
533C23—1882no—
534C23—1882no—
535C23—1882no—
536C23—1882no—
537C23—1882no—
538C23—1882no—
539C23C22×C4×D4711yescharacteristic
540C23C2×D4×D4711yescharacteristic
541C23C22×C22⋊D4711yescharacteristic
542C23C2×Q8×D4711yescharacteristic
543C23#41×D4712yes—
544C23C2×#1062535012yes—
545C23C2×#1062535012yes—
546C23C2×#1062535011yescharacteristic
547C23#1309406911yescharacteristic
548C23#1309407012yes—
549C23#1309407211yescharacteristic
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—474no—
579C2×C4—474no—
580C2×C4—944no—
581C2×C4—944no—
582C2×C4—944no—
583C2×C4—944no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—942no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—942no—
590C2×C4—942no—
591C2×C4—18816no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4—1882no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4C4⋊C4×D4714yes—
609C2×C4#1309398114yes—
610C2×C4C4×C22⋊D4714yes—
611C22—944no—
612C22—944no—
613C22—944no—
614C22—1882no—
615C22—1882no—
616C22—1884no—
617C22—1884no—
618C22—1882no—
619C22—1884no—
620C22—1884no—
621C22—1882no—
622C22C2×C4⋊C4×D4711yescharacteristic
623C22#117×D4711yescharacteristic
624C22C2×#1309398111yescharacteristic
625C22C2×C4×C22⋊D4711yescharacteristic
626C22C4×#1062535011yescharacteristic
627C22#1568959711yescharacteristic
628C22#1568960411yescharacteristic
629C22#1568963412yes—
630C22#1568965712yes—
631C22#1568967012yes—
632C22#1568967712yes—
633C22#1568972412yes—
634C22#1568974512yes—
635C22D4×C22⋊D4711yescharacteristic
636C22#1568976112yes—
637C22#1568976912yes—
638C22#1568977411yescharacteristic
639C22Q8×C22⋊D4711yescharacteristic
640C22#1568979712yes—
641C22#1568982212yes—
642C22#1568982811yescharacteristic
643C22#1568983312yes—
644C22#1568984512yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—944no—
651C4—944no—
652C4—3762no—
653C4—3762no—
654C4—1882no—
655C2—1884no—
656C2#2024579211yescharacteristic
657C2#2024583111yescharacteristic
658C2#2024602611yescharacteristic
659C2#2024623811yescharacteristic
660C2#2024628211yescharacteristic
661C2C4⋊C4×C22⋊D4711yescharacteristic
662C2#2024629911yescharacteristic
663C2#2024738812yes—
664C2#2024740312yes—
665C2#2024808512yes—
666C2#2024810012yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C4×C4⋊C4×C47C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×C4×C4⋊C4D47
hypocenterC47#13891
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #20241651C2semidirect
6. #20241621C2semidirect
7. Dic47⋊#95C2semidirect
8. #20241615C2semidirect
9. #20241613C2semidirect
11. C2×C4×C4⋊C4×C47C2semidirect
22. C4×(C2×D47)⋊C4C4semidirect
178. (C2×D47)⋊C4C4⋊C4semidirect
324. C2×C4×C4⋊C4D47semidirect
372. C47#13891semidirect

Maximal subgroups

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

subgroupindex
C2×#156890142
C2×#156889822
C2×C4×(C2×D47)⋊C42
#202416512
#202416302
#202416212
Dic47⋊#952
#202416152
#202416132
(C2×D47)⋊#952
C2×C4×C4⋊C4×C472
#1389147