Finite Group Databasebeta

← #26226312

Subgroups — #26226312

Order 12032.

1094 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Dic47C211yesmaximal, characteristic
2#20244004C212yesmaximal, complemented
3#20243998C212yesmaximal, complemented
4#20243997C212yesmaximal, complemented
5#20243991C212yesmaximal, complemented
6#20243366C211yesmaximal, characteristic, complemented
7#20243366C211yesmaximal, characteristic, complemented
8#20243298C211yesmaximal, characteristic
9#20243152C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic47C211yesmaximal, characteristic, complemented
11C47×#1779C211yesmaximal, characteristic
12C22×C22⋊Dic47C411yescharacteristic
13C22×C22⋊Dic47C411yescharacteristic
14C22×C4×Dic47C2211yescharacteristic
15C22×C4×Dic47C2211yescharacteristic
16C2×Dic47⋊C8C2212yescomplemented
17C2×Dic47⋊C8C2212yescomplemented
18C2×C22⋊(C47⋊C8)C2211yescharacteristic
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20C2×C22⋊(C47⋊C8)C2212yes—
21C2×C22⋊(C47⋊C8)C2212yes—
22C2×#13093934C2212yescomplemented
23C2×C8×Dic47C2212yescomplemented
24C4×C22⋊Dic47C411yescharacteristic
25C4×C22⋊Dic47C411yescharacteristic
26C4×C22⋊Dic47C411yescharacteristic
27C4×C22⋊Dic47C411yescharacteristic
28C4×C22⋊Dic47C2211yescharacteristic
29C4×C22⋊Dic47C2211yescharacteristic
30C4×C22⋊Dic47C2211yescharacteristic
31C4×C22⋊Dic47C2211yescharacteristic
32C2×C47⋊#95C411yescharacteristic
33C2×C47⋊#95C411yescharacteristic
34#15688877C2212yescomplemented
35#15688877C2212yescomplemented
36#15688877C2212yescomplemented
37#15688877C2212yescomplemented
38C24×C4×C47C2211yescharacteristic
39C2×C22⋊C8×C47C2211yescharacteristic
40C2×C22⋊C8×C47C2211yescharacteristic
41C2×C22⋊C8×C47C2212yes—
42C2×C22⋊C8×C47C2212yes—
43C23×Dic47C2×C411yescharacteristic
44C23×Dic47C2×C411yescharacteristic
45C23×Dic47C2×C411yescharacteristic
46C23×Dic47C2×C411yescharacteristic
47C22×C47⋊C8C2312yes—
48C22×C47⋊C8C2312yes—
49C2×C22⋊Dic47—24no—
50C2×C22⋊Dic47—24no—
51C2×C22⋊Dic47C2×C411yescharacteristic
52C2×C22⋊Dic47C2×C411yescharacteristic
53C2×C22⋊Dic47C2×C411yescharacteristic
54C2×C22⋊Dic47C2×C411yescharacteristic
55C2×C22⋊Dic47C2×C411yescharacteristic
56C2×C22⋊Dic47C2×C411yescharacteristic
57C2×C22⋊Dic47C2×C411yescharacteristic
58C2×C22⋊Dic47C2×C411yescharacteristic
59C2×C4×Dic47—42no—
60C2×C4×Dic47—42no—
61C2×C4×Dic47C2×C411yescharacteristic
62C2×C4×Dic47C2×C411yescharacteristic
63C2×C4×Dic47C2×C411yescharacteristic
64C2×C4×Dic47C2×C411yescharacteristic
65C2×C4×Dic47C2311yescharacteristic
66C2×C4×Dic47C2311yescharacteristic
67C2×C4×Dic47C2311yescharacteristic
68C2×C4×Dic47C2311yescharacteristic
69Dic47⋊C8—42no—
70Dic47⋊C8—42no—
71C22⋊(C47⋊C8)—22no—
72C22⋊(C47⋊C8)—22no—
73C22⋊(C47⋊C8)—42no—
74C22⋊(C47⋊C8)—42no—
75#13093934—42no—
76C8×Dic47—42no—
77C47⋊#95C2×C411yescharacteristic
78C47⋊#95C2×C411yescharacteristic
79C47⋊#95C2×C411yescharacteristic
80C47⋊#95C2×C411yescharacteristic
81C47⋊#95C2×C411yescharacteristic
82C47⋊#95C2×C411yescharacteristic
83C47⋊#95C2×C411yescharacteristic
84C47⋊#95C2×C411yescharacteristic
85C25×C47C2×C411yescharacteristic
86C23×C4×C47—22no—
87C23×C4×C47—22no—
88C23×C4×C47—42no—
89C23×C4×C47—42no—
90C23×C4×C47C2×C411yescharacteristic
91C23×C4×C47C2×C411yescharacteristic
92C23×C4×C47C2×C411yescharacteristic
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2311yescharacteristic
96C22×C8×C47C2312yes—
97C22×C8×C47C2312yes—
98C22⋊C8×C47—22no—
99C22⋊C8×C47—22no—
100C22⋊C8×C47—42no—
101C22⋊C8×C47—42no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47—22no—
106C24×C47—22no—
107C24×C47—41no—
108C24×C47—41no—
109C24×C47C22×C411yescharacteristic
110C24×C47C22×C411yescharacteristic
111C24×C47C22×C411yescharacteristic
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—41no—
126C22×C4×C47—41no—
127C22×C4×C47—41no—
128C22×C4×C47—41no—
129C22×C4×C47—41no—
130C22×C4×C47—41no—
131C22×C4×C47—48no—
132C22×C4×C47—48no—
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47#4112yes—
137C22×C4×C47#4112yes—
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—41no—
148C22×Dic47—41no—
149C22×Dic47—41no—
150C22×Dic47—41no—
151C22×Dic47C22×C411yescharacteristic
152C22×Dic47C22×C411yescharacteristic
153C22×Dic47C22×C411yescharacteristic
154C22×Dic47C22×C411yescharacteristic
155C22×Dic47C22×C411yescharacteristic
156C22×Dic47C22×C411yescharacteristic
157C22×Dic47C22×C411yescharacteristic
158C22×Dic47C22×C411yescharacteristic
159C2×C8×C47—22no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C22⋊Dic47—22no—
166C22⋊Dic47—22no—
167C22⋊Dic47—22no—
168C22⋊Dic47—22no—
169C22⋊Dic47—22no—
170C22⋊Dic47—22no—
171C22⋊Dic47M1614yes—
172C22⋊Dic47M1614yes—
173C4×Dic47—22no—
174C4×Dic47—22no—
175C4×Dic47—22no—
176C4×Dic47—22no—
177C4×Dic47—41no—
178C4×Dic47—41no—
179C4×Dic47—41no—
180C4×Dic47—41no—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C23×C47—21no—
188C23×C47—21no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—28no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47—21no—
201C23×C47—21no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—44no—
205C23×C47—44no—
206C23×C47—44no—
207C23×C47—44no—
208C23×C47—44no—
209C23×C47—44no—
210C23×C47#11712yes—
211C23×C47C2×M1612yes—
212C23×C47C2×M1612yes—
213C23×C47#13112yes—
214C23×C47#13112yes—
215C23×C47C23×C411yescharacteristic
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—24no—
219C2×C4×C47—24no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—44no—
227C2×C4×C47—44no—
228C2×C4×C47—44no—
229C2×C4×C47—42no—
230C2×C4×C47—42no—
231C2×C4×C47—42no—
232C2×C4×C47—42no—
233C2×C4×C47#11712yes—
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47H(5,𝔽2)12yes—
237C2×C4×C47H(5,𝔽2)12yes—
238C2×Dic47—21no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—21no—
242C2×Dic47—21no—
243C2×Dic47—21no—
244C2×Dic47—21no—
245C2×Dic47—21no—
246C2×Dic47—21no—
247C2×Dic47—21no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—21no—
257C2×Dic47—21no—
258C2×Dic47—44no—
259C2×Dic47—44no—
260C2×Dic47—44no—
261C2×Dic47—44no—
262C2×Dic47C2×M1612yes—
263C2×Dic47C2×M1612yes—
264C2×Dic47C2×M1612yes—
265C2×Dic47C2×M1612yes—
266C8×C47—42no—
267C8×C47—42no—
268C47⋊C8—42no—
269C47⋊C8—42no—
270#21904—471nomaximal, Sylow
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—22no—
278C22×C47—22no—
279C22×C47—22no—
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—416no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47—41no—
290C22×C47—41no—
291C22×C47—41no—
292C22×C47#43314yes—
293C22×C47C2×#11711yescharacteristic
294C22×C47#51811yescharacteristic
295C22×C47#51811yescharacteristic
296C22×C47C22×M1611yescharacteristic
297C22×C47C22×M1611yescharacteristic
298C22×C47C2×#13111yescharacteristic
299C22×C47C2×#13111yescharacteristic
300C4×C47—22no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47#53812yes—
304C4×C47#53812yes—
305Dic47—22no—
306Dic47—22no—
307Dic47—22no—
308Dic47—22no—
309Dic47—41no—
310Dic47—41no—
311Dic47—41no—
312Dic47—41no—
313C2×C4×C22⋊C4—471no—
314#1924—472no—
315#1924—472no—
316#1923—472no—
317#1923—472no—
318#1797—471no—
319#1797—471no—
320#1779—471no—
321#1779D4711yescharacteristic
322#1749—471no—
323C4×C22⋊C8—471no—
324C2×C47—24no—
325C2×C47—42no—
326C2×C47—42no—
327C2×C47#231211yescharacteristic
328C2×C47C2×#43311yescharacteristic
329C2×C47#292011yescharacteristic
330C2×C47#297411yescharacteristic
331C2×C47#297411yescharacteristic
332C2×C47#297411yescharacteristic
333C2×C47#297411yescharacteristic
334C24×C4C2×D4711yescharacteristic
335C22×C22⋊C4—471no—
336C22×C22⋊C4—471no—
337C22×C42—471no—
338C22×C42—471no—
339C2×C4⋊C8—472no—
340C2×C4⋊C8—472no—
341C2×C22⋊C8—471no—
342C2×C22⋊C8—471no—
343C2×C22⋊C8—472no—
344C2×C22⋊C8—472no—
345C2×C22⋊C8C2×D4711yescharacteristic
346C2×C22⋊C8C2×D4711yescharacteristic
347C2×C22⋊C8C2×D4712yes—
348C2×C22⋊C8C2×D4712yes—
349C2×#97—472no—
350C2×C4×C8—472no—
351C4×C22⋊C4—471no—
352C4×C22⋊C4—471no—
353C4×C22⋊C4—471no—
354C4×C22⋊C4—471no—
355C4×C22⋊C4—471no—
356C4×C22⋊C4—471no—
357C4×C22⋊C4—471no—
358C4×C22⋊C4—471no—
359C2×#95—471no—
360C2×#95—471no—
361#336—472no—
362#336—472no—
363#336—472no—
364#336—472no—
365C47#2190411yescharacteristic, complemented, Sylow
366C25C4×D4711yescharacteristic
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—42no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—471no—
375C23×C4C4×D4711yescharacteristic
376C23×C4C4×D4711yescharacteristic
377C23×C4C4×D4711yescharacteristic
378C23×C4C22×D4711yescharacteristic
379C23×C4C22×D4711yescharacteristic
380C23×C4C22×D4711yescharacteristic
381C22×C8—472no—
382C22×C8—472no—
383C22×C8C22×D4712yes—
384C22×C8C22×D4712yes—
385C2×C22⋊C4—471no—
386C2×C22⋊C4—471no—
387C2×C22⋊C4—471no—
388C2×C22⋊C4—471no—
389C2×C22⋊C4—471no—
390C2×C22⋊C4—471no—
391C2×C22⋊C4—471no—
392C2×C22⋊C4—471no—
393C2×C22⋊C4—944no—
394C2×C22⋊C4—944no—
395C2×C42—471no—
396C2×C42—471no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—471no—
401C2×C42—471no—
402C2×C42—471no—
403C2×C42—1882no—
404C2×C42—1882no—
405C4⋊C8—1882no—
406C4⋊C8—1882no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—42no—
410C22⋊C8—42no—
411C22⋊C8—942no—
412C22⋊C8—942no—
413C22⋊C8—1882no—
414C22⋊C8—1882no—
415#97—1882no—
416C4×C8—1882no—
417#95—471no—
418#95—471no—
419#95—471no—
420#95—471no—
421#95—471no—
422#95—471no—
423#95—471no—
424#95—471no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—41no—
431C24—41no—
432C24C2×C4×D4711yescharacteristic
433C24C2×C4×D4711yescharacteristic
434C24C2×C4×D4711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—471no—
457C22×C4—471no—
458C22×C4—471no—
459C22×C4—471no—
460C22×C4—471no—
461C22×C4—471no—
462C22×C4—471no—
463C22×C4—471no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—1881no—
473C22×C4—1881no—
474C22×C4—1881no—
475C22×C4—1881no—
476C22×C4C2×C4×D4711yescharacteristic
477C22×C4C2×C4×D4711yescharacteristic
478C22×C4C2×C4×D4711yescharacteristic
479C22×C4#1062535212yes—
480C22×C4#1062535212yes—
481C22×C4C23×D4711yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—942no—
489C2×C8—942no—
490C2×C8—942no—
491C2×C8—942no—
492C2×C8—942no—
493C2×C8—942no—
494C22⋊C4—474no—
495C22⋊C4—474no—
496C22⋊C4—942no—
497C22⋊C4—942no—
498C22⋊C4—942no—
499C22⋊C4—942no—
500C22⋊C4—942no—
501C22⋊C4—942no—
502C42—942no—
503C42—942no—
504C42—942no—
505C42—942no—
506C42—1881no—
507C42—1881no—
508C42—1881no—
509C42—1881no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—28no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23#1309398012yes—
534C23M16×D4712yes—
535C23M16×D4712yes—
536C23#1309403012yes—
537C23#1309403212yes—
538C23C22×C4×D4711yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—472no—
557C2×C4—472no—
558C2×C4—472no—
559C2×C4—472no—
560C2×C4—941no—
561C2×C4—941no—
562C2×C4—941no—
563C2×C4—941no—
564C2×C4—941no—
565C2×C4—941no—
566C2×C4—941no—
567C2×C4—941no—
568C2×C4—941no—
569C2×C4—941no—
570C2×C4—941no—
571C2×C4—941no—
572C2×C4—941no—
573C2×C4—941no—
574C2×C4—941no—
575C2×C4—941no—
576C2×C4—941no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—941no—
580C2×C4—1884no—
581C2×C4—1884no—
582C2×C4—1884no—
583C2×C4—1884no—
584C2×C4#1309398012yes—
585C2×C4C2×#1062535211yescharacteristic
586C2×C4C2×#1062535211yescharacteristic
587C2×C4#1309406912yes—
588C2×C4#1309406912yes—
589C8—42no—
590C8—42no—
591C8—1882no—
592C8—1882no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—416no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22#1568918114yes—
615C22C2×#1309398011yescharacteristic
616C22#1568959611yescharacteristic
617C22#1568959611yescharacteristic
618C22C2×M16×D4711yescharacteristic
619C22C2×M16×D4711yescharacteristic
620C22C2×#1309403011yescharacteristic
621C22C2×#1309403211yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—942no—
626C4—942no—
627C4—942no—
628C4—942no—
629C4—1881no—
630C4—1881no—
631C4—1881no—
632C4—1881no—
633C4#1568970812yes—
634C4#1568970812yes—
635C2—24no—
636C2—42no—
637C2—42no—
638C2#2024598411yescharacteristic
639C2C2×#1568918111yescharacteristic
640C2#2025201711yescharacteristic
641C2#2025241911yescharacteristic
642C2#2025241911yescharacteristic
643C2#2025242411yescharacteristic
644C2#2025242411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1779C2
prime radicalC25×C47C2×C4
cosocleC22×C4×C47C24
hypercenter#1779D47
hypocenterC47#21904
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #20244004C2semidirect
3. #20243998C2semidirect
4. #20243997C2semidirect
5. #20243991C2semidirect
6. #20243366C2semidirect
7. #20243366C2semidirect
9. #20243152C2semidirect
10. C22⋊C8×Dic47C2semidirect
16. C2×Dic47⋊C8C22semidirect
17. C2×Dic47⋊C8C22semidirect
22. C2×#13093934C22semidirect
23. C2×C8×Dic47C22semidirect
34. #15688877C22semidirect
35. #15688877C22semidirect
36. #15688877C22semidirect
37. #15688877C22semidirect
365. C47#21904semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic472
#202440042
#202439982
#202439972
#202439912
#202433662
#202432982
#202431522
C22⋊C8×Dic472
C47×#17792
#2190447