Finite Group Databasebeta

← #26226314

Subgroups — #26226314

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×(C2×D47)⋊C4C211yesmaximal, characteristic
2#20244002C212yesmaximal, complemented
3#20244000C212yesmaximal, complemented
4#20243995C212yesmaximal, complemented
5#20243993C212yesmaximal, complemented
6#20243368C211yesmaximal, characteristic, complemented
7#20243297C211yesmaximal, characteristic
8#20243297C211yesmaximal, characteristic
9#20243152C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic47C211yesmaximal, characteristic, complemented
11C47×#1797C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×(C2×D47)⋊C4C411yescharacteristic
14C22×(C2×D47)⋊C4C411yescharacteristic
15C22×C4×Dic47C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2212yescomplemented
17C2×C22⋊(C47⋊C8)C2211yescharacteristic
18C2×C22⋊(C47⋊C8)C2211yescharacteristic
19C2×(C2×D47)⋊C8C2212yes—
20C2×(C2×D47)⋊C8C2212yes—
21C2×(C2×D47)⋊C8C2212yes—
22C2×(C2×D47)⋊C8C2212yes—
23C2×#13093934C2212yescomplemented
24C2×C8×Dic47C2212yescomplemented
25C4×(C2×D47)⋊C4C411yescharacteristic
26C4×(C2×D47)⋊C4C411yescharacteristic
27C4×(C2×D47)⋊C4C411yescharacteristic
28C4×(C2×D47)⋊C4C411yescharacteristic
29C4×(C2×D47)⋊C4C2211yescharacteristic
30C4×(C2×D47)⋊C4C2211yescharacteristic
31C4×(C2×D47)⋊C4C2211yescharacteristic
32C4×(C2×D47)⋊C4C2211yescharacteristic
33C2×C47⋊#95C411yescharacteristic
34C2×C47⋊#95C411yescharacteristic
35#15688878C2212yescomplemented
36#15688877C2212yescomplemented
37#15688877C2212yescomplemented
38C22×C42×C47C2211yescharacteristic
39C2×C4⋊C8×C47C2212yescomplemented
40C2×C22⋊C8×C47C2211yescharacteristic
41C2×C22⋊C8×C47C2211yescharacteristic
42C47×#336C2212yescomplemented
43C24×D47C2×C411yescharacteristic
44C23×Dic47C2×C411yescharacteristic
45C23×Dic47C2×C411yescharacteristic
46C23×Dic47C2×C411yescharacteristic
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—42no—
50C22×C4×D47—42no—
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2311yescharacteristic
54C22×C4×D47C2311yescharacteristic
55C22×C47⋊C8C2312yes—
56C22×C47⋊C8C2312yes—
57C2×(C2×D47)⋊C4—24no—
58C2×(C2×D47)⋊C4—24no—
59C2×(C2×D47)⋊C4C2×C411yescharacteristic
60C2×(C2×D47)⋊C4C2×C411yescharacteristic
61C2×(C2×D47)⋊C4C2×C411yescharacteristic
62C2×(C2×D47)⋊C4C2×C411yescharacteristic
63C2×(C2×D47)⋊C4C2×C411yescharacteristic
64C2×(C2×D47)⋊C4C2×C411yescharacteristic
65C2×(C2×D47)⋊C4C2×C411yescharacteristic
66C2×(C2×D47)⋊C4C2×C411yescharacteristic
67C2×C4×Dic47—42no—
68C2×C4×Dic47C2×C411yescharacteristic
69C2×C4×Dic47C2×C411yescharacteristic
70C2×C4×Dic47C2311yescharacteristic
71C2×C4×Dic47C2311yescharacteristic
72C4⋊(C47⋊C8)—42no—
73C22⋊(C47⋊C8)—22no—
74C22⋊(C47⋊C8)—22no—
75(C2×D47)⋊C8—42no—
76(C2×D47)⋊C8—42no—
77(C2×D47)⋊C8—42no—
78(C2×D47)⋊C8—42no—
79#13093934—42no—
80C8×Dic47—42no—
81C47⋊#95C2×C411yescharacteristic
82C47⋊#95C2×C411yescharacteristic
83C47⋊#95C2×C411yescharacteristic
84C47⋊#95C2×C411yescharacteristic
85C47⋊#95C2×C411yescharacteristic
86C47⋊#95C2×C411yescharacteristic
87C47⋊#95C2×C411yescharacteristic
88C47⋊#95C2×C411yescharacteristic
89C23×C4×C47C2×C411yescharacteristic
90C23×C4×C47C2×C411yescharacteristic
91C23×C4×C47C2311yescharacteristic
92C22×C8×C47C2312yes—
93C22×C8×C47C2312yes—
94C2×C42×C47—42no—
95C2×C42×C47C2×C411yescharacteristic
96C2×C42×C47C2×C411yescharacteristic
97C2×C42×C47C2311yescharacteristic
98C2×C42×C47C2311yescharacteristic
99C4⋊C8×C47—42no—
100C22⋊C8×C47—22no—
101C22⋊C8×C47—22no—
102C24×C47C22×C411yescharacteristic
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47—22no—
107C23×D47—22no—
108C23×D47—41no—
109C23×D47—41no—
110C23×D47C22×C411yescharacteristic
111C23×D47C22×C411yescharacteristic
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—41no—
119C22×C4×C47—41no—
120C22×C4×C47C22×C411yescharacteristic
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C22×C411yescharacteristic
124C22×C4×C47C22×C411yescharacteristic
125C22×C4×C47#4112yes—
126C22×C4×C47#4112yes—
127C22×C4×C47C2411yescharacteristic
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—41no—
134C22×Dic47—41no—
135C22×Dic47—41no—
136C22×Dic47—41no—
137C22×Dic47C22×C411yescharacteristic
138C22×Dic47C22×C411yescharacteristic
139C22×Dic47C22×C411yescharacteristic
140C22×Dic47C22×C411yescharacteristic
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—21no—
146C2×C4×D47—21no—
147C2×C4×D47—21no—
148C2×C4×D47—21no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—41no—
154C2×C4×D47—41no—
155C2×C4×D47—41no—
156C2×C4×D47—41no—
157C2×C4×D47—48no—
158C2×C4×D47—48no—
159C2×C8×C47—22no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C42×C47—22no—
166C42×C47—22no—
167C42×C47—41no—
168C42×C47—41no—
169(C2×D47)⋊C4—22no—
170(C2×D47)⋊C4—22no—
171(C2×D47)⋊C4—22no—
172(C2×D47)⋊C4—22no—
173(C2×D47)⋊C4—22no—
174(C2×D47)⋊C4—22no—
175(C2×D47)⋊C4M1614yes—
176(C2×D47)⋊C4M1614yes—
177C4×Dic47—22no—
178C4×Dic47—22no—
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#11712yes—
192C23×C47#13112yes—
193C23×C47#13112yes—
194C23×C47C23×C411yescharacteristic
195C22×D47—28no—
196C22×D47—21no—
197C22×D47—21no—
198C22×D47—21no—
199C22×D47—21no—
200C22×D47—21no—
201C22×D47—21no—
202C22×D47—21no—
203C22×D47—21no—
204C22×D47—21no—
205C22×D47—21no—
206C22×D47—21no—
207C22×D47—21no—
208C22×D47—44no—
209C22×D47—44no—
210C22×D47—44no—
211C22×D47—44no—
212C22×D47—44no—
213C22×D47—44no—
214C22×D47C2×M1612yes—
215C22×D47C2×M1612yes—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—24no—
223C2×C4×C47—24no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—44no—
231C2×C4×C47—44no—
232C2×C4×C47#11712yes—
233C2×C4×C47C2×M1612yes—
234C2×C4×C47C2×M1612yes—
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47H(5,𝔽2)12yes—
238C2×C4×C47H(5,𝔽2)12yes—
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—44no—
256C2×Dic47—44no—
257C2×Dic47—44no—
258C2×Dic47C2×M1612yes—
259C2×Dic47C2×M1612yes—
260C4×D47—44no—
261C4×D47—44no—
262C4×D47—42no—
263C4×D47—42no—
264C4×D47—42no—
265C4×D47—42no—
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#43314yes—
278C22×C47C2×#11711yescharacteristic
279C22×C47#51811yescharacteristic
280C22×C47#51811yescharacteristic
281C22×C47C22×M1611yescharacteristic
282C22×C47C22×M1611yescharacteristic
283C22×C47C2×#13111yescharacteristic
284C22×C47C2×#13111yescharacteristic
285C2×D47—22no—
286C2×D47—22no—
287C2×D47—22no—
288C2×D47—22no—
289C2×D47—22no—
290C2×D47—22no—
291C2×D47—416no—
292C2×D47—41no—
293C2×D47—41no—
294C2×D47—41no—
295C2×D47—41no—
296C2×D47—41no—
297C2×D47—41no—
298C2×D47—41no—
299C2×D47—41no—
300C4×C47—22no—
301C4×C47—22no—
302C4×C47—22no—
303C4×C47—41no—
304C4×C47—41no—
305C4×C47#53812yes—
306C4×C47#53812yes—
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#1797D4711yescharacteristic, complemented
320#1779—471no—
321#1779—471no—
322#1749—471no—
323C4×C22⋊C8—471no—
324C2×C47—24no—
325C2×C47#231211yescharacteristic
326C2×C47C2×#43311yescharacteristic
327C2×C47#292011yescharacteristic
328C2×C47#297411yescharacteristic
329C2×C47#297411yescharacteristic
330C2×C47#297411yescharacteristic
331C2×C47#297411yescharacteristic
332D47—42no—
333D47—42no—
334C24×C4—471no—
335C22×C22⋊C4—471no—
336C22×C22⋊C4—471no—
337C22×C42—471no—
338C22×C42C2×D4711yescharacteristic
339C2×C4⋊C8—472no—
340C2×C4⋊C8C2×D4712yescomplemented
341C2×C22⋊C8—471no—
342C2×C22⋊C8—471no—
343C2×C22⋊C8—472no—
344C2×C22⋊C8—472no—
345C2×C22⋊C8—472no—
346C2×C22⋊C8—472no—
347C2×C22⋊C8C2×D4711yescharacteristic
348C2×C22⋊C8C2×D4711yescharacteristic
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#336C2×D4712yescomplemented
365C47#2190411yescharacteristic, complemented, Sylow
366C25—471no—
367C23×C4—471no—
368C23×C4—471no—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—942no—
375C23×C4—942no—
376C23×C4—1882no—
377C23×C4—1882no—
378C23×C4C4×D4711yescharacteristic
379C23×C4C4×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—42no—
396C2×C42—471no—
397C2×C42—471no—
398C2×C42—471no—
399C2×C42—471no—
400C2×C42—1882no—
401C2×C42C4×D4711yescharacteristic
402C2×C42C4×D4711yescharacteristic
403C2×C42C22×D4711yescharacteristic
404C2×C42C22×D4711yescharacteristic
405C4⋊C8—42no—
406C4⋊C8—1882no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—942no—
410C22⋊C8—942no—
411C22⋊C8—1882no—
412C22⋊C8—1882no—
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—471no—
426C24—471no—
427C24—942no—
428C24—942no—
429C24—942no—
430C24—942no—
431C24—942no—
432C24—1881no—
433C24—1881no—
434C24C2×C4×D4711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—471no—
444C22×C4—471no—
445C22×C4—471no—
446C22×C4—471no—
447C22×C4—471no—
448C22×C4—471no—
449C22×C4—942no—
450C22×C4—942no—
451C22×C4—942no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—1881no—
465C22×C4—1881no—
466C22×C4—1881no—
467C22×C4—1881no—
468C22×C4—1888no—
469C22×C4—1888no—
470C22×C4—1881no—
471C22×C4—1881no—
472C22×C4—1881no—
473C22×C4—1881no—
474C22×C4C2×C4×D4711yescharacteristic
475C22×C4C2×C4×D4711yescharacteristic
476C22×C4C2×C4×D4711yescharacteristic
477C22×C4C2×C4×D4711yescharacteristic
478C22×C4C2×C4×D4711yescharacteristic
479C22×C4#1062535212yes—
480C22×C4#1062535412yes—
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—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C42—942no—
507C42—942no—
508C42—1881no—
509C42—1881no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—472no—
515C23—472no—
516C23—948no—
517C23—941no—
518C23—941no—
519C23—941no—
520C23—941no—
521C23—941no—
522C23—941no—
523C23—941no—
524C23—941no—
525C23—941no—
526C23—941no—
527C23—941no—
528C23—941no—
529C23—1884no—
530C23—1884no—
531C23—1884no—
532C23—1884no—
533C23—1884no—
534C23—1884no—
535C23#1309398212yes—
536C23#1309403012yes—
537C23#1309403212yes—
538C23C22×C4×D4711yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—472no—
556C2×C4—472no—
557C2×C4—941no—
558C2×C4—941no—
559C2×C4—941no—
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—1884no—
574C2×C4—1884no—
575C2×C4—1884no—
576C2×C4—1884no—
577C2×C4—1884no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4#1309398212yes—
583C2×C4M16×D4712yes—
584C2×C4M16×D4712yes—
585C2×C4C2×#1062535211yescharacteristic
586C2×C4C2×#1062535411yescharacteristic
587C2×C4#1309406912yes—
588C2×C4#1309407012yes—
589C8—42no—
590C8—42no—
591C8—1882no—
592C8—1882no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—942no—
600C22—942no—
601C22—942no—
602C22—942no—
603C22—942no—
604C22—942no—
605C22—18816no—
606C22—1881no—
607C22—1881no—
608C22—1881no—
609C22—1881no—
610C22—1881no—
611C22—1881no—
612C22—1881no—
613C22—1881no—
614C22#1568918314yes—
615C22C2×#1309398211yescharacteristic
616C22#1568959611yescharacteristic
617C22#1568959711yescharacteristic
618C22C2×M16×D4711yescharacteristic
619C22C2×M16×D4711yescharacteristic
620C22C2×#1309403011yescharacteristic
621C22C2×#1309403211yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—942no—
628C4—942no—
629C4—1881no—
630C4—1881no—
631C4—1881no—
632C4—1881no—
633C4#1568971012yes—
634C4#1568971312yes—
635C2—24no—
636C2—1882no—
637C2—1882no—
638C2#2024598511yescharacteristic
639C2C2×#1568918311yescharacteristic
640C2#2025202011yescharacteristic
641C2#2025241911yescharacteristic
642C2#2025242211yescharacteristic
643C2#2025242411yescharacteristic
644C2#2025242511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1797C2
prime radicalC24×D47C2×C4
cosocleC22×C4×C47C24
hypercenter#1797D47
hypocenterC47#21904
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #20244002C2semidirect
3. #20244000C2semidirect
4. #20243995C2semidirect
5. #20243993C2semidirect
6. #20243368C2semidirect
9. #20243152C2semidirect
10. C22⋊C8×Dic47C2semidirect
11. C47×#1797C2semidirect
16. C2×C4⋊(C47⋊C8)C22semidirect
23. C2×#13093934C22semidirect
24. C2×C8×Dic47C22semidirect
35. #15688878C22semidirect
36. #15688877C22semidirect
37. #15688877C22semidirect
39. C2×C4⋊C8×C47C22semidirect
42. C47×#336C22semidirect
319. #1797D47semidirect
340. C2×C4⋊C8C2×D47semidirect
364. #336C2×D47semidirect
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×(C2×D47)⋊C42
#202440022
#202440002
#202439952
#202439932
#202433682
#202432972
#202431522
C22⋊C8×Dic472
C47×#17972
#2190447