Finite Group Databasebeta

← #26216175

Subgroups — #26216175

Order 12032.

1426 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×#117×D47C211yesmaximal, characteristic
2C2×#15689178C211yesmaximal, characteristic, complemented
3C2×#15689174C211yesmaximal, characteristic, complemented
4#20243455C212yesmaximal, complemented
5#20243371C212yesmaximal, complemented
6#20243370C212yesmaximal, complemented
7#20243297C212yesmaximal
8#20243157C211yesmaximal, characteristic, complemented
9#20243155C211yesmaximal, characteristic, complemented
10#20243154C211yesmaximal, characteristic, complemented
11C47×#1750C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14C2×#13093982C411yescharacteristic
15C2×#13093982C411yescharacteristic
16C2×#13093981C2211yescharacteristic
17C2×#13093981C2211yescharacteristic
18C2×#13093980C411yescharacteristic
19C2×#13093980C411yescharacteristic
20#117×D47—24no—
21#117×D47—24no—
22C2×C4⋊C4×D47C411yescharacteristic
23C2×C4⋊C4×D47C411yescharacteristic
24C2×C22⋊C4×D47C411yescharacteristic
25C2×C22⋊C4×D47C411yescharacteristic
26C2×C42×D47C2211yescharacteristic
27C2×C42×D47C2211yescharacteristic
28#15689178—24no—
29#15689174—24no—
30C2×Dic47⋊C8C2212yescomplemented
31C2×Dic47⋊C8C2212yescomplemented
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×(C2×D47)⋊C8C2212yes—
34C2×(C2×D47)⋊C8C2212yes—
35C2×(C2×D47)⋊C8C2212yes—
36C2×(C2×D47)⋊C8C2212yes—
37C2×#13093935C2211yescharacteristic, complemented
38C2×C4×C47⋊C8C2211yescharacteristic, complemented
39#15688878C2212yescomplemented
40#15688877C2212yescomplemented
41#15688877C2212yescomplemented
42C2×#117×C47C2211yescharacteristic
43C2×C22⋊C8×C47C2212yes—
44C2×#97×C47C2211yescharacteristic, complemented
45C2×C4×C8×C47C2211yescharacteristic, complemented
46C47×#336C2212yescomplemented
47C24×D47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47—24no—
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—24no—
53C22×C4×D47C2×C411yescharacteristic
54C22×C4×D47C2×C411yescharacteristic
55C22×C4×D47C2×C411yescharacteristic
56C22×C4×D47C2×C411yescharacteristic
57C22×C4×D47C2×C411yescharacteristic
58C22×C4×D47C2×C411yescharacteristic
59C22×C4×D47C2311yescharacteristic
60C22×C4×D47C2311yescharacteristic
61C22×C47⋊C8C2312yes—
62C22×C47⋊C8C2312yes—
63#13093983—22no—
64#13093983—22no—
65#13093982—24no—
66#13093982—24no—
67#13093981—22no—
68#13093981—22no—
69#13093981—22no—
70#13093981—22no—
71#13093980—24no—
72#13093980—24no—
73C2×C4⋊Dic47C2×C411yescharacteristic
74C2×C4⋊Dic47C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×Dic47⋊C4C2×C411yescharacteristic
78C2×Dic47⋊C4C2×C411yescharacteristic
79C4⋊C4×D47—24no—
80C4⋊C4×D47—24no—
81C2×C22⋊Dic47C2×C411yescharacteristic
82C2×C22⋊Dic47C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2×C411yescharacteristic
84C2×(C2×D47)⋊C4C2×C411yescharacteristic
85C2×(C2×D47)⋊C4C2×C411yescharacteristic
86C2×(C2×D47)⋊C4C2×C411yescharacteristic
87C22⋊C4×D47—24no—
88C22⋊C4×D47—24no—
89C2×C4×Dic47C2×C411yescharacteristic
90C2×C4×Dic47C2×C411yescharacteristic
91C2×C4×Dic47C2×C411yescharacteristic
92C2×C4×Dic47C2×C411yescharacteristic
93C2×C4×Dic47C2311yescharacteristic
94C2×C4×Dic47C2311yescharacteristic
95C42×D47—22no—
96C42×D47—22no—
97C42×D47D414yes—
98C42×D47D414yes—
99Dic47⋊C8—24no—
100Dic47⋊C8—24no—
101C22⋊(C47⋊C8)—24no—
102(C2×D47)⋊C8—24no—
103(C2×D47)⋊C8—24no—
104(C2×D47)⋊C8—24no—
105(C2×D47)⋊C8—24no—
106#13093935—22no—
107C4×C47⋊C8—22no—
108C23×C4×C47C2311yescharacteristic
109C22×C8×C47C2312yes—
110C22×C8×C47C2312yes—
111#117×C47—22no—
112#117×C47—22no—
113C2×C4⋊C4×C47C2×C411yescharacteristic
114C2×C4⋊C4×C47C2×C411yescharacteristic
115C2×C22⋊C4×C47C2×C411yescharacteristic
116C2×C22⋊C4×C47C2×C411yescharacteristic
117C2×C42×C47C2311yescharacteristic
118C2×C42×C47C2311yescharacteristic
119C22⋊C8×C47—24no—
120#97×C47—22no—
121C4×C8×C47—22no—
122C24×C47C22×C411yescharacteristic
123C23×D47—22no—
124C23×D47—22no—
125C23×D47—22no—
126C23×D47—24no—
127C23×D47—42no—
128C23×D47C22×C411yescharacteristic
129C23×D47C22×C411yescharacteristic
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—22no—
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47#4114yes—
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47C22×C411yescharacteristic
144C22×Dic47C22×C411yescharacteristic
145C22×Dic47C22×C411yescharacteristic
146C22×Dic47C22×C411yescharacteristic
147C22×Dic47C22×C411yescharacteristic
148C22×Dic47C22×C411yescharacteristic
149C2×C4×D47—216no—
150C2×C4×D47—216no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—24no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—21no—
157C2×C4×D47—21no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—22no—
163C2×C4×D47—22no—
164C2×C4×D47—22no—
165C2×C4×D47—24no—
166C2×C4×D47—42no—
167C2×C4×D47—42no—
168C2×C4×D47C22⋊C414yes—
169C2×C4×D47C22⋊C414yes—
170C2×C4×D47C2×D412yes—
171C2×C4×D47C2×D412yes—
172C2×C4×D47#4114yes—
173C2×C8×C47—22no—
174C2×C8×C47—22no—
175C2×C8×C47—22no—
176C2×C8×C47—22no—
177C2×C8×C47—22no—
178C2×C8×C47—22no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—22no—
181C22⋊C4×C47—22no—
182C22⋊C4×C47—22no—
183C42×C47—21no—
184C42×C47—21no—
185C42×C47C2×D412yes—
186C42×C47C2×D412yes—
187C22⋊Dic47—22no—
188C22⋊Dic47—22no—
189(C2×D47)⋊C4—22no—
190(C2×D47)⋊C4—22no—
191(C2×D47)⋊C4—22no—
192(C2×D47)⋊C4—22no—
193C4⋊Dic47—22no—
194C4⋊Dic47—22no—
195Dic47⋊C4—22no—
196Dic47⋊C4—22no—
197Dic47⋊C4—22no—
198Dic47⋊C4—22no—
199C4×Dic47—21no—
200C4×Dic47—21no—
201C4×Dic47—22no—
202C4×Dic47—22no—
203C4×Dic47C22⋊C414yes—
204C4×Dic47C22⋊C414yes—
205C4×Dic47C2×D412yes—
206C4×Dic47C2×D412yes—
207C2×C47⋊C8—22no—
208C2×C47⋊C8—22no—
209C2×C47⋊C8—22no—
210C2×C47⋊C8—22no—
211C2×C47⋊C8—22no—
212C2×C47⋊C8—22no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47#11712yes—
218C23×C47C23×C411yescharacteristic
219C22×D47—21no—
220C22×D47—21no—
221C22×D47—22no—
222C22×D47—28no—
223C22×D47—21no—
224C22×D47—21no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—48no—
228C22×D47—48no—
229C22×D47—44no—
230C22×D47—44no—
231C22×D47C2×C22⋊C412yes—
232C22×D47#11712yes—
233C22×D47#13114yes—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—22no—
237C2×C4×C47—22no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—22no—
243C2×C4×C47—22no—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47#11712yes—
247C2×C4×C47C22×D411yescharacteristic
248C2×C4×C47C22×D411yescharacteristic
249C2×C4×C47C2×#4112yes—
250C2×C4×C47C2×#4112yes—
251C2×Dic47—28no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—21no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—22no—
261C2×Dic47—22no—
262C2×Dic47C2×C22⋊C412yes—
263C2×Dic47C2×C22⋊C412yes—
264C2×Dic47C2×C22⋊C412yes—
265C2×Dic47#11712yes—
266C4×D47—28no—
267C4×D47—28no—
268C4×D47—28no—
269C4×D47—24no—
270C4×D47—24no—
271C4×D47—24no—
272C4×D47—24no—
273C4×D47—44no—
274C4×D47—44no—
275C4×D47—44no—
276C8×C47—42no—
277C8×C47—42no—
278C47⋊C8—42no—
279C47⋊C8—42no—
280#20717—471nomaximal, Sylow
281C22×C47—24no—
282C22×C47—22no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—42no—
286C22×C47C22×C22⋊C411yescharacteristic
287C22×C47C2×#11711yescharacteristic
288C22×C47C2×#11711yescharacteristic
289C22×C47C2×#13112yes—
290C22×C47#56812yes—
291C2×D47—22no—
292C2×D47—24no—
293C2×D47—24no—
294C2×D47—22no—
295C2×D47—24no—
296C2×D47—22no—
297C2×D47—416no—
298C2×D47—41no—
299C2×D47—41no—
300C2×D47—42no—
301C2×D47#40814yes—
302C4×C47—22no—
303C4×C47—22no—
304C4×C47—22no—
305C4×C47—41no—
306C4×C47—41no—
307C4×C47#52512yes—
308C4×C47#52512yes—
309Dic47—22no—
310Dic47—22no—
311Dic47—22no—
312Dic47—22no—
313Dic47—41no—
314Dic47—41no—
315C22×#117—471no—
316C2×#432—471no—
317C2×#431—471no—
318#1817—472no—
319#1798—472no—
320#1798—472no—
321#1779—472no—
322#1750—471no—
323#1750—471no—
324#1750—471no—
325#1750D4711yescharacteristic, complemented
326C2×C47—42no—
327C2×C47#228011yescharacteristic
328C2×C47C2×#40811yescharacteristic
329C2×C47#287911yescharacteristic
330C2×C47#292412yes—
331C2×C47#292412yes—
332D47—24no—
333D47—42no—
334C24×C4—471no—
335C2×#117—471no—
336C2×#117—471no—
337C2×#117—471no—
338C2×#117—471no—
339C2×#117—471no—
340C2×#117—471no—
341C2×#117—471no—
342C2×#117—944no—
343C2×#117—944no—
344C2×#117C2×D4711yescharacteristic
345C22×C4⋊C4—471no—
346C22×C4⋊C4—471no—
347C22×C22⋊C4—471no—
348C22×C22⋊C4—471no—
349C22×C42—471no—
350C22×C42—471no—
351#432—944no—
352#431—944no—
353C2×C4⋊C8—472no—
354C2×C4⋊C8—472no—
355C2×C22⋊C8—472no—
356C2×C22⋊C8—472no—
357C2×C22⋊C8—472no—
358C2×C22⋊C8—472no—
359C2×C22⋊C8—472no—
360C2×C22⋊C8C2×D4712yes—
361C2×#97—471no—
362C2×#97C2×D4711yescharacteristic, complemented
363C2×C4×C8—471no—
364C2×C4×C8C2×D4711yescharacteristic, complemented
365#336—472no—
366#336—472no—
367#336—472no—
368#336C2×D4712yescomplemented
369C47#2071711yescharacteristic, complemented, Sylow
370C25—471no—
371C23×C4—471no—
372C23×C4—471no—
373C23×C4—471no—
374C23×C4—471no—
375C23×C4—471no—
376C23×C4—471no—
377C23×C4—471no—
378C23×C4—471no—
379C23×C4—471no—
380C23×C4—942no—
381C23×C4—942no—
382C23×C4—944no—
383C23×C4—944no—
384C23×C4C22×D4711yescharacteristic
385C22×C8—472no—
386C22×C8—472no—
387C22×C8C22×D4712yes—
388C22×C8C22×D4712yes—
389#117—22no—
390#117—22no—
391#117—942no—
392#117—942no—
393#117—944no—
394#117—944no—
395#117—944no—
396#117—944no—
397#117—942no—
398#117—942no—
399#117—942no—
400#117—942no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—471no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—471no—
406C2×C4⋊C4—471no—
407C2×C4⋊C4—944no—
408C2×C4⋊C4—944no—
409C2×C4⋊C4C4×D4711yescharacteristic
410C2×C4⋊C4C4×D4711yescharacteristic
411C2×C22⋊C4—471no—
412C2×C22⋊C4—471no—
413C2×C22⋊C4—471no—
414C2×C22⋊C4—471no—
415C2×C22⋊C4—471no—
416C2×C22⋊C4—471no—
417C2×C22⋊C4—944no—
418C2×C22⋊C4—944no—
419C2×C22⋊C4C4×D4711yescharacteristic
420C2×C22⋊C4C4×D4711yescharacteristic
421C2×C42—471no—
422C2×C42—471no—
423C2×C42—471no—
424C2×C42—474no—
425C2×C42—474no—
426C2×C42—471no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—942no—
430C2×C42—942no—
431C2×C42C22×D4711yescharacteristic
432C2×C42C22×D4711yescharacteristic
433C4⋊C8—944no—
434C4⋊C8—944no—
435C22⋊C8—24no—
436C22⋊C8—944no—
437C22⋊C8—944no—
438C22⋊C8—944no—
439C22⋊C8—944no—
440C22⋊C8—944no—
441#97—22no—
442#97—942no—
443C4×C8—22no—
444C4×C8—942no—
445C24—471no—
446C24—471no—
447C24—942no—
448C24—942no—
449C24—944no—
450C24—942no—
451C24—1882no—
452C24C2×C4×D4711yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—471no—
457C22×C4—471no—
458C22×C4—471no—
459C22×C4—471no—
460C22×C4—474no—
461C22×C4—472no—
462C22×C4—472no—
463C22×C4—474no—
464C22×C4—474no—
465C22×C4—471no—
466C22×C4—471no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—944no—
471C22×C4—942no—
472C22×C4—9416no—
473C22×C4—9416no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—944no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—941no—
483C22×C4—941no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—1882no—
488C22×C4—1882no—
489C22×C4C2×C4×D4711yescharacteristic
490C22×C4C2×C4×D4711yescharacteristic
491C22×C4C2×C4×D4711yescharacteristic
492C22×C4C2×C4×D4711yescharacteristic
493C22×C4C2×C4×D4711yescharacteristic
494C22×C4#1062535014yes—
495C22×C4C23×D4711yescharacteristic
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—942no—
503C2×C8—942no—
504C2×C8—942no—
505C2×C8—942no—
506C2×C8—942no—
507C2×C8—942no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—942no—
511C4⋊C4—942no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—942no—
515C4⋊C4—942no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—942no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—942no—
522C22⋊C4—942no—
523C22⋊C4—942no—
524C42—21no—
525C42—21no—
526C42—474no—
527C42—474no—
528C42—472no—
529C42—472no—
530C42—942no—
531C42—942no—
532C42—941no—
533C42—941no—
534C42D4×D4712yes—
535C42D4×D4712yes—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—472no—
541C23—474no—
542C23—472no—
543C23—948no—
544C23—941no—
545C23—941no—
546C23—942no—
547C23—941no—
548C23—941no—
549C23—942no—
550C23—942no—
551C23—1888no—
552C23—1884no—
553C23—1888no—
554C23—1884no—
555C23#1309398112yes—
556C23C22×C4×D4711yescharacteristic
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—472no—
568C2×C4—472no—
569C2×C4—472no—
570C2×C4—472no—
571C2×C4—948no—
572C2×C4—948no—
573C2×C4—948no—
574C2×C4—948no—
575C2×C4—941no—
576C2×C4—941no—
577C2×C4—941no—
578C2×C4—941no—
579C2×C4—942no—
580C2×C4—942no—
581C2×C4—944no—
582C2×C4—944no—
583C2×C4—944no—
584C2×C4—944no—
585C2×C4—941no—
586C2×C4—941no—
587C2×C4—942no—
588C2×C4—942no—
589C2×C4—1884no—
590C2×C4—1884no—
591C2×C4—1884no—
592C2×C4C22⋊C4×D4712yes—
593C2×C4C22⋊C4×D4712yes—
594C2×C4#1309398112yes—
595C2×C4C2×D4×D4711yescharacteristic
596C2×C4C2×D4×D4711yescharacteristic
597C2×C4#41×D4712yes—
598C2×C4C2×#1062535012yes—
599C8—42no—
600C8—42no—
601C8—1882no—
602C8—1882no—
603C22—22no—
604C22—24no—
605C22—42no—
606C22—41no—
607C22—41no—
608C22—474no—
609C22—942no—
610C22—944no—
611C22—944no—
612C22—942no—
613C22—942no—
614C22—944no—
615C22—18816no—
616C22—1882no—
617C22—1881no—
618C22—1881no—
619C22C2×C22⋊C4×D4711yescharacteristic
620C22#117×D4711yescharacteristic
621C22C2×#1309398111yescharacteristic
622C22#131×D4712yes—
623C22#1568988812yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—942no—
630C4—942no—
631C4—942no—
632C4—942no—
633C4—1881no—
634C4—1881no—
635C4#1568963212yes—
636C4#1568963212yes—
637C2—42no—
638C2—944no—
639C2—1882no—
640C2#2024576911yescharacteristic
641C2#408×D4711yescharacteristic
642C2#2025183011yescharacteristic
643C2#2025204312yes—
644C2#2025204412yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1750C2
prime radicalC24×D47C2×C4
cosocleC22×C4×C47C24
hypercenter#1750D47
hypocenterC47#20717
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#15689178C2semidirect
3. C2×#15689174C2semidirect
4. #20243455C2semidirect
5. #20243371C2semidirect
6. #20243370C2semidirect
8. #20243157C2semidirect
9. #20243155C2semidirect
10. #20243154C2semidirect
11. C47×#1750C2semidirect
30. C2×Dic47⋊C8C22semidirect
31. C2×Dic47⋊C8C22semidirect
37. C2×#13093935C22semidirect
38. C2×C4×C47⋊C8C22semidirect
39. #15688878C22semidirect
40. #15688877C22semidirect
41. #15688877C22semidirect
44. C2×#97×C47C22semidirect
45. C2×C4×C8×C47C22semidirect
46. C47×#336C22semidirect
325. #1750D47semidirect
362. C2×#97C2×D47semidirect
364. C2×C4×C8C2×D47semidirect
368. #336C2×D47semidirect
369. C47#20717semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D472
C2×#156891782
C2×#156891742
#202434552
#202433712
#202433702
#202432972
#202431572
#202431552
#202431542
C47×#17502
#2071747