Finite Group Databasebeta

← #26221251

Subgroups — #26221251

Order 12032.

1158 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689653C211yesmaximal, characteristic
2C2×#15689177C211yesmaximal, characteristic, complemented
3C2×M16×Dic47C211yesmaximal, characteristic, complemented
4#20244326C212yesmaximal, complemented
5#20244184C212yesmaximal, complemented
6#20243686C211yesmaximal, characteristic, complemented
7#20243685C211yesmaximal, characteristic, complemented
8#20243683C211yesmaximal, characteristic, complemented
9#20243156C212yesmaximal, complemented
10#20243153C212yesmaximal, complemented
11C47×#1870C211yesmaximal, characteristic, complemented
12C22×#10625329C2211yescharacteristic
13#15689653—24no—
14#15689653—24no—
15C2×#13094019C411yescharacteristic
16C2×#13094017C411yescharacteristic
17C2×#13094014C411yescharacteristic
18C2×#13094010C411yescharacteristic
19C2×#13094008C411yescharacteristic
20C2×#13094005C411yescharacteristic
21C2×#13094001C411yescharacteristic
22C2×#13093996C411yescharacteristic
23C2×C4×Dic94C2211yescharacteristic
24C2×C4×C22⋊D47C2211yescharacteristic
25C2×C4×C22⋊D47C2211yescharacteristic
26C2×#13093983C2211yescharacteristic
27C2×#13093981C2211yescharacteristic
28C22×C4×Dic47C2211yescharacteristic
29#15689177—24no—
30C2×Dic47⋊C8C2212yes—
31C2×C22⋊(C47⋊C8)C2211yescharacteristic
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×(C2×D47)⋊C8C2212yes—
34C2×(C2×D47)⋊C8C2212yes—
35M16×Dic47—24no—
36C2×#13093934C2212yes—
37C2×#13093934C2212yes—
38C2×C8×Dic47C2212yes—
39#15688878C2212yes—
40#15688877C2212yes—
41#15688877C2212yes—
42C22×M16×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C2×C22⋊C8×C47C2211yescharacteristic
45C2×C22⋊C8×C47C2212yes—
46C47×#336C2212yes—
47C22×Dic94C2×C411yescharacteristic
48C22×C22⋊D47C2×C411yescharacteristic
49C22×C22⋊D47C2×C411yescharacteristic
50C23×Dic47C2×C411yescharacteristic
51C23×Dic47C2×C411yescharacteristic
52C22×C4×D47C2311yescharacteristic
53C2×#10625329—22no—
54C2×#10625329—24no—
55C22×C47⋊C8C2312yes—
56C22×C47⋊C8C2312yes—
57#13094019—24no—
58#13094017—24no—
59#13094014—24no—
60#13094010—24no—
61#13094008—24no—
62#13094005—24no—
63#13094001—24no—
64#13093996—24no—
65C4×Dic94—22no—
66C4×Dic94—22no—
67C4×C22⋊D47—22no—
68C4×C22⋊D47—22no—
69C4×C22⋊D47—22no—
70C4×C22⋊D47—22no—
71#13093983—22no—
72#13093983—22no—
73#13093981—22no—
74#13093981—22no—
75C2×C4⋊Dic47C2×C411yescharacteristic
76C2×C4⋊Dic47C2×C411yescharacteristic
77C2×C4⋊Dic47C2×C411yescharacteristic
78C2×Dic47⋊C4C2×C411yescharacteristic
79C2×Dic47⋊C4C2×C411yescharacteristic
80C2×Dic47⋊C4C2×C411yescharacteristic
81C2×Dic47⋊C4C2×C411yescharacteristic
82C2×Dic47⋊C4C2×C411yescharacteristic
83C2×Dic47⋊C4C2×C411yescharacteristic
84C2×C22⋊Dic47C2×C411yescharacteristic
85C2×C22⋊Dic47C2×C411yescharacteristic
86C2×C22⋊Dic47C2×C411yescharacteristic
87C2×C22⋊Dic47C2×C411yescharacteristic
88C2×(C2×D47)⋊C4C2×C411yescharacteristic
89C2×(C2×D47)⋊C4C2×C411yescharacteristic
90C2×(C2×D47)⋊C4C2×C411yescharacteristic
91C2×(C2×D47)⋊C4C2×C411yescharacteristic
92C2×C4×Dic47C2×C411yescharacteristic
93C2×C4×Dic47C2×C411yescharacteristic
94C2×C4×Dic47D414yes—
95C2×C4×Dic47D414yes—
96C2×C4×Dic47C2311yescharacteristic
97C2×C4×Dic47C2311yescharacteristic
98C2×C4×Dic47C2311yescharacteristic
99Dic47⋊C8—24no—
100C22⋊(C47⋊C8)—22no—
101C22⋊(C47⋊C8)—24no—
102(C2×D47)⋊C8—24no—
103(C2×D47)⋊C8—24no—
104#13093934—24no—
105#13093934—24no—
106C8×Dic47—24no—
107C22×D4×C47C2×C411yescharacteristic
108C23×C4×C47C2311yescharacteristic
109C23×C4×C47C2311yescharacteristic
110C2×M16×C47—22no—
111C2×M16×C47—24no—
112C22×C8×C47C2312yes—
113C22×C8×C47C2312yes—
114C4×D4×C47—22no—
115C4×D4×C47—22no—
116C2×C4⋊C4×C47C2×C411yescharacteristic
117C2×C22⋊C4×C47C2×C411yescharacteristic
118C2×C22⋊C4×C47C2×C411yescharacteristic
119C2×C42×C47C2311yescharacteristic
120C22⋊C8×C47—22no—
121C22⋊C8×C47—24no—
122C24×C47C22×C411yescharacteristic
123C24×C47C22×C411yescharacteristic
124C23×D47C22×C411yescharacteristic
125C2×D4×C47—22no—
126C2×D4×C47—41no—
127C2×D4×C47—41no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C22×C411yescharacteristic
137C22×C4×C47C22×C411yescharacteristic
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2411yescharacteristic
141C2×C22⋊D47—22no—
142C2×C22⋊D47—22no—
143C2×C22⋊D47—42no—
144C2×C22⋊D47—41no—
145C2×C22⋊D47—41no—
146C22×Dic47—42no—
147C22×Dic47—41no—
148C22×Dic47—41no—
149C22×Dic47C22⋊C414yes—
150C22×Dic47C22⋊C414yes—
151C22×Dic47C22×C411yescharacteristic
152C22×Dic47C22×C411yescharacteristic
153C22×Dic47C22×C411yescharacteristic
154C22×Dic47C22×C411yescharacteristic
155C22×Dic47C22×C411yescharacteristic
156C22×Dic47C22×C411yescharacteristic
157C22×Dic47C22×C411yescharacteristic
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×Dic94—22no—
163C2×Dic94—42no—
164M16×C47—28no—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47#4114yes—
171C4⋊C4×C47—22no—
172C22⋊C4×C47—22no—
173C22⋊C4×C47—22no—
174C42×C47—21no—
175C42×C47—21no—
176C22⋊Dic47—22no—
177C22⋊Dic47—22no—
178C22⋊Dic47—22no—
179C22⋊Dic47—22no—
180(C2×D47)⋊C4—22no—
181(C2×D47)⋊C4—22no—
182(C2×D47)⋊C4—22no—
183(C2×D47)⋊C4—22no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47—22no—
187Dic47⋊C4—22no—
188Dic47⋊C4—22no—
189Dic47⋊C4—22no—
190Dic47⋊C4—22no—
191Dic47⋊C4—22no—
192Dic47⋊C4—22no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C4×Dic47C22⋊C414yes—
196C4×Dic47C22⋊C414yes—
197C4×Dic47C2×D412yes—
198C4×Dic47C2×D412yes—
199C4×Dic47C2×D412yes—
200C4×Dic47C2×D412yes—
201#10625329—28no—
202C2×C47⋊C8—22no—
203C2×C47⋊C8—22no—
204C2×C47⋊C8—22no—
205C2×C47⋊C8—22no—
206C2×C47⋊C8—22no—
207C2×C47⋊C8#4114yes—
208C23×C47—22no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—22no—
212C23×C47—21no—
213C23×C47—22no—
214C23×C47—22no—
215C23×C47—21no—
216C23×C47—21no—
217C23×C47C2×C22⋊C412yes—
218C23×C47C23×C411yescharacteristic
219C22×D47—21no—
220C22×D47—22no—
221C22×D47—22no—
222C22×D47—21no—
223C22×D47—21no—
224D4×C47—44no—
225C2×C4×C47—22no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—24no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—42no—
239C2×C4×C47C2×C22⋊C412yes—
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C2×#4112yes—
243C2×C4×C47C2×#4112yes—
244C22⋊D47—44no—
245C22⋊D47—44no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—21no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—21no—
254C2×Dic47—21no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—44no—
260C2×Dic47—44no—
261C2×Dic47C2×C22⋊C412yes—
262C2×Dic47C2×C22⋊C412yes—
263C2×Dic47C2×C22⋊C412yes—
264C2×Dic47C2×C22⋊C412yes—
265C4×D47—41no—
266C4×D47—41no—
267C4×D47—42no—
268Dic94—44no—
269C8×C47—24no—
270C8×C47—42no—
271C47⋊C8—24no—
272C47⋊C8—42no—
273#21376—471nomaximal, Sylow
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—22no—
277C22×C47—42no—
278C22×C47—42no—
279C22×C47—41no—
280C22×C47—41no—
281C22×C47—42no—
282C22×C47—42no—
283C22×C47—41no—
284C22×C47C22×C22⋊C411yescharacteristic
285C22×C47C4×#4112yes—
286C22×C47#56812yes—
287C22×C47#56811yescharacteristic
288C22×C47#56811yescharacteristic
289C2×D47—41no—
290C2×D47—41no—
291C2×D47—42no—
292C2×D47—42no—
293C2×D47—41no—
294C4×C47—22no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47—41no—
298C4×C47#52511yescharacteristic
299C4×C47#52511yescharacteristic
300C4×C47#53212yes—
301Dic47—22no—
302Dic47—22no—
303Dic47—22no—
304Dic47—22no—
305Dic47—41no—
306Dic47—41no—
307Dic47—41no—
308C2×#529—471no—
309C2×#432—471no—
310C2×C4×M16—471no—
311#1985—472no—
312#1955—472no—
313#1870—471no—
314#1870—471no—
315#1870—471no—
316#1870D4711yescharacteristic, complemented
317#1750—472no—
318#1750—472no—
319C2×C47—42no—
320C2×C47—81no—
321C2×C47#229711yescharacteristic
322C2×C47#287911yescharacteristic
323C2×C47#287911yescharacteristic
324C2×C47#296712yes—
325C2×C47#296712yes—
326D47—81no—
327C22×M16—471no—
328C22×M16C2×D4711yescharacteristic
329#529—944no—
330#529—944no—
331C2×#126—471no—
332C2×#126—471no—
333C2×#125—471no—
334C2×#124—471no—
335C2×#123—471no—
336C2×#123—471no—
337C2×C22⋊Q8—471no—
338C2×#121—471no—
339C2×C4×Q8—471no—
340C2×C4×D4—471no—
341C2×C4×D4—471no—
342C2×C4×D4C2×D4711yescharacteristic
343C2×#117—471no—
344C2×#117—471no—
345C22×C42—471no—
346#432—944no—
347C2×C4⋊C8—472no—
348C2×C22⋊C8—471no—
349C2×C22⋊C8—472no—
350C2×C22⋊C8—472no—
351C2×C22⋊C8—472no—
352C2×C22⋊C8C2×D4711yescharacteristic
353C2×C22⋊C8C2×D4712yes—
354C4×M16—944no—
355C2×#97—472no—
356C2×#97—472no—
357C2×C4×C8—472no—
358#336—472no—
359#336—472no—
360#336—472no—
361#336C2×D4712yes—
362C47#2137611yescharacteristic, complemented, Sylow
363C22×Q8—471no—
364C22×D4—471no—
365C22×D4—471no—
366C22×D4C4×D4711yescharacteristic
367C23×C4—471no—
368C23×C4—471no—
369C23×C4—471no—
370C23×C4C22×D4711yescharacteristic
371C23×C4C22×D4711yescharacteristic
372C2×M16—22no—
373C2×M16—24no—
374C2×M16—944no—
375C2×M16—942no—
376C22×C8—472no—
377C22×C8—472no—
378C22×C8C22×D4712yes—
379C22×C8C22×D4712yes—
380#126—944no—
381#126—944no—
382#125—944no—
383#124—944no—
384#123—944no—
385#123—944no—
386C22⋊Q8—944no—
387#121—944no—
388C4×Q8—942no—
389C4×Q8—942no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—942no—
393C4×D4—942no—
394C4×D4—942no—
395C4×D4—942no—
396#117—942no—
397#117—942no—
398#117—942no—
399#117—942no—
400C2×C4⋊C4—471no—
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—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4C4×D4711yescharacteristic
410C2×C22⋊C4—471no—
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—471no—
418C2×C22⋊C4C4×D4711yescharacteristic
419C2×C22⋊C4C4×D4711yescharacteristic
420C2×C42—471no—
421C2×C42—471no—
422C2×C42—474no—
423C2×C42—474no—
424C2×C42—471no—
425C2×C42—471no—
426C2×C42—471no—
427C2×C42C22×D4711yescharacteristic
428C4⋊C8—944no—
429C22⋊C8—22no—
430C22⋊C8—24no—
431C22⋊C8—944no—
432C22⋊C8—944no—
433C22⋊C8—942no—
434C22⋊C8—944no—
435#97—944no—
436#97—944no—
437C4×C8—944no—
438C24—471no—
439C24C2×C4×D4711yescharacteristic
440C24C2×C4×D4711yescharacteristic
441C2×Q8—942no—
442C2×Q8—1882no—
443C2×D4—22no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—942no—
447C2×D4—942no—
448C2×D4—1882no—
449C2×D4—1881no—
450C2×D4—1881no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—471no—
458C22×C4—471no—
459C22×C4—471no—
460C22×C4—471no—
461C22×C4—474no—
462C22×C4—474no—
463C22×C4—471no—
464C22×C4—471no—
465C22×C4—471no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—1882no—
471C22×C4—1881no—
472C22×C4—1881no—
473C22×C4C2×C4×D4711yescharacteristic
474C22×C4C2×C4×D4711yescharacteristic
475C22×C4C2×C4×D4711yescharacteristic
476C22×C4C2×C4×D4711yescharacteristic
477C22×C4D4×D4712yes—
478C22×C4D4×D4712yes—
479C22×C4C23×D4711yescharacteristic
480M16—28no—
481M16—948no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—474no—
488C2×C8—942no—
489C2×C8—942no—
490C2×C8—942no—
491C2×C8—942no—
492C2×C8—942no—
493C2×C8#1062535214yes—
494C4⋊C4—22no—
495C4⋊C4—942no—
496C4⋊C4—942no—
497C4⋊C4—942no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—942no—
501C4⋊C4—942no—
502C4⋊C4—942no—
503C4⋊C4—942no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—942no—
507C22⋊C4—942no—
508C22⋊C4—942no—
509C22⋊C4—942no—
510C22⋊C4—942no—
511C22⋊C4—942no—
512C22⋊C4—942no—
513C22⋊C4—942no—
514C42—21no—
515C42—21no—
516C42—474no—
517C42—474no—
518C42—472no—
519C42—472no—
520C42—472no—
521C42—472no—
522C42—941no—
523C42—941no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—941no—
534C23—942no—
535C23—942no—
536C23—941no—
537C23—941no—
538C23C22⋊C4×D4712yes—
539C23C22×C4×D4711yescharacteristic
540Q8—1884no—
541D4—44no—
542D4—1884no—
543D4—1884no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—472no—
559C2×C4—472no—
560C2×C4—472no—
561C2×C4—472no—
562C2×C4—942no—
563C2×C4—942no—
564C2×C4—941no—
565C2×C4—941no—
566C2×C4—941no—
567C2×C4—941no—
568C2×C4—941no—
569C2×C4—941no—
570C2×C4—941no—
571C2×C4—942no—
572C2×C4—942no—
573C2×C4—941no—
574C2×C4—941no—
575C2×C4—1884no—
576C2×C4—1884no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1882no—
580C2×C4C22⋊C4×D4712yes—
581C2×C4C2×D4×D4711yescharacteristic
582C2×C4C2×D4×D4711yescharacteristic
583C2×C4#41×D4712yes—
584C2×C4C2×#1062535212yes—
585C8—24no—
586C8—42no—
587C8—944no—
588C8—1882no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—42no—
599C22—1881no—
600C22—1881no—
601C22—1881no—
602C22—1882no—
603C22—1882no—
604C22C2×C22⋊C4×D4711yescharacteristic
605C22C4×#1062535212yes—
606C22#1568988812yes—
607C22#1568988911yescharacteristic
608C22#1568988911yescharacteristic
609C4—22no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—942no—
614C4—942no—
615C4—942no—
616C4—942no—
617C4—1881no—
618C4—1881no—
619C4—1881no—
620C4#1568962911yescharacteristic
621C4#1568963111yescharacteristic
622C4#1568966812yes—
623C2—42no—
624C2—81no—
625C2—3761no—
626C2#2024587311yescharacteristic
627C2#2025183011yescharacteristic
628C2#2025183111yescharacteristic
629C2#2025236712yes—
630C2#2025237412yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1870C2
prime radicalC2×#13093996C4
cosocleC22×C4×C47C24
hypercenter#1870D47
hypocenterC47#21376
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. C2×#15689177C2semidirect
3. C2×M16×Dic47C2semidirect
4. #20244326C2semidirect
5. #20244184C2semidirect
6. #20243686C2semidirect
7. #20243685C2semidirect
8. #20243683C2semidirect
9. #20243156C2semidirect
10. #20243153C2semidirect
11. C47×#1870C2semidirect
316. #1870D47semidirect
362. C47#21376semidirect

Maximal subgroups

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

subgroupindex
C2×#156896532
C2×#156891772
C2×M16×Dic472
#202443262
#202441842
#202436862
#202436852
#202436832
#202431562
#202431532
C47×#18702
#2137647