Finite Group Databasebeta

← #26227448

Subgroups — #26227448

Order 12032.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689276C211yesmaximal, characteristic
2C4×#13094003C211yesmaximal, characteristic, complemented
3C2×#15689025C211yesmaximal, characteristic
4#20244242C212yesmaximal, complemented
5#20244071C212yesmaximal, complemented
6#20244016C212yesmaximal, complemented
7#20243742C212yesmaximal, complemented
8#20243693C211yesmaximal, characteristic, complemented
9#20243583C211yesmaximal, characteristic, complemented
10#20243320C211yesmaximal, characteristic
11C4×C22⋊C8×C47C211yesmaximal, characteristic, complemented
12C23×D188C2211yescharacteristic
13C22×#10625325C2212yes—
14C2×#13094003C2211yescharacteristic
15C2×C4×Dic94C2211yescharacteristic, complemented
16C22×C4⋊Dic47C2211yescharacteristic
17C22×(C2×D47)⋊C4—21no—
18C22×(C2×D47)⋊C4—21no—
19#15689276C418yescomplemented
20C2×#13093956C2212yescomplemented
21C2×#13093949C2212yescomplemented
22C2×#13093946C2212yes—
23C2×#13093946C2212yes—
24C2×#13093946C2212yes—
25#15689029C2211yescharacteristic, complemented
26#15689025—21no—
27#15689025—21no—
28#15689025C2212yes—
29#15689025C2212yes—
30#15689021C2211yescharacteristic, complemented
31#15689007—21no—
32#15688996—21no—
33#15688983C2211yescharacteristic
34C4×Dic47⋊C4—21no—
35C4×C4⋊Dic47C2211yescharacteristic, complemented
36C4×C22⋊Dic47—21no—
37#15688886C2212yescomplemented
38#15688886C2212yescomplemented
39C22×C42×C47C2211yescharacteristic
40C2×C22⋊C8×C47C2211yescharacteristic
41C2×C22⋊C8×C47C2211yescharacteristic
42C2×C4×C8×C47C2212yescomplemented
43C47×#336C2212yescomplemented
44C24×D47—21no—
45C22×Dic94C2311yescharacteristic
46C22×D188—21no—
47C22×D188—22no—
48C22×D188—42no—
49C22×D188—42no—
50C22×D188C2×C414yes—
51C22×D188C2312yes—
52C23×Dic47—21no—
53C2×#10625325—42no—
54C2×#10625325—42no—
55C2×#10625325C2×C418yescomplemented
56#13094003—22no—
57#13094003C2×C414yes—
58C4×Dic94—42no—
59C2×C4⋊Dic47—21no—
60C2×C4⋊Dic47—42no—
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C4⋊Dic47C2311yescharacteristic
63C2×C4⋊Dic47C2312yes—
64C2×C4⋊Dic47C2311yescharacteristic
65C2×Dic47⋊C4—21no—
66C2×Dic47⋊C4—21no—
67C2×Dic47⋊C4—21no—
68C2×C22⋊Dic47—21no—
69C2×C22⋊Dic47—21no—
70C2×(C2×D47)⋊C4—22no—
71C2×(C2×D47)⋊C4—22no—
72C2×(C2×D47)⋊C4—22no—
73C2×(C2×D47)⋊C4—22no—
74C2×(C2×D47)⋊C4—44no—
75C2×(C2×D47)⋊C4—44no—
76C2×C4×Dic47—21no—
77C2×C4×Dic47—21no—
78#13093956—42no—
79#13093949—42no—
80#13093946—42no—
81#13093946—42no—
82#13093946C2×C418yescomplemented
83C47⋊#95—21no—
84C47⋊#95—21no—
85C47⋊#95—21no—
86C47⋊#95—21no—
87C47⋊#95—21no—
88C23×C4×C47D411yescharacteristic
89C23×C4×C47D411yescharacteristic
90C23×C4×C47C2311yescharacteristic
91C22×C8×C47C2312yes—
92C22×C8×C47C2312yes—
93C2×C42×C47—42no—
94C2×C42×C47D411yescharacteristic
95C2×C42×C47D411yescharacteristic
96C2×C42×C47C2311yescharacteristic
97C2×C42×C47C2311yescharacteristic
98C22⋊C8×C47—22no—
99C22⋊C8×C47C2×C414yes—
100C4×C8×C47—42no—
101C24×C47C2×D411yescharacteristic
102C23×D47—24no—
103C23×D47—22no—
104C23×D47—42no—
105C23×D47—42no—
106C23×D47—42no—
107C23×D47—42no—
108C23×D47—42no—
109C23×D47—42no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—41no—
115C22×C4×C47—41no—
116C22×C4×C47—41no—
117C22×C4×C47SD1614yescomplemented
118C22×C4×C47C22×C412yes—
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47#4112yes—
125C22×C4×C47C2411yescharacteristic
126C22×Dic47—21no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—22no—
131C22×Dic47—42no—
132C22×Dic47—42no—
133C22×Dic47—42no—
134C2×D188—22no—
135C2×D188—22no—
136C2×D188—22no—
137C2×D188—22no—
138C2×D188—24no—
139C2×D188—22no—
140C2×D188—48no—
141C2×D188—42no—
142C2×D188—42no—
143C2×D188—48no—
144C2×D188C22×C414yes—
145C2×Dic94—22no—
146C2×Dic94—22no—
147C2×Dic94—21no—
148C2×Dic94C22×C412yes—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C2×C8×C47C22×C414yes—
155C42×C47—22no—
156C42×C47—22no—
157C42×C47—41no—
158C42×C47—41no—
159C22⋊Dic47—24no—
160C22⋊Dic47—42no—
161(C2×D47)⋊C4—44no—
162(C2×D47)⋊C4—44no—
163(C2×D47)⋊C4—44no—
164(C2×D47)⋊C4—44no—
165C4⋊Dic47—21no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—22no—
170C4⋊Dic47—42no—
171C4⋊Dic47C22×C412yes—
172Dic47⋊C4—24no—
173Dic47⋊C4—42no—
174Dic47⋊C4—42no—
175C4×Dic47—42no—
176C4×Dic47—42no—
177#10625325—48no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47C4×D412yes—
183C23×C47C2×SD1612yes—
184C23×C47#13512yes—
185C23×C47C22×D411yescharacteristic
186C22×D47—24no—
187C22×D47—48no—
188C22×D47—48no—
189C22×D47—42no—
190C22×D47—42no—
191C22×D47—42no—
192C22×D47—42no—
193C22×D47—42no—
194C22×D47—42no—
195C22×D47—48no—
196C22×D47—84no—
197C22×D47—84no—
198C22×D47—84no—
199C22×D47—84no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—44no—
214C2×C4×C47—44no—
215C2×C4×C47C4×D412yes—
216C2×C4×C47C2×SD1612yes—
217C2×C4×C47C2×SD1612yes—
218C2×C4×C47AGL(1,ℤ/8)12yes—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47H(5,𝔽2)12yes—
222C2×C4×C47H(5,𝔽2)12yes—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—42no—
231C2×Dic47—41no—
232C2×Dic47—42no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—42no—
237C2×Dic47—41no—
238C2×Dic47—84no—
239D188—24no—
240D188—24no—
241D188—48no—
242D188—42no—
243D188—42no—
244Dic94—24no—
245Dic94—42no—
246C8×C47—24no—
247C8×C47—42no—
248#22043—471nomaximal, Sylow
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47C4×SD1614yes—
256C22×C47C2×C4×D411yescharacteristic
257C22×C47#53411yescharacteristic
258C22×C47#53511yescharacteristic
259C22×C47C22×SD1611yescharacteristic
260C22×C47C2×#13511yescharacteristic
261C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C47#57511yescharacteristic
263C2×D47—44no—
264C2×D47—44no—
265C2×D47—44no—
266C2×D47—816no—
267C2×D47—82no—
268C2×D47—82no—
269C2×D47—82no—
270C2×D47—82no—
271C4×C47—22no—
272C4×C47—22no—
273C4×C47—22no—
274C4×C47—22no—
275C4×C47—41no—
276C4×C47#51812yes—
277C4×C47#53812yes—
278Dic47—42no—
279Dic47—42no—
280Dic47—81no—
281Dic47—82no—
282Dic47—81no—
283C2×#450—471no—
284C4×C22⋊Q8—471no—
285C2×#390—471no—
286#1968—472no—
287#1934—472no—
288#1926—472no—
289#1880—472no—
290#1872—471no—
291#1853—471no—
292#1786—471no—
293C4×C22⋊C8D4711yescharacteristic, complemented
294C2×C47—24no—
295C2×C47#232111yescharacteristic
296C2×C47C2×C4×SD1611yescharacteristic
297C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
298C2×C47#318711yescharacteristic
299C2×C47#318811yescharacteristic
300C2×C47#320811yescharacteristic
301C2×C47#321111yescharacteristic
302D47—84no—
303C23×D4—471no—
304C22×SD16—472no—
305C2×C22⋊Q8—471no—
306C2×C4×Q8—471no—
307C22×C4⋊C4—471no—
308C22×C22⋊C4—941no—
309C22×C22⋊C4—941no—
310C22×C42C2×D4711yescharacteristic
311#450—478no—
312C2×#106—472no—
313C2×#103—472no—
314C2×#102—472no—
315C2×#102—472no—
316C2×#102—472no—
317C2×C22⋊C8C2×D4711yescharacteristic
318C2×C22⋊C8C2×D4711yescharacteristic
319C2×C4×C8C2×D4712yescomplemented
320#391—471no—
321#390—472no—
322#390—472no—
323#390—941no—
324#390—941no—
325#389—471no—
326#387—941no—
327#385—941no—
328#380—471no—
329C4×C4⋊C4—471no—
330C4×C4⋊C4—941no—
331C4×C22⋊C4—941no—
332#340—472no—
333#340—472no—
334#336C2×D4712yescomplemented
335C47#2204311yescharacteristic, complemented, Sylow
336C25—941no—
337C22×Q8—471no—
338C22×D4—474no—
339C22×D4—472no—
340C22×D4—941no—
341C22×D4—942no—
342C22×D4—1882no—
343C22×D4—1882no—
344C23×C4—941no—
345C23×C4D18811yescharacteristic
346C23×C4D18811yescharacteristic
347C23×C4C22×D4711yescharacteristic
348C2×SD16—478no—
349C2×SD16—1882no—
350C2×SD16—1882no—
351C22×C8C22×D4712yes—
352C22×C8C22×D4712yes—
353C22⋊Q8—474no—
354C22⋊Q8—942no—
355C4×Q8—1882no—
356C2×C4⋊C4—471no—
357C2×C4⋊C4—472no—
358C2×C4⋊C4—471no—
359C2×C4⋊C4—471no—
360C2×C4⋊C4—941no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—941no—
363C2×C4⋊C4—941no—
364C2×C4⋊C4—1882no—
365C2×C22⋊C4—941no—
366C2×C22⋊C4—941no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—942no—
370C2×C22⋊C4—942no—
371C2×C22⋊C4—1884no—
372C2×C22⋊C4—1884no—
373C2×C42—42no—
374C2×C42—941no—
375C2×C42—941no—
376C2×C42D18811yescharacteristic
377C2×C42D18811yescharacteristic
378C2×C42C22×D4711yescharacteristic
379C2×C42C22×D4711yescharacteristic
380#106—1882no—
381#103—1882no—
382#102—478no—
383#102—1882no—
384#102—1882no—
385C22⋊C8—22no—
386C22⋊C8C4×D4714yes—
387C4×C8—42no—
388#95—941no—
389#95—941no—
390#95—941no—
391#95—941no—
392#95—941no—
393C24—944no—
394C24—942no—
395C24—1882no—
396C24—1882no—
397C24—1882no—
398C24—1882no—
399C24—1882no—
400C24—1882no—
401C24C2×D18811yescharacteristic
402C2×Q8—472no—
403C2×Q8—941no—
404C2×Q8—942no—
405C2×Q8—942no—
406C2×D4—474no—
407C2×D4—944no—
408C2×D4—942no—
409C2×D4—942no—
410C2×D4—942no—
411C2×D4—942no—
412C2×D4—942no—
413C2×D4—1888no—
414C2×D4—1882no—
415C2×D4—1882no—
416C2×D4—1888no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—941no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—942no—
429C22×C4—1882no—
430C22×C4—1882no—
431C22×C4—1882no—
432C22×C4#1062532514yescomplemented
433C22×C4C2×C4×D4712yes—
434C22×C4C2×D18811yescharacteristic
435C22×C4C2×D18811yescharacteristic
436C22×C4C2×D18811yescharacteristic
437C22×C4C2×D18811yescharacteristic
438C22×C4C2×D18811yescharacteristic
439C22×C4#1062535012yes—
440C22×C4C23×D4711yescharacteristic
441SD16—1888no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8C2×C4×D4714yes—
448C4⋊C4—472no—
449C4⋊C4—944no—
450C4⋊C4—941no—
451C4⋊C4—942no—
452C4⋊C4—942no—
453C4⋊C4—942no—
454C4⋊C4—942no—
455C4⋊C4—1882no—
456C4⋊C4—1882no—
457C4⋊C4—1882no—
458C22⋊C4—944no—
459C22⋊C4—1884no—
460C22⋊C4—1882no—
461C22⋊C4—1884no—
462C22⋊C4—1884no—
463C22⋊C4—1884no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—1882no—
469C42—1882no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—944no—
475C23—3764no—
476C23—3764no—
477C23—3764no—
478C23—3764no—
479C23—1888no—
480C23—1888no—
481C23—1882no—
482C23—1882no—
483C23—1882no—
484C23—1882no—
485C23—1882no—
486C23—1882no—
487C23—1888no—
488C23C4×D18812yes—
489C23C2×#1062532512yes—
490C23#1309404612yes—
491C23C22×D18811yescharacteristic
492Q8—944no—
493Q8—1882no—
494D4—944no—
495D4—944no—
496D4—1888no—
497D4—1882no—
498D4—1882no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—3764no—
517C2×C4—1882no—
518C2×C4—1881no—
519C2×C4—1881no—
520C2×C4—1882no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1882no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4C4×D18812yes—
531C2×C4C2×#1062532512yes—
532C2×C4C2×#1062532512yes—
533C2×C4#1309405212yes—
534C2×C4C22×C4×D4711yescharacteristic
535C2×C4C2×#1062535011yescharacteristic
536C2×C4#1309406912yes—
537C2×C4#1309407012yes—
538C8—24no—
539C8—42no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—37616no—
547C22—3762no—
548C22—3762no—
549C22—3762no—
550C22—3762no—
551C22—1884no—
552C22—1884no—
553C22—1884no—
554C22C4×#1062532514yes—
555C22C2×C4×D18811yescharacteristic
556C22#1568968511yescharacteristic
557C22#1568969211yescharacteristic
558C22C22×#1062532511yescharacteristic
559C22C2×#1309404611yescharacteristic
560C22C2×#1309405211yescharacteristic
561C22#1568992311yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—3762no—
568C4—3761no—
569C4—3761no—
570C4—1882no—
571C4—1882no—
572C4#1568959512yes—
573C4#1568970612yes—
574C2—24no—
575C2—3764no—
576C2#2024604811yescharacteristic
577C2C2×C4×#1062532511yescharacteristic
578C2C4×#1309405211yescharacteristic
579C2#2025414611yescharacteristic
580C2#2025415511yescharacteristic
581C2#2025433111yescharacteristic
582C2#2025435511yescharacteristic
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×C22⋊C8×C47C2
socleC23×C47C22×D4
prime radicalC23×D188C22
cosocleC22×C4×C47C24
hypercenterC4×C22⋊C8D47
hypocenterC47#22043
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×#13094003C2semidirect
4. #20244242C2semidirect
5. #20244071C2semidirect
6. #20244016C2semidirect
7. #20243742C2semidirect
8. #20243693C2semidirect
9. #20243583C2semidirect
11. C4×C22⋊C8×C47C2semidirect
15. C2×C4×Dic94C22semidirect
19. #15689276C4semidirect
20. C2×#13093956C22semidirect
21. C2×#13093949C22semidirect
25. #15689029C22semidirect
30. #15689021C22semidirect
35. C4×C4⋊Dic47C22semidirect
37. #15688886C22semidirect
38. #15688886C22semidirect
42. C2×C4×C8×C47C22semidirect
43. C47×#336C22semidirect
55. C2×#10625325C2×C4semidirect
82. #13093946C2×C4semidirect
117. C22×C4×C47SD16semidirect
293. C4×C22⋊C8D47semidirect
319. C2×C4×C8C2×D47semidirect
334. #336C2×D47semidirect
335. C47#22043semidirect
432. C22×C4#10625325semidirect

Maximal subgroups

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

subgroupindex
C2×#156892762
C4×#130940032
C2×#156890252
#202442422
#202440712
#202440162
#202437422
#202436932
#202435832
#202433202
C4×C22⋊C8×C472
#2204347