Finite Group Databasebeta

← #26227449

Subgroups — #26227449

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
1C22⋊Q8×Dic47C211yesmaximal, characteristic, complemented
2C2×#15689026C211yesmaximal, characteristic
3#20244244C212yesmaximal, complemented
4#20244073C212yesmaximal, complemented
5#20244017C212yesmaximal, complemented
6#20243745C212yesmaximal, complemented
7#20243696C211yesmaximal, characteristic, complemented
8#20243583C211yesmaximal, characteristic, complemented
9#20243319C211yesmaximal, characteristic
10C22⋊C8×Dic47C211yesmaximal, characteristic, complemented
11C2×C47×#450C211yesmaximal, characteristic
12C2×Q8×Dic47C2211yescharacteristic, complemented
13C22×C4⋊Dic47C2211yescharacteristic
14C22×C22⋊Dic47—21no—
15C22×C22⋊Dic47—21no—
16C22×C4×Dic47C2211yescharacteristic
17C2×#13093956C2212yescomplemented
18C2×#13093948C2212yescomplemented
19C2×#13093945C2212yes—
20C2×#13093945C2212yes—
21C2×C22⋊(C47⋊C8)C2211yescharacteristic
22C2×C8×Dic47C2212yescomplemented
23#15689030C2211yescharacteristic, complemented
24#15689026—21no—
25#15689026—21no—
26#15689026C2212yes—
27#15689026C2212yes—
28#15689022C2211yescharacteristic, complemented
29#15689009—21no—
30#15688998—21no—
31#15688983C2211yescharacteristic
32C4⋊C4×Dic47—21no—
33C4⋊C4×Dic47C2211yescharacteristic, complemented
34C22⋊C4×Dic47—21no—
35#15688886C2212yescomplemented
36#15688885C2212yescomplemented
37#15688877C2212yescomplemented
38C23×D4×C47C2211yescharacteristic
39C22×SD16×C47C2212yes—
40C2×C22⋊Q8×C47C2211yescharacteristic
41C47×#450C418yescomplemented
42C2×#102×C47C2212yes—
43C2×C22⋊C8×C47C2211yescharacteristic
44C23×Dic47—21no—
45C23×Dic47D411yescharacteristic
46C23×Dic47D411yescharacteristic
47C22×C47⋊C8C2312yes—
48Q8×Dic47—42no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—42no—
53C2×C4⋊Dic47C2311yescharacteristic
54C2×C4⋊Dic47C2311yescharacteristic
55C2×C4⋊Dic47C2312yes—
56C2×C22⋊Dic47—21no—
57C2×C22⋊Dic47—22no—
58C2×C22⋊Dic47—22no—
59C2×C22⋊Dic47—22no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—44no—
62C2×C22⋊Dic47—44no—
63C2×C4×Dic47—21no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47—42no—
66C2×C4×Dic47D411yescharacteristic
67C2×C4×Dic47D411yescharacteristic
68C2×C4×Dic47C2311yescharacteristic
69C2×C4×Dic47C2311yescharacteristic
70#13093956—42no—
71#13093948—42no—
72#13093945—42no—
73#13093945—42no—
74C22⋊(C47⋊C8)—22no—
75C8×Dic47—42no—
76C47⋊#95—21no—
77C47⋊#95—21no—
78C47⋊#95—21no—
79C47⋊#95—21no—
80C47⋊#95—21no—
81C25×C47—21no—
82C22×Q8×C47C2311yescharacteristic
83C22×D4×C47—21no—
84C22×D4×C47—22no—
85C22×D4×C47—42no—
86C22×D4×C47—42no—
87C22×D4×C47C2×C414yes—
88C22×D4×C47C2312yes—
89C23×C4×C47C2311yescharacteristic
90C2×SD16×C47—42no—
91C2×SD16×C47—42no—
92C2×SD16×C47C2×C418yescomplemented
93C22×C8×C47C2312yes—
94C22⋊Q8×C47—22no—
95C22⋊Q8×C47C2×C414yes—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C22⋊C4×C47—21no—
99#102×C47C2×C418yescomplemented
100C22⋊C8×C47C2×C414yes—
101C24×C47—24no—
102C24×C47—22no—
103C24×C47—42no—
104C24×C47—42no—
105C24×C47—42no—
106C24×C47—42no—
107C24×C47—42no—
108C24×C47—42no—
109C24×C47C2×D411yescharacteristic
110C2×Q8×C47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47C22×C412yes—
114C2×D4×C47—24no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47—22no—
120C2×D4×C47—48no—
121C2×D4×C47—42no—
122C2×D4×C47—42no—
123C2×D4×C47—48no—
124C2×D4×C47C22×C414yes—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47#4112yes—
132C22×C4×C47C2411yescharacteristic
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—41no—
139C22×Dic47—41no—
140C22×Dic47—41no—
141C22×Dic47—42no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144C22×Dic47SD1614yescomplemented
145C22×Dic47C2×D411yescharacteristic
146C22×Dic47C2×D411yescharacteristic
147C22×Dic47C2×D411yescharacteristic
148C22×Dic47C2×D411yescharacteristic
149SD16×C47—48no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C2×C8×C47C22×C414yes—
153C4⋊C4×C47—24no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47C22×C412yes—
156C22⋊C4×C47—24no—
157C22⋊Dic47—44no—
158C22⋊Dic47—42no—
159C22⋊Dic47—44no—
160C22⋊Dic47—44no—
161C22⋊Dic47—44no—
162C4⋊Dic47—22no—
163C4⋊Dic47—22no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47—42no—
167C4⋊Dic47—42no—
168C4⋊Dic47—42no—
169C4×Dic47—22no—
170C4×Dic47—22no—
171C4×Dic47—41no—
172C4×Dic47—41no—
173C4×Dic47—42no—
174C4×Dic47—42no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—22no—
178C23×C47—21no—
179C23×C47—21no—
180C23×C47—21no—
181C23×C47—21no—
182C23×C47—24no—
183C23×C47—48no—
184C23×C47—48no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47—48no—
192C23×C47—84no—
193C23×C47—84no—
194C23×C47—84no—
195C23×C47—84no—
196C23×C47C4×D412yes—
197C23×C47C2×SD1612yes—
198C23×C47#13512yes—
199C23×C47C22×D411yescharacteristic
200Q8×C47—24no—
201Q8×C47—42no—
202D4×C47—24no—
203D4×C47—24no—
204D4×C47—48no—
205D4×C47—42no—
206D4×C47—42no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—22no—
212C2×C4×C47—22no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47C4×D412yes—
218C2×C4×C47C23×C411yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47H(5,𝔽2)12yes—
221C2×C4×C47H(5,𝔽2)12yes—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—44no—
232C2×Dic47—44no—
233C2×Dic47—41no—
234C2×Dic47—42no—
235C2×Dic47—41no—
236C2×Dic47—42no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—42no—
241C2×Dic47—41no—
242C2×Dic47—84no—
243C2×Dic47C2×SD1612yes—
244C2×Dic47C2×SD1612yes—
245C2×Dic47AGL(1,ℤ/8)12yes—
246C8×C47—24no—
247C47⋊C8—42no—
248#22043—471nomaximal, Sylow
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—44no—
256C22×C47—44no—
257C22×C47—44no—
258C22×C47—816no—
259C22×C47—82no—
260C22×C47—82no—
261C22×C47—82no—
262C22×C47—82no—
263C22×C47C4×SD1614yes—
264C22×C47C2×C4×D411yescharacteristic
265C22×C47#53411yescharacteristic
266C22×C47#53511yescharacteristic
267C22×C47C22×SD1611yescharacteristic
268C22×C47C2×#13511yescharacteristic
269C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C47#57511yescharacteristic
271C4×C47—22no—
272C4×C47—42no—
273C4×C47—42no—
274C4×C47#51812yes—
275C4×C47#53812yes—
276Dic47—22no—
277Dic47—22no—
278Dic47—22no—
279Dic47—41no—
280Dic47—81no—
281Dic47—82no—
282Dic47—81no—
283C2×#450D4711yescharacteristic
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⋊C8—471no—
294C2×C47—24no—
295C2×C47—84no—
296C2×C47#232111yescharacteristic
297C2×C47C2×C4×SD1611yescharacteristic
298C2×C47C4×AGL(1,ℤ/8)11yescharacteristic
299C2×C47#318711yescharacteristic
300C2×C47#318811yescharacteristic
301C2×C47#320811yescharacteristic
302C2×C47#321111yescharacteristic
303C23×D4C2×D4711yescharacteristic
304C22×SD16C2×D4712yes—
305C2×C22⋊Q8C2×D4711yescharacteristic
306C2×C4×Q8—471no—
307C22×C4⋊C4—471no—
308C22×C22⋊C4—941no—
309C22×C22⋊C4—941no—
310C22×C42—471no—
311#450Dic4718yescomplemented
312C2×#106—472no—
313C2×#103—472no—
314C2×#102—472no—
315C2×#102—472no—
316C2×#102C2×D4712yes—
317C2×C22⋊C8—471no—
318C2×C22⋊C8C2×D4711yescharacteristic
319C2×C4×C8—472no—
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#336—472no—
335C47#2204311yescharacteristic, complemented, Sylow
336C25—21no—
337C22×Q8C22×D4711yescharacteristic
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C2×Dic4714yes—
343C22×D4C22×D4712yes—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—941no—
347C23×C4C22×D4711yescharacteristic
348C2×SD16—42no—
349C2×SD16—42no—
350C2×SD16C2×Dic4718yescomplemented
351C22×C8—472no—
352C22×C8C22×D4712yes—
353C22⋊Q8—22no—
354C22⋊Q8C2×Dic4714yes—
355C4×Q8—1882no—
356C2×C4⋊C4—21no—
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—1882no—
364C2×C4⋊C4C22×D4711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—942no—
367C2×C22⋊C4—942no—
368C2×C22⋊C4—942no—
369C2×C22⋊C4—942no—
370C2×C22⋊C4—941no—
371C2×C22⋊C4—1884no—
372C2×C22⋊C4—1884no—
373C2×C42—471no—
374C2×C42—471no—
375C2×C42—471no—
376C2×C42—471no—
377C2×C42—941no—
378C2×C42—941no—
379C2×C42—1882no—
380#106—1882no—
381#103—1882no—
382#102—1882no—
383#102—1882no—
384#102C2×Dic4718yescomplemented
385C22⋊C8—942no—
386C22⋊C8C2×Dic4714yes—
387C4×C8—1882no—
388#95—941no—
389#95—941no—
390#95—941no—
391#95—941no—
392#95—941no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24D4×D4711yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C22×Dic4712yes—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C22×Dic4714yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—474no—
425C22×C4—471no—
426C22×C4—942no—
427C22×C4—942no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—942no—
431C22×C4—1881no—
432C22×C4—1881no—
433C22×C4—1881no—
434C22×C4—1882no—
435C22×C4—1882no—
436C22×C4—1882no—
437C22×C4D4×D4711yescharacteristic
438C22×C4#1062535212yes—
439C22×C4C22×Dic4712yes—
440C22×C4C23×D4711yescharacteristic
441SD16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—942no—
445C2×C8—942no—
446C2×C8—942no—
447C2×C8C22×Dic4714yes—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—942no—
451C4⋊C4—942no—
452C4⋊C4—942no—
453C4⋊C4—942no—
454C4⋊C4—1882no—
455C4⋊C4—1882no—
456C4⋊C4—1882no—
457C4⋊C4C22×Dic4712yes—
458C22⋊C4—24no—
459C22⋊C4—1884no—
460C22⋊C4—1884no—
461C22⋊C4—1884no—
462C22⋊C4—1884no—
463C22⋊C4—1882no—
464C42—942no—
465C42—942no—
466C42—1881no—
467C42—1881no—
468C42—1882no—
469C42—1882no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23D4×Dic4712yes—
489C23SD16×D4712yes—
490C23#1309404512yes—
491C23C2×D4×D4711yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—472no—
510C2×C4—472no—
511C2×C4—472no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—3764no—
522C2×C4—1884no—
523C2×C4—1884no—
524C2×C4—1882no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1882no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1882no—
532C2×C4—1881no—
533C2×C4D4×Dic4712yes—
534C2×C4C23×Dic4711yescharacteristic
535C2×C4C2×#1062535211yescharacteristic
536C2×C4#1309406912yes—
537C2×C4#1309406912yes—
538C8—24no—
539C8—1882no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22SD16×Dic4714yes—
555C22C2×D4×Dic4711yescharacteristic
556C22#1568968311yescharacteristic
557C22#1568968811yescharacteristic
558C22C2×SD16×D4711yescharacteristic
559C22C2×#1309404511yescharacteristic
560C22AGL(1,ℤ/8)×D4711yescharacteristic
561C22#1568991911yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—942no—
566C4—942no—
567C4—942no—
568C4—3762no—
569C4—3761no—
570C4—3761no—
571C4—1881no—
572C4#1568959812yes—
573C4#1568970812yes—
574C2—24no—
575C2—84no—
576C2#2024604911yescharacteristic
577C2C2×SD16×Dic4711yescharacteristic
578C2AGL(1,ℤ/8)×Dic4711yescharacteristic
579C2#2025414211yescharacteristic
580C2#2025415611yescharacteristic
581C2#2025432711yescharacteristic
582C2#2025435611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#450C2
socleC23×C47C22×D4
prime radicalC23×D4×C47C22
cosocleC22×C4×C47C24
hypercenterC2×#450D47
hypocenterC47#22043
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×Dic47C2semidirect
3. #20244244C2semidirect
4. #20244073C2semidirect
5. #20244017C2semidirect
6. #20243745C2semidirect
7. #20243696C2semidirect
8. #20243583C2semidirect
10. C22⋊C8×Dic47C2semidirect
12. C2×Q8×Dic47C22semidirect
17. C2×#13093956C22semidirect
18. C2×#13093948C22semidirect
22. C2×C8×Dic47C22semidirect
23. #15689030C22semidirect
28. #15689022C22semidirect
33. C4⋊C4×Dic47C22semidirect
35. #15688886C22semidirect
36. #15688885C22semidirect
37. #15688877C22semidirect
41. C47×#450C4semidirect
92. C2×SD16×C47C2×C4semidirect
99. #102×C47C2×C4semidirect
144. C22×Dic47SD16semidirect
311. #450Dic47semidirect
335. C47#22043semidirect
350. C2×SD16C2×Dic47semidirect
384. #102C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic472
C2×#156890262
#202442442
#202440732
#202440172
#202437452
#202436962
#202435832
#202433192
C22⋊C8×Dic472
C2×C47×#4502
#2204347