Finite Group Databasebeta

← #26217649

Subgroups — #26217649

Order 12032.

1802 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic47C211yesmaximal, characteristic, complemented
2C2×#15689124C211yesmaximal, characteristic
3C2×#15689132C211yesmaximal, characteristic, complemented
4#20245879C211yesmaximal, characteristic
5#20243745C214yesmaximal, complemented
6#20243731C214yesmaximal, complemented
7#20243108C211yesmaximal, characteristic, complemented
8#20243081C211yesmaximal, characteristic, complemented
9C22×#137×C47C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C2×Q8×Dic47—24no—
12C2×Q8×Dic47—22no—
13C2×Q8×Dic47—24no—
14C2×Q8×Dic47C2214yes—
15C2×#13093983C2211yescharacteristic
16C22×C4⋊Dic47—21no—
17C22×C4⋊Dic47C2211yescharacteristic
18C22×C4×Dic47—21no—
19C22×C4×Dic47C2211yescharacteristic
20#15689132—24no—
21#15689124—24no—
22C2×#13093948C2212yes—
23C2×#13093948C2214yes—
24C2×#13093945C2212yes—
25#15689030C2212yes—
26#15689026C2212yes—
27#15689003—21no—
28#15689003—21no—
29C4×C22⋊Dic47—22no—
30#15688886C2214yes—
31#15688877C2214yes—
32C22×#41×C47C2211yescharacteristic
33C23×Q8×C47C2211yescharacteristic
34C2×#137×C47—24no—
35C2×#137×C47—24no—
36C2×#137×C47C418yescomplemented
37C22×Q16×C47C2212yes—
38C22×SD16×C47C2212yes—
39C22×M16×C47C2211yescharacteristic
40C23×Dic47—21no—
41C23×Dic47D411yescharacteristic
42C23×Dic47D411yescharacteristic
43C2×#10625329—22no—
44C2×#10625329—22no—
45C2×#10625329—22no—
46C22×C47⋊C8C2312yes—
47Q8×Dic47—216no—
48Q8×Dic47—28no—
49Q8×Dic47—28no—
50#13093983—22no—
51#13093983—22no—
52C2×C4⋊Dic47—28no—
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47—22no—
55C2×C4⋊Dic47—24no—
56C2×C4⋊Dic47D414yes—
57C2×C4⋊Dic47C2312yes—
58C2×C4⋊Dic47C2312yes—
59C2×C22⋊Dic47—21no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C4×Dic47—28no—
63C2×C4×Dic47—22no—
64C2×C4×Dic47—22no—
65C2×C4×Dic47—24no—
66C2×C4×Dic47—21no—
67C2×C4×Dic47D412yes—
68C2×C4×Dic47C2312yes—
69#13093948—28no—
70#13093948—24no—
71#13093945—24no—
72C47⋊#95—22no—
73C47⋊#95—22no—
74C2×#41×C47—22no—
75C2×#41×C47—22no—
76C2×#41×C47—22no—
77C2×#41×C47—22no—
78C2×#41×C47—22no—
79C2×#41×C47C2×C414yes—
80C22×Q8×C47—22no—
81C22×Q8×C47—21no—
82C22×Q8×C47—22no—
83C22×Q8×C47—22no—
84C22×Q8×C47C2×C414yes—
85C22×Q8×C47D414yes—
86C22×Q8×C47D414yes—
87C22×Q8×C47C2311yescharacteristic
88C22×Q8×C47C2312yes—
89C22×D4×C47—21no—
90C22×D4×C47C2311yescharacteristic
91C23×C4×C47—21no—
92C23×C4×C47—21no—
93C23×C4×C47C2311yescharacteristic
94#137×C47—232no—
95C2×Q16×C47—24no—
96C2×Q16×C47—24no—
97C2×Q16×C47C2×C418yescomplemented
98C2×SD16×C47—24no—
99C2×SD16×C47—24no—
100C2×SD16×C47C2×C418yescomplemented
101C2×M16×C47—22no—
102C2×M16×C47—22no—
103C2×M16×C47C2×C414yes—
104C22×C8×C47C2312yes—
105C24×C47—21no—
106C24×C47C2×D411yescharacteristic
107#41×C47—28no—
108#41×C47—28no—
109#41×C47—44no—
110#41×C47—44no—
111C2×Q8×C47—24no—
112C2×Q8×C47—28no—
113C2×Q8×C47—24no—
114C2×Q8×C47—24no—
115C2×Q8×C47—28no—
116C2×Q8×C47—28no—
117C2×Q8×C47—22no—
118C2×Q8×C47—22no—
119C2×Q8×C47—22no—
120C2×Q8×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47C22⋊C4116yescomplemented
126C2×Q8×C47C22×C412yes—
127C2×Q8×C47C22×C414yes—
128C2×Q8×C47C2×D414yes—
129C2×Q8×C47C2×D414yes—
130C2×D4×C47—24no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C2×D4×C47—21no—
134C2×D4×C47—21no—
135C2×D4×C47—21no—
136C2×D4×C47—42no—
137C2×D4×C47—42no—
138C2×D4×C47C22×C412yes—
139C22×C4×C47—24no—
140C22×C4×C47—24no—
141C22×C4×C47—24no—
142C22×C4×C47—22no—
143C22×C4×C47—21no—
144C22×C4×C47—24no—
145C22×C4×C47—42no—
146C22×C4×C47—42no—
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D412yes—
152C22×C4×C47#4112yes—
153C22×C4×C47#4112yes—
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—22no—
156C22×Dic47—24no—
157C22×Dic47—22no—
158C22×Dic47—24no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—42no—
162C22×Dic47—44no—
163C22×Dic47C2×D412yes—
164C22×Dic47C2×D412yes—
165Q16×C47—216no—
166SD16×C47—216no—
167M16×C47—28no—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47C22×C414yes—
171C22⋊Dic47—44no—
172C22⋊Dic47—42no—
173C22⋊Dic47—44no—
174C4⋊Dic47—216no—
175C4⋊Dic47—24no—
176C4⋊Dic47—22no—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47C2×D414yes—
180C4×Dic47—216no—
181C4×Dic47—24no—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47—42no—
185#10625329—44no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47C4×D412yes—
197C23×C47#12012yes—
198C23×C47#12112yes—
199C23×C47C22×D411yescharacteristic
200Q8×C47—24no—
201Q8×C47—24no—
202Q8×C47—24no—
203Q8×C47—216no—
204Q8×C47—22no—
205Q8×C47—22no—
206Q8×C47—41no—
207Q8×C47—41no—
208Q8×C47C2×C22⋊C418yes—
209D4×C47—22no—
210D4×C47—22no—
211D4×C47—48no—
212D4×C47—41no—
213D4×C47—41no—
214C2×C4×C47—24no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—216no—
218C2×C4×C47—22no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—48no—
222C2×C4×C47C2×C22⋊C414yes—
223C2×C4×C47C4×D414yes—
224C2×C4×C47C4×D412yes—
225C2×C4×C47#12012yes—
226C2×C4×C47#12114yes—
227C2×C4×C47#12112yes—
228C2×C4×C47#12314yes—
229C2×C4×C47C23×C411yescharacteristic
230C2×C4×C47C22×D411yescharacteristic
231C2×C4×C47C22×D411yescharacteristic
232C2×C4×C47C22×D411yescharacteristic
233C2×C4×C47C2×#4111yescharacteristic
234C2×C4×C47C2×#4111yescharacteristic
235C2×Dic47—28no—
236C2×Dic47—22no—
237C2×Dic47—22no—
238C2×Dic47—22no—
239C2×Dic47—24no—
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—44no—
243C2×Dic47—48no—
244C2×Dic47—44no—
245C2×Dic47—42no—
246C2×Dic47—42no—
247C2×Dic47—42no—
248C2×Dic47#13714yescomplemented
249C8×C47—24no—
250C47⋊C8—42no—
251#20926—471nomaximal, Sylow
252C22×C47—24no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—24no—
256C22×C47—44no—
257C22×C47—42no—
258C22×C47—81no—
259C22×C47—81no—
260C22×C47—82no—
261C22×C47#38614yes—
262C22×C47C2×C4×D411yescharacteristic
263C22×C47C2×#12011yescharacteristic
264C22×C47C2×#12111yescharacteristic
265C22×C47C2×#13712yes—
266C22×C47#57512yes—
267C4×C47—28no—
268C4×C47—42no—
269C4×C47#38614yes—
270C4×C47C22×C22⋊C411yescharacteristic
271C4×C47C2×C4×D411yescharacteristic
272C4×C47C2×#12111yescharacteristic
273C4×C47C2×#12311yescharacteristic
274Dic47—24no—
275Dic47—42no—
276Dic47—44no—
277Dic47—82no—
278C22×#137D4711yescharacteristic
279C22×C4×Q8—471no—
280C2×#417—471no—
281C2×#419—471no—
282#2297—471no—
283#1880—474no—
284#1878—474no—
285#1736—471no—
286#1728—471no—
287C2×C47—24no—
288C2×C47—82no—
289C2×C47C2×#38611yescharacteristic
290C2×C47C4×#13712yes—
291C2×C47#300411yescharacteristic
292C2×C47#300511yescharacteristic
293C2×C47#303712yes—
294C22×#41C2×D4711yescharacteristic
295C23×Q8C2×D4711yescharacteristic
296C2×#137—24no—
297C2×#137—24no—
298C2×#137Dic4718yescomplemented
299C22×Q16C2×D4712yes—
300C22×SD16C2×D4712yes—
301C22×M16—471no—
302C22×M16C2×D4711yescharacteristic
303C2×C4×Q8—474no—
304C2×C4×Q8—942no—
305C2×C4×Q8—944no—
306C2×C4×Q8—944no—
307C2×#117—471no—
308C22×C4⋊C4—471no—
309C22×C4⋊C4—941no—
310C22×C42—471no—
311C22×C42—941no—
312#419—944no—
313#417—944no—
314C2×#103—474no—
315C2×#103—472no—
316C2×#102—472no—
317#391—472no—
318#390—472no—
319#386—941no—
320#386—941no—
321C4×C22⋊C4—942no—
322#340—474no—
323#336—474no—
324C47#2092611yescharacteristic, complemented, Sylow
325C2×#41—22no—
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41C2×Dic4714yes—
331C22×Q8—22no—
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8C2×Dic4714yes—
336C22×Q8C22⋊D4714yes—
337C22×Q8C22⋊D4714yes—
338C22×Q8C22×D4711yescharacteristic
339C22×Q8C22×D4712yes—
340C22×D4—21no—
341C22×D4C22×D4711yescharacteristic
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—941no—
347C23×C4C22×D4711yescharacteristic
348#137—232no—
349C2×Q16—24no—
350C2×Q16—24no—
351C2×Q16C2×Dic4718yescomplemented
352C2×SD16—24no—
353C2×SD16—24no—
354C2×SD16C2×Dic4718yescomplemented
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—942no—
358C2×M16—942no—
359C2×M16—942no—
360C2×M16C2×Dic4714yes—
361C22×C8—472no—
362C22×C8C22×D4712yes—
363C4×Q8—948no—
364C4×Q8—9416no—
365C4×Q8—948no—
366#117—942no—
367#117—942no—
368C2×C4⋊C4—474no—
369C2×C4⋊C4—472no—
370C2×C4⋊C4—472no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—948no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—942no—
375C2×C22⋊C4—942no—
376C2×C22⋊C4—942no—
377C2×C22⋊C4—941no—
378C2×C42—472no—
379C2×C42—472no—
380C2×C42—944no—
381C2×C42—948no—
382C2×C42—942no—
383C2×C42—942no—
384C2×C42—941no—
385#103—944no—
386#103—948no—
387#102—944no—
388#95—942no—
389#95—942no—
390C24—21no—
391C24D4×D4711yescharacteristic
392#41—28no—
393#41—28no—
394#41—44no—
395#41—44no—
396C2×Q8—28no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8C22⋊Dic47116yescomplemented
411C2×Q8C22×Dic4712yes—
412C2×Q8C22×Dic4714yes—
413C2×Q8C2×C22⋊D4714yes—
414C2×Q8C2×C22⋊D4714yes—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C22×Dic4712yes—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—472no—
433C22×C4—472no—
434C22×C4—944no—
435C22×C4—942no—
436C22×C4—944no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—1882no—
441C22×C4—1884no—
442C22×C4D4×D4711yescharacteristic
443C22×C4D4×D4712yes—
444C22×C4#1062535212yes—
445C22×C4#1062535212yes—
446C22×C4C22×Dic4712yes—
447C22×C4C2×C22⋊D4712yes—
448C22×C4C2×C22⋊D4712yes—
449C22×C4C23×D4711yescharacteristic
450Q16—216no—
451SD16—216no—
452M16—28no—
453M16—1884no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—942no—
457C2×C8—942no—
458C2×C8—942no—
459C2×C8C22×Dic4714yes—
460C4⋊C4—474no—
461C4⋊C4—9416no—
462C4⋊C4—944no—
463C4⋊C4—942no—
464C4⋊C4—942no—
465C4⋊C4—942no—
466C22⋊C4—1884no—
467C22⋊C4—1884no—
468C22⋊C4—1882no—
469C42—9416no—
470C42—944no—
471C42—942no—
472C42—942no—
473C42—1882no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23D4×Dic4712yes—
482C23#1309399212yes—
483C23#1309399612yes—
484C23C2×D4×D4711yescharacteristic
485Q8—216no—
486Q8—24no—
487Q8—24no—
488Q8—24no—
489Q8—22no—
490Q8—22no—
491Q8—41no—
492Q8—41no—
493Q8C2×C22⋊Dic4718yes—
494D4—22no—
495D4—22no—
496D4—48no—
497D4—41no—
498D4—41no—
499C2×C4—216no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—48no—
507C2×C4—474no—
508C2×C4—948no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—944no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1884no—
519C2×C4—1888no—
520C2×C4—1884no—
521C2×C4C2×C22⋊Dic4714yes—
522C2×C4D4×Dic4712yes—
523C2×C4D4×Dic4714yes—
524C2×C4#1309399212yes—
525C2×C4#1309399612yes—
526C2×C4#1309399614yes—
527C2×C4#1309400514yes—
528C2×C4C23×Dic4711yescharacteristic
529C2×C4C2×D4×D4711yescharacteristic
530C2×C4C22×C22⋊D4711yescharacteristic
531C2×C4C22×C22⋊D4711yescharacteristic
532C2×C4C2×#1062535211yescharacteristic
533C2×C4C2×#1062535211yescharacteristic
534C8—24no—
535C8—1882no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—44no—
542C22—82no—
543C22—81no—
544C22—81no—
545C22#1568900314yes—
546C22C2×D4×Dic4711yescharacteristic
547C22C2×#1309399211yescharacteristic
548C22C2×#1309399611yescharacteristic
549C22#137×D4712yes—
550C22#1568992212yes—
551C4—28no—
552C4—42no—
553C4—944no—
554C4—3762no—
555C4—1882no—
556C4—1884no—
557C4#1568900314yes—
558C4C22×C22⋊Dic4711yescharacteristic
559C4C2×D4×Dic4711yescharacteristic
560C4C2×#1309399611yescharacteristic
561C4C2×#1309400511yescharacteristic
562C2—24no—
563C2—82no—
564C2C2×#1568900311yescharacteristic
565C2#137×Dic4712yes—
566C2#2025262011yescharacteristic
567C2#2025262911yescharacteristic
568C2#2025285712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#137×C47C2
socleC23×C47C22×D4
prime radicalC22×#41×C47C22
cosocleC22×C4×C47C24
hypercenterC22×#137D47
hypocenterC47#20926
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic47C2semidirect
3. C2×#15689132C2semidirect
5. #20243745C2semidirect
6. #20243731C2semidirect
7. #20243108C2semidirect
8. #20243081C2semidirect
36. C2×#137×C47C4semidirect
97. C2×Q16×C47C2×C4semidirect
100. C2×SD16×C47C2×C4semidirect
125. C2×Q8×C47C22⋊C4semidirect
248. C2×Dic47#137semidirect
298. C2×#137Dic47semidirect
324. C47#20926semidirect
351. C2×Q16C2×Dic47semidirect
354. C2×SD16C2×Dic47semidirect
410. C2×Q8C22⋊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×#156891242
C2×#156891322
#202458792
#202437452
#202437312
#202431082
#202430812
C22×#137×C472
#2092647