Finite Group Databasebeta

← #26217695

Subgroups — #26217695

Order 12032.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094055C211yesmaximal, characteristic
2C2×#15689603C211yesmaximal, characteristic
3C2×#15689117C211yesmaximal, characteristic
4C22×#13093948C211yesmaximal, characteristic, complemented
5#20243732C214yesmaximal, complemented
6#20243725C214yesmaximal, complemented
7#20243109C211yesmaximal, characteristic, complemented
8#20243085C211yesmaximal, characteristic, complemented
9C47×#2310C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C2×#13094055—24no—
12C2×#13094055—24no—
13C2×#13094055C418yescomplemented
14C22×#10625337C2212yes—
15C22×#10625336C2212yes—
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18#15689603—24no—
19#15689603—24no—
20C2×C4×Dic94C2212yes—
21C2×C4×D188C2212yes—
22C2×(C22⋊D47)⋊C4—22no—
23C2×#13093982—21no—
24C22×C4⋊Dic47C2211yescharacteristic
25C2×C4⋊C4×D47—21no—
26#15689117—24no—
27C2×#13093948—24no—
28C2×#13093948C2214yes—
29C2×#13093947C2212yes—
30C2×#13093944C2212yes—
31#15688885C2214yes—
32#15688878C2214yes—
33C23×Q8×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C2×#117×C47C2211yescharacteristic
36C47×#391—21no—
37C47×#391—21no—
38C47×#391C2214yes—
39C47×#381—22no—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350C2×C414yes—
44C2×#10625350D414yes—
45C2×#10625350D414yes—
46C22×Dic94C2311yescharacteristic
47C22×D188C2311yescharacteristic
48C22×C22⋊D47—21no—
49C23×Dic47—21no—
50C22×C4×D47—21no—
51C22×C4×D47—22no—
52#13094055—232no—
53C2×#10625337—24no—
54C2×#10625337—24no—
55C2×#10625337C2×C418yescomplemented
56C2×#10625336—24no—
57C2×#10625336—24no—
58C2×#10625336C2×C418yescomplemented
59C2×#10625329—22no—
60C2×#10625329—22no—
61C2×#10625329C2×C414yescomplemented
62C22×C47⋊C8—22no—
63C22×C47⋊C8—22no—
64C22×C47⋊C8—22no—
65C22×C47⋊C8C2312yes—
66C22×C47⋊C8C2312yes—
67C4×Dic94—24no—
68C4×Dic94—24no—
69C4×D188—24no—
70C4×D188—24no—
71(C22⋊D47)⋊C4—216no—
72#13093982—28no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47—22no—
75C2×C4⋊Dic47D414yes—
76C2×C4⋊Dic47C2312yes—
77C2×Dic47⋊C4—22no—
78C4⋊C4×D47—28no—
79C2×(C2×D47)⋊C4—22no—
80C2×C4×Dic47—22no—
81#13093948—28no—
82#13093947—24no—
83#13093944—24no—
84C22×Q8×C47—21no—
85C22×Q8×C47—22no—
86C22×Q8×C47—22no—
87C22×Q8×C47—22no—
88C22×Q8×C47—22no—
89C22×Q8×C47—22no—
90C22×Q8×C47C2×C414yescomplemented
91C22×Q8×C47C2312yes—
92C23×C4×C47—21no—
93C23×C4×C47C2311yescharacteristic
94#117×C47—22no—
95#117×C47—22no—
96#117×C47—22no—
97#117×C47—22no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47D412yes—
103C2×C42×C47—21no—
104C2×C42×C47C2312yes—
105#95×C47—24no—
106#95×C47—24no—
107C24×C47C2×D411yescharacteristic
108C23×D47—21no—
109C2×Q8×C47—24no—
110C2×Q8×C47—22no—
111C2×Q8×C47—28no—
112C2×Q8×C47—28no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47—22no—
117C2×Q8×C47—42no—
118C2×Q8×C47—42no—
119C2×Q8×C47—44no—
120C2×Q8×C47—44no—
121C2×Q8×C47C22×C414yes—
122C22×C4×C47—24no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D412yes—
134C22×C4×C47#4112yes—
135C22×C4×C47#4112yes—
136C22×C4×C47C2411yescharacteristic
137C2×C22⋊D47—24no—
138C2×C22⋊D47—24no—
139C2×C22⋊D47—24no—
140C22×Dic47—21no—
141C22×Dic47—22no—
142C22×Dic47—24no—
143C22×Dic47—82no—
144#10625350—28no—
145#10625350—28no—
146#10625350—28no—
147#10625350C22⋊C4116yes—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188—21no—
151C2×D188C22×C412yes—
152C2×D188C2×D412yes—
153C2×D188C2×D412yes—
154C2×C4×D47—24no—
155C2×C4×D47—24no—
156C2×C4×D47—24no—
157C2×C4×D47—28no—
158C2×C4×D47—84no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94C22×C412yes—
163C2×Dic94C2×D412yes—
164C2×Dic94C2×D412yes—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C22⋊C4×C47—24no—
170C22⋊C4×C47—42no—
171C42×C47—22no—
172C42×C47—22no—
173C42×C47—42no—
174#10625337—216no—
175#10625336—216no—
176(C2×D47)⋊C4—28no—
177C4⋊Dic47—24no—
178C4⋊Dic47—22no—
179C4⋊Dic47C2×D414yes—
180Dic47⋊C4—28no—
181C4×Dic47—28no—
182#10625329—28no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—44no—
189C2×C47⋊C8C22×C414yes—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47C4×D412yes—
193C23×C47#12012yes—
194C23×C47#12112yes—
195C23×C47C22×D411yescharacteristic
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—42no—
200C22×D47—42no—
201Q8×C47—24no—
202Q8×C47—24no—
203Q8×C47—42no—
204Q8×C47—42no—
205Q8×C47—48no—
206C2×C4×C47—24no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—48no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47C2×C22⋊C414yes—
218C2×C4×C47C4×D414yes—
219C2×C4×C47C4×D412yes—
220C2×C4×C47#12012yes—
221C2×C4×C47#12112yes—
222C2×C4×C47#12114yes—
223C2×C4×C47#12314yes—
224C2×C4×C47C23×C411yescharacteristic
225C2×C4×C47C22×D411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C2×#4111yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C22⋊D47—216no—
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—22no—
234C2×Dic47—24no—
235C2×Dic47—44no—
236C2×Dic47—84no—
237D188—22no—
238D188—22no—
239D188—22no—
240D188C2×C22⋊C414yes—
241C4×D47—216no—
242C4×D47—88no—
243Dic94—22no—
244Dic94—22no—
245Dic94—22no—
246Dic94C2×C22⋊C414yes—
247C47⋊C8—24no—
248C47⋊C8—42no—
249#20931—471nomaximal, Sylow
250C22×C47—24no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—24no—
254C22×C47#38614yes—
255C22×C47C2×C4×D411yescharacteristic
256C22×C47C2×#12011yescharacteristic
257C22×C47C2×#12111yescharacteristic
258C22×C47#57712yes—
259C22×C47#57812yes—
260C2×D47—24no—
261C2×D47—44no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—44no—
265C4×C47—42no—
266C4×C47—42no—
267C4×C47—44no—
268C4×C47—82no—
269C4×C47#38614yes—
270C4×C47C22×C22⋊C411yescharacteristic
271C4×C47C2×C4×D411yescharacteristic
272C4×C47C2×#12111yescharacteristic
273C4×C47C2×#12311yescharacteristic
274Dic47—24no—
275Dic47—44no—
276C22×#137—471no—
277C2×#520—471no—
278C2×#416—471no—
279C22×#103—471no—
280#2310D4711yescharacteristic, complemented
281#1878—474no—
282#1877—474no—
283#1736—471no—
284#1729—471no—
285C2×C47—24no—
286C2×C47C2×#38611yescharacteristic
287C2×C47#294712yes—
288C2×C47#300911yescharacteristic
289C2×C47#301111yescharacteristic
290C2×C47#304612yes—
291D47—44no—
292C22×#41—471no—
293C23×Q8C2×D4711yescharacteristic
294C2×#137—478no—
295C2×#137—944no—
296C2×#137—944no—
297C22×Q16—472no—
298C22×SD16—472no—
299C22×M16—471no—
300C23×C8—471no—
301#520—944no—
302#520—944no—
303C2×C4×Q8—472no—
304C2×C4×D4—472no—
305C2×C4×D4—942no—
306C2×#117—941no—
307C2×#117C2×D4711yescharacteristic
308C2×#117C2×D4711yescharacteristic
309C22×C4⋊C4—471no—
310C22×C4⋊C4—941no—
311#416—944no—
312C2×#103—474no—
313C2×#103—472no—
314C2×#103—944no—
315C2×#102—472no—
316#391—21no—
317#391—21no—
318#391C2×D4714yes—
319#381—22no—
320#340—474no—
321#336—474no—
322C47#2093111yescharacteristic, complemented, Sylow
323C2×#41—474no—
324C2×#41—474no—
325C2×#41—474no—
326C2×#41—942no—
327C2×#41—942no—
328C2×#41—942no—
329C22×Q8—22no—
330C22×Q8—21no—
331C22×Q8—22no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—471no—
336C22×Q8C4×D4714yescomplemented
337C22×Q8C22×D4712yes—
338C22×D4—471no—
339C22×D4—941no—
340C23×C4—21no—
341C23×C4—941no—
342C23×C4—941no—
343C23×C4—942no—
344C23×C4C22×D4711yescharacteristic
345#137—9432no—
346C2×Q16—478no—
347C2×Q16—944no—
348C2×Q16—944no—
349C2×SD16—478no—
350C2×SD16—944no—
351C2×SD16—944no—
352C2×M16—474no—
353C2×M16—942no—
354C2×M16—942no—
355C22×C8—472no—
356C22×C8—472no—
357C22×C8—942no—
358C22×C8—942no—
359C22×C8—942no—
360C4×Q8—944no—
361C4×Q8—944no—
362C4×D4—944no—
363C4×D4—944no—
364C4×D4—9416no—
365#117—22no—
366#117—22no—
367#117—22no—
368#117—22no—
369#117—948no—
370C2×C4⋊C4—474no—
371C2×C4⋊C4—472no—
372C2×C4⋊C4—948no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4C22⋊D4711yescharacteristic
377C2×C4⋊C4C22⋊D4711yescharacteristic
378C2×C4⋊C4C22×D4712yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—942no—
381C2×C22⋊C4C22⋊D4712yes—
382C2×C42—21no—
383C2×C42—942no—
384C2×C42C22×D4712yes—
385#103—944no—
386#103—948no—
387#102—944no—
388#95—24no—
389#95—24no—
390C24—941no—
391C24C2×C22⋊D4711yescharacteristic
392#41—4716no—
393#41—948no—
394#41—948no—
395#41—948no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—472no—
409C2×Q8—472no—
410C2×Q8—472no—
411C2×Q8—941no—
412C2×Q8—941no—
413C2×Q8—941no—
414C2×Q8C2×C4×D4714yes—
415C2×D4—472no—
416C2×D4—472no—
417C2×D4—472no—
418C2×D4—944no—
419C2×D4—944no—
420C2×D4—944no—
421C2×D4—941no—
422C2×D4—941no—
423C2×D4—941no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—948no—
430C22×C4—941no—
431C22×C4—944no—
432C22×C4—944no—
433C22×C4—944no—
434C22×C4—944no—
435C22×C4—942no—
436C22×C4—3764no—
437C22×C4—3762no—
438C22×C4C2×C4×D4712yes—
439C22×C4#1062535012yes—
440C22×C4D4×D4712yes—
441C22×C4D4×D4712yes—
442C22×C4D4×D4712yes—
443C22×C4#1062535212yes—
444C22×C4C2×C22⋊D4712yes—
445C22×C4C2×C22⋊D4712yes—
446C22×C4C2×C22⋊D4711yescharacteristic
447C22×C4C23×D4711yescharacteristic
448Q16—9416no—
449SD16—9416no—
450M16—948no—
451C2×C8—474no—
452C2×C8—942no—
453C2×C8—942no—
454C2×C8—942no—
455C2×C8—942no—
456C2×C8—942no—
457C2×C8—1884no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—474no—
463C4⋊C4—948no—
464C4⋊C4—944no—
465C4⋊C4—942no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—948no—
469C42—22no—
470C42—22no—
471C42—42no—
472C42—948no—
473C23—22no—
474C23—22no—
475C23—942no—
476C23—942no—
477C23—942no—
478C23—1882no—
479C23—1882no—
480C23C4×C22⋊D4712yes—
481C23#1309399212yes—
482C23#1309399512yes—
483C23C22×C22⋊D4711yescharacteristic
484Q8—24no—
485Q8—24no—
486Q8—48no—
487Q8—42no—
488Q8—42no—
489Q8—474no—
490Q8—942no—
491Q8—942no—
492Q8—942no—
493D4—474no—
494D4—9416no—
495D4—942no—
496D4—942no—
497D4—942no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—942no—
512C2×C4—9416no—
513C2×C4—944no—
514C2×C4—3768no—
515C2×C4—3764no—
516C2×C4—1884no—
517C2×C4C22⋊C4×D4714yes—
518C2×C4(C22⋊D47)⋊C414yes—
519C2×C4C4×C22⋊D4712yes—
520C2×C4#1309399212yes—
521C2×C4#1309399512yes—
522C2×C4#1309399714yes—
523C2×C4#1309400614yes—
524C2×C4C22×C4×D4711yescharacteristic
525C2×C4C2×D4×D4711yescharacteristic
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4C2×D4×D4711yescharacteristic
528C2×C4C2×#1062535211yescharacteristic
529C2×C4C2×#1062535011yescharacteristic
530C8—944no—
531C8—1882no—
532C22—24no—
533C22—22no—
534C22—24no—
535C22—22no—
536C22—944no—
537C22—1884no—
538C22—1882no—
539C22—1884no—
540C22—1882no—
541C22#1568899914yes—
542C22C2×C4×C22⋊D4711yescharacteristic
543C22C2×#1309399211yescharacteristic
544C22C2×#1309399511yescharacteristic
545C22#1568993112yes—
546C22#1568993812yes—
547C4—42no—
548C4—44no—
549C4—42no—
550C4—82no—
551C4—944no—
552C4—1884no—
553C4#1568899914yes—
554C4C2×C22⋊C4×D4711yescharacteristic
555C4C2×(C22⋊D47)⋊C411yescharacteristic
556C4C2×#1309399711yescharacteristic
557C4C2×#1309400611yescharacteristic
558C2—24no—
559C2—1884no—
560C2C2×#1568899911yescharacteristic
561C2#2025223112yes—
562C2#2025266311yescharacteristic
563C2#2025268111yescharacteristic
564C2#2025293412yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2310C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#2310D47
hypocenterC47#20931
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. C22×#13093948C2semidirect
5. #20243732C2semidirect
6. #20243725C2semidirect
7. #20243109C2semidirect
8. #20243085C2semidirect
9. C47×#2310C2semidirect
13. C2×#13094055C4semidirect
55. C2×#10625337C2×C4semidirect
58. C2×#10625336C2×C4semidirect
61. C2×#10625329C2×C4semidirect
90. C22×Q8×C47C2×C4semidirect
280. #2310D47semidirect
322. C47#20931semidirect
336. C22×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×#130940552
C2×#156896032
C2×#156891172
C22×#130939482
#202437322
#202437252
#202431092
#202430852
C47×#23102
#2093147