Finite Group Databasebeta

← #26217972

Subgroups — #26217972

Order 12032.

1914 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625324C211yesmaximal, characteristic, complemented
2C2×#15689635C211yesmaximal, characteristic
3C2×#15689120C211yesmaximal, characteristic, complemented
4C2×#15689117C211yesmaximal, characteristic, complemented
5#20245767C211yesmaximal, characteristic, complemented
6#20243825C214yesmaximal, complemented
7#20243820C214yesmaximal, complemented
8#20243104C211yesmaximal, characteristic, complemented
9C47×#1735C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C22×#10625324—24no—
13C22×#10625324—24no—
14C22×#10625324—24no—
15C22×#10625324C2214yes—
16C23×C47⋊C8C2211yescharacteristic
17#15689635—24no—
18#15689635C418yescomplemented
19C2×#13094008—21no—
20C2×#13094004C2212yes—
21C2×#13093999C2212yes—
22C2×#13093994—21no—
23C2×C4×D188—22no—
24C2×#13093983C2211yescharacteristic
25#15689120—24no—
26#15689117—24no—
27C2×#13093949C2212yescomplemented
28C2×#13093947C2212yescomplemented
29C2×#13093946C2212yescomplemented
30C2×#13093944C2212yescomplemented
31#15688984—21no—
32#15688980C2214yes—
33#15688978—21no—
34C4×(C2×D47)⋊C4—22no—
35#15688885C2214yescomplemented
36C23×C8×C47C2211yescharacteristic
37C2×#117×C47C2211yescharacteristic
38C2×#117×C47C2211yescharacteristic
39C47×#340C2214yescomplemented
40C24×D47D411yescharacteristic
41C2×#10625350—22no—
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350C2×C414yes—
47C22×Dic94C2311yescharacteristic
48C22×D188—22no—
49C22×D188C2311yescharacteristic
50C22×C22⋊D47—21no—
51C23×Dic47D411yescharacteristic
52C22×C4×D47—22no—
53C22×C4×D47—22no—
54C22×C4×D47—22no—
55C22×C4×D47—22no—
56C22×C4×D47—21no—
57C22×C4×D47C2×C414yes—
58C22×C4×D47D412yes—
59C22×C4×D47D414yes—
60C22×C4×D47C2312yes—
61C2×#10625324—232no—
62C2×#10625324—28no—
63C2×#10625324—28no—
64C2×#10625324—28no—
65C22×C47⋊C8—22no—
66C22×C47⋊C8—22no—
67C22×C47⋊C8D414yescomplemented
68C22×C47⋊C8C2312yes—
69#13094008—28no—
70#13094004—24no—
71#13094004C2×C418yes—
72#13093999—24no—
73#13093999C2×C418yes—
74#13093994—28no—
75C4×D188—216no—
76#13093983—22no—
77#13093983—22no—
78C2×C4⋊Dic47—22no—
79C2×C4⋊Dic47C2312yes—
80C2×C22⋊Dic47—21no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×(C2×D47)⋊C4—24no—
84C2×C4×Dic47—21no—
85#13093949—24no—
86#13093947—24no—
87#13093946—24no—
88#13093944—24no—
89C47⋊#95—24no—
90C23×C4×C47C2311yescharacteristic
91C22×C8×C47—22no—
92C22×C8×C47—22no—
93C22×C8×C47D414yescomplemented
94C22×C8×C47C2312yes—
95#117×C47—22no—
96#117×C47—22no—
97#117×C47—22no—
98#117×C47C2×C414yes—
99C2×C4⋊C4×C47C2312yes—
100C2×C4⋊C4×C47C2312yes—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—21no—
103C2×C42×C47—21no—
104C2×C42×C47—21no—
105C24×C47C2×D411yescharacteristic
106C23×D47—22no—
107C23×D47—22no—
108C23×D47—21no—
109C23×D47—44no—
110C23×D47C22⋊C414yes—
111C23×D47C2×D412yes—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C2×D411yescharacteristic
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47#4112yes—
120C22×C4×C47#4112yes—
121C22×C4×C47C2411yescharacteristic
122C2×C22⋊D47—24no—
123C2×C22⋊D47—42no—
124C2×C22⋊D47—42no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—21no—
128C22×Dic47—22no—
129C22×Dic47C22⋊C414yes—
130C22×Dic47C2×D412yes—
131#10625350—28no—
132#10625350—28no—
133#10625350—44no—
134#10625350—44no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—28no—
140C2×D188—21no—
141C2×D188—48no—
142C2×D188C22×C412yes—
143C2×C4×D47—28no—
144C2×C4×D47—28no—
145C2×C4×D47—28no—
146C2×C4×D47—24no—
147C2×C4×D47—24no—
148C2×C4×D47—28no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—24no—
154C2×C4×D47—42no—
155C2×C4×D47—42no—
156C2×C4×D47C22⋊C418yes—
157C2×C4×D47C22×C414yes—
158C2×C4×D47C2×D414yes—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94C22×C412yes—
165C2×C8×C47—28no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C2×C8×C47C2×D414yes—
169C4⋊C4×C47—22no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47C22×C414yes—
173C22⋊C4×C47—24no—
174C22⋊C4×C47—42no—
175C42×C47—24no—
176C42×C47—42no—
177C22⋊Dic47—42no—
178(C2×D47)⋊C4—28no—
179(C2×D47)⋊C4—28no—
180(C2×D47)⋊C4—48no—
181C4⋊Dic47—28no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4×Dic47—42no—
185C2×C47⋊C8—28no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8C2×D414yes—
189#10625324—232no—
190C23×C47—24no—
191C23×C47C2×C22⋊C412yes—
192C23×C47#12112yes—
193C23×C47#12112yes—
194C23×C47C22×D411yescharacteristic
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—24no—
198C22×D47—24no—
199C22×D47—28no—
200C22×D47—22no—
201C22×D47—48no—
202C22×D47—48no—
203C22×D47—48no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—44no—
207C22×D47C2×C22⋊C414yes—
208C2×C4×C47—24no—
209C2×C4×C47—44no—
210C2×C4×C47—42no—
211C2×C4×C47—44no—
212C2×C4×C47C2×C22⋊C412yes—
213C2×C4×C47C4×D414yes—
214C2×C4×C47C4×D414yes—
215C2×C4×C47#12014yes—
216C2×C4×C47#12112yes—
217C2×C4×C47#12112yes—
218C2×C4×C47#12314yes—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C2×C4×C47C2×#4111yescharacteristic
225C22⋊D47—48no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—28no—
229C2×Dic47—22no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—42no—
233C2×Dic47—44no—
234C2×Dic47C2×C22⋊C414yes—
235D188—22no—
236D188—22no—
237D188—416no—
238D188—41no—
239D188—41no—
240C4×D47—24no—
241C4×D47—24no—
242C4×D47—24no—
243C4×D47—24no—
244C4×D47—216no—
245C4×D47—48no—
246Dic94—22no—
247Dic94—22no—
248Dic94—41no—
249Dic94—41no—
250C8×C47—24no—
251C47⋊C8—24no—
252#20970—471nomaximal, Sylow
253C22×C47—24no—
254C22×C47—24no—
255C22×C47—24no—
256C22×C47#38614yes—
257C22×C47C22×C22⋊C411yescharacteristic
258C22×C47C2×#12111yescharacteristic
259C22×C47C2×#12111yescharacteristic
260C22×C47#57514yes—
261C2×D47—24no—
262C2×D47—24no—
263C2×D47—24no—
264C2×D47—416no—
265C2×D47—44no—
266C2×D47—44no—
267C2×D47—44no—
268C2×D47—42no—
269C2×D47—82no—
270C2×D47—82no—
271C4×C47—44no—
272C4×C47—82no—
273C4×C47#38614yes—
274C4×C47C2×C4×D411yescharacteristic
275C4×C47C2×C4×D411yescharacteristic
276C4×C47C2×#12011yescharacteristic
277C4×C47C2×#12311yescharacteristic
278Dic47—24no—
279Dic47—42no—
280Dic47—82no—
281C23×M16—471no—
282C2×#525—471no—
283C2×#416—471no—
284C2×#416—471no—
285#2280—471no—
286#1895—474no—
287#1894—474no—
288#1735—471no—
289#1735D4711yescharacteristic, complemented
290C2×C47—24no—
291C2×C47C2×#38611yescharacteristic
292C2×C47#289612yes—
293C2×C47#305412yes—
294C2×C47#305412yes—
295D47—44no—
296D47—82no—
297C22×#41—471no—
298C24×C4—471no—
299C22×M16—474no—
300C22×M16—944no—
301C22×M16—944no—
302C22×M16—944no—
303C23×C8—471no—
304C23×C8C2×D4711yescharacteristic
305#525—478no—
306#525—944no—
307C2×#123—941no—
308C2×C22⋊Q8—472no—
309C2×#121—472no—
310C2×#120—941no—
311C2×C4×D4—942no—
312C2×#117—471no—
313C2×#117C2×D4711yescharacteristic
314C2×#117C2×D4711yescharacteristic
315#416—944no—
316#416—944no—
317C2×#103—472no—
318C2×#103—472no—
319C2×#102—472no—
320C2×#102—472no—
321#381—941no—
322#380—474no—
323#379—941no—
324C4×C22⋊C4—942no—
325#340—474no—
326#340C2×D4714yescomplemented
327C47#2097011yescharacteristic, complemented, Sylow
328C25—471no—
329C2×#41—474no—
330C2×#41—942no—
331C2×#41—942no—
332C2×#41—942no—
333C2×#41—942no—
334C2×#41—942no—
335C22×Q8—471no—
336C22×D4—471no—
337C22×D4—942no—
338C22×D4—941no—
339C23×C4—471no—
340C23×C4—474no—
341C23×C4—474no—
342C23×C4—472no—
343C23×C4—472no—
344C23×C4—942no—
345C23×C4—942no—
346C23×C4—942no—
347C23×C4—942no—
348C23×C4—941no—
349C23×C4C22×D4711yescharacteristic
350C2×M16—948no—
351C2×M16—948no—
352C2×M16—9432no—
353C2×M16—948no—
354C22×C8—22no—
355C22×C8—22no—
356C22×C8—474no—
357C22×C8—472no—
358C22×C8—942no—
359C22×C8—942no—
360C22×C8D18814yescomplemented
361C22×C8C22×D4712yes—
362#123—948no—
363C22⋊Q8—478no—
364C22⋊Q8—944no—
365#121—478no—
366#121—944no—
367#120—948no—
368C4×D4—9416no—
369#117—22no—
370#117—22no—
371#117—22no—
372#117—942no—
373#117—942no—
374#117C4×D4714yes—
375C2×C4⋊C4—472no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4C22×D4712yes—
378C2×C4⋊C4C22×D4712yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—944no—
384C2×C22⋊C4—941no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—941no—
388#103—944no—
389#103—944no—
390#102—944no—
391#102—944no—
392#95—944no—
393C24—472no—
394C24—474no—
395C24—942no—
396C24—942no—
397C24—941no—
398C24—1884no—
399C24D4×D4711yescharacteristic
400#41—948no—
401#41—948no—
402#41—1884no—
403#41—1884no—
404C2×Q8—472no—
405C2×Q8—941no—
406C2×Q8—941no—
407C2×Q8—941no—
408C2×Q8—941no—
409C2×Q8—941no—
410C2×D4—472no—
411C2×D4—948no—
412C2×D4—944no—
413C2×D4—941no—
414C2×D4—941no—
415C2×D4—941no—
416C2×D4—941no—
417C2×D4—941no—
418C2×D4—1888no—
419C2×D4—1882no—
420C2×D4—1882no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—474no—
424C22×C4—472no—
425C22×C4—478no—
426C22×C4—474no—
427C22×C4—474no—
428C22×C4—942no—
429C22×C4—942no—
430C22×C4—948no—
431C22×C4—948no—
432C22×C4—948no—
433C22×C4—944no—
434C22×C4—944no—
435C22×C4—948no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—941no—
441C22×C4—944no—
442C22×C4—942no—
443C22×C4—1882no—
444C22×C4—1882no—
445C22×C4C2×C4×D4712yes—
446C22×C4C2×D18812yes—
447C22×C4#1062535012yes—
448C22×C4D4×D4712yes—
449C22×C4D4×D4712yes—
450C22×C4D4×D4711yescharacteristic
451C22×C4#1062535412yes—
452C22×C4C23×D4711yescharacteristic
453M16—9432no—
454C2×C8—28no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—474no—
458C2×C8—948no—
459C2×C8—942no—
460C2×C8—942no—
461C2×C8C2×D18814yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—948no—
466C4⋊C4—942no—
467C4⋊C4—942no—
468C4⋊C4C2×C4×D4714yes—
469C22⋊C4—24no—
470C22⋊C4—42no—
471C22⋊C4—948no—
472C22⋊C4—948no—
473C22⋊C4—1888no—
474C22⋊C4—1882no—
475C42—24no—
476C42—42no—
477C42—1882no—
478C23—24no—
479C23—474no—
480C23—948no—
481C23—942no—
482C23—942no—
483C23—944no—
484C23—944no—
485C23—942no—
486C23—1888no—
487C23—1888no—
488C23—1888no—
489C23—1881no—
490C23—1881no—
491C23—1884no—
492C23C22⋊C4×D4712yes—
493C23#1309399712yes—
494C23#1309399912yes—
495C23C2×D4×D4711yescharacteristic
496Q8—942no—
497Q8—942no—
498Q8—1881no—
499Q8—1881no—
500D4—942no—
501D4—942no—
502D4—18816no—
503D4—1888no—
504D4—1881no—
505D4—1881no—
506C2×C4—24no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—474no—
511C2×C4—948no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—9416no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—942no—
520C2×C4—1882no—
521C2×C4—1884no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1888no—
525C2×C4C22⋊C4×D4712yes—
526C2×C4#1309398614yes—
527C2×C4C4×D18814yes—
528C2×C4#1309399414yes—
529C2×C4#1309399712yes—
530C2×C4#1309399912yes—
531C2×C4#1309400914yes—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4C22×D18811yescharacteristic
536C2×C4C2×#1062535011yescharacteristic
537C2×C4C2×#1062535411yescharacteristic
538C8—24no—
539C8—944no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—944no—
544C22—944no—
545C22—944no—
546C22—3762no—
547C22—3762no—
548C22—18816no—
549C22—1882no—
550C22—1884no—
551C22—1884no—
552C22—1884no—
553C22#1568900114yes—
554C22C2×C22⋊C4×D4711yescharacteristic
555C22C2×#1309399711yescharacteristic
556C22C2×#1309399911yescharacteristic
557C22#1568992014yes—
558C4—44no—
559C4—82no—
560C4—944no—
561C4—3762no—
562C4—1882no—
563C4#1568900114yes—
564C4C2×#1309398611yescharacteristic
565C4C2×C4×D18811yescharacteristic
566C4C2×#1309399411yescharacteristic
567C4C2×#1309400911yescharacteristic
568C2—24no—
569C2—3762no—
570C2—1884no—
571C2C2×#1568900111yescharacteristic
572C2#2025190312yes—
573C2#2025299612yes—
574C2#2025299812yes—
subgroupG/H
centerC22×C4D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1735C2
socleC23×C47C22×D4
prime radicalC2×#15689635C2
cosocleC22×C4×C47C24
hypercenter#1735D47
hypocenterC47#20970
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
1. C23×#10625324C2semidirect
3. C2×#15689120C2semidirect
4. C2×#15689117C2semidirect
5. #20245767C2semidirect
6. #20243825C2semidirect
7. #20243820C2semidirect
8. #20243104C2semidirect
9. C47×#1735C2semidirect
18. #15689635C4semidirect
27. C2×#13093949C22semidirect
28. C2×#13093947C22semidirect
29. C2×#13093946C22semidirect
30. C2×#13093944C22semidirect
35. #15688885C22semidirect
39. C47×#340C22semidirect
67. C22×C47⋊C8D4semidirect
93. C22×C8×C47D4semidirect
289. #1735D47semidirect
326. #340C2×D47semidirect
327. C47#20970semidirect
360. C22×C8D188semidirect

Maximal subgroups

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

subgroupindex
C23×#106253242
C2×#156896352
C2×#156891202
C2×#156891172
#202457672
#202438252
#202438202
#202431042
C47×#17352
#2097047