Finite Group Databasebeta

← #26225966

Subgroups — #26225966

Order 12032.

1166 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689659C211yesmaximal, characteristic
2C2×#15689173C211yesmaximal, characteristic, complemented
3C2×#15688988C211yesmaximal, characteristic, complemented
4#20244351C212yesmaximal, complemented
5#20244196C212yesmaximal, complemented
6#20243705C211yesmaximal, characteristic
7#20243703C211yesmaximal, characteristic
8#20243696C211yesmaximal, characteristic, complemented
9#20243401C212yesmaximal, complemented
10#20243397C212yesmaximal, complemented
11C47×#1872C211yesmaximal, characteristic, complemented
12#15689659—24no—
13#15689659C418yescomplemented
14C2×#13094019—21no—
15C2×#13094015C2211yescharacteristic
16C2×#13094013C2211yescharacteristic
17C2×#13094008—21no—
18C2×#13093998C2211yescharacteristic
19C2×Q8×Dic47C2211yescharacteristic
20C2×#13093986C2211yescharacteristic
21C2×(C22⋊D47)⋊C4—21no—
22C2×#13093982—21no—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4×Dic47C2211yescharacteristic
25#15689173—24no—
26C2×Dic47⋊C8C2212yes—
27C2×#13093948C2212yes—
28C2×#13093946C2212yes—
29C2×#13093944C2212yes—
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×C8×Dic47C2212yes—
32#15688988—21no—
33#15688988—21no—
34#15688988C2212yes—
35#15688988C2212yes—
36C2×C47⋊#95—21no—
37C2×C47⋊#95—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C22⋊Q8×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C2×C22⋊C8×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×D188C2311yescharacteristic
47C22×C22⋊D47—21no—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C22×C4×D47—21no—
52C22×C47⋊C8C2312yes—
53#13094019—28no—
54#13094015—22no—
55#13094015C2×C414yescomplemented
56#13094013—22no—
57#13094013C2×C414yescomplemented
58#13094008—28no—
59#13093998—22no—
60#13093998C2×C414yes—
61Q8×Dic47—22no—
62Q8×Dic47C2×C414yescomplemented
63#13093986—22no—
64#13093986C2×C414yescomplemented
65(C22⋊D47)⋊C4—28no—
66#13093982—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—42no—
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×(C2×D47)⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47C2×C414yes—
80C2×C4×Dic47D411yescharacteristic
81C2×C4×Dic47D411yescharacteristic
82C2×C4×Dic47D414yes—
83C2×C4×Dic47C2311yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85Dic47⋊C8—24no—
86#13093948—42no—
87#13093946—42no—
88#13093944—42no—
89C22⋊(C47⋊C8)—22no—
90C8×Dic47—24no—
91C47⋊#95—22no—
92C47⋊#95—22no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C22×Q8×C47C2311yescharacteristic
96C23×C4×C47—21no—
97C23×C4×C47C2311yescharacteristic
98C22×C8×C47C2312yes—
99C22⋊Q8×C47—22no—
100C22⋊Q8×C47C2×C414yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—42no—
104C2×C4⋊C4×C47C2312yes—
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C22⋊C4×C47—21no—
107#103×C47—42no—
108C22⋊C8×C47—22no—
109C24×C47C2×D411yescharacteristic
110C23×D47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47C22×C412yes—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47—42no—
123C22×C4×C47C22×C412yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2411yescharacteristic
127C2×C22⋊D47—24no—
128C2×C22⋊D47—42no—
129C2×C22⋊D47—42no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—21no—
133C22×Dic47—22no—
134C22×Dic47—41no—
135C22×Dic47—41no—
136C22×Dic47—42no—
137C22×Dic47—42no—
138C22×Dic47—42no—
139C22×Dic47C22⋊C414yes—
140C22×Dic47C22⋊C414yes—
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C2×D188—22no—
146C2×D188—22no—
147C2×D188—21no—
148C2×D188C22×C412yes—
149C2×C4×D47—24no—
150C2×C4×D47—42no—
151C2×C4×D47—42no—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C4⋊C4×C47—24no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47—42no—
160C4⋊C4×C47C22×C412yes—
161C22⋊C4×C47—24no—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—24no—
165(C2×D47)⋊C4—24no—
166C4⋊Dic47—21no—
167C4⋊Dic47—24no—
168C4⋊Dic47—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47—42no—
172C4⋊Dic47C22×C412yes—
173Dic47⋊C4—24no—
174C4×Dic47—24no—
175C4×Dic47C22⋊C414yes—
176C4×Dic47C22⋊C414yes—
177C4×Dic47C22×C412yes—
178C4×Dic47C22×C412yes—
179C4×Dic47C2×D412yes—
180C4×Dic47C2×D412yes—
181C2×C47⋊C8—22no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47—21no—
187C23×C47—21no—
188C23×C47C2×C22⋊C412yes—
189C23×C47#13512yes—
190C23×C47#13512yes—
191C23×C47C22×D411yescharacteristic
192C22×D47—22no—
193C22×D47—41no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—41no—
199Q8×C47—24no—
200Q8×C47—42no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—84no—
215C2×C4×C47C2×C22⋊C412yes—
216C2×C4×C47C23×C411yescharacteristic
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47H(5,𝔽2)12yes—
219C2×C4×C472−1+412yes—
220C22⋊D47—48no—
221C2×Dic47—21no—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—28no—
230C2×Dic47—22no—
231C2×Dic47—44no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—84no—
238C2×Dic47C2×C22⋊C412yes—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47C2×C22⋊C412yes—
241C2×Dic47C2×C22⋊C412yes—
242D188—24no—
243D188—42no—
244C4×D47—48no—
245C8×C47—42no—
246C47⋊C8—42no—
247#21861—471nomaximal, Sylow
248C22×C47—22no—
249C22×C47—22no—
250C22×C47—22no—
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47#41614yes—
255C22×C47C22×C22⋊C411yescharacteristic
256C22×C47#53511yescharacteristic
257C22×C47#53611yescharacteristic
258C22×C47C2×#13511yescharacteristic
259C22×C47C2×#13511yescharacteristic
260C22×C47#57511yescharacteristic
261C22×C47#57511yescharacteristic
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—42no—
265C2×D47—81no—
266C2×D47—81no—
267C2×D47—81no—
268C2×D47—81no—
269C4×C47—22no—
270C4×C47—42no—
271C4×C47—42no—
272C4×C47—82no—
273C4×C47#52012yes—
274C4×C47#53712yes—
275Dic47—22no—
276Dic47—22no—
277Dic47—22no—
278Dic47—22no—
279Dic47—42no—
280Dic47—82no—
281C2×#529—471no—
282C2×#431—471no—
283C2×#382—471no—
284#1991—472no—
285#1958—472no—
286#1874—471no—
287#1874—471no—
288#1872—471no—
289#1872D4711yescharacteristic, complemented
290#1805—472no—
291#1804—472no—
292C2×C47—24no—
293C2×C47#231111yescharacteristic
294C2×C47C2×#41611yescharacteristic
295C2×C47#289611yescharacteristic
296C2×C47#321111yescharacteristic
297C2×C47#321111yescharacteristic
298C2×C47#323211yescharacteristic
299C2×C47#323211yescharacteristic
300D47—82no—
301#529—478no—
302#529—944no—
303C2×#126—941no—
304C2×#125—471no—
305C2×#124—471no—
306C2×#123—941no—
307C2×C22⋊Q8C2×D4711yescharacteristic
308C2×#121—471no—
309C2×C4×Q8—471no—
310C2×C4×D4—471no—
311C2×C4×D4—941no—
312C2×#117—941no—
313C22×C4⋊C4—471no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C42—471no—
316#431—944no—
317C2×C4⋊C8—472no—
318C2×#103—472no—
319C2×#103C2×D4712yes—
320C2×#102—472no—
321C2×#102—472no—
322C2×C22⋊C8—471no—
323C2×C22⋊C8C2×D4711yescharacteristic
324C2×C4×C8—472no—
325#382—472no—
326#382—472no—
327#382—941no—
328#382—941no—
329C2×#95—941no—
330C2×#95—941no—
331#340—472no—
332#340—472no—
333#340—472no—
334#340C2×D4712yes—
335C47#2186111yescharacteristic, complemented, Sylow
336C22×Q8C22×D4711yescharacteristic
337C22×D4—471no—
338C22×D4—941no—
339C23×C4—21no—
340C23×C4—471no—
341C23×C4—471no—
342C23×C4—941no—
343C23×C4—941no—
344C23×C4C22×D4711yescharacteristic
345C22×C8—472no—
346C22×C8C22×D4712yes—
347#126—948no—
348#125—474no—
349#125—942no—
350#124—474no—
351#124—942no—
352#123—948no—
353C22⋊Q8—22no—
354C22⋊Q8C4×D4714yes—
355#121—474no—
356#121—942no—
357C4×Q8—474no—
358C4×Q8—942no—
359C4×D4—474no—
360C4×D4—942no—
361C4×D4—948no—
362#117—948no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—472no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—1882no—
373C2×C4⋊C4C22×D4712yes—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—941no—
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C42—474no—
381C2×C42—471no—
382C2×C42—471no—
383C2×C42—474no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—941no—
387C4⋊C8—944no—
388#103—42no—
389#103—1882no—
390#102—1882no—
391#102—1882no—
392C22⋊C8—22no—
393C22⋊C8—942no—
394C4×C8—944no—
395#95—942no—
396#95—942no—
397#95—942no—
398#95—942no—
399C24—941no—
400C24D4×D4711yescharacteristic
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8C2×C4×D4712yes—
405C2×D4—472no—
406C2×D4—944no—
407C2×D4—941no—
408C2×D4—942no—
409C2×D4—942no—
410C2×D4—1882no—
411C2×D4—1882no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—474no—
421C22×C4—471no—
422C22×C4—471no—
423C22×C4—471no—
424C22×C4—471no—
425C22×C4—474no—
426C22×C4—944no—
427C22×C4—942no—
428C22×C4—942no—
429C22×C4—941no—
430C22×C4—942no—
431C22×C4—1882no—
432C22×C4—1882no—
433C22×C4—1881no—
434C22×C4—1881no—
435C22×C4—1882no—
436C22×C4—1882no—
437C22×C4—1882no—
438C22×C4C2×C4×D4712yes—
439C22×C4D4×D4711yescharacteristic
440C22×C4D4×D4712yes—
441C22×C4C23×D4711yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—942no—
446C2×C8—942no—
447C2×C8—942no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—472no—
454C4⋊C4—944no—
455C4⋊C4—944no—
456C4⋊C4—944no—
457C4⋊C4—941no—
458C4⋊C4—942no—
459C4⋊C4—942no—
460C4⋊C4—1882no—
461C4⋊C4C2×C4×D4712yes—
462C22⋊C4—24no—
463C22⋊C4—944no—
464C22⋊C4—944no—
465C22⋊C4—944no—
466C22⋊C4—944no—
467C42—474no—
468C42—474no—
469C42—472no—
470C42—472no—
471C42—472no—
472C42—472no—
473C42—944no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—942no—
479C23—1881no—
480C23—1881no—
481C23—1881no—
482C23—1881no—
483C23—1881no—
484C23—1881no—
485C23C22⋊C4×D4712yes—
486C23#1309404512yes—
487C23#1309404712yes—
488C23C2×D4×D4711yescharacteristic
489Q8—24no—
490Q8—42no—
491D4—944no—
492D4—1888no—
493D4—1882no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4—472no—
509C2×C4—472no—
510C2×C4—472no—
511C2×C4—472no—
512C2×C4—948no—
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—942no—
522C2×C4—3764no—
523C2×C4—1888no—
524C2×C4—1884no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4C22⋊C4×D4712yes—
531C2×C4C22×C4×D4711yescharacteristic
532C2×C4C2×D4×D4711yescharacteristic
533C2×C4#1309406912yes—
534C2×C4#1309407112yes—
535C8—42no—
536C8—1882no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—3761no—
544C22—3761no—
545C22—3761no—
546C22—3761no—
547C22—1882no—
548C22—1882no—
549C22—1882no—
550C22#1568911814yes—
551C22C2×C22⋊C4×D4711yescharacteristic
552C22#1568968811yescharacteristic
553C22#1568969511yescharacteristic
554C22C2×#1309404511yescharacteristic
555C22C2×#1309404711yescharacteristic
556C22#1568991911yescharacteristic
557C22#1568992211yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—82no—
562C4—942no—
563C4—942no—
564C4—942no—
565C4—942no—
566C4—3762no—
567C4—1882no—
568C4#1568960212yes—
569C4#1568970012yes—
570C2—24no—
571C2—3762no—
572C2#2024597511yescharacteristic
573C2C2×#1568911811yescharacteristic
574C2#2025190411yescharacteristic
575C2#2025435611yescharacteristic
576C2#2025435811yescharacteristic
577C2#2025455911yescharacteristic
578C2#2025456111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1872C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenter#1872D47
hypocenterC47#21861
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#15689173C2semidirect
3. C2×#15688988C2semidirect
4. #20244351C2semidirect
5. #20244196C2semidirect
8. #20243696C2semidirect
9. #20243401C2semidirect
10. #20243397C2semidirect
11. C47×#1872C2semidirect
13. #15689659C4semidirect
55. #13094015C2×C4semidirect
57. #13094013C2×C4semidirect
62. Q8×Dic47C2×C4semidirect
64. #13093986C2×C4semidirect
289. #1872D47semidirect
335. C47#21861semidirect

Maximal subgroups

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

subgroupindex
C2×#156896592
C2×#156891732
C2×#156889882
#202443512
#202441962
#202437052
#202437032
#202436962
#202434012
#202433972
C47×#18722
#2186147