Finite Group Databasebeta

← #26250189

Subgroups — #26250189

Order 12032.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689830C211yesmaximal, characteristic, complemented
2C2×#15689413C211yesmaximal, characteristic, complemented
3C2×C4×#10625335C211yesmaximal, characteristic, complemented
4#20244584C212yesmaximal, complemented
5#20244391C212yesmaximal, complemented
6#20244348C211yesmaximal, characteristic, complemented
7#20244183C211yesmaximal, characteristic, complemented
8#20243706C212yesmaximal, complemented
9#20243580C212yesmaximal, complemented
10#20243390C211yesmaximal, characteristic, complemented
11C47×#2537C211yesmaximal, characteristic
12C22×#10625335C2211yescharacteristic
13#15689830—24no—
14#15689830—24no—
15C2×#13094024C2211yescharacteristic
16C2×#13094015—21no—
17C2×#13094003C2212yes—
18C2×#13094001—22no—
19C2×C4×Dic94C2211yescharacteristic
20C2×D4×Dic47—21no—
21C22×C4⋊Dic47C2212yes—
22#15689413—24no—
23C4×#10625335—24no—
24C2×#13093955C2212yes—
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×#13093947C2212yes—
29C2×#13093945C2212yes—
30C2×#13093945C2211yescharacteristic
31C2×C22⋊(C47⋊C8)C2212yes—
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×C4×C47⋊C8C2211yescharacteristic
34#15688885C2212yes—
35#15688885C2212yes—
36#15688885C2212yes—
37C2×#121×C47—21no—
38C2×#121×C47C2212yes—
39C2×C4×D4×C47C2211yescharacteristic
40C2×C4×D4×C47C2211yescharacteristic
41C47×#394—22no—
42C47×#384C2211yescharacteristic
43C47×#380—21no—
44C47×#380C2212yes—
45C22×Dic94—21no—
46C22×Dic94C2311yescharacteristic
47C23×Dic47—22no—
48C2×#10625335—22no—
49C2×#10625335—22no—
50C2×#10625335—22no—
51C22×C47⋊C8C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53C22×C47⋊C8C2311yescharacteristic
54#13094024—22no—
55#13094024D414yescomplemented
56#13094015—28no—
57#13094003—24no—
58#13094003—24no—
59#13094001—216no—
60C4×Dic94—22no—
61C4×Dic94D414yescomplemented
62D4×Dic47—28no—
63C2×C4⋊Dic47—22no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47—24no—
66C2×C4⋊Dic47—24no—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2312yes—
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—22no—
71C2×C22⋊Dic47—22no—
72C2×C4×Dic47—21no—
73#13093955—24no—
74#13093955—22no—
75C4⋊(C47⋊C8)—22no—
76#13093947—22no—
77#13093947—42no—
78#13093945—24no—
79#13093945—22no—
80C22⋊(C47⋊C8)—24no—
81C22⋊(C47⋊C8)—42no—
82C4×C47⋊C8—22no—
83C22×D4×C47—22no—
84C22×D4×C47D411yescharacteristic
85C22×D4×C47C2311yescharacteristic
86C23×C4×C47—21no—
87C23×C4×C47C2312yes—
88#121×C47—24no—
89#121×C47—42no—
90#121×C47—44no—
91C4×D4×C47—22no—
92C4×D4×C47—22no—
93C4×D4×C47—22no—
94C4×D4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C4⋊C4×C47C2311yescharacteristic
99C2×C4⋊C4×C47C2312yes—
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47D412yes—
104C2×C42×C47C2311yescharacteristic
105#95×C47—22no—
106C24×C47—21no—
107C24×C47C2×D412yes—
108C2×D4×C47—22no—
109C2×D4×C47—22no—
110C2×D4×C47—21no—
111C2×D4×C47—21no—
112C2×D4×C47—21no—
113C2×D4×C47—21no—
114C2×D4×C47—21no—
115C2×D4×C47—44no—
116C2×D4×C47—44no—
117C2×D4×C47—82no—
118C2×D4×C47SD1614yescomplemented
119C2×D4×C47#4112yes—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47—21no—
127C22×C4×C47—22no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47—42no—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—28no—
139C22×Dic47—82no—
140C22×Dic47—82no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—24no—
147C2×Dic94—81no—
148C2×Dic94—81no—
149C2×Dic94C2×D412yes—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—22no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—42no—
156C4⋊C4×C47—81no—
157C22⋊C4×C47—24no—
158C22⋊C4×C47—44no—
159C22⋊C4×C47—44no—
160C22⋊C4×C47—42no—
161C42×C47C2×D412yes—
162C42×C47C2×D412yes—
163C22⋊Dic47—28no—
164#10625335—28no—
165C4⋊Dic47—28no—
166C4⋊Dic47—24no—
167C4⋊Dic47—21no—
168C4⋊Dic47—22no—
169C4⋊Dic47C2×D412yes—
170C4⋊Dic47C2×D414yes—
171Dic47⋊C4—28no—
172Dic47⋊C4—24no—
173C4×Dic47—24no—
174C2×C47⋊C8—21no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8#4112yes—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47C2×SD1614yes—
197C23×C47C22×D411yescharacteristic
198D4×C47—22no—
199D4×C47—22no—
200D4×C47—28no—
201D4×C47—41no—
202D4×C47—41no—
203D4×C47—88no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—42no—
217C2×C4×C47—41no—
218C2×C4×C47—42no—
219C2×C4×C47—84no—
220C2×C4×C47#12012yes—
221C2×C4×C47#12012yes—
222C2×C4×C47C2×SD1612yes—
223C2×C4×C47#13712yes—
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C22×D411yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×C4×C47H(5,𝔽2)11yescharacteristic
229C2×C4×C472−1+411yescharacteristic
230C2×Dic47—22no—
231C2×Dic47—22no—
232C2×Dic47—24no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—42no—
236C2×Dic47—41no—
237C2×Dic47—42no—
238C2×Dic47—41no—
239C2×Dic47—88no—
240Dic94—22no—
241Dic94—22no—
242Dic94—41no—
243Dic94—41no—
244Dic94—84no—
245C47⋊C8—22no—
246C47⋊C8—41no—
247C47⋊C8—42no—
248#24118—471nomaximal, Sylow
249C22×C47—24no—
250C22×C47—24no—
251C22×C47—24no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—81no—
260C22×C47—81no—
261C22×C47—81no—
262C22×C47—81no—
263C22×C47C2×#12011yescharacteristic
264C22×C47#54611yescharacteristic
265C22×C47#54711yescharacteristic
266C22×C47C22×SD1611yescharacteristic
267C22×C47C2×#13711yescharacteristic
268C22×C47#57611yescharacteristic
269C22×C47#57711yescharacteristic
270C4×C47—22no—
271C4×C47—42no—
272C4×C47—81no—
273C4×C47—82no—
274C4×C47#44814yes—
275C4×C47#52511yescharacteristic
276C4×C47#53711yescharacteristic
277C4×C47#54811yescharacteristic
278C4×C47#55011yescharacteristic
279Dic47—42no—
280Dic47—44no—
281Dic47—42no—
282C2×#554—471no—
283C2×#476—471no—
284C2×C4×SD16—471no—
285#2537D4711yescharacteristic
286#2035—472no—
287#2000—472no—
288#1990—471no—
289#1954—471no—
290#1874—472no—
291#1853—472no—
292#1803—471no—
293C2×C47—44no—
294C2×C47—82no—
295C2×C47#240711yescharacteristic
296C2×C47C2×#44811yescharacteristic
297C2×C47#301211yescharacteristic
298C2×C47#329511yescharacteristic
299C2×C47#333111yescharacteristic
300C2×C47#333511yescharacteristic
301C2×C47#335711yescharacteristic
302C22×SD16—471no—
303#554—944no—
304#554—944no—
305C2×C4⋊Q8—471no—
306C2×#125—941no—
307C2×C22⋊Q8—472no—
308C2×C22⋊Q8—942no—
309C2×#121—21no—
310C2×#121C2×D4712yes—
311C2×C4×Q8—471no—
312C2×C4×D4—941no—
313C2×C4×D4C2×D4711yescharacteristic
314C2×C4×D4C2×D4711yescharacteristic
315C22×C4⋊C4—472no—
316#476—944no—
317C4×SD16—944no—
318C2×#106—471no—
319C2×#106—472no—
320C2×C4⋊C8—471no—
321C2×#103—471no—
322C2×#103—472no—
323C2×#102—471no—
324C2×#102—472no—
325C2×C22⋊C8—472no—
326C2×C22⋊C8—472no—
327C2×C4×C8—471no—
328#394—22no—
329#384C2×D4711yescharacteristic
330#380—21no—
331#380C2×D4712yes—
332#340—472no—
333#340—472no—
334#340—472no—
335C47#2411811yescharacteristic, complemented, Sylow
336C22×Q8—471no—
337C22×Q8—941no—
338C22×D4—22no—
339C22×D4C22⋊D4711yescharacteristic
340C22×D4C22×D4711yescharacteristic
341C23×C4—21no—
342C23×C4—942no—
343C23×C4C22×D4712yes—
344C2×SD16—942no—
345C2×SD16—942no—
346C2×SD16—942no—
347C22×C8—471no—
348C22×C8—471no—
349C22×C8—472no—
350C4⋊Q8—474no—
351C4⋊Q8—942no—
352#125—948no—
353C22⋊Q8—9416no—
354C22⋊Q8—944no—
355C22⋊Q8—944no—
356#121—24no—
357#121—42no—
358#121—44no—
359C4×Q8—474no—
360C4×Q8—942no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—948no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—472no—
370C2×C4⋊C4—944no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—942no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4C22⋊D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4712yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4C22⋊D4712yes—
384C2×C42—941no—
385C2×C42C22×D4711yescharacteristic
386#106—942no—
387#106—944no—
388C4⋊C8—942no—
389#103—942no—
390#103—1882no—
391#102—942no—
392#102—944no—
393C22⋊C8—944no—
394C22⋊C8—1882no—
395C4×C8—942no—
396#95—22no—
397C24—21no—
398C24C2×C22⋊D4712yes—
399C2×Q8—472no—
400C2×Q8—944no—
401C2×Q8—941no—
402C2×Q8—941no—
403C2×Q8—941no—
404C2×Q8—941no—
405C2×Q8—941no—
406C2×Q8—3761no—
407C2×Q8—3761no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—82no—
418C2×D4#1062533514yescomplemented
419C2×D4#1062535212yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—948no—
432C22×C4—941no—
433C22×C4—941no—
434C22×C4—942no—
435C22×C4—3762no—
436C22×C4—3762no—
437C22×C4C2×C22⋊D4712yes—
438C22×C4C2×C22⋊D4711yescharacteristic
439C22×C4C2×C22⋊D4711yescharacteristic
440C22×C4C23×D4711yescharacteristic
441SD16—948no—
442C2×C8—472no—
443C2×C8—941no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—942no—
447C2×C8—942no—
448C2×C8—941no—
449C2×C8—942no—
450C2×C8—941no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—472no—
459C4⋊C4—474no—
460C4⋊C4—948no—
461C4⋊C4—948no—
462C4⋊C4—944no—
463C4⋊C4—944no—
464C4⋊C4—941no—
465C4⋊C4—942no—
466C22⋊C4—24no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—948no—
471C42—944no—
472C42D4×D4712yes—
473C42D4×D4712yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C2×#1062533514yes—
488C23C22×C22⋊D4711yescharacteristic
489Q8—942no—
490Q8—942no—
491Q8—3764no—
492Q8—1881no—
493Q8—1881no—
494D4—28no—
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499D4—88no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—84no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—944no—
519C2×C4—3768no—
520C2×C4—1881no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1882no—
524C2×C4—1881no—
525C2×C4—1882no—
526C2×C4#1309399212yes—
527C2×C4#1309399212yes—
528C2×C4C2×#1062533512yes—
529C2×C4#1309405412yes—
530C2×C4C2×D4×D4711yescharacteristic
531C2×C4C2×D4×D4711yescharacteristic
532C2×C4#41×D4711yescharacteristic
533C2×C4C2×#1062535211yescharacteristic
534C2×C4#1309406911yescharacteristic
535C2×C4#1309407111yescharacteristic
536C8—942no—
537C8—1882no—
538C8—1881no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1309399211yescharacteristic
554C22#1568976711yescharacteristic
555C22#1568977211yescharacteristic
556C22C22×#1062533511yescharacteristic
557C22C2×#1309405411yescharacteristic
558C22#1568992611yescharacteristic
559C22#1568993111yescharacteristic
560C4—22no—
561C4—42no—
562C4—81no—
563C4—82no—
564C4—1882no—
565C4—1884no—
566C4—1882no—
567C4#1568926114yes—
568C4#1568963311yescharacteristic
569C4#1568970211yescharacteristic
570C4#1568977911yescharacteristic
571C4#1568979111yescharacteristic
572C2—44no—
573C2—82no—
574C2#2024672911yescharacteristic
575C2C2×#1568926111yescharacteristic
576C2#2025268911yescharacteristic
577C2#2025521211yescharacteristic
578C2#2025570411yescharacteristic
579C2#2025574511yescharacteristic
580C2#2025600811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC47×#2537C2
socleC23×C47C22×D4
hypercenter#2537D47
hypocenterC47#24118
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#15689830C2semidirect
2. C2×#15689413C2semidirect
3. C2×C4×#10625335C2semidirect
4. #20244584C2semidirect
5. #20244391C2semidirect
6. #20244348C2semidirect
7. #20244183C2semidirect
8. #20243706C2semidirect
9. #20243580C2semidirect
10. #20243390C2semidirect
55. #13094024D4semidirect
61. C4×Dic94D4semidirect
118. C2×D4×C47SD16semidirect
335. C47#24118semidirect
418. C2×D4#10625335semidirect

Maximal subgroups

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

subgroupindex
C2×#156898302
C2×#156894132
C2×C4×#106253352
#202445842
#202443912
#202443482
#202441832
#202437062
#202435802
#202433902
C47×#25372
#2411847