Finite Group Databasebeta

← #26221938

Subgroups — #26221938

Order 12032.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic47C211yesmaximal, characteristic, complemented
2C2×#15689262C211yesmaximal, characteristic
3C2×#15689226C212yesmaximal, complemented
4#20245894C211yesmaximal, characteristic
5#20243741C212yesmaximal, complemented
6#20243703C211yesmaximal, characteristic
7#20243640C212yesmaximal, complemented
8#20243575C211yesmaximal, characteristic, complemented
9#20243220C212yesmaximal, complemented
10#20243152C211yesmaximal, characteristic, complemented
11C47×#1787C211yesmaximal, characteristic, complemented
12C22×#10625336C2212yes—
13C2×#13093998C2211yescharacteristic
14C2×Q8×Dic47—28no—
15C2×Q8×Dic47—21no—
16C2×Q8×Dic47—21no—
17C2×Q8×Dic47C2212yes—
18C2×Q8×Dic47C2212yes—
19C2×#13093986C2211yescharacteristic
20C2×(C22⋊D47)⋊C4—21no—
21C22×C4⋊Dic47—21no—
22C22×C4⋊Dic47C2211yescharacteristic
23C22×C4×Dic47—21no—
24C22×C4×Dic47C2211yescharacteristic
25#15689262C418yescomplemented
26#15689226—28no—
27C2×#13093957C2212yes—
28C2×#13093948C2212yes—
29C2×#13093946C2212yes—
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×#13093934C2212yes—
32#15689024C2211yescharacteristic
33#15689019C2211yescharacteristic
34#15689012—21no—
35#15689000—21no—
36C4⋊C4×Dic47C2211yescharacteristic
37#15688970—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688877C2212yes—
41C23×Q8×C47C2211yescharacteristic
42C2×#103×C47C2212yes—
43C2×#103×C47C2212yes—
44C2×C22⋊C8×C47C2211yescharacteristic
45C47×#380C2211yescharacteristic
46C22×D188C2311yescharacteristic
47C22×C22⋊D47—21no—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C22×C4×D47—21no—
52C2×#10625336—24no—
53C2×#10625336—24no—
54C2×#10625336C2×C418yescomplemented
55C22×C47⋊C8C2312yes—
56#13093998—22no—
57#13093998C2×C414yes—
58Q8×Dic47—28no—
59Q8×Dic47—28no—
60Q8×Dic47—24no—
61Q8×Dic47—24no—
62Q8×Dic47—24no—
63Q8×Dic47D418yescomplemented
64#13093986—24no—
65(C22⋊D47)⋊C4—44no—
66C2×C4⋊Dic47—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—22no—
69C2×C4⋊Dic47—22no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2312yes—
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—28no—
79C2×C4×Dic47—22no—
80C2×C4×Dic47—22no—
81C2×C4×Dic47—24no—
82C2×C4×Dic47—21no—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86C2×C4×Dic47C2311yescharacteristic
87#13093957—24no—
88#13093948C2×C418yescomplemented
89#13093946—24no—
90C22⋊(C47⋊C8)C2×C414yescomplemented
91#13093934—24no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C22×Q8×C47—24no—
95C22×Q8×C47—24no—
96C22×Q8×C47—21no—
97C22×Q8×C47—22no—
98C22×Q8×C47C2×C414yescomplemented
99C22×Q8×C47C2312yes—
100C23×C4×C47—21no—
101C23×C4×C47C2311yescharacteristic
102C22×C8×C47C2312yes—
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C22⋊C4×C47—21no—
106#103×C47—24no—
107#103×C47—24no—
108C22⋊C8×C47—22no—
109#95×C47—21no—
110C24×C47C2×D411yescharacteristic
111C23×D47—21no—
112C2×Q8×C47—216no—
113C2×Q8×C47—24no—
114C2×Q8×C47—24no—
115C2×Q8×C47—216no—
116C2×Q8×C47—24no—
117C2×Q8×C47—22no—
118C2×Q8×C47—22no—
119C2×Q8×C47—22no—
120C2×Q8×C47C22×C414yes—
121C2×Q8×C47C2×D414yes—
122C2×Q8×C47#4114yes—
123C22×C4×C47—24no—
124C22×C4×C47—24no—
125C22×C4×C47—24no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47C22×C412yes—
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47#4112yes—
134C22×C4×C47C2411yescharacteristic
135C2×C22⋊D47—24no—
136C2×C22⋊D47—42no—
137C2×C22⋊D47—42no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—24no—
142C22×Dic47—22no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—41no—
147C22×Dic47—42no—
148C22×Dic47—42no—
149C22×Dic47C2×D411yescharacteristic
150C22×Dic47C2×D411yescharacteristic
151C22×Dic47C2×D411yescharacteristic
152C22×Dic47C2×D411yescharacteristic
153C2×D188—22no—
154C2×D188—22no—
155C2×D188—21no—
156C2×D188C22×C412yes—
157C2×C4×D47—42no—
158C2×C4×D47—42no—
159C2×C4×D47—42no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C22⋊C4×C47—42no—
166#10625336—216no—
167(C2×D47)⋊C4—24no—
168(C2×D47)⋊C4—42no—
169(C2×D47)⋊C4—42no—
170(C2×D47)⋊C4—42no—
171C4⋊Dic47—28no—
172C4⋊Dic47—24no—
173C4⋊Dic47—22no—
174C4⋊Dic47—21no—
175C4⋊Dic47—28no—
176C4⋊Dic47C22×C412yes—
177C4⋊Dic47C2×D414yes—
178Dic47⋊C4—42no—
179C4×Dic47—28no—
180C4×Dic47—22no—
181C4×Dic47—28no—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47—42no—
185C4×Dic47C2×D414yes—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8C22×C414yes—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47C4×D412yes—
194C23×C47AGL(1,ℤ/8)12yes—
195C23×C47#13712yes—
196C23×C47C22×D411yescharacteristic
197C22×D47—22no—
198C22×D47—41no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202C22×D47—41no—
203C22×D47—41no—
204Q8×C47—216no—
205Q8×C47—24no—
206Q8×C47—24no—
207Q8×C47—24no—
208Q8×C47C4×D418yescomplemented
209C2×C4×C47—216no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—21no—
213C2×C4×C47—24no—
214C2×C4×C47—21no—
215C2×C4×C47—24no—
216C2×C4×C47—24no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47C4×D412yes—
224C2×C4×C47C23×C411yescharacteristic
225C2×C4×C47C22×D412yes—
226C2×C4×C47C2×#4112yes—
227C2×C4×C47C2×#4111yescharacteristic
228C22⋊D47—48no—
229C2×Dic47—28no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—22no—
237C2×Dic47—21no—
238C2×Dic47—24no—
239C2×Dic47—21no—
240C2×Dic47—21no—
241C2×Dic47—44no—
242C2×Dic47—48no—
243C2×Dic47—41no—
244C2×Dic47—42no—
245C2×Dic47—41no—
246C2×Dic47—42no—
247C2×Dic47#12012yes—
248C2×Dic47#12012yes—
249C2×Dic47#13712yes—
250D188—24no—
251D188—42no—
252C4×D47—84no—
253C8×C47—42no—
254C47⋊C8—24no—
255#21454—471nomaximal, Sylow
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47—22no—
262C22×C47#44014yescomplemented
263C22×C47C2×C4×D411yescharacteristic
264C22×C47C2×#12011yescharacteristic
265C22×C47#52511yescharacteristic
266C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C47C2×#13711yescharacteristic
268C22×C47C2×#13711yescharacteristic
269C22×C47#57511yescharacteristic
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—81no—
274C2×D47—81no—
275C2×D47—81no—
276C2×D47—81no—
277C4×C47—28no—
278C4×C47—22no—
279C4×C47—81no—
280C4×C47—81no—
281C4×C47C2×C4×D412yes—
282C4×C47#52512yes—
283Dic47—22no—
284Dic47—22no—
285Dic47—22no—
286Dic47—42no—
287Dic47—41no—
288Dic47—44no—
289C22×C4×Q8—471no—
290C2×#448—471no—
291C2×#440—472no—
292C4×#121—471no—
293#1880—472no—
294#1874—471no—
295#1863—472no—
296#1852—471no—
297#1787D4711yescharacteristic, complemented
298#1763—472no—
299#1749—471no—
300C2×C47—24no—
301C2×C47C4×#12011yescharacteristic
302C2×C47C2×#44011yescharacteristic
303C2×C47C4×#13711yescharacteristic
304C2×C47#300411yescharacteristic
305C2×C47#300611yescharacteristic
306C2×C47#311211yescharacteristic
307C2×C47#311711yescharacteristic
308D47—82no—
309C23×Q8C2×D4711yescharacteristic
310C22×SD16—472no—
311C2×#121—471no—
312C2×C4×Q8—472no—
313C2×C4×Q8—472no—
314C2×C4×Q8—941no—
315C2×C4×Q8—941no—
316C2×C4×Q8—948no—
317C2×C4×D4—471no—
318C2×C4×D4—941no—
319C22×C4⋊C4—471no—
320C22×C4⋊C4—941no—
321C22×C42—471no—
322C22×C42—941no—
323#448—478no—
324#440—948no—
325C2×#107—472no—
326C2×#103—472no—
327C2×#103C2×D4712yes—
328C2×#103C2×D4712yes—
329C2×#102—472no—
330C2×C22⋊C8—471no—
331C2×C22⋊C8C2×D4711yescharacteristic
332C2×#97—472no—
333#390—471no—
334#389—471no—
335#388—941no—
336#386—941no—
337#380C2×D4711yescharacteristic
338C4×C4⋊C4—471no—
339C4×C22⋊C4—941no—
340#340—472no—
341#340—472no—
342#336—472no—
343C47#2145411yescharacteristic, complemented, Sylow
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C4×D4714yescomplemented
349C22×Q8C22×D4712yes—
350C22×D4—471no—
351C22×D4—941no—
352C23×C4—21no—
353C23×C4—471no—
354C23×C4—471no—
355C23×C4—941no—
356C23×C4—941no—
357C23×C4C22×D4711yescharacteristic
358C2×SD16—478no—
359C2×SD16—944no—
360C2×SD16—944no—
361C22×C8—472no—
362C22×C8C22×D4712yes—
363#121—474no—
364#121—942no—
365C4×Q8—478no—
366C4×Q8—944no—
367C4×Q8—944no—
368C4×Q8—948no—
369C4×Q8—948no—
370C4×Q8—944no—
371C4×D4—944no—
372C4×D4—1884no—
373C2×C4⋊C4—472no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—944no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—948no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4C22×D4711yescharacteristic
382C2×C4⋊C4C22×D4711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4—941no—
387C2×C22⋊C4—941no—
388C2×C42—471no—
389C2×C42—471no—
390C2×C42—471no—
391C2×C42—471no—
392C2×C42—944no—
393C2×C42—942no—
394C2×C42—942no—
395C2×C42—948no—
396C2×C42—941no—
397#107—944no—
398#103—24no—
399#103—24no—
400#103—478no—
401#102—944no—
402C22⋊C8—22no—
403C22⋊C8—474no—
404#97—944no—
405#95—21no—
406#95—941no—
407#95—941no—
408C24—941no—
409C24D4×D4711yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8C2×C4×D4714yes—
419C2×Q8D4×D4714yes—
420C2×Q8#1062535214yes—
421C2×D4—472no—
422C2×D4—944no—
423C2×D4—942no—
424C2×D4—942no—
425C2×D4—941no—
426C2×D4—1882no—
427C2×D4—1882no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—471no—
439C22×C4—471no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—944no—
445C22×C4—942no—
446C22×C4—941no—
447C22×C4—942no—
448C22×C4—1882no—
449C22×C4—1882no—
450C22×C4—1882no—
451C22×C4—1881no—
452C22×C4—1882no—
453C22×C4—1882no—
454C22×C4C2×C4×D4712yes—
455C22×C4D4×D4711yescharacteristic
456C22×C4#1062535212yes—
457C22×C4C23×D4711yescharacteristic
458SD16—9416no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—474no—
463C2×C8—942no—
464C2×C8—942no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—472no—
468C4⋊C4—474no—
469C4⋊C4—948no—
470C4⋊C4—948no—
471C4⋊C4—944no—
472C4⋊C4—942no—
473C4⋊C4—941no—
474C4⋊C4—1882no—
475C22⋊C4—42no—
476C22⋊C4—944no—
477C22⋊C4—1882no—
478C22⋊C4—1882no—
479C22⋊C4—1882no—
480C42—474no—
481C42—942no—
482C42—942no—
483C42—948no—
484C42—948no—
485C42—942no—
486C42—1882no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—942no—
492C23—1881no—
493C23—1881no—
494C23—1881no—
495C23—1881no—
496C23—1881no—
497C23—1881no—
498C23(C22⋊D47)⋊C412yes—
499C23#1309405112yes—
500C23#1309405612yes—
501C23C2×D4×D4711yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8(C22⋊D47)⋊C418yescomplemented
507D4—944no—
508D4—1888no—
509D4—1882no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—472no—
525C2×C4—472no—
526C2×C4—472no—
527C2×C4—948no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—942no—
535C2×C4—944no—
536C2×C4—941no—
537C2×C4—941no—
538C2×C4—941no—
539C2×C4—3764no—
540C2×C4—1884no—
541C2×C4—1888no—
542C2×C4—1881no—
543C2×C4—1882no—
544C2×C4—1881no—
545C2×C4—1882no—
546C2×C4(C22⋊D47)⋊C412yes—
547C2×C4C22×C4×D4711yescharacteristic
548C2×C4C2×D4×D4712yes—
549C2×C4C2×#1062535212yes—
550C2×C4C2×#1062535211yescharacteristic
551C8—42no—
552C8—944no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—3761no—
560C22—3761no—
561C22—3761no—
562C22—3761no—
563C22—1882no—
564C22—1882no—
565C22—1882no—
566C22#1568922614yescomplemented
567C22C2×(C22⋊D47)⋊C411yescharacteristic
568C22#120×D4711yescharacteristic
569C22#1568962911yescharacteristic
570C22C2×#1309405111yescharacteristic
571C22#137×D4711yescharacteristic
572C22C2×#1309405611yescharacteristic
573C22#1568992211yescharacteristic
574C4—28no—
575C4—22no—
576C4—81no—
577C4—81no—
578C4—942no—
579C4—942no—
580C4—942no—
581C4—1882no—
582C4—1881no—
583C4—1884no—
584C4C2×(C22⋊D47)⋊C412yes—
585C4#1568962912yes—
586C2—24no—
587C2—3762no—
588C2#2024588711yescharacteristic
589C2C2×#1568922611yescharacteristic
590C2#2025220311yescharacteristic
591C2#2025262311yescharacteristic
592C2#2025263911yescharacteristic
593C2#2025350511yescharacteristic
594C2#2025356011yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1787C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenter#1787D47
hypocenterC47#21454
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic47C2semidirect
3. C2×#15689226C2semidirect
5. #20243741C2semidirect
7. #20243640C2semidirect
8. #20243575C2semidirect
9. #20243220C2semidirect
10. #20243152C2semidirect
11. C47×#1787C2semidirect
25. #15689262C4semidirect
54. C2×#10625336C2×C4semidirect
63. Q8×Dic47D4semidirect
88. #13093948C2×C4semidirect
90. C22⋊(C47⋊C8)C2×C4semidirect
98. C22×Q8×C47C2×C4semidirect
208. Q8×C47C4×D4semidirect
262. C22×C47#440semidirect
297. #1787D47semidirect
343. C47#21454semidirect
348. C22×Q8C4×D47semidirect
506. Q8(C22⋊D47)⋊C4semidirect
566. C22#15689226semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic472
C2×#156892622
C2×#156892262
#202458942
#202437412
#202437032
#202436402
#202435752
#202432202
#202431522
C47×#17872
#2145447