Finite Group Databasebeta

← #26226003

Subgroups — #26226003

Order 12032.

1114 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689675C211yesmaximal, characteristic
2C2×#15689177C211yesmaximal, characteristic
3C2×#15688988C211yesmaximal, characteristic
4#20244359C212yesmaximal, complemented
5#20244206C212yesmaximal, complemented
6#20243696C211yesmaximal, characteristic
7#20243695C211yesmaximal, characteristic
8#20243693C211yesmaximal, characteristic
9#20243402C212yesmaximal, complemented
10#20243401C212yesmaximal, complemented
11C47×#1872C211yesmaximal, characteristic
12#15689675—24no—
13#15689675C418yescomplemented
14C2×#13094023C2211yescharacteristic
15C2×#13094022C2211yescharacteristic
16C2×#13094014—21no—
17C2×#13094003C2211yescharacteristic
18C2×#13094001—21no—
19C2×Q8×Dic47C2211yescharacteristic
20C2×#13093989—21no—
21C2×#13093989C2211yescharacteristic
22C2×#13093980—21no—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4×Dic47C2211yescharacteristic
25#15689177—24no—
26C2×Dic47⋊C8C2212yes—
27C2×#13093949C2212yes—
28C2×#13093948C2212yes—
29C2×#13093947C2212yes—
30C2×C22⋊(C47⋊C8)C2211yescharacteristic
31C2×#13093934C2212yes—
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×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C22×C47⋊C8C2312yes—
52#13094023—22no—
53#13094023C2×C414yescomplemented
54#13094022—22no—
55#13094022C2×C414yescomplemented
56#13094014—28no—
57#13094003—22no—
58#13094003C2×C414yes—
59#13094001—28no—
60Q8×Dic47—22no—
61Q8×Dic47C2×C414yescomplemented
62#13093989—28no—
63#13093989—22no—
64#13093989C2×C414yescomplemented
65#13093980—28no—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—42no—
69C2×C4⋊Dic47C2312yes—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×C22⋊Dic47—21no—
77C2×C4×Dic47—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#13093949—42no—
87#13093948—42no—
88#13093947—42no—
89C22⋊(C47⋊C8)—22no—
90#13093934—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
110C2×Q8×C47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47C22×C412yes—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—22no—
119C22×C4×C47—42no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2411yescharacteristic
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—22no—
131C22×Dic47—41no—
132C22×Dic47—41no—
133C22×Dic47—42no—
134C22×Dic47—42no—
135C22×Dic47—42no—
136C22×Dic47C22⋊C414yes—
137C22×Dic47C22⋊C414yes—
138C22×Dic47C2×D411yescharacteristic
139C22×Dic47C2×D411yescharacteristic
140C22×Dic47C2×D411yescharacteristic
141C22×Dic47C2×D411yescharacteristic
142C2×Dic94—22no—
143C2×Dic94—22no—
144C2×Dic94—21no—
145C2×Dic94—24no—
146C2×Dic94—42no—
147C2×Dic94—42no—
148C2×Dic94C22×C412yes—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C2×C8×C47—22no—
152C4⋊C4×C47—24no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—22no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—42no—
157C4⋊C4×C47C22×C412yes—
158C22⋊C4×C47—24no—
159C22⋊Dic47—24no—
160C4⋊Dic47—21no—
161C4⋊Dic47—24no—
162C4⋊Dic47—22no—
163C4⋊Dic47—22no—
164C4⋊Dic47—42no—
165C4⋊Dic47C22×C412yes—
166Dic47⋊C4—24no—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169Dic47⋊C4—24no—
170Dic47⋊C4—24no—
171C4×Dic47—24no—
172C4×Dic47—24no—
173C4×Dic47C22⋊C414yes—
174C4×Dic47C22⋊C414yes—
175C4×Dic47C22×C412yes—
176C4×Dic47C22×C412yes—
177C4×Dic47C2×D412yes—
178C4×Dic47C2×D412yes—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C23×C47—21no—
183C23×C47—21no—
184C23×C47—21no—
185C23×C47—21no—
186C23×C47C2×C22⋊C412yes—
187C23×C47#13712yes—
188C23×C47#13712yes—
189C23×C47C22×D411yescharacteristic
190Q8×C47—24no—
191Q8×C47—42no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—24no—
195C2×C4×C47—24no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—84no—
206C2×C4×C47C2×C22⋊C412yes—
207C2×C4×C47C23×C411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47H(5,𝔽2)12yes—
210C2×C4×C472−1+412yes—
211C2×Dic47—21no—
212C2×Dic47—21no—
213C2×Dic47—21no—
214C2×Dic47—21no—
215C2×Dic47—21no—
216C2×Dic47—21no—
217C2×Dic47—21no—
218C2×Dic47—21no—
219C2×Dic47—28no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—44no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—84no—
231C2×Dic47C2×C22⋊C412yes—
232C2×Dic47C2×C22⋊C412yes—
233C2×Dic47C2×C22⋊C412yes—
234C2×Dic47C2×C22⋊C412yes—
235Dic94—24no—
236Dic94—48no—
237Dic94—42no—
238C8×C47—42no—
239C47⋊C8—42no—
240#21866—471nomaximal, Sylow
241C22×C47—22no—
242C22×C47—22no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—22no—
246C22×C47—22no—
247C22×C47#41914yes—
248C22×C47C22×C22⋊C411yescharacteristic
249C22×C47#53511yescharacteristic
250C22×C47#53611yescharacteristic
251C22×C47C2×#13711yescharacteristic
252C22×C47C2×#13711yescharacteristic
253C22×C47#57511yescharacteristic
254C22×C47#57511yescharacteristic
255C4×C47—22no—
256C4×C47—42no—
257C4×C47—42no—
258C4×C47—82no—
259C4×C47#52012yes—
260C4×C47#53712yes—
261Dic47—22no—
262Dic47—22no—
263Dic47—22no—
264Dic47—22no—
265Dic47—42no—
266Dic47—42no—
267Dic47—82no—
268C2×#533—471no—
269C2×#432—471no—
270C2×#382—471no—
271#1993—472no—
272#1960—472no—
273#1872—471no—
274#1872—471no—
275#1872—471no—
276#1872D4711yescharacteristic
277#1805—472no—
278#1805—472no—
279C2×C47—24no—
280C2×C47#231111yescharacteristic
281C2×C47C2×#41911yescharacteristic
282C2×C47#289611yescharacteristic
283C2×C47#321511yescharacteristic
284C2×C47#321511yescharacteristic
285C2×C47#323611yescharacteristic
286C2×C47#323611yescharacteristic
287#533—478no—
288#533—944no—
289C2×C4⋊Q8—471no—
290C2×C4⋊Q8—471no—
291C2×#125—941no—
292C2×C22⋊Q8—471no—
293C2×C22⋊Q8—941no—
294C2×C22⋊Q8C2×D4711yescharacteristic
295C2×C4×Q8—471no—
296C2×C4×Q8—471no—
297C2×C4×Q8—941no—
298C2×#117—941no—
299C22×C4⋊C4—471no—
300C22×C4⋊C4C2×D4711yescharacteristic
301C22×C42—471no—
302#432—944no—
303C2×C4⋊C8—472no—
304C2×#103—472no—
305C2×#103—472no—
306C2×#103—472no—
307C2×#103C2×D4712yes—
308C2×C22⋊C8—471no—
309C2×C22⋊C8C2×D4711yescharacteristic
310C2×#97—472no—
311#382—472no—
312#382—472no—
313#382—941no—
314#382—941no—
315C2×#95—941no—
316C2×#95—941no—
317#340—472no—
318#340—472no—
319#340—472no—
320#340C2×D4712yes—
321C47#2186611yescharacteristic, complemented, Sylow
322C22×Q8—471no—
323C22×Q8—941no—
324C22×Q8C22×D4711yescharacteristic
325C23×C4—21no—
326C23×C4—471no—
327C23×C4—471no—
328C23×C4—941no—
329C23×C4C22×D4711yescharacteristic
330C22×C8—472no—
331C22×C8C22×D4712yes—
332C4⋊Q8—474no—
333C4⋊Q8—474no—
334C4⋊Q8—942no—
335C4⋊Q8—942no—
336#125—948no—
337C22⋊Q8—22no—
338C22⋊Q8—474no—
339C22⋊Q8—948no—
340C22⋊Q8—942no—
341C22⋊Q8C4×D4714yes—
342C4×Q8—474no—
343C4×Q8—474no—
344C4×Q8—942no—
345C4×Q8—948no—
346C4×Q8—942no—
347#117—948no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—471no—
352C2×C4⋊C4—472no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—941no—
355C2×C4⋊C4—941no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4—941no—
358C2×C4⋊C4—941no—
359C2×C4⋊C4—941no—
360C2×C4⋊C4—1882no—
361C2×C4⋊C4C22×D4712yes—
362C2×C4⋊C4C22×D4711yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—941no—
365C2×C42—474no—
366C2×C42—471no—
367C2×C42—471no—
368C2×C42—474no—
369C2×C42—471no—
370C2×C42—471no—
371C2×C42—941no—
372C2×C42—941no—
373C4⋊C8—944no—
374#103—42no—
375#103—1882no—
376#103—1882no—
377#103—1882no—
378C22⋊C8—22no—
379C22⋊C8—942no—
380#97—944no—
381#95—942no—
382#95—942no—
383#95—942no—
384#95—942no—
385C24D4×D4711yescharacteristic
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—472no—
390C2×Q8—944no—
391C2×Q8—941no—
392C2×Q8—942no—
393C2×Q8—942no—
394C2×Q8—1882no—
395C2×Q8—1882no—
396C2×Q8C2×C4×D4712yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—474no—
406C22×C4—471no—
407C22×C4—471no—
408C22×C4—471no—
409C22×C4—471no—
410C22×C4—474no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—941no—
414C22×C4—941no—
415C22×C4—942no—
416C22×C4—1881no—
417C22×C4—1881no—
418C22×C4—1882no—
419C22×C4—1882no—
420C22×C4—1882no—
421C22×C4C2×C4×D4712yes—
422C22×C4D4×D4711yescharacteristic
423C22×C4D4×D4712yes—
424C22×C4C23×D4711yescharacteristic
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—942no—
429C2×C8—942no—
430C2×C8—942no—
431C4⋊C4—24no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—472no—
437C4⋊C4—944no—
438C4⋊C4—944no—
439C4⋊C4—944no—
440C4⋊C4—944no—
441C4⋊C4—944no—
442C4⋊C4—944no—
443C4⋊C4—941no—
444C4⋊C4—942no—
445C4⋊C4—942no—
446C4⋊C4—1882no—
447C4⋊C4C2×C4×D4712yes—
448C22⋊C4—24no—
449C22⋊C4—944no—
450C42—474no—
451C42—474no—
452C42—472no—
453C42—472no—
454C42—472no—
455C42—472no—
456C42—944no—
457C42—944no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23C22⋊C4×D4712yes—
463C23#1309405312yes—
464C23#1309405612yes—
465C23C2×D4×D4711yescharacteristic
466Q8—24no—
467Q8—42no—
468Q8—944no—
469Q8—1888no—
470Q8—1882no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—472no—
486C2×C4—472no—
487C2×C4—472no—
488C2×C4—472no—
489C2×C4—948no—
490C2×C4—941no—
491C2×C4—941no—
492C2×C4—941no—
493C2×C4—941no—
494C2×C4—941no—
495C2×C4—941no—
496C2×C4—941no—
497C2×C4—941no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—3764no—
501C2×C4—1884no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1881no—
506C2×C4—1881no—
507C2×C4—1881no—
508C2×C4—1881no—
509C2×C4C22⋊C4×D4712yes—
510C2×C4C22×C4×D4711yescharacteristic
511C2×C4C2×D4×D4711yescharacteristic
512C2×C4#1309406912yes—
513C2×C4#1309407112yes—
514C8—42no—
515C8—1882no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22#1568913014yes—
523C22C2×C22⋊C4×D4711yescharacteristic
524C22#1568968811yescharacteristic
525C22#1568969511yescharacteristic
526C22C2×#1309405311yescharacteristic
527C22C2×#1309405611yescharacteristic
528C22#1568991911yescharacteristic
529C22#1568992211yescharacteristic
530C4—22no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—942no—
535C4—942no—
536C4—942no—
537C4—942no—
538C4—3762no—
539C4—1882no—
540C4—1882no—
541C4#1568960212yes—
542C4#1568970012yes—
543C2—24no—
544C2#2024597511yescharacteristic
545C2C2×#1568913011yescharacteristic
546C2#2025190411yescharacteristic
547C2#2025438811yescharacteristic
548C2#2025439011yescharacteristic
549C2#2025459111yescharacteristic
550C2#2025459311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1872C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#1872D47
hypocenterC47#21866
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
4. #20244359C2semidirect
5. #20244206C2semidirect
9. #20243402C2semidirect
10. #20243401C2semidirect
13. #15689675C4semidirect
53. #13094023C2×C4semidirect
55. #13094022C2×C4semidirect
61. Q8×Dic47C2×C4semidirect
64. #13093989C2×C4semidirect
321. C47#21866semidirect

Maximal subgroups

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

subgroupindex
C2×#156896752
C2×#156891772
C2×#156889882
#202443592
#202442062
#202436962
#202436952
#202436932
#202434022
#202434012
C47×#18722
#2186647