Finite Group Databasebeta

← #26184826

Subgroups — #26184826

Order 12032.

952 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689678C211yesmaximal, characteristic
2#20252018C211yesmaximal, characteristic
3#432×D47C211yesmaximal, characteristic
4#20242117C214yesmaximal, complemented
5#20242113C211yesmaximal, characteristic, complemented
6#20242113C211yesmaximal, characteristic, complemented
7#20242105C211yesmaximal, characteristic, complemented
8#20242044C214yesmaximal, complemented
9C47×#1556C211yesmaximal, characteristic, complemented
10#15689678—21no—
11#15689678—21no—
12#15689678—21no—
13#15689678—21no—
14#15689678C412yes—
15#15689678C412yes—
16#15689678C2212yes—
17#15689678C2212yes—
18C2×#13094024C2211yescharacteristic
19C2×#13094015—21no—
20C2×#13094004C412yes—
21C2×#13094000—21no—
22C2×Q8×Dic47—21no—
23C2×#13093989C412yes—
24C2×#13093982—21no—
25C2×C42×D47C2211yescharacteristic
26#15689180C2211yescharacteristic
27#15689180C2211yescharacteristic
28#15689179C2212yes—
29#15689178—21no—
30#15689178—21no—
31#15689177—21no—
32#15689175C2212yes—
33#15689173—21no—
34#15689145C2212yes—
35C4⋊C8×D47C2212yes—
36#15689085—21no—
37C22⋊C8×D47—21no—
38#15689075—21no—
39#97×D47—21no—
40#15688871C2214yescomplemented
41#15688871C2214yescomplemented
42#15688869C2214yescomplemented
43C2×C4⋊Q8×C47C2211yescharacteristic
44C47×#432C2211yescharacteristic
45C47×#432C2211yescharacteristic
46C47×#332C2214yescomplemented
47C22×Dic94C2×C412yes—
48C22×C4×D47C2×C411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C2×#10625324—22no—
52C2×C8×D47—22no—
53#13094024—21no—
54#13094024—21no—
55#13094024C2×C412yes—
56#13094024C2312yes—
57#13094015—22no—
58#13094015—22no—
59#13094015—22no—
60#13094015—22no—
61#13094004—22no—
62#13094004—22no—
63#13094004C2×C414yes—
64#13094004C2×C414yes—
65#13094000—22no—
66#13094000—22no—
67#13094000—22no—
68#13094000—22no—
69Q8×Dic47—22no—
70Q8×Dic47—22no—
71Q8×Dic47—22no—
72Q8×Dic47—22no—
73#13093989—22no—
74#13093989—22no—
75#13093989C2×C414yes—
76#13093989C2×C414yes—
77#13093982—22no—
78#13093982—22no—
79#13093982—22no—
80#13093982—22no—
81C2×C4⋊Dic47—21no—
82C2×C4⋊Dic47—21no—
83C2×C4⋊Dic47—22no—
84C2×Dic47⋊C4—22no—
85C2×(C2×D47)⋊C4—22no—
86C2×C4×Dic47—22no—
87C2×C4×Dic47C2×C411yescharacteristic
88C2×C4×Dic47D411yescharacteristic
89C2×C4×Dic47D411yescharacteristic
90C42×D47C2×C411yescharacteristic
91C42×D47C2×C411yescharacteristic
92C42×D47D411yescharacteristic
93C42×D47D411yescharacteristic
94C42×D47D411yescharacteristic
95C42×D47D411yescharacteristic
96C42×D47C2311yescharacteristic
97C42×D47C2311yescharacteristic
98C4⋊(C47⋊C8)C2312yes—
99C4⋊(C47⋊C8)C2312yes—
100Dic47⋊C8—22no—
101Dic47⋊C8—22no—
102C22⋊(C47⋊C8)—21no—
103C22⋊(C47⋊C8)—21no—
104(C2×D47)⋊C8—22no—
105(C2×D47)⋊C8—22no—
106#13093935—21no—
107#13093935—21no—
108#13093934—22no—
109C8×Dic47—22no—
110C22×Q8×C47—21no—
111C4⋊Q8×C47—21no—
112C4⋊Q8×C47—21no—
113C4⋊Q8×C47C2×C412yes—
114C4⋊Q8×C47C2312yes—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47C2×C412yes—
117C2×C42×C47C2311yescharacteristic
118C4⋊C8×C47C2312yes—
119C4⋊C8×C47C2312yes—
120C22⋊C8×C47—21no—
121C22⋊C8×C47—21no—
122#97×C47—21no—
123#97×C47—21no—
124C23×D47C22⋊C411yescharacteristic
125C2×Q8×C47—22no—
126C2×Q8×C47—22no—
127C2×Q8×C47—44no—
128C22×C4×C47—22no—
129C22×C4×C47C22×C411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×Dic47—22no—
133C22×Dic47C22⋊C411yescharacteristic
134C22×Dic47C22⋊C411yescharacteristic
135C22×Dic47C22⋊C411yescharacteristic
136C2×C4×D47—24no—
137C2×C4×D47—42no—
138C2×C4×D47—42no—
139C2×C4×D47C22⋊C411yescharacteristic
140C2×C4×D47C22⋊C411yescharacteristic
141C2×C4×D47C22⋊C411yescharacteristic
142C2×C4×D47C22⋊C411yescharacteristic
143C2×C4×D47C22⋊C411yescharacteristic
144C2×C4×D47C22⋊C411yescharacteristic
145C2×C4×D47C22×C411yescharacteristic
146C2×C4×D47C22×C411yescharacteristic
147C2×C4×D47C2×D411yescharacteristic
148C2×C4×D47C2×D411yescharacteristic
149C2×C4×D47C2×D411yescharacteristic
150C2×C4×D47C2×D411yescharacteristic
151C2×Dic94—22no—
152C2×Dic94—44no—
153C2×Dic94C22×C414yes—
154C2×C8×C47—22no—
155C2×C8×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—22no—
159C4⋊C4×C47C22×C414yes—
160C42×C47C22×C411yescharacteristic
161C42×C47C2×D411yescharacteristic
162C42×C47C2×D411yescharacteristic
163C42×C47C2411yescharacteristic
164(C2×D47)⋊C4—24no—
165(C2×D47)⋊C4—24no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47—22no—
169C4⋊Dic47—22no—
170C4⋊Dic47—24no—
171C4⋊Dic47—24no—
172Dic47⋊C4—24no—
173Dic47⋊C4—24no—
174C4×Dic47—24no—
175C4×Dic47—24no—
176C4×Dic47C22⋊C411yescharacteristic
177C4×Dic47C22⋊C411yescharacteristic
178C4×Dic47C22⋊C411yescharacteristic
179C4×Dic47C22⋊C411yescharacteristic
180C4×Dic47C22⋊C411yescharacteristic
181C4×Dic47C22⋊C411yescharacteristic
182C4×Dic47C22×C411yescharacteristic
183C4×Dic47C22×C411yescharacteristic
184C4×Dic47C2×D411yescharacteristic
185C4×Dic47C2×D411yescharacteristic
186C4×Dic47C2×D411yescharacteristic
187C4×Dic47C2×D411yescharacteristic
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190#10625324—44no—
191C8×D47—44no—
192C23×C47C2×C22⋊C411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—41no—
196C22×D47C2×C22⋊C411yescharacteristic
197C22×D47C2×C22⋊C411yescharacteristic
198Q8×C47—44no—
199Q8×C47—44no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—24no—
204C2×C4×C47—44no—
205C2×C4×C47C2×C22⋊C411yescharacteristic
206C2×C4×C47C2×C22⋊C411yescharacteristic
207C2×C4×C47C2×C22⋊C411yescharacteristic
208C2×C4×C47C23×C411yescharacteristic
209C2×C4×C47C22×D411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×Dic47—22no—
212C2×Dic47—22no—
213C2×Dic47—24no—
214C2×Dic47—41no—
215C2×Dic47—44no—
216C2×Dic47C2×C22⋊C411yescharacteristic
217C2×Dic47C2×C22⋊C411yescharacteristic
218C2×Dic47C2×C22⋊C411yescharacteristic
219C2×Dic47C2×C22⋊C411yescharacteristic
220C2×Dic47C2×C22⋊C411yescharacteristic
221C2×Dic47C2×C22⋊C411yescharacteristic
222C4×D47—22no—
223C4×D47—22no—
224C4×D47—22no—
225C4×D47—22no—
226C4×D47—22no—
227C4×D47—41no—
228C4×D47—41no—
229C4×D47—41no—
230C4×D47—41no—
231C4×D47—41no—
232C4×D47—41no—
233C4×D47#13714yes—
234Dic94—44no—
235Dic94—44no—
236C8×C47—42no—
237C8×C47—42no—
238C47⋊C8—42no—
239C47⋊C8—42no—
240#17734—471nomaximal, Sylow
241C22×C47—21no—
242C22×C47—21no—
243C22×C47#41312yes—
244C22×C47C22×C22⋊C411yescharacteristic
245C2×D47—21no—
246C2×D47—21no—
247C2×D47—21no—
248C2×D47—21no—
249C2×D47—21no—
250C2×D47—44no—
251C2×D47#41912yes—
252C4×C47—21no—
253C4×C47—21no—
254C4×C47—21no—
255C4×C47—21no—
256C4×C47—44no—
257C4×C47C2×#13712yes—
258C4×C47#57512yes—
259Dic47—21no—
260Dic47—21no—
261Dic47—21no—
262Dic47—21no—
263Dic47—21no—
264Dic47—44no—
265Dic47#41912yes—
266C2×#533—471no—
267#2920—471no—
268C2×#432—471no—
269#1559—474no—
270#1558—471no—
271#1558—471no—
272#1556—471no—
273#1556D4711yescharacteristic, complemented
274#1542—474no—
275C2×C47—22no—
276C2×C47C2×#41311yescharacteristic
277C2×C47C2×#41911yescharacteristic
278C2×C47#289611yescharacteristic
279D47—22no—
280D47—41no—
281#533—472no—
282#533—472no—
283#533—472no—
284#533—472no—
285#533—941no—
286#533—941no—
287#533—941no—
288#533—941no—
289C2×C4⋊Q8—471no—
290C2×C4⋊Q8C2×D4711yescharacteristic
291C2×#125—941no—
292C2×C22⋊Q8—472no—
293C2×C22⋊Q8—941no—
294C2×C4×Q8—472no—
295C2×C4×Q8—941no—
296C2×#117—941no—
297C22×C42—471no—
298#432—471no—
299#432—471no—
300#432—472no—
301#432—941no—
302#432—941no—
303#432—941no—
304#432C2×D4711yescharacteristic
305#432C2×D4711yescharacteristic
306#431—472no—
307#431—941no—
308#423—472no—
309C2×C4⋊C8—472no—
310#407—941no—
311C2×C22⋊C8—941no—
312C4×M16—941no—
313C2×#97—941no—
314#333—474no—
315#333—474no—
316#332—474no—
317#332C2×D4714yescomplemented
318C47#1773411yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×Q8—472no—
321C23×C4—471no—
322C23×C4—471no—
323C23×C4—471no—
324C2×M16—942no—
325C22×C8—942no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—472no—
329C4⋊Q8—472no—
330C4⋊Q8—941no—
331C4⋊Q8—941no—
332C4⋊Q8C4×D4712yes—
333C4⋊Q8C22×D4712yes—
334#125—942no—
335#125—942no—
336#125—942no—
337#125—942no—
338C22⋊Q8—474no—
339C22⋊Q8—474no—
340C22⋊Q8—942no—
341C22⋊Q8—942no—
342C22⋊Q8—942no—
343C22⋊Q8—942no—
344C22⋊Q8—942no—
345C22⋊Q8—942no—
346C4×Q8—474no—
347C4×Q8—474no—
348C4×Q8—942no—
349C4×Q8—942no—
350C4×Q8—942no—
351C4×Q8—942no—
352C4×Q8—942no—
353C4×Q8—942no—
354#117—942no—
355#117—942no—
356#117—942no—
357#117—942no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4—941no—
362C2×C4⋊C4—941no—
363C2×C4⋊C4C4×D4712yes—
364C2×C22⋊C4—942no—
365C2×C42—471no—
366C2×C42—471no—
367C2×C42—471no—
368C2×C42—471no—
369C2×C42—471no—
370C2×C42—471no—
371C2×C42—471no—
372C2×C42—471no—
373C2×C42—471no—
374C2×C42—471no—
375C2×C42—471no—
376C2×C42—942no—
377C2×C42C22×D4711yescharacteristic
378C4⋊C8—472no—
379C4⋊C8—472no—
380C4⋊C8—942no—
381C4⋊C8—942no—
382C4⋊C8C22×D4712yes—
383C4⋊C8C22×D4712yes—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8—942no—
387C22⋊C8—942no—
388C22⋊C8—941no—
389C22⋊C8—941no—
390#97—21no—
391#97—21no—
392#97—942no—
393#97—941no—
394#97—941no—
395C4×C8—942no—
396C24—471no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—474no—
401C2×Q8—942no—
402C2×Q8—1884no—
403C22×C4—22no—
404C22×C4—471no—
405C22×C4—471no—
406C22×C4—471no—
407C22×C4—471no—
408C22×C4—471no—
409C22×C4—471no—
410C22×C4—471no—
411C22×C4—471no—
412C22×C4—471no—
413C22×C4—471no—
414C22×C4—471no—
415C22×C4—471no—
416C22×C4—471no—
417C22×C4—471no—
418C22×C4—471no—
419C22×C4—944no—
420C22×C4—942no—
421C22×C4—1882no—
422C22×C4—1882no—
423C22×C4C2×C4×D4711yescharacteristic
424C22×C4D4×D4711yescharacteristic
425C22×C4D4×D4711yescharacteristic
426M16—1884no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—942no—
430C2×C8—942no—
431C2×C8—1884no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—944no—
436C4⋊C4—944no—
437C4⋊C4—944no—
438C4⋊C4—944no—
439C4⋊C4—942no—
440C4⋊C4—942no—
441C4⋊C4—942no—
442C4⋊C4—942no—
443C4⋊C4C2×C4×D4714yes—
444C22⋊C4—944no—
445C22⋊C4—944no—
446C42—471no—
447C42—471no—
448C42—471no—
449C42—471no—
450C42—471no—
451C42—471no—
452C42—471no—
453C42—471no—
454C42—471no—
455C42—471no—
456C42—471no—
457C42—471no—
458C42—944no—
459C42—944no—
460C42C2×C4×D4711yescharacteristic
461C42D4×D4711yescharacteristic
462C42D4×D4711yescharacteristic
463C42C23×D4711yescharacteristic
464C23—471no—
465C23—471no—
466C23—942no—
467C23—942no—
468C23—1881no—
469C23C22⋊C4×D4711yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—1884no—
473Q8—1884no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—44no—
479C2×C4—471no—
480C2×C4—474no—
481C2×C4—471no—
482C2×C4—471no—
483C2×C4—471no—
484C2×C4—471no—
485C2×C4—471no—
486C2×C4—942no—
487C2×C4—942no—
488C2×C4—942no—
489C2×C4—942no—
490C2×C4—942no—
491C2×C4—942no—
492C2×C4—942no—
493C2×C4—944no—
494C2×C4—1881no—
495C2×C4—1881no—
496C2×C4—1881no—
497C2×C4—1881no—
498C2×C4—1881no—
499C2×C4—1881no—
500C2×C4—1881no—
501C2×C4—1884no—
502C2×C4C22⋊C4×D4711yescharacteristic
503C2×C4C22⋊C4×D4711yescharacteristic
504C2×C4C22⋊C4×D4711yescharacteristic
505C2×C4C22×C4×D4711yescharacteristic
506C2×C4C2×D4×D4711yescharacteristic
507C2×C4C2×D4×D4711yescharacteristic
508C8—42no—
509C8—42no—
510C8—1882no—
511C8—1882no—
512C22—21no—
513C22—21no—
514C22—472no—
515C22—941no—
516C22—941no—
517C22—941no—
518C22—941no—
519C22—941no—
520C22—1884no—
521C22#1568910612yes—
522C22C2×C22⋊C4×D4711yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—44no—
528C4—472no—
529C4—941no—
530C4—941no—
531C4—941no—
532C4—941no—
533C4—941no—
534C4—1884no—
535C4#137×D4712yes—
536C4#1568991912yes—
537C2—22no—
538C2—942no—
539C2—1881no—
540C2C2×#1568910611yescharacteristic
541C2#419×D4711yescharacteristic
542C2#2025190211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1556C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1556D47
hypocenterC47#17734
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
4. #20242117C2semidirect
5. #20242113C2semidirect
6. #20242113C2semidirect
7. #20242105C2semidirect
8. #20242044C2semidirect
9. C47×#1556C2semidirect
40. #15688871C22semidirect
41. #15688871C22semidirect
42. #15688869C22semidirect
46. C47×#332C22semidirect
273. #1556D47semidirect
317. #332C2×D47semidirect
318. C47#17734semidirect

Maximal subgroups

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

subgroupindex
C2×#156896782
#202520182
#432×D472
#202421172
#202421132
#202421052
#202420442
C47×#15562
#1773447