Finite Group Databasebeta

← #26225789

Subgroups — #26225789

Order 12032.

1066 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D47C211yesmaximal, characteristic
2C2×#15689178C211yesmaximal, characteristic
3#20245793C211yesmaximal, characteristic
4#20244358C211yesmaximal, characteristic, complemented
5#20244357C211yesmaximal, characteristic, complemented
6#20244356C211yesmaximal, characteristic, complemented
7#20244344C212yesmaximal, complemented
8#20244342C212yesmaximal, complemented
9#20243477C212yesmaximal
10#20243473C212yesmaximal
11C47×#1992C211yesmaximal, characteristic, complemented
12C2×#13094016C2211yescharacteristic
13C2×#13094015C2211yescharacteristic
14C2×#13094015C2211yescharacteristic
15C2×#13094014—21no—
16C2×#13094014—21no—
17#125×D47—24no—
18#125×D47C418yescomplemented
19C2×#13093982—21no—
20C2×C4⋊C4×D47—21no—
21C2×C4⋊C4×D47—21no—
22C2×C4⋊C4×D47C2211yescharacteristic
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C42×D47C2211yescharacteristic
25#15689178—24no—
26C2×Dic47⋊C8C2212yes—
27C2×(C2×D47)⋊C8C2212yes—
28C2×#13093935C2211yescharacteristic
29#15688993C2211yescharacteristic
30#15688988—21no—
31#15688980—21no—
32#15688980C2212yes—
33C4⋊C4×Dic47—21no—
34C4⋊C4×Dic47C2212yes—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2212yes—
38#15688885C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×#125×C47C2211yescharacteristic
42C2×#97×C47C2211yescharacteristic
43C47×#384C2211yescharacteristic
44C47×#340C2212yes—
45C47×#340C2212yes—
46C22×C4×D47—21no—
47C22×C4×D47—21no—
48C22×C4×D47D411yescharacteristic
49C22×C4×D47D411yescharacteristic
50C22×C4×D47C2311yescharacteristic
51C22×C47⋊C8C2312yes—
52#13094016—22no—
53#13094016C2×C414yes—
54#13094015—22no—
55#13094015—22no—
56#13094015C2×C414yes—
57#13094015C2×C414yes—
58#13094014—28no—
59#13094014—28no—
60#13093982—44no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C2×C4⋊Dic47—21no—
64C2×C4⋊Dic47—21no—
65C2×C4⋊Dic47C2312yes—
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×Dic47⋊C4—21no—
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C4⋊C4×D47—28no—
75C4⋊C4×D47—28no—
76C4⋊C4×D47—22no—
77C4⋊C4×D47—22no—
78C4⋊C4×D47C2×C414yes—
79C4⋊C4×D47C2×C414yes—
80C2×(C2×D47)⋊C4—22no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47C2×C414yes—
86C42×D47D414yes—
87Dic47⋊C8—24no—
88(C2×D47)⋊C8—24no—
89#13093935—22no—
90C47⋊#95—22no—
91C47⋊#95—22no—
92C22×C8×C47C2312yes—
93#125×C47—22no—
94#125×C47C2×C414yes—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47C2312yes—
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C42×C47C2311yescharacteristic
103#97×C47—22no—
104C23×D47C2×D411yescharacteristic
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—22no—
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×D411yescharacteristic
110C22×C4×C47C2411yescharacteristic
111C22×Dic47—21no—
112C22×Dic47—21no—
113C22×Dic47—22no—
114C22×Dic47C2×D411yescharacteristic
115C22×Dic47C2×D411yescharacteristic
116C22×Dic47C2×D411yescharacteristic
117C2×C4×D47—22no—
118C2×C4×D47—22no—
119C2×C4×D47—22no—
120C2×C4×D47—22no—
121C2×C4×D47—24no—
122C2×C4×D47—24no—
123C2×C4×D47—41no—
124C2×C4×D47—41no—
125C2×C4×D47—42no—
126C2×C4×D47—42no—
127C2×C4×D47—42no—
128C2×C4×D47—42no—
129C2×C4×D47C22⋊C414yes—
130C2×C4×D47C22⋊C414yes—
131C2×C4×D47C22×C412yes—
132C2×C4×D47C2×D412yes—
133C2×C8×C47—22no—
134C2×C8×C47—22no—
135C2×C8×C47—22no—
136C4⋊C4×C47—24no—
137C4⋊C4×C47—24no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—21no—
140C4⋊C4×C47—24no—
141C4⋊C4×C47—42no—
142C4⋊C4×C47—42no—
143C4⋊C4×C47C22×C412yes—
144C4⋊C4×C47C22×C412yes—
145C42×C47C22×C412yes—
146C42×C47C2×D412yes—
147(C2×D47)⋊C4—44no—
148C4⋊Dic47—21no—
149C4⋊Dic47—21no—
150C4⋊Dic47—24no—
151C4⋊Dic47—24no—
152C4⋊Dic47—24no—
153C4⋊Dic47—42no—
154C4⋊Dic47—42no—
155C4⋊Dic47C22×C412yes—
156C4⋊Dic47C22×C412yes—
157Dic47⋊C4—24no—
158Dic47⋊C4—24no—
159Dic47⋊C4—24no—
160Dic47⋊C4—24no—
161Dic47⋊C4—24no—
162Dic47⋊C4—24no—
163C4×Dic47—44no—
164C4×Dic47C22⋊C414yes—
165C4×Dic47C22⋊C414yes—
166C4×Dic47C22×C412yes—
167C4×Dic47C2×D412yes—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C23×C47C22×D411yescharacteristic
172C22×D47—21no—
173C22×D47—21no—
174C22×D47—21no—
175C22×D47—21no—
176C22×D47C2×C22⋊C412yes—
177C22×D47AGL(1,ℤ/8)12yes—
178C22×D47#13712yes—
179C2×C4×C47—21no—
180C2×C4×C47—21no—
181C2×C4×C47—21no—
182C2×C4×C47—21no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—41no—
186C2×C4×C47—41no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—42no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47C2×C22⋊C412yes—
193C2×C4×C47C2×C22⋊C412yes—
194C2×C4×C47C23×C411yescharacteristic
195C2×C4×C47C22×D411yescharacteristic
196C2×C4×C472−1+412yes—
197C2×C4×C472−1+412yes—
198C2×Dic47—21no—
199C2×Dic47—21no—
200C2×Dic47—21no—
201C2×Dic47—21no—
202C2×Dic47—21no—
203C2×Dic47—21no—
204C2×Dic47—22no—
205C2×Dic47—22no—
206C2×Dic47—41no—
207C2×Dic47—41no—
208C2×Dic47—41no—
209C2×Dic47—41no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47C2×C22⋊C412yes—
214C2×Dic47C2×C22⋊C412yes—
215C2×Dic47C2×C22⋊C412yes—
216C4×D47—28no—
217C4×D47—24no—
218C4×D47—24no—
219C4×D47—44no—
220C4×D47—48no—
221C4×D47—48no—
222C8×C47—42no—
223C47⋊C8—42no—
224#21838—471nomaximal, Sylow
225C22×C47C22×C22⋊C411yescharacteristic
226C22×C47#53611yescharacteristic
227C22×C47#53611yescharacteristic
228C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
229C22×C47C2×#13711yescharacteristic
230C22×C47#57511yescharacteristic
231C22×C47#57511yescharacteristic
232C2×D47—22no—
233C2×D47—22no—
234C2×D47—22no—
235C2×D47—22no—
236C2×D47—22no—
237C2×D47—22no—
238C2×D47#41714yes—
239C4×C47—22no—
240C4×C47—22no—
241C4×C47—42no—
242C4×C47—42no—
243C4×C47—82no—
244C4×C47#51912yes—
245C4×C47#53612yes—
246Dic47—22no—
247Dic47—22no—
248Dic47—22no—
249Dic47—42no—
250Dic47—42no—
251Dic47—82no—
252C22×#125—471no—
253C2×#432—471no—
254#2285—471no—
255#1992—471no—
256#1992—471no—
257#1992—471no—
258#1992D4711yescharacteristic, complemented
259#1989—472no—
260#1989—472no—
261#1824—472no—
262#1823—472no—
263C2×C47#231011yescharacteristic
264C2×C47C2×#41711yescharacteristic
265C2×C47#289611yescharacteristic
266C2×C47#323411yescharacteristic
267C2×C47#323411yescharacteristic
268C2×C47#323611yescharacteristic
269C2×C47#323611yescharacteristic
270D47—24no—
271C2×#125—478no—
272C2×#125—471no—
273C2×#125—471no—
274C2×#125—471no—
275C2×#125—941no—
276C2×#125—941no—
277C2×#125—944no—
278C2×#125C2×D4711yescharacteristic
279C2×#117—941no—
280C22×C4⋊C4—471no—
281C22×C4⋊C4—471no—
282C22×C4⋊C4—941no—
283C22×C4⋊C4—941no—
284C22×C42—471no—
285#432—944no—
286C2×C4⋊C8—472no—
287C2×C22⋊C8—472no—
288C2×#97—471no—
289C2×#97C2×D4711yescharacteristic
290#384—471no—
291#384C2×D4711yescharacteristic
292#382—941no—
293#380—472no—
294#380—941no—
295C4×C4⋊C4—472no—
296C4×C4⋊C4—941no—
297#340—472no—
298#340—472no—
299#340—472no—
300#340—472no—
301#340—472no—
302#340—472no—
303#340C2×D4712yes—
304#340C2×D4712yes—
305C47#2183811yescharacteristic, complemented, Sylow
306C23×C4—471no—
307C23×C4—471no—
308C23×C4—471no—
309C23×C4—941no—
310C23×C4—941no—
311C22×C8—472no—
312C22×C8C22×D4712yes—
313#125—22no—
314#125—474no—
315#125—474no—
316#125—474no—
317#125—948no—
318#125—948no—
319#125—942no—
320#125—942no—
321#125—942no—
322#125C4×D4714yes—
323#117—1884no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—471no—
329C2×C4⋊C4—471no—
330C2×C4⋊C4—474no—
331C2×C4⋊C4—474no—
332C2×C4⋊C4—472no—
333C2×C4⋊C4—942no—
334C2×C4⋊C4—942no—
335C2×C4⋊C4—948no—
336C2×C4⋊C4—948no—
337C2×C4⋊C4—941no—
338C2×C4⋊C4—941no—
339C2×C4⋊C4—941no—
340C2×C4⋊C4—941no—
341C2×C4⋊C4—941no—
342C2×C4⋊C4—941no—
343C2×C4⋊C4—941no—
344C2×C4⋊C4—941no—
345C2×C4⋊C4—941no—
346C2×C4⋊C4—941no—
347C2×C4⋊C4C22×D4712yes—
348C2×C4⋊C4C22×D4711yescharacteristic
349C2×C4⋊C4C22×D4711yescharacteristic
350C2×C22⋊C4—942no—
351C2×C42—471no—
352C2×C42—471no—
353C2×C42—471no—
354C2×C42—474no—
355C2×C42—474no—
356C2×C42—942no—
357C2×C42C22×D4711yescharacteristic
358C4⋊C8—944no—
359C22⋊C8—944no—
360#97—22no—
361#97—942no—
362#95—942no—
363#95—942no—
364C24—471no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—471no—
369C22×C4—471no—
370C22×C4—471no—
371C22×C4—474no—
372C22×C4—472no—
373C22×C4—472no—
374C22×C4—474no—
375C22×C4—942no—
376C22×C4—942no—
377C22×C4—944no—
378C22×C4—944no—
379C22×C4—942no—
380C22×C4—942no—
381C22×C4—941no—
382C22×C4—941no—
383C22×C4—942no—
384C22×C4—1881no—
385C22×C4—1881no—
386C22×C4—1882no—
387C22×C4—1882no—
388C22×C4—1882no—
389C22×C4—1882no—
390C22×C4D4×D4711yescharacteristic
391C22×C4D4×D4711yescharacteristic
392C22×C4C23×D4711yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—942no—
397C2×C8—942no—
398C2×C8—942no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—472no—
407C4⋊C4—472no—
408C4⋊C4—944no—
409C4⋊C4—944no—
410C4⋊C4—944no—
411C4⋊C4—944no—
412C4⋊C4—944no—
413C4⋊C4—944no—
414C4⋊C4—944no—
415C4⋊C4—944no—
416C4⋊C4—941no—
417C4⋊C4—941no—
418C4⋊C4—944no—
419C4⋊C4—1882no—
420C4⋊C4—1882no—
421C4⋊C4C2×C4×D4712yes—
422C4⋊C4C2×C4×D4712yes—
423C22⋊C4—1884no—
424C42—474no—
425C42—474no—
426C42—472no—
427C42—472no—
428C42—1884no—
429C42C2×C4×D4712yes—
430C42D4×D4712yes—
431C23—472no—
432C23—472no—
433C23—472no—
434C23—941no—
435C23—941no—
436C23—941no—
437C23—941no—
438C23C2×D4×D4711yescharacteristic
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—472no—
453C2×C4—472no—
454C2×C4—472no—
455C2×C4—948no—
456C2×C4—941no—
457C2×C4—941no—
458C2×C4—941no—
459C2×C4—941no—
460C2×C4—941no—
461C2×C4—941no—
462C2×C4—944no—
463C2×C4—944no—
464C2×C4—942no—
465C2×C4—942no—
466C2×C4—1884no—
467C2×C4—1888no—
468C2×C4—1888no—
469C2×C4—1882no—
470C2×C4—1882no—
471C2×C4—1882no—
472C2×C4—1881no—
473C2×C4—1881no—
474C2×C4—1881no—
475C2×C4—1881no—
476C2×C4C22⋊C4×D4712yes—
477C2×C4C22⋊C4×D4712yes—
478C2×C4C22×C4×D4711yescharacteristic
479C2×C4C2×D4×D4711yescharacteristic
480C2×C4#1309407112yes—
481C2×C4#1309407212yes—
482C8—42no—
483C8—1882no—
484C22—474no—
485C22—942no—
486C22—942no—
487C22—942no—
488C22—942no—
489C22—942no—
490C22—942no—
491C22C2×C22⋊C4×D4711yescharacteristic
492C22#1568969311yescharacteristic
493C22#1568969811yescharacteristic
494C22AGL(1,ℤ/8)×D4711yescharacteristic
495C22#137×D4711yescharacteristic
496C22#1568991911yescharacteristic
497C22#1568992211yescharacteristic
498C4—22no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—942no—
504C4—942no—
505C4—942no—
506C4—3762no—
507C4—1882no—
508C4—1882no—
509C4#1568959912yes—
510C4#1568969612yes—
511C2—944no—
512C2#2024597111yescharacteristic
513C2#417×D4711yescharacteristic
514C2#2025190411yescharacteristic
515C2#2025457311yescharacteristic
516C2#2025457911yescharacteristic
517C2#2025458911yescharacteristic
518C2#2025459511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1992C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1992D47
hypocenterC47#21838
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
4. #20244358C2semidirect
5. #20244357C2semidirect
6. #20244356C2semidirect
7. #20244344C2semidirect
8. #20244342C2semidirect
11. C47×#1992C2semidirect
18. #125×D47C4semidirect
258. #1992D47semidirect
305. C47#21838semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D472
C2×#156891782
#202457932
#202443582
#202443572
#202443562
#202443442
#202443422
#202434772
#202434732
C47×#19922
#2183847