Finite Group Databasebeta

← #26221348

Subgroups — #26221348

Order 12032.

1290 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689652C211yesmaximal, characteristic
2C2×#15689175C211yesmaximal, characteristic, complemented
3C4×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#20244345C211yesmaximal, characteristic, complemented
5#20244343C211yesmaximal, characteristic, complemented
6#20244321C211yesmaximal, characteristic, complemented
7#20243702C212yesmaximal
8#20243692C212yesmaximal, complemented
9#103×Dic47C212yesmaximal, complemented
10#102×Dic47C212yesmaximal, complemented
11C47×#1983C211yesmaximal, characteristic, complemented
12#15689652—24no—
13#15689652C418yescomplemented
14C2×#13094017—21no—
15C2×#13094016C2211yescharacteristic
16C2×#13094006—21no—
17C2×#13094000C2211yescharacteristic
18C2×#13093995C2211yescharacteristic
19C2×Q8×Dic47C2211yescharacteristic
20C2×D4×Dic47C2211yescharacteristic
21C2×(C22⋊D47)⋊C4—21no—
22C2×#13093980—21no—
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689175—24no—
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×#13093948C2212yes—
29C2×#13093945C2212yes—
30C2×(C2×D47)⋊C8C2212yes—
31C2×C8×Dic47C2212yes—
32C4⋊C4×Dic47C2212yes—
33C4×Dic47⋊C4—21no—
34C4×C4⋊Dic47C2211yescharacteristic
35#15688974—21no—
36#95×D47—21no—
37#15688886C2212yes—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×#124×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C2×#102×C47C2212yes—
45C4×C4⋊C4×C47C2211yescharacteristic
46C22×C22⋊D47—21no—
47C23×Dic47—21no—
48C22×C4×D47—22no—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53#13094017—28no—
54#13094016—22no—
55#13094016C2×C414yescomplemented
56#13094006—28no—
57#13094000—22no—
58#13094000C2×C414yes—
59#13093995—22no—
60#13093995C2×C414yes—
61Q8×Dic47—22no—
62Q8×Dic47C2×C414yescomplemented
63D4×Dic47—22no—
64D4×Dic47C2×C414yescomplemented
65(C22⋊D47)⋊C4—28no—
66#13093980—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2312yes—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—22no—
74C2×Dic47⋊C4—21no—
75C4⋊C4×D47—24no—
76C4⋊C4×D47—24no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47—22no—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47C2311yescharacteristic
86C42×D47—44no—
87C42×D47C2×C414yes—
88C42×D47D414yes—
89C4⋊(C47⋊C8)—22no—
90#13093948—24no—
91#13093945—24no—
92(C2×D47)⋊C8—24no—
93C8×Dic47—24no—
94C47⋊#95—21no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C22×D4×C47C2311yescharacteristic
98C22×C8×C47C2312yes—
99#124×C47—22no—
100#124×C47C2×C414yescomplemented
101C2×C4⋊C4×C47C2312yes—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—21no—
104C2×C42×C47—21no—
105C2×C42×C47C2311yescharacteristic
106C4⋊C8×C47—22no—
107#103×C47—24no—
108#102×C47—24no—
109#95×C47—21no—
110C24×C47—21no—
111C23×D47C2×D411yescharacteristic
112C2×Q8×C47—21no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47C22×C412yes—
116C2×D4×C47—21no—
117C2×D4×C47—22no—
118C2×D4×C47—22no—
119C2×D4×C47C22×C412yes—
120C22×C4×C47—22no—
121C22×C4×C47—21no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2411yescharacteristic
125C2×C22⋊D47—24no—
126C2×C22⋊D47—44no—
127C22×Dic47—22no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47—24no—
131C22×Dic47—42no—
132C22×Dic47—42no—
133C22×Dic47C2×D411yescharacteristic
134C22×Dic47C2×D411yescharacteristic
135C22×Dic47C2×D411yescharacteristic
136C2×C4×D47—24no—
137C2×C4×D47—22no—
138C2×C4×D47—22no—
139C2×C4×D47—22no—
140C2×C4×D47—22no—
141C2×C4×D47—28no—
142C2×C4×D47—44no—
143C2×C4×D47—44no—
144C2×C4×D47C22⋊C414yes—
145C2×C4×D47C22⋊C414yes—
146C2×C4×D47C22×C412yes—
147C2×C4×D47C2×D412yes—
148C2×C8×C47—22no—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47#4114yes—
153C22⋊C4×C47—24no—
154C22⋊C4×C47—24no—
155C42×C47—42no—
156C42×C47C22×C412yes—
157C42×C47C2×D412yes—
158C22⋊Dic47—24no—
159C22⋊Dic47—24no—
160(C2×D47)⋊C4—24no—
161C4⋊Dic47—21no—
162C4⋊Dic47—21no—
163C4⋊Dic47—22no—
164C4⋊Dic47—24no—
165C4⋊Dic47C22×C412yes—
166C4⋊Dic47C22×C412yes—
167C4⋊Dic47#4114yes—
168Dic47⋊C4—28no—
169Dic47⋊C4—24no—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172C4×Dic47—24no—
173C4×Dic47—42no—
174C4×Dic47—44no—
175C4×Dic47C22⋊C414yes—
176C4×Dic47C22⋊C414yes—
177C4×Dic47C22×C412yes—
178C4×Dic47C2×D412yes—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—22no—
182C23×C47—22no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47C22×D411yescharacteristic
188C22×D47—22no—
189C22×D47—21no—
190C22×D47—21no—
191C22×D47C2×C22⋊C412yes—
192C22×D47#13514yes—
193Q8×C47—24no—
194Q8×C47—41no—
195Q8×C47—41no—
196D4×C47—24no—
197D4×C47—41no—
198D4×C47—41no—
199C2×C4×C47—22no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—22no—
203C2×C4×C47—24no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47C2×C22⋊C412yes—
208C2×C4×C47C2×C22⋊C412yes—
209C2×C4×C47C23×C411yescharacteristic
210C2×C4×C47C22×D411yescharacteristic
211C2×C4×C47C2×#4112yes—
212C2×C4×C47C2×#4112yes—
213C22⋊D47—48no—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—24no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—48no—
225C2×Dic47C2×C22⋊C412yes—
226C2×Dic47C2×C22⋊C412yes—
227C2×Dic47C2×C22⋊C412yes—
228C2×Dic47AGL(1,ℤ/8)12yes—
229C2×Dic47#13712yes—
230C4×D47—28no—
231C4×D47—24no—
232C4×D47—22no—
233C4×D47—22no—
234C4×D47—28no—
235C4×D47—416no—
236C8×C47—42no—
237C47⋊C8—42no—
238#21388—471nomaximal, Sylow
239C22×C47—42no—
240C22×C47—42no—
241C22×C47—42no—
242C22×C47—82no—
243C22×C47—82no—
244C22×C47C22×C22⋊C411yescharacteristic
245C22×C47#52512yes—
246C22×C47C2×#13512yes—
247C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C47C2×#13711yescharacteristic
249C2×D47—24no—
250C2×D47—24no—
251C2×D47—22no—
252C2×D47—22no—
253C2×D47#41614yes—
254C4×C47—22no—
255C4×C47—22no—
256C4×C47—44no—
257C4×C47—42no—
258C4×C47C4×#4112yes—
259C4×C47#53211yescharacteristic
260C4×C47#53211yescharacteristic
261Dic47—22no—
262Dic47—22no—
263Dic47—44no—
264Dic47—42no—
265Dic47—42no—
266Dic47#41714yes—
267C2×#529—471no—
268C2×#431—471no—
269C2×C4×C4⋊C4—471no—
270#1989—471no—
271#1989—471no—
272#1983—471no—
273#1983D4711yescharacteristic, complemented
274#1874—472no—
275#1872—472no—
276C4×#103—472no—
277C4×#102—472no—
278C2×C47—82no—
279C2×C47#229711yescharacteristic
280C2×C47C2×#41611yescharacteristic
281C2×C47C2×#41711yescharacteristic
282C2×C47#310512yes—
283C2×C47#310812yes—
284D47—24no—
285#529—478no—
286#529—944no—
287C2×#126—941no—
288C2×#125—471no—
289C2×#124C2×D4711yescharacteristic
290C2×#123—941no—
291C2×C22⋊Q8—471no—
292C2×#121—471no—
293C2×C4×Q8—471no—
294C2×C4×D4—471no—
295C2×C4×D4—941no—
296C2×#117—941no—
297C22×C4⋊C4—472no—
298C22×C42—471no—
299C22×C42—941no—
300#431—944no—
301C2×C4⋊C8—471no—
302C2×C4⋊C8C2×D4711yescharacteristic
303C2×#103—472no—
304C2×#103C2×D4712yes—
305C2×#102—472no—
306C2×#102C2×D4712yes—
307C2×C22⋊C8—472no—
308C2×C4×C8—472no—
309C4×C4⋊C4—472no—
310C4×C4⋊C4—471no—
311C4×C4⋊C4—941no—
312C4×C4⋊C4—941no—
313C4×C4⋊C4C2×D4711yescharacteristic
314C2×#95—941no—
315#340—472no—
316#340—472no—
317#340—472no—
318#340—472no—
319C47#2138811yescharacteristic, complemented, Sylow
320C22×Q8C22×D4711yescharacteristic
321C22×D4—941no—
322C22×D4C22×D4711yescharacteristic
323C23×C4—471no—
324C23×C4—471no—
325C23×C4—471no—
326C23×C4—942no—
327C23×C4—941no—
328C22×C8—472no—
329C22×C8C22×D4712yes—
330#126—948no—
331#125—474no—
332#125—942no—
333#124—22no—
334#124C4×D4714yescomplemented
335#123—948no—
336C22⋊Q8—474no—
337C22⋊Q8—942no—
338#121—474no—
339#121—942no—
340C4×Q8—474no—
341C4×Q8—942no—
342C4×D4—474no—
343C4×D4—942no—
344C4×D4—948no—
345#117—948no—
346C2×C4⋊C4—472no—
347C2×C4⋊C4—471no—
348C2×C4⋊C4—471no—
349C2×C4⋊C4—944no—
350C2×C4⋊C4—942no—
351C2×C4⋊C4—941no—
352C2×C4⋊C4—941no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—941no—
355C2×C4⋊C4—944no—
356C2×C4⋊C4C22×D4712yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—941no—
360C2×C22⋊C4—941no—
361C2×C22⋊C4—941no—
362C2×C42—21no—
363C2×C42—471no—
364C2×C42—471no—
365C2×C42—471no—
366C2×C42—474no—
367C2×C42—474no—
368C2×C42—941no—
369C2×C42—942no—
370C2×C42—941no—
371C2×C42—1884no—
372C2×C42C22×D4711yescharacteristic
373C4⋊C8—22no—
374C4⋊C8—942no—
375#103—24no—
376#103—944no—
377#102—24no—
378#102—944no—
379C22⋊C8—944no—
380C4×C8—944no—
381#95—21no—
382#95—942no—
383#95—941no—
384C24—21no—
385C24—471no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8C2×C4×D4712yes—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—944no—
394C2×D4—1884no—
395C2×D4C2×C4×D4712yes—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—471no—
399C22×C4—471no—
400C22×C4—471no—
401C22×C4—474no—
402C22×C4—474no—
403C22×C4—472no—
404C22×C4—472no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4—944no—
408C22×C4—948no—
409C22×C4—944no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—941no—
414C22×C4—941no—
415C22×C4—1884no—
416C22×C4—1884no—
417C22×C4—1882no—
418C22×C4—1882no—
419C22×C4D4×D4711yescharacteristic
420C22×C4D4×D4711yescharacteristic
421C22×C4C23×D4711yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—942no—
426C2×C8—942no—
427C2×C8—942no—
428C4⋊C4—22no—
429C4⋊C4—472no—
430C4⋊C4—472no—
431C4⋊C4—474no—
432C4⋊C4—944no—
433C4⋊C4—944no—
434C4⋊C4—948no—
435C4⋊C4—944no—
436C4⋊C4—944no—
437C4⋊C4—941no—
438C4⋊C4—941no—
439C4⋊C4—942no—
440C4⋊C4#1062535214yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—944no—
444C22⋊C4—944no—
445C22⋊C4—944no—
446C42—42no—
447C42—474no—
448C42—474no—
449C42—472no—
450C42—472no—
451C42—944no—
452C42—1882no—
453C42—1884no—
454C42C2×C4×D4712yes—
455C42D4×D4712yes—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—472no—
462C23—474no—
463C23—942no—
464C23—941no—
465C23—941no—
466C23C2×D4×D4711yescharacteristic
467Q8—24no—
468Q8—41no—
469Q8—41no—
470D4—24no—
471D4—41no—
472D4—41no—
473D4—1888no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—472no—
483C2×C4—472no—
484C2×C4—472no—
485C2×C4—472no—
486C2×C4—472no—
487C2×C4—948no—
488C2×C4—948no—
489C2×C4—942no—
490C2×C4—942no—
491C2×C4—944no—
492C2×C4—942no—
493C2×C4—942no—
494C2×C4—942no—
495C2×C4—942no—
496C2×C4—944no—
497C2×C4—18816no—
498C2×C4—1888no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1881no—
503C2×C4—1881no—
504C2×C4C22⋊C4×D4712yes—
505C2×C4C22⋊C4×D4712yes—
506C2×C4C22×C4×D4711yescharacteristic
507C2×C4C2×D4×D4711yescharacteristic
508C2×C4#41×D4712yes—
509C2×C4C2×#1062535212yes—
510C8—42no—
511C8—1882no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—82no—
516C22—82no—
517C22—474no—
518C22—944no—
519C22—942no—
520C22—942no—
521C22—944no—
522C22C2×C22⋊C4×D4711yescharacteristic
523C22#1568963312yes—
524C22#135×D4712yes—
525C22AGL(1,ℤ/8)×D4711yescharacteristic
526C22#137×D4711yescharacteristic
527C4—22no—
528C4—22no—
529C4—42no—
530C4—44no—
531C4—474no—
532C4—942no—
533C4—942no—
534C4—1882no—
535C4—1882no—
536C4—1884no—
537C4C4×#1062535212yes—
538C4#1568967111yescharacteristic
539C4#1568967211yescharacteristic
540C2—82no—
541C2—944no—
542C2#2024587511yescharacteristic
543C2#416×D4711yescharacteristic
544C2#417×D4711yescharacteristic
545C2#2025343412yes—
546C2#2025346712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1983C2
socleC23×C47C22×D4
prime radicalC2×#13093995C22
cosocleC22×C4×C47C24
hypercenter#1983D47
hypocenterC47#21388
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#15689175C2semidirect
3. C4×C4⋊C4×D47C2semidirect
4. #20244345C2semidirect
5. #20244343C2semidirect
6. #20244321C2semidirect
8. #20243692C2semidirect
9. #103×Dic47C2semidirect
10. #102×Dic47C2semidirect
11. C47×#1983C2semidirect
13. #15689652C4semidirect
55. #13094016C2×C4semidirect
62. Q8×Dic47C2×C4semidirect
64. D4×Dic47C2×C4semidirect
100. #124×C47C2×C4semidirect
273. #1983D47semidirect
319. C47#21388semidirect
334. #124C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896522
C2×#156891752
C4×C4⋊C4×D472
#202443452
#202443432
#202443212
#202437022
#202436922
#103×Dic472
#102×Dic472
C47×#19832
#2138847