Finite Group Databasebeta

← #26221374

Subgroups — #26221374

Order 12032.

1270 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×#15689655C211yesmaximal, characteristic
2C2×#15689178C211yesmaximal, characteristic, complemented
3C4×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#20244344C212yesmaximal, complemented
5#20244319C212yesmaximal, complemented
6#20243702C212yesmaximal
7#20243694C212yesmaximal, complemented
8#20243221C211yesmaximal, characteristic, complemented
9#20243219C211yesmaximal, characteristic, complemented
10#20243213C211yesmaximal, characteristic, complemented
11C47×#1762C211yesmaximal, characteristic, complemented
12#15689655—24no—
13#15689655C418yescomplemented
14C2×#13094017—21no—
15C2×#13094015C2211yescharacteristic
16C2×#13094011C2211yescharacteristic
17C2×#13094006—21no—
18C2×#13094004C2211yescharacteristic
19C2×#13093995C2211yescharacteristic
20C2×C4×Dic94C2211yescharacteristic
21C2×C4×C22⋊D47—21no—
22C2×#13093983—21no—
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689178—24no—
27C2×Dic47⋊C8C2212yes—
28C2×#13093949C2212yes—
29C2×#13093947C2212yes—
30C2×#13093945C2212yes—
31C2×(C2×D47)⋊C8C2212yes—
32C2×#13093935C2211yescharacteristic
33C4⋊C4×Dic47C2212yes—
34C4×Dic47⋊C4—21no—
35C4×C4⋊Dic47C2211yescharacteristic
36#15688974—21no—
37#95×D47—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×D4×C47C2211yescharacteristic
42C2×#102×C47C2212yes—
43C2×#97×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94C2311yescharacteristic
47C22×C22⋊D47—21no—
48C22×C4×D47—22no—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53#13094017—28no—
54#13094015—22no—
55#13094015C2×C414yescomplemented
56#13094011—22no—
57#13094011C2×C414yescomplemented
58#13094006—28no—
59#13094004—22no—
60#13094004C2×C414yes—
61#13093995—22no—
62#13093995C2×C414yes—
63C4×Dic94—22no—
64C4×Dic94C2×C414yescomplemented
65C4×C22⋊D47—28no—
66#13093983—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2312yes—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—22no—
73C2×Dic47⋊C4—21no—
74C4⋊C4×D47—24no—
75C4⋊C4×D47—24no—
76C2×C22⋊Dic47—21no—
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—
89Dic47⋊C8—24no—
90#13093949—24no—
91#13093947—24no—
92#13093945—24no—
93(C2×D47)⋊C8—24no—
94#13093935—22no—
95C47⋊#95—21no—
96C47⋊#95—22no—
97C22×D4×C47C2311yescharacteristic
98C23×C4×C47—21no—
99C22×C8×C47C2312yes—
100C4×D4×C47—22no—
101C4×D4×C47C2×C414yescomplemented
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2312yes—
104C2×C22⋊C4×C47—21no—
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107#102×C47—24no—
108#97×C47—22no—
109#95×C47—21no—
110C24×C47—21no—
111C23×D47C2×D411yescharacteristic
112C2×D4×C47—21no—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47C22×C412yes—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—24no—
119C22×C4×C47—44no—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2411yescharacteristic
123C2×C22⋊D47—24no—
124C2×C22⋊D47—42no—
125C2×C22⋊D47—42no—
126C22×Dic47—22no—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47C2×D411yescharacteristic
130C22×Dic47C2×D411yescharacteristic
131C22×Dic47C2×D411yescharacteristic
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—22no—
136C2×C4×D47—28no—
137C2×C4×D47—42no—
138C2×C4×D47—44no—
139C2×C4×D47—44no—
140C2×C4×D47C22⋊C414yes—
141C2×C4×D47C22⋊C414yes—
142C2×C4×D47C22×C412yes—
143C2×C4×D47C2×D412yes—
144C2×Dic94—22no—
145C2×Dic94—22no—
146C2×Dic94—21no—
147C2×Dic94C22×C412yes—
148C2×C8×C47—22no—
149C2×C8×C47—22no—
150C2×C8×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47C22×C412yes—
154C4⋊C4×C47#4114yes—
155C22⋊C4×C47—24no—
156C42×C47—42no—
157C42×C47C22×C412yes—
158C42×C47C2×D412yes—
159C22⋊Dic47—24no—
160C22⋊Dic47—24no—
161C22⋊Dic47—24no—
162(C2×D47)⋊C4—24no—
163C4⋊Dic47—21no—
164C4⋊Dic47—22no—
165C4⋊Dic47—24no—
166C4⋊Dic47—24no—
167C4⋊Dic47C22×C412yes—
168C4⋊Dic47#4114yes—
169Dic47⋊C4—28no—
170Dic47⋊C4—24no—
171Dic47⋊C4—24no—
172C4×Dic47—24no—
173C4×Dic47—44no—
174C4×Dic47—42no—
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—41no—
185C23×C47—41no—
186C23×C47—42no—
187C23×C47C22×D411yescharacteristic
188C22×D47—22no—
189C22×D47—22no—
190C22×D47C2×C22⋊C412yes—
191C22×D47AGL(1,ℤ/8)12yes—
192C22×D47#13712yes—
193D4×C47—22no—
194D4×C47—22no—
195D4×C47—42no—
196C2×C4×C47—21no—
197C2×C4×C47—21no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—24no—
201C2×C4×C47—48no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47C2×C22⋊C412yes—
206C2×C4×C47C2×C22⋊C412yes—
207C2×C4×C47C23×C411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C2×#4112yes—
210C2×C4×C47C2×#4112yes—
211C22⋊D47—48no—
212C2×Dic47—22no—
213C2×Dic47—21no—
214C2×Dic47—21no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—24no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—41no—
222C2×Dic47—41no—
223C2×Dic47—42no—
224C2×Dic47C2×C22⋊C412yes—
225C2×Dic47C2×C22⋊C412yes—
226C2×Dic47C2×C22⋊C412yes—
227C4×D47—28no—
228C4×D47—22no—
229C4×D47—22no—
230C4×D47—24no—
231C4×D47—44no—
232C4×D47—416no—
233Dic94—22no—
234Dic94—22no—
235Dic94—42no—
236C8×C47—42no—
237C47⋊C8—42no—
238#21391—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×AGL(1,ℤ/8)11yescharacteristic
247C22×C47C2×#13711yescharacteristic
248C22×C47#57512yes—
249C2×D47—24no—
250C2×D47—24no—
251C2×D47—22no—
252C2×D47—22no—
253C2×D47#41714yes—
254C4×C47—22no—
255C4×C47—22no—
256C4×C47—44no—
257C4×C47—42no—
258C4×C47C4×#4111yescharacteristic
259C4×C47C4×#4111yescharacteristic
260C4×C47#53212yes—
261Dic47—22no—
262Dic47—22no—
263Dic47—22no—
264Dic47—44no—
265Dic47—42no—
266Dic47—42no—
267C2×#529—471no—
268C2×#432—471no—
269C2×C4×C4⋊C4—471no—
270#1989—472no—
271#1983—472no—
272#1874—472no—
273#1872—472no—
274#1763—471no—
275#1763—471no—
276#1762—471no—
277#1762D4711yescharacteristic, complemented
278C2×C47—82no—
279C2×C47#229711yescharacteristic
280C2×C47C2×#41711yescharacteristic
281C2×C47#289611yescharacteristic
282C2×C47#311412yes—
283C2×C47#311712yes—
284D47—24no—
285#529—478no—
286#529—944no—
287C2×#126—941no—
288C2×#125—471no—
289C2×#124—471no—
290C2×#123—941no—
291C2×C22⋊Q8—471no—
292C2×#121—471no—
293C2×C4×Q8—471no—
294C2×C4×D4—941no—
295C2×C4×D4C2×D4711yescharacteristic
296C2×#117—941no—
297C22×C4⋊C4—472no—
298C22×C42—471no—
299C22×C42—941no—
300#432—944no—
301C2×C4⋊C8—472no—
302C2×#103—472no—
303C2×#103—472no—
304C2×#102—472no—
305C2×#102C2×D4712yes—
306C2×C22⋊C8—472no—
307C2×#97—471no—
308C2×#97C2×D4711yescharacteristic
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#340C2×D4712yes—
319C47#2139111yescharacteristic, complemented, Sylow
320C22×Q8—471no—
321C22×D4—941no—
322C22×D4C22×D4711yescharacteristic
323C23×C4—21no—
324C23×C4—471no—
325C23×C4—471no—
326C23×C4—471no—
327C23×C4—942no—
328C22×C8—472no—
329C22×C8C22×D4712yes—
330#126—948no—
331#125—474no—
332#125—942no—
333#124—474no—
334#124—942no—
335#123—948no—
336C22⋊Q8—474no—
337C22⋊Q8—942no—
338#121—474no—
339#121—942no—
340C4×Q8—474no—
341C4×Q8—942no—
342C4×D4—22no—
343C4×D4—948no—
344C4×D4C4×D4714yescomplemented
345#117—948no—
346C2×C4⋊C4—472no—
347C2×C4⋊C4—471no—
348C2×C4⋊C4—944no—
349C2×C4⋊C4—942no—
350C2×C4⋊C4—941no—
351C2×C4⋊C4—941no—
352C2×C4⋊C4—941no—
353C2×C4⋊C4—941no—
354C2×C4⋊C4—944no—
355C2×C4⋊C4C22×D4711yescharacteristic
356C2×C4⋊C4C22×D4712yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—941no—
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—944no—
374#103—944no—
375#103—944no—
376#102—24no—
377#102—944no—
378C22⋊C8—944no—
379#97—22no—
380#97—942no—
381#95—21no—
382#95—942no—
383#95—941no—
384C24—21no—
385C24—471no—
386C2×Q8—472no—
387C2×Q8—941no—
388C2×Q8—942no—
389C2×Q8—942no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—944no—
394C2×D4—1882no—
395C2×D4—1882no—
396C2×D4C2×C4×D4712yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—44no—
401C22×C4—471no—
402C22×C4—471no—
403C22×C4—471no—
404C22×C4—474no—
405C22×C4—472no—
406C22×C4—472no—
407C22×C4—474no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—948no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—942no—
414C22×C4—941no—
415C22×C4—941no—
416C22×C4—1882no—
417C22×C4—1884no—
418C22×C4—1884no—
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—21no—
429C4⋊C4—22no—
430C4⋊C4—472no—
431C4⋊C4—474no—
432C4⋊C4—944no—
433C4⋊C4—944no—
434C4⋊C4—944no—
435C4⋊C4—948no—
436C4⋊C4—944no—
437C4⋊C4—941no—
438C4⋊C4—942no—
439C4⋊C4C2×C4×D4712yes—
440C4⋊C4#1062535014yes—
441C22⋊C4—24no—
442C22⋊C4—944no—
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—41no—
459C23—41no—
460C23—42no—
461C23—472no—
462C23—472no—
463C23—472no—
464C23—942no—
465C23—942no—
466C23C2×D4×D4711yescharacteristic
467Q8—942no—
468Q8—942no—
469Q8—1882no—
470D4—22no—
471D4—22no—
472D4—42no—
473D4—1888no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—48no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—472no—
484C2×C4—472no—
485C2×C4—472no—
486C2×C4—948no—
487C2×C4—942no—
488C2×C4—941no—
489C2×C4—941no—
490C2×C4—942no—
491C2×C4—942no—
492C2×C4—942no—
493C2×C4—944no—
494C2×C4—942no—
495C2×C4—942no—
496C2×C4—944no—
497C2×C4—1884no—
498C2×C4—18816no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1881no—
502C2×C4—1881no—
503C2×C4—1882no—
504C2×C4C22⋊C4×D4712yes—
505C2×C4C22⋊C4×D4712yes—
506C2×C4C22×C4×D4711yescharacteristic
507C2×C4C2×D4×D4711yescharacteristic
508C2×C4#41×D4712yes—
509C2×C4C2×#1062535012yes—
510C8—42no—
511C8—1882no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—82no—
516C22—82no—
517C22—474no—
518C22—944no—
519C22—942no—
520C22—944no—
521C22—942no—
522C22C2×C22⋊C4×D4711yescharacteristic
523C22#1568963212yes—
524C22AGL(1,ℤ/8)×D4711yescharacteristic
525C22#137×D4711yescharacteristic
526C22#1568991912yes—
527C4—22no—
528C4—22no—
529C4—44no—
530C4—42no—
531C4—942no—
532C4—942no—
533C4—942no—
534C4—1884no—
535C4—1882no—
536C4—1882no—
537C4#1568959111yescharacteristic
538C4C4×#1062535011yescharacteristic
539C4#1568967012yes—
540C2—82no—
541C2—944no—
542C2#2024587411yescharacteristic
543C2#417×D4711yescharacteristic
544C2#2025190211yescharacteristic
545C2#2025352412yes—
546C2#2025355712yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1762C2
socleC23×C47C22×D4
prime radicalC2×#13093995C22
cosocleC22×C4×C47C24
hypercenter#1762D47
hypocenterC47#21391
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#15689178C2semidirect
3. C4×C4⋊C4×D47C2semidirect
4. #20244344C2semidirect
5. #20244319C2semidirect
7. #20243694C2semidirect
8. #20243221C2semidirect
9. #20243219C2semidirect
10. #20243213C2semidirect
11. C47×#1762C2semidirect
13. #15689655C4semidirect
55. #13094015C2×C4semidirect
57. #13094011C2×C4semidirect
64. C4×Dic94C2×C4semidirect
101. C4×D4×C47C2×C4semidirect
277. #1762D47semidirect
319. C47#21391semidirect
344. C4×D4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896552
C2×#156891782
C4×C4⋊C4×D472
#202443442
#202443192
#202437022
#202436942
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C47×#17622
#2139147