Finite Group Databasebeta

← #26221393

Subgroups — #26221393

Order 12032.

1238 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689677C211yesmaximal, characteristic
2C2×#15689178C211yesmaximal, characteristic
3C4×C4⋊C4×D47C211yesmaximal, characteristic
4#20244338C212yesmaximal, complemented
5#20244309C212yesmaximal, complemented
6#20243694C212yesmaximal
7#20243692C212yesmaximal
8#20243221C211yesmaximal, characteristic, complemented
9#20243219C211yesmaximal, characteristic, complemented
10#20243218C211yesmaximal, characteristic, complemented
11C47×#1763C211yesmaximal, characteristic, complemented
12#15689677—24no—
13#15689677C418yescomplemented
14C2×#13094023C2211yescharacteristic
15C2×#13094022C2211yescharacteristic
16C2×#13094014—21no—
17C2×#13094004C2211yescharacteristic
18C2×#13094002—21no—
19C2×#13094000C2211yescharacteristic
20C2×C4×Dic94—21no—
21C2×C4×Dic94C2211yescharacteristic
22C2×#13093981—21no—
23C2×C4⋊C4×D47C2212yes—
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26#15689178—24no—
27C2×Dic47⋊C8C2212yes—
28C2×#13093949C2212yes—
29C2×#13093948C2212yes—
30C2×#13093947C2212yes—
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×Q8×C47C2211yescharacteristic
42C2×#103×C47C2212yes—
43C2×#97×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×Dic94—21no—
47C22×Dic94C2311yescharacteristic
48C22×C4×D47—22no—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53#13094023—22no—
54#13094023C2×C414yescomplemented
55#13094022—22no—
56#13094022C2×C414yescomplemented
57#13094014—28no—
58#13094004—22no—
59#13094004C2×C414yes—
60#13094002—28no—
61#13094000—22no—
62#13094000C2×C414yes—
63C4×Dic94—22no—
64C4×Dic94—28no—
65C4×Dic94C2×C414yescomplemented
66#13093981—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2312yes—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—22no—
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C4⋊C4×D47—24no—
77C4⋊C4×D47—24no—
78C2×(C2×D47)⋊C4—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#13093948—24no—
92#13093947—24no—
93(C2×D47)⋊C8—24no—
94#13093935—22no—
95C47⋊#95—21no—
96C47⋊#95—22no—
97C22×Q8×C47C2311yescharacteristic
98C22×C8×C47C2312yes—
99C4×Q8×C47—22no—
100C4×Q8×C47C2×C414yescomplemented
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2312yes—
104C2×C42×C47—21no—
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107#103×C47—24no—
108#97×C47—22no—
109#95×C47—21no—
110C23×D47C2×D411yescharacteristic
111C2×Q8×C47—21no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47C22×C412yes—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—22no—
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2411yescharacteristic
121C22×Dic47—22no—
122C22×Dic47—21no—
123C22×Dic47—21no—
124C22×Dic47C2×D411yescharacteristic
125C22×Dic47C2×D411yescharacteristic
126C22×Dic47C2×D411yescharacteristic
127C2×C4×D47—24no—
128C2×C4×D47—22no—
129C2×C4×D47—22no—
130C2×C4×D47—22no—
131C2×C4×D47—22no—
132C2×C4×D47—28no—
133C2×C4×D47—44no—
134C2×C4×D47—44no—
135C2×C4×D47C22⋊C414yes—
136C2×C4×D47C22⋊C414yes—
137C2×C4×D47C22×C412yes—
138C2×C4×D47C2×D412yes—
139C2×Dic94—22no—
140C2×Dic94—22no—
141C2×Dic94—21no—
142C2×Dic94—24no—
143C2×Dic94—42no—
144C2×Dic94—42no—
145C2×Dic94C22×C412yes—
146C2×C8×C47—22no—
147C2×C8×C47—22no—
148C2×C8×C47—22no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—22no—
152C4⋊C4×C47C22×C412yes—
153C4⋊C4×C47#4114yes—
154C42×C47—24no—
155C42×C47—42no—
156C42×C47C22×C412yes—
157C42×C47C2×D412yes—
158(C2×D47)⋊C4—24no—
159(C2×D47)⋊C4—24no—
160C4⋊Dic47—21no—
161C4⋊Dic47—22no—
162C4⋊Dic47—24no—
163C4⋊Dic47C22×C412yes—
164C4⋊Dic47#4114yes—
165Dic47⋊C4—28no—
166Dic47⋊C4—24no—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169Dic47⋊C4—24no—
170Dic47⋊C4—24no—
171C4×Dic47—24no—
172C4×Dic47—42no—
173C4×Dic47—44no—
174C4×Dic47C22⋊C414yes—
175C4×Dic47C22⋊C414yes—
176C4×Dic47C22×C412yes—
177C4×Dic47C2×D412yes—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C23×C47C22×D411yescharacteristic
182C22×D47—22no—
183C22×D47—21no—
184C22×D47—21no—
185C22×D47C2×C22⋊C412yes—
186C22×D47#13714yes—
187Q8×C47—22no—
188Q8×C47—22no—
189Q8×C47—42no—
190C2×C4×C47—21no—
191C2×C4×C47—22no—
192C2×C4×C47—21no—
193C2×C4×C47—22no—
194C2×C4×C47—22no—
195C2×C4×C47—24no—
196C2×C4×C47—42no—
197C2×C4×C47—42no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—42no—
201C2×C4×C47C2×C22⋊C412yes—
202C2×C4×C47C2×C22⋊C412yes—
203C2×C4×C47C23×C411yescharacteristic
204C2×C4×C47C22×D411yescharacteristic
205C2×C4×C47C2×#4112yes—
206C2×C4×C47C2×#4112yes—
207C2×Dic47—22no—
208C2×Dic47—22no—
209C2×Dic47—22no—
210C2×Dic47—22no—
211C2×Dic47—24no—
212C2×Dic47—42no—
213C2×Dic47—42no—
214C2×Dic47—41no—
215C2×Dic47—41no—
216C2×Dic47—42no—
217C2×Dic47C2×C22⋊C412yes—
218C2×Dic47C2×C22⋊C412yes—
219C2×Dic47C2×C22⋊C412yes—
220C2×Dic47AGL(1,ℤ/8)12yes—
221C2×Dic47#13712yes—
222C4×D47—28no—
223C4×D47—22no—
224C4×D47—22no—
225C4×D47—24no—
226C4×D47—28no—
227C4×D47—416no—
228Dic94—22no—
229Dic94—22no—
230Dic94—42no—
231Dic94—48no—
232C8×C47—42no—
233C47⋊C8—42no—
234#21394—471nomaximal, Sylow
235C22×C47C22×C22⋊C411yescharacteristic
236C22×C47#52512yes—
237C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C47C2×#13712yes—
239C22×C47C2×#13711yescharacteristic
240C2×D47—24no—
241C2×D47—24no—
242C2×D47—22no—
243C2×D47—22no—
244C2×D47#41914yes—
245C4×C47—22no—
246C4×C47—22no—
247C4×C47—44no—
248C4×C47—42no—
249C4×C47—42no—
250C4×C47C4×#4111yescharacteristic
251C4×C47C4×#4111yescharacteristic
252C4×C47#53212yes—
253Dic47—22no—
254Dic47—22no—
255Dic47—44no—
256Dic47—42no—
257Dic47—42no—
258Dic47#41714yes—
259C2×#533—471no—
260C2×#432—471no—
261C2×C4×C4⋊C4—471no—
262#1988—472no—
263#1981—472no—
264#1872—472no—
265#1872—472no—
266#1763—471no—
267#1763—471no—
268#1763—471no—
269#1763D4711yescharacteristic, complemented
270C2×C47#229711yescharacteristic
271C2×C47C2×#41911yescharacteristic
272C2×C47C2×#41711yescharacteristic
273C2×C47#311012yes—
274C2×C47#311312yes—
275D47—24no—
276#533—478no—
277#533—944no—
278C2×C4⋊Q8—471no—
279C2×C4⋊Q8—471no—
280C2×#125—941no—
281C2×C22⋊Q8—471no—
282C2×C22⋊Q8—471no—
283C2×C22⋊Q8—941no—
284C2×C4×Q8—471no—
285C2×C4×Q8—941no—
286C2×C4×Q8C2×D4711yescharacteristic
287C2×#117—941no—
288C22×C4⋊C4—472no—
289C22×C42—471no—
290C22×C42—941no—
291#432—944no—
292C2×C4⋊C8—472no—
293C2×#103—472no—
294C2×#103—472no—
295C2×#103—472no—
296C2×#103C2×D4712yes—
297C2×C22⋊C8—472no—
298C2×#97—471no—
299C2×#97C2×D4711yescharacteristic
300C4×C4⋊C4—472no—
301C4×C4⋊C4—471no—
302C4×C4⋊C4—941no—
303C4×C4⋊C4—941no—
304C4×C4⋊C4C2×D4711yescharacteristic
305C2×#95—941no—
306#340—472no—
307#340—472no—
308#340—472no—
309#340C2×D4712yes—
310C47#2139411yescharacteristic, complemented, Sylow
311C22×Q8—471no—
312C22×Q8—941no—
313C22×Q8C22×D4711yescharacteristic
314C23×C4—471no—
315C23×C4—471no—
316C23×C4—471no—
317C23×C4—942no—
318C22×C8—472no—
319C22×C8C22×D4712yes—
320C4⋊Q8—474no—
321C4⋊Q8—474no—
322C4⋊Q8—942no—
323C4⋊Q8—942no—
324#125—948no—
325C22⋊Q8—474no—
326C22⋊Q8—474no—
327C22⋊Q8—942no—
328C22⋊Q8—948no—
329C22⋊Q8—942no—
330C4×Q8—22no—
331C4×Q8—474no—
332C4×Q8—948no—
333C4×Q8—942no—
334C4×Q8C4×D4714yescomplemented
335#117—948no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—472no—
338C2×C4⋊C4—471no—
339C2×C4⋊C4—944no—
340C2×C4⋊C4—942no—
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⋊C4—944no—
348C2×C4⋊C4C22×D4711yescharacteristic
349C2×C4⋊C4C22×D4712yes—
350C2×C22⋊C4—941no—
351C2×C22⋊C4—941no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—471no—
355C2×C42—471no—
356C2×C42—471no—
357C2×C42—474no—
358C2×C42—474no—
359C2×C42—941no—
360C2×C42—942no—
361C2×C42—941no—
362C2×C42—1884no—
363C2×C42C22×D4711yescharacteristic
364C4⋊C8—944no—
365#103—24no—
366#103—944no—
367#103—944no—
368#103—944no—
369C22⋊C8—944no—
370#97—22no—
371#97—942no—
372#95—21no—
373#95—942no—
374#95—941no—
375C24—471no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—472no—
380C2×Q8—944no—
381C2×Q8—941no—
382C2×Q8—942no—
383C2×Q8—942no—
384C2×Q8—1882no—
385C2×Q8—1882no—
386C2×Q8C2×C4×D4712yes—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—471no—
391C22×C4—471no—
392C22×C4—471no—
393C22×C4—474no—
394C22×C4—474no—
395C22×C4—472no—
396C22×C4—472no—
397C22×C4—942no—
398C22×C4—942no—
399C22×C4—944no—
400C22×C4—948no—
401C22×C4—942no—
402C22×C4—942no—
403C22×C4—942no—
404C22×C4—941no—
405C22×C4—941no—
406C22×C4—1884no—
407C22×C4—1884no—
408C22×C4D4×D4711yescharacteristic
409C22×C4D4×D4711yescharacteristic
410C22×C4C23×D4711yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—942no—
415C2×C8—942no—
416C2×C8—942no—
417C4⋊C4—24no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—472no—
421C4⋊C4—474no—
422C4⋊C4—944no—
423C4⋊C4—944no—
424C4⋊C4—944no—
425C4⋊C4—944no—
426C4⋊C4—944no—
427C4⋊C4—948no—
428C4⋊C4—944no—
429C4⋊C4—941no—
430C4⋊C4—942no—
431C4⋊C4C2×C4×D4712yes—
432C4⋊C4#1062535014yes—
433C22⋊C4—944no—
434C22⋊C4—944no—
435C42—24no—
436C42—42no—
437C42—474no—
438C42—474no—
439C42—472no—
440C42—472no—
441C42—944no—
442C42—1882no—
443C42—1884no—
444C42C2×C4×D4712yes—
445C42D4×D4712yes—
446C23—472no—
447C23—474no—
448C23—942no—
449C23—941no—
450C23—941no—
451C23C2×D4×D4711yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—42no—
455Q8—942no—
456Q8—942no—
457Q8—1888no—
458Q8—1882no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—472no—
471C2×C4—472no—
472C2×C4—472no—
473C2×C4—472no—
474C2×C4—472no—
475C2×C4—948no—
476C2×C4—948no—
477C2×C4—942no—
478C2×C4—942no—
479C2×C4—942no—
480C2×C4—942no—
481C2×C4—944no—
482C2×C4—942no—
483C2×C4—942no—
484C2×C4—944no—
485C2×C4—18816no—
486C2×C4—1882no—
487C2×C4—1882no—
488C2×C4—1881no—
489C2×C4—1882no—
490C2×C4—1881no—
491C2×C4C22⋊C4×D4712yes—
492C2×C4C22⋊C4×D4712yes—
493C2×C4C22×C4×D4711yescharacteristic
494C2×C4C2×D4×D4711yescharacteristic
495C2×C4#41×D4712yes—
496C2×C4C2×#1062535012yes—
497C8—42no—
498C8—1882no—
499C22—474no—
500C22—944no—
501C22—944no—
502C22—942no—
503C22—942no—
504C22C2×C22⋊C4×D4711yescharacteristic
505C22#1568963212yes—
506C22AGL(1,ℤ/8)×D4711yescharacteristic
507C22#137×D4712yes—
508C22#137×D4711yescharacteristic
509C4—22no—
510C4—22no—
511C4—44no—
512C4—42no—
513C4—42no—
514C4—474no—
515C4—942no—
516C4—942no—
517C4—1884no—
518C4—1882no—
519C4—1882no—
520C4#1568959111yescharacteristic
521C4C4×#1062535011yescharacteristic
522C4#1568967012yes—
523C2—944no—
524C2#2024587411yescharacteristic
525C2#419×D4711yescharacteristic
526C2#417×D4711yescharacteristic
527C2#2025348512yes—
528C2#2025351612yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1763C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1763D47
hypocenterC47#21394
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. #20244338C2semidirect
5. #20244309C2semidirect
8. #20243221C2semidirect
9. #20243219C2semidirect
10. #20243218C2semidirect
11. C47×#1763C2semidirect
13. #15689677C4semidirect
54. #13094023C2×C4semidirect
56. #13094022C2×C4semidirect
65. C4×Dic94C2×C4semidirect
100. C4×Q8×C47C2×C4semidirect
269. #1763D47semidirect
310. C47#21394semidirect
334. C4×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896772
C2×#156891782
C4×C4⋊C4×D472
#202443382
#202443092
#202436942
#202436922
#202432212
#202432192
#202432182
C47×#17632
#2139447