Finite Group Databasebeta

← #26221532

Subgroups — #26221532

Order 12032.

1174 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689660C211yesmaximal, characteristic
2C2×#15689075C211yesmaximal, characteristic, complemented
3#20245784C211yesmaximal, characteristic, complemented
4#20244351C212yesmaximal, complemented
5#20244196C212yesmaximal, complemented
6#20243825C212yesmaximal
7#20243818C212yesmaximal, complemented
8#20243218C211yesmaximal, characteristic, complemented
9#20243216C211yesmaximal, characteristic, complemented
10#20243214C211yesmaximal, characteristic, complemented
11C47×#1763C211yesmaximal, characteristic, complemented
12C22×#10625324C2212yes—
13#15689660—24no—
14#15689660C418yescomplemented
15C2×#13094019—21no—
16C2×#13094015C2211yescharacteristic
17C2×#13094013C2211yescharacteristic
18C2×#13094009—21no—
19C2×#13094000C2211yescharacteristic
20C2×#13093999C2211yescharacteristic
21C2×C4×D188—21no—
22C2×C4×D188C2211yescharacteristic
23C2×#13093981—21no—
24C2×C42×D47C2211yescharacteristic
25C2×#13093948C2212yes—
26C2×#13093946C2212yes—
27C2×#13093944C2212yes—
28#15689075—24no—
29C2×#13093935C2211yescharacteristic
30C2×C8×Dic47C2212yes—
31#15688990—21no—
32#15688988C2212yes—
33#15688985—21no—
34#15688980C2212yes—
35C4×C4⋊Dic47C2211yescharacteristic
36C4×(C2×D47)⋊C4—21no—
37#15688969—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×#103×C47C2212yes—
43C2×#97×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×D188—21no—
47C22×D188C2311yescharacteristic
48C22×C4×D47—21no—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47C2311yescharacteristic
52C2×#10625324—24no—
53C2×#10625324—24no—
54C2×#10625324—24no—
55C22×C47⋊C8C2312yes—
56#13094019—28no—
57#13094015—22no—
58#13094015C2×C414yes—
59#13094013—22no—
60#13094013C2×C414yes—
61#13094009—28no—
62#13094000—22no—
63#13094000C2×C414yes—
64#13093999—22no—
65#13093999C2×C414yes—
66C4×D188—28no—
67C4×D188—22no—
68C4×D188C2×C414yes—
69#13093981—28no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47C2312yes—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×(C2×D47)⋊C4—22no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47C2×C414yes—
86C42×D47D414yes—
87#13093948—24no—
88#13093946—24no—
89#13093944—24no—
90#13093935—22no—
91C8×Dic47—24no—
92C47⋊#95—21no—
93C47⋊#95—22no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C22×C8×C47C2312yes—
98C4×Q8×C47—22no—
99C4×Q8×C47C2×C414yes—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2312yes—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C42×C47—21no—
104C2×C42×C47—21no—
105C2×C42×C47C2311yescharacteristic
106#103×C47—24no—
107#97×C47—22no—
108#95×C47—21no—
109C23×D47—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—21no—
122C22×Dic47—22no—
123C22×Dic47C2×D411yescharacteristic
124C22×Dic47C2×D411yescharacteristic
125C22×Dic47C2×D411yescharacteristic
126C2×D188—22no—
127C2×D188—22no—
128C2×D188—24no—
129C2×D188—21no—
130C2×D188—42no—
131C2×D188—42no—
132C2×D188C22×C412yes—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—24no—
136C2×C4×D47—41no—
137C2×C4×D47—41no—
138C2×C4×D47—42no—
139C2×C4×D47—44no—
140C2×C4×D47C22⋊C414yes—
141C2×C4×D47C22⋊C414yes—
142C2×C4×D47C22×C412yes—
143C2×C4×D47C2×D412yes—
144C2×C8×C47—22no—
145C2×C8×C47—22no—
146C2×C8×C47#4114yes—
147C4⋊C4×C47—24no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47C22×C412yes—
152C42×C47—24no—
153C42×C47—42no—
154C42×C47C22×C412yes—
155C42×C47C2×D412yes—
156(C2×D47)⋊C4—24no—
157(C2×D47)⋊C4—24no—
158(C2×D47)⋊C4—24no—
159(C2×D47)⋊C4—24no—
160(C2×D47)⋊C4—24no—
161(C2×D47)⋊C4—44no—
162C4⋊Dic47—21no—
163C4⋊Dic47—24no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47C22×C412yes—
167Dic47⋊C4—24no—
168Dic47⋊C4—24no—
169C4×Dic47—42no—
170C4×Dic47C22⋊C414yes—
171C4×Dic47C22⋊C414yes—
172C4×Dic47C22×C412yes—
173C4×Dic47C2×D412yes—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8#4114yes—
177#10625324—216no—
178C23×C47C22×D411yescharacteristic
179C22×D47—22no—
180C22×D47—22no—
181C22×D47—21no—
182C22×D47—21no—
183C22×D47—22no—
184C22×D47—42no—
185C22×D47—42no—
186C22×D47—41no—
187C22×D47—41no—
188C22×D47C2×C22⋊C412yes—
189Q8×C47—22no—
190Q8×C47—22no—
191Q8×C47—42no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—22no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—42no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—42no—
203C2×C4×C47C2×C22⋊C412yes—
204C2×C4×C47C2×C22⋊C412yes—
205C2×C4×C47C23×C411yescharacteristic
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C2×#4112yes—
208C2×C4×C47C2×#4112yes—
209C2×Dic47—22no—
210C2×Dic47—21no—
211C2×Dic47—21no—
212C2×Dic47—22no—
213C2×Dic47—22no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47C2×C22⋊C412yes—
219C2×Dic47C2×C22⋊C412yes—
220C2×Dic47C2×C22⋊C412yes—
221D188—22no—
222D188—22no—
223D188—48no—
224D188—42no—
225C4×D47—22no—
226C4×D47—22no—
227C4×D47—24no—
228C4×D47—44no—
229C4×D47—44no—
230C4×D47—48no—
231C8×C47—24no—
232C47⋊C8—24no—
233#21411—471nomaximal, Sylow
234C22×C47C22×C22⋊C411yescharacteristic
235C22×C47#53212yes—
236C22×C47#57511yescharacteristic
237C22×C47#57511yescharacteristic
238C22×C47#57512yes—
239C2×D47—22no—
240C2×D47—22no—
241C2×D47—22no—
242C2×D47—42no—
243C2×D47—42no—
244C2×D47—42no—
245C2×D47—42no—
246C2×D47—42no—
247C2×D47—82no—
248C2×D47—82no—
249C4×C47—22no—
250C4×C47—22no—
251C4×C47—42no—
252C4×C47—42no—
253C4×C47—82no—
254C4×C47C4×#4111yescharacteristic
255C4×C47C4×#4111yescharacteristic
256C4×C47#52512yes—
257Dic47—22no—
258Dic47—22no—
259Dic47—22no—
260Dic47—42no—
261Dic47—82no—
262C2×#529—471no—
263C2×C4×M16—471no—
264#2283—471no—
265#1991—472no—
266#1958—472no—
267#1895—472no—
268#1894—472no—
269#1763—471no—
270#1763D4711yescharacteristic, complemented
271#1762—471no—
272#1762—471no—
273C2×C47#229711yescharacteristic
274C2×C47#289611yescharacteristic
275C2×C47#289611yescharacteristic
276C2×C47#318312yes—
277C2×C47#318312yes—
278D47—42no—
279D47—82no—
280C22×M16—472no—
281#529—478no—
282#529—944no—
283C2×#126—941no—
284C2×#125—471no—
285C2×#124—471no—
286C2×#123—941no—
287C2×C22⋊Q8—471no—
288C2×#121—471no—
289C2×C4×Q8C2×D4711yescharacteristic
290C2×C4×D4—471no—
291C2×C4×D4—941no—
292C2×#117—941no—
293C22×C42—471no—
294C2×#103—472no—
295C2×#103C2×D4712yes—
296C2×#102—472no—
297C2×#102—472no—
298C4×M16—944no—
299C2×#97—471no—
300C2×#97C2×D4711yescharacteristic
301C2×C4×C8—472no—
302#383—941no—
303#382—472no—
304#381—941no—
305#380—472no—
306C4×C4⋊C4—471no—
307C4×C4⋊C4C2×D4711yescharacteristic
308C4×C22⋊C4—941no—
309#376—941no—
310#340—472no—
311#340—472no—
312#340—472no—
313#340C2×D4712yes—
314C47#2141111yescharacteristic, complemented, Sylow
315C22×Q8C22×D4711yescharacteristic
316C22×D4—471no—
317C22×D4—941no—
318C23×C4—471no—
319C23×C4—471no—
320C23×C4—471no—
321C23×C4—941no—
322C2×M16—944no—
323C2×M16—944no—
324C2×M16—944no—
325C22×C8—472no—
326C22×C8C22×D4712yes—
327#126—948no—
328#125—474no—
329#125—942no—
330#124—474no—
331#124—942no—
332#123—948no—
333C22⋊Q8—474no—
334C22⋊Q8—942no—
335#121—474no—
336#121—942no—
337C4×Q8—22no—
338C4×Q8C4×D4714yes—
339C4×D4—474no—
340C4×D4—948no—
341C4×D4—942no—
342#117—948no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—471no—
345C2×C4⋊C4—472no—
346C2×C4⋊C4—941no—
347C2×C4⋊C4—941no—
348C2×C4⋊C4—941no—
349C2×C4⋊C4C22×D4712yes—
350C2×C4⋊C4C22×D4711yescharacteristic
351C2×C22⋊C4—941no—
352C2×C22⋊C4—941no—
353C2×C22⋊C4—941no—
354C2×C22⋊C4—941no—
355C2×C22⋊C4—941no—
356C2×C22⋊C4—942no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—471no—
360C2×C42—471no—
361C2×C42—471no—
362C2×C42—474no—
363C2×C42—474no—
364C2×C42—941no—
365C2×C42C22×D4711yescharacteristic
366#103—24no—
367#103—944no—
368#102—944no—
369#102—944no—
370#97—22no—
371#97—942no—
372C4×C8—944no—
373#95—21no—
374#95—942no—
375#95—942no—
376#95—942no—
377#95—941no—
378C24—471no—
379C24—941no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8C2×C4×D4712yes—
384C2×D4—472no—
385C2×D4—944no—
386C2×D4—941no—
387C2×D4—942no—
388C2×D4—942no—
389C2×D4—1882no—
390C2×D4—1882no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—471no—
395C22×C4—471no—
396C22×C4—471no—
397C22×C4—472no—
398C22×C4—472no—
399C22×C4—474no—
400C22×C4—474no—
401C22×C4—942no—
402C22×C4—942no—
403C22×C4—941no—
404C22×C4—944no—
405C22×C4—942no—
406C22×C4—1881no—
407C22×C4—1881no—
408C22×C4—1882no—
409C22×C4—1884no—
410C22×C4D4×D4711yescharacteristic
411C22×C4D4×D4711yescharacteristic
412C22×C4C23×D4711yescharacteristic
413M16—9416no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—474no—
417C2×C8—942no—
418C2×C8—942no—
419C2×C8#1062535014yes—
420C4⋊C4—24no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—472no—
425C4⋊C4—944no—
426C4⋊C4—944no—
427C4⋊C4—944no—
428C4⋊C4—941no—
429C4⋊C4—942no—
430C4⋊C4—942no—
431C4⋊C4C2×C4×D4712yes—
432C22⋊C4—944no—
433C22⋊C4—944no—
434C22⋊C4—944no—
435C22⋊C4—944no—
436C22⋊C4—944no—
437C22⋊C4—1884no—
438C42—24no—
439C42—42no—
440C42—474no—
441C42—474no—
442C42—472no—
443C42—472no—
444C42—1882no—
445C42C2×C4×D4712yes—
446C42D4×D4712yes—
447C23—472no—
448C23—941no—
449C23—942no—
450C23—941no—
451C23—942no—
452C23—942no—
453C23—1881no—
454C23—1882no—
455C23—1881no—
456C23—1882no—
457C23C2×D4×D4711yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—42no—
461D4—942no—
462D4—942no—
463D4—1888no—
464D4—1882no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—472no—
477C2×C4—472no—
478C2×C4—472no—
479C2×C4—941no—
480C2×C4—941no—
481C2×C4—942no—
482C2×C4—942no—
483C2×C4—942no—
484C2×C4—942no—
485C2×C4—944no—
486C2×C4—942no—
487C2×C4—1884no—
488C2×C4—1884no—
489C2×C4—1888no—
490C2×C4—1882no—
491C2×C4—1882no—
492C2×C4—1882no—
493C2×C4—1882no—
494C2×C4C22⋊C4×D4712yes—
495C2×C4C22⋊C4×D4712yes—
496C2×C4C22×C4×D4711yescharacteristic
497C2×C4C2×D4×D4711yescharacteristic
498C2×C4#41×D4712yes—
499C2×C4C2×#1062535012yes—
500C8—24no—
501C8—944no—
502C22—942no—
503C22—942no—
504C22—942no—
505C22—3762no—
506C22—3762no—
507C22—1882no—
508C22—1882no—
509C22—1882no—
510C22—1882no—
511C22—1882no—
512C22C2×C22⋊C4×D4711yescharacteristic
513C22#1568967012yes—
514C22#1568991911yescharacteristic
515C22#1568992211yescharacteristic
516C22#1568992212yes—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—942no—
523C4—942no—
524C4—942no—
525C4—3762no—
526C4—1882no—
527C4#1568959111yescharacteristic
528C4C4×#1062535011yescharacteristic
529C4#1568963212yes—
530C2—3762no—
531C2—1882no—
532C2#2024587411yescharacteristic
533C2#2025190411yescharacteristic
534C2#2025190611yescharacteristic
535C2#2025411312yes—
536C2#2025412012yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1763C2
socleC23×C47C22×D4
prime radicalC2×#13093999C22
cosocleC22×C4×C47C24
hypercenter#1763D47
hypocenterC47#21411
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
2. C2×#15689075C2semidirect
3. #20245784C2semidirect
4. #20244351C2semidirect
5. #20244196C2semidirect
7. #20243818C2semidirect
8. #20243218C2semidirect
9. #20243216C2semidirect
10. #20243214C2semidirect
11. C47×#1763C2semidirect
14. #15689660C4semidirect
270. #1763D47semidirect
314. C47#21411semidirect

Maximal subgroups

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

subgroupindex
C2×#156896602
C2×#156890752
#202457842
#202443512
#202441962
#202438252
#202438182
#202432182
#202432162
#202432142
C47×#17632
#2141147