Finite Group Databasebeta

← #26231713

Subgroups — #26231713

Order 12032.

1530 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689788C211yesmaximal, characteristic
2C2×#15689188C211yesmaximal, characteristic
3C2×#103×D47C212yesmaximal, complemented
4C4⋊C4×D188C211yesmaximal, characteristic
5#20244191C211yesmaximal, characteristic, complemented
6#20243808C212yesmaximal, complemented
7#20243692C212yesmaximal, complemented
8#20243394C211yesmaximal, characteristic, complemented
9#20243321C212yesmaximal, complemented
10C4×#13093947C211yesmaximal, characteristic, complemented
11C47×#1981C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2212yes—
13C22×C8×D47C2212yes—
14#15689788—24no—
15#15689788C418yescomplemented
16C2×#13094023—21no—
17C2×#13094002—22no—
18C2×#13094000C2212yes—
19C2×C4×Dic94C2211yescharacteristic
20C2×C4×D188—21no—
21C2×C4×D188C2211yescharacteristic
22C2×#13093986—21no—
23C2×C4⋊C4×D47C2212yes—
24#15689188—24no—
25C2×#13093949C2212yes—
26C2×#13093948C2212yes—
27C2×#13093948C2212yes—
28C2×#13093947C2211yescharacteristic
29C2×#13093947C2211yescharacteristic
30#103×D47—28no—
31C2×(C2×D47)⋊C8C2212yes—
32C2×C4×C47⋊C8C2211yescharacteristic
33#15689021—21no—
34#15688997—22no—
35#15688993C2211yescharacteristic
36#15688980C2212yes—
37#15688886C2212yes—
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40C2×C4⋊Q8×C47C2211yescharacteristic
41C2×C4⋊C8×C47C2211yescharacteristic
42C2×#103×C47C2212yes—
43C2×#103×C47C2212yes—
44C4×C4⋊C4×C47C2211yescharacteristic
45C22×Dic94—22no—
46C22×Dic94C2311yescharacteristic
47C2×Q8×D47—24no—
48C2×Q8×D47—24no—
49C2×Q8×D47—24no—
50C2×Q8×D47—42no—
51C2×Q8×D47—42no—
52C2×Q8×D47C2×C418yescomplemented
53C22×D188D411yescharacteristic
54C22×C4×D47—22no—
55C22×C4×D47—22no—
56C22×C4×D47C2312yes—
57C22×C47⋊C8C2311yescharacteristic
58C22×C47⋊C8C2311yescharacteristic
59C2×C8×D47—24no—
60C2×C8×D47—24no—
61C2×C8×D47—42no—
62#13094023—28no—
63#13094002—216no—
64#13094000—24no—
65#13094000C2×C418yescomplemented
66C4×Dic94—22no—
67C4×Dic94C2×C414yescomplemented
68C4×D188—44no—
69C4×D188C2×C414yes—
70C4×D188D414yes—
71#13093986—28no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—21no—
74C2×C4⋊Dic47D411yescharacteristic
75C2×C4⋊Dic47C2312yes—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—22no—
79C2×Dic47⋊C4—22no—
80C4⋊C4×D47—24no—
81C4⋊C4×D47—42no—
82C2×(C2×D47)⋊C4—22no—
83C2×(C2×D47)⋊C4—22no—
84C2×(C2×D47)⋊C4D412yes—
85C2×C4×Dic47—21no—
86#13093949—24no—
87#13093948—24no—
88#13093948—42no—
89#13093947—22no—
90#13093947D414yescomplemented
91(C2×D47)⋊C8—24no—
92C4×C47⋊C8—22no—
93C47⋊#95—22no—
94C22×Q8×C47C2312yes—
95C22×C8×C47C2312yes—
96C4⋊Q8×C47—22no—
97C4⋊Q8×C47C2×C414yescomplemented
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C42×C47—21no—
103C2×C42×C47C2311yescharacteristic
104C4⋊C8×C47—22no—
105#103×C47—24no—
106#103×C47—42no—
107#95×C47—21no—
108C23×D47C2×D412yes—
109C2×Q8×C47—22no—
110C2×Q8×C47—22no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47C22×C414yes—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47C2×D411yescharacteristic
119C22×C4×C47C2×D411yescharacteristic
120C22×C4×C47C2411yescharacteristic
121C22×Dic47—21no—
122C22×Dic47—21no—
123C22×Dic47—22no—
124C22×Dic47C2×D412yes—
125Q8×D47—216no—
126Q8×D47—48no—
127Q8×D47—48no—
128Q8×D47—48no—
129C2×D188C22⋊C414yes—
130C2×D188SD1614yes—
131C2×D188Q1614yes—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—22no—
136C2×C4×D47—28no—
137C2×C4×D47—44no—
138C2×C4×D47—44no—
139C2×C4×D47—44no—
140C2×C4×D47—44no—
141C2×C4×D47—82no—
142C2×C4×D47C22×C414yes—
143C2×C4×D47C2×D414yes—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—21no—
147C2×Dic94—28no—
148C2×Dic94—44no—
149C2×Dic94—44no—
150C2×Dic94C22×C412yes—
151C2×Dic94C2×D412yes—
152C2×Dic94#4112yes—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C2×C8×C47—22no—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—24no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—21no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47C2×D412yes—
162C42×C47—42no—
163C42×C47C22×C412yes—
164C42×C47C2×D412yes—
165(C2×D47)⋊C4—28no—
166(C2×D47)⋊C4—44no—
167(C2×D47)⋊C4C22⋊C418yes—
168C4⋊Dic47—21no—
169C4⋊Dic47—22no—
170C4⋊Dic47—22no—
171C4⋊Dic47—42no—
172C4⋊Dic47—81no—
173C4⋊Dic47C22⋊C414yes—
174C4⋊Dic47C22×C412yes—
175Dic47⋊C4—24no—
176Dic47⋊C4—28no—
177Dic47⋊C4—44no—
178C4×Dic47—24no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8C2×D412yes—
185C8×D47—48no—
186C23×C47C22×D411yescharacteristic
187C22×D47—22no—
188C22×D47—22no—
189C22×D47—22no—
190C22×D47—22no—
191C22×D47C2×C22⋊C414yes—
192C22×D47C2×SD1614yes—
193C22×D47C2×Q1614yes—
194Q8×C47—24no—
195Q8×C47—24no—
196Q8×C47—42no—
197Q8×C47—42no—
198C2×C4×C47—21no—
199C2×C4×C47—21no—
200C2×C4×C47—24no—
201C2×C4×C47—22no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—41no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47C2×C22⋊C412yes—
210C2×C4×C47C2×C22⋊C412yes—
211C2×C4×C47C2×SD1612yes—
212C2×C4×C47C2×Q1612yes—
213C2×C4×C47C23×C411yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×C4×C47H(5,𝔽2)11yescharacteristic
218C2×C4×C47H(5,𝔽2)11yescharacteristic
219C2×Dic47—22no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—22no—
223C2×Dic47—22no—
224C2×Dic47—22no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47C2×C22⋊C414yes—
231D188#103116yescomplemented
232C4×D47—24no—
233C4×D47—24no—
234C4×D47—24no—
235C4×D47—24no—
236C4×D47—416no—
237C4×D47—88no—
238Dic94—22no—
239Dic94—22no—
240Dic94—22no—
241Dic94—416no—
242Dic94C4×D414yescomplemented
243C8×C47—42no—
244C47⋊C8—22no—
245C47⋊C8—41no—
246D4×#103—471nomaximal, Sylow
247C22×C47C22×C22⋊C411yescharacteristic
248C22×C47D4211yescharacteristic
249C22×C47#54611yescharacteristic
250C22×C47C22×SD1611yescharacteristic
251C22×C47C22×Q1611yescharacteristic
252C22×C47#57711yescharacteristic
253C22×C47#57811yescharacteristic
254C2×D47—24no—
255C2×D47—24no—
256C2×D47—24no—
257C2×D47—24no—
258C2×D47—24no—
259C2×D47—24no—
260C2×D47C2×#10318yescomplemented
261C4×C47—22no—
262C4×C47—44no—
263C4×C47—42no—
264C4×C47—81no—
265C4×C47C2×#10314yescomplemented
266C4×C47C2×C4×D411yescharacteristic
267C4×C47#51811yescharacteristic
268C4×C47D4211yescharacteristic
269C4×C47#54611yescharacteristic
270Dic47—24no—
271Dic47—24no—
272Dic47—42no—
273Dic47—82no—
274C2×D4×Q8—471no—
275C2×C8×D4—471no—
276C22×#103—472no—
277D4×C4⋊C4—471no—
278#1981D4711yescharacteristic, complemented
279#1956—471no—
280#1892—472no—
281#1872—472no—
282#1803—471no—
283#1787—472no—
284C4×#103—471no—
285C2×C47D4×C22⋊C411yescharacteristic
286C2×C47C22×#10311yescharacteristic
287C2×C47#289811yescharacteristic
288C2×C47D4×SD1611yescharacteristic
289C2×C47D4×Q1611yescharacteristic
290C2×C47#329811yescharacteristic
291C2×C47#329911yescharacteristic
292D47—28no—
293C23×Q8—472no—
294C23×C8—472no—
295D4×Q8—478no—
296D4×Q8—944no—
297C2×C4⋊Q8—941no—
298C2×C4⋊Q8C2×D4711yescharacteristic
299C2×C22⋊Q8—472no—
300C2×C22⋊Q8—942no—
301C2×C4×Q8—471no—
302C2×C4×D4—471no—
303C2×C4×D4—941no—
304C2×C4×D4—941no—
305C22×C4⋊C4—472no—
306C8×D4—944no—
307C2×C4⋊C8C2×D4711yescharacteristic
308C2×#103—471no—
309C2×#103—472no—
310C2×#103—472no—
311C2×#103—471no—
312C2×#103—472no—
313C2×#103—948no—
314C2×#103C2×D4712yes—
315C2×#103C2×D4712yes—
316C2×C22⋊C8—472no—
317C2×C4×C8—471no—
318#389—941no—
319#385—942no—
320#384—471no—
321#380—472no—
322C4×C4⋊C4C2×D4711yescharacteristic
323#340—472no—
324#340—471no—
325#340—471no—
326C47D4×#10311yescharacteristic, complemented, Sylow
327C22×Q8—478no—
328C22×Q8—471no—
329C22×Q8—942no—
330C22×Q8—944no—
331C22×Q8—944no—
332C22×Q8—944no—
333C22×Q8—1882no—
334C22×Q8—1882no—
335C22×Q8C22×D4712yes—
336C22×D4—471no—
337C23×C4—472no—
338C23×C4—942no—
339C23×C4—942no—
340C22×C8—471no—
341C22×C8—471no—
342C22×C8—944no—
343C22×C8—944no—
344C22×C8—1882no—
345C22×C8C22×D4712yes—
346C4⋊Q8—22no—
347C4⋊Q8—948no—
348C4⋊Q8C4×D4714yescomplemented
349C22⋊Q8—478no—
350C22⋊Q8—9416no—
351C22⋊Q8—944no—
352C4×Q8—474no—
353C4×Q8—942no—
354C4×D4—474no—
355C4×D4—474no—
356C4×D4—948no—
357C4×D4—1884no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—471no—
361C2×C4⋊C4—471no—
362C2×C4⋊C4—472no—
363C2×C4⋊C4—942no—
364C2×C4⋊C4—941no—
365C2×C4⋊C4—944no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—1882no—
370C2×C4⋊C4C22×D4711yescharacteristic
371C2×C4⋊C4C22×D4711yescharacteristic
372C2×C22⋊C4—472no—
373C2×C22⋊C4—942no—
374C2×C22⋊C4—942no—
375C2×C42—21no—
376C2×C42—941no—
377C2×C42C22×D4711yescharacteristic
378C4⋊C8—22no—
379#103—24no—
380#103—42no—
381#103—474no—
382#103—942no—
383#103—944no—
384#103—944no—
385#103—1882no—
386C22⋊C8—944no—
387C4×C8—942no—
388#95—21no—
389#95—942no—
390C24—472no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—472no—
397C2×Q8—472no—
398C2×Q8—472no—
399C2×Q8—948no—
400C2×Q8—941no—
401C2×Q8—941no—
402C2×Q8—941no—
403C2×Q8—9416no—
404C2×Q8—1884no—
405C2×Q8—1884no—
406C2×Q8—1888no—
407C2×Q8—1888no—
408C2×Q8—1888no—
409C2×Q8C2×C4×D4714yes—
410C2×D4—474no—
411C2×D4—474no—
412C2×D4—474no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—472no—
417C22×C4—474no—
418C22×C4—474no—
419C22×C4—948no—
420C22×C4—941no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—941no—
426C22×C4—942no—
427C22×C4—3762no—
428C22×C4—1884no—
429C22×C4—1884no—
430C22×C4—1884no—
431C22×C4—1884no—
432C22×C4D4×D4711yescharacteristic
433C22×C4D4×D4711yescharacteristic
434C22×C4C23×D4711yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—472no—
439C2×C8—941no—
440C2×C8—941no—
441C2×C8—941no—
442C2×C8—941no—
443C2×C8—941no—
444C2×C8—1888no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—474no—
451C4⋊C4—472no—
452C4⋊C4—948no—
453C4⋊C4—944no—
454C4⋊C4—941no—
455C4⋊C4—942no—
456C4⋊C4—942no—
457C4⋊C4—3761no—
458C4⋊C4—1884no—
459C4⋊C4—1882no—
460C4⋊C4D4×D4712yes—
461C22⋊C4—478no—
462C22⋊C4—948no—
463C22⋊C4—1884no—
464C42—42no—
465C42—944no—
466C42C2×C4×D4712yes—
467C42D4×D4712yes—
468C23—474no—
469C23—474no—
470C23—474no—
471C23—942no—
472C23—942no—
473C23—942no—
474C23—942no—
475C23C2×D4×D4711yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—474no—
481Q8—942no—
482Q8—942no—
483Q8—942no—
484Q8—18816no—
485D4—4716no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—474no—
498C2×C4—942no—
499C2×C4—942no—
500C2×C4—942no—
501C2×C4—942no—
502C2×C4—942no—
503C2×C4—944no—
504C2×C4—944no—
505C2×C4—944no—
506C2×C4—944no—
507C2×C4—942no—
508C2×C4—3768no—
509C2×C4—18816no—
510C2×C4—1882no—
511C2×C4—1882no—
512C2×C4—1882no—
513C2×C4—1881no—
514C2×C4—1881no—
515C2×C4C22⋊C4×D4712yes—
516C2×C4C22⋊C4×D4712yes—
517C2×C4SD16×D4712yes—
518C2×C4Q16×D4712yes—
519C2×C4C22×C4×D4711yescharacteristic
520C2×C4C2×D4×D4711yescharacteristic
521C2×C4C2×D4×D4711yescharacteristic
522C2×C4#41×D4711yescharacteristic
523C2×C4#1309406911yescharacteristic
524C2×C4#1309407011yescharacteristic
525C8—42no—
526C8—942no—
527C8—1881no—
528C22—478no—
529C22—944no—
530C22—944no—
531C22—944no—
532C22—944no—
533C22—944no—
534C22—944no—
535C22C2×C22⋊C4×D4711yescharacteristic
536C22#1568975711yescharacteristic
537C22#1568977011yescharacteristic
538C22C2×SD16×D4711yescharacteristic
539C22C2×Q16×D4711yescharacteristic
540C22#1568993211yescharacteristic
541C22#1568993911yescharacteristic
542C4—22no—
543C4—42no—
544C4—44no—
545C4—81no—
546C4—944no—
547C4—944no—
548C4—3762no—
549C4—1882no—
550C4#103×D4714yescomplemented
551C4C4×D4×D4711yescharacteristic
552C4#1568959511yescharacteristic
553C4#1568975811yescharacteristic
554C4#1568976611yescharacteristic
555C2—948no—
556C2#2024620911yescharacteristic
557C2C2×#103×D4711yescharacteristic
558C2#2025192011yescharacteristic
559C2#2025508711yescharacteristic
560C2#2025513511yescharacteristic
561C2#2025525611yescharacteristic
562C2#2025526711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1981C2
socleC23×C47C22×D4
prime radicalC22×D188D4
cosocleC22×C4×C47C24
hypercenter#1981D47
hypocenterC47D4×#103
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. C2×#103×D47C2semidirect
5. #20244191C2semidirect
6. #20243808C2semidirect
7. #20243692C2semidirect
8. #20243394C2semidirect
9. #20243321C2semidirect
10. C4×#13093947C2semidirect
11. C47×#1981C2semidirect
15. #15689788C4semidirect
52. C2×Q8×D47C2×C4semidirect
65. #13094000C2×C4semidirect
67. C4×Dic94C2×C4semidirect
90. #13093947D4semidirect
97. C4⋊Q8×C47C2×C4semidirect
231. D188#103semidirect
242. Dic94C4×D4semidirect
260. C2×D47C2×#103semidirect
265. C4×C47C2×#103semidirect
278. #1981D47semidirect
326. C47D4×#103semidirect
348. C4⋊Q8C4×D47semidirect
550. C4#103×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156897882
C2×#156891882
C2×#103×D472
C4⋊C4×D1882
#202441912
#202438082
#202436922
#202433942
#202433212
C4×#130939472
C47×#19812
D4×#10347