Finite Group Databasebeta

← #26231733

Subgroups — #26231733

Order 12032.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689777C211yesmaximal, characteristic
2C2×#15689188C211yesmaximal, characteristic, complemented
3C2×#15689116C212yesmaximal, complemented
4#20246339C211yesmaximal, characteristic, complemented
5#20244310C211yesmaximal, characteristic, complemented
6#20244189C211yesmaximal, characteristic, complemented
7#20243810C212yesmaximal, complemented
8#20243699C212yesmaximal, complemented
9#20243325C212yesmaximal, complemented
10C4×#13093945C211yesmaximal, characteristic, complemented
11C47×#1802C211yesmaximal, characteristic, complemented
12C22×#10625352C2212yes—
13C22×C8×D47C2212yes—
14#15689777—24no—
15#15689777C418yescomplemented
16C2×#13094024C2211yescharacteristic
17C2×#13094008—22no—
18C2×#13094004C2212yes—
19C2×#13093998—21no—
20C2×C4×D188C2211yescharacteristic
21C2×#13093982C2212yes—
22C22×C4⋊Dic47—21no—
23#15689188—24no—
24#15689116—28no—
25C2×#13093949C2212yes—
26C2×#13093949C2212yes—
27C2×#13093947C2212yes—
28C2×#13093947C2212yes—
29C2×#13093945C2211yescharacteristic
30C2×#13093945C2211yescharacteristic
31C2×(C2×D47)⋊C8C2212yes—
32C2×C4×C47⋊C8C2211yescharacteristic
33#15689024—21no—
34#15689013—22no—
35#15688980C2212yes—
36C4⋊C4×Dic47—21no—
37C4×C4⋊Dic47C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40C2×C4×D4×C47C2211yescharacteristic
41C2×C4⋊C8×C47C2211yescharacteristic
42C2×#102×C47C2212yes—
43C47×#384C2211yescharacteristic
44C47×#340C2212yes—
45C2×#10625352—24no—
46C2×#10625352—24no—
47C2×#10625352—24no—
48C2×#10625352—42no—
49C2×#10625352—42no—
50C2×#10625352C2×C418yes—
51C22×Dic94C2312yes—
52C22×D188D411yescharacteristic
53C22×C22⋊D47—22no—
54C23×Dic47—22no—
55C22×C4×D47C2312yes—
56C22×C47⋊C8C2311yescharacteristic
57C22×C47⋊C8C2311yescharacteristic
58C2×C8×D47—24no—
59C2×C8×D47—24no—
60C2×C8×D47—42no—
61#13094024—22no—
62#13094024C2×C414yes—
63#13094008—216no—
64#13094004—24no—
65#13094004C2×C418yes—
66#13093998—28no—
67C4×D188C2×C414yes—
68C4×D188D414yes—
69#13093982—24no—
70#13093982—42no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47—22no—
74C2×C4⋊Dic47—28no—
75C2×C4⋊Dic47D411yescharacteristic
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×(C2×D47)⋊C4—22no—
79C2×(C2×D47)⋊C4—22no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4D412yes—
82C2×C4×Dic47—21no—
83C2×C4×Dic47—22no—
84#13093949—24no—
85#13093949—42no—
86#13093947—24no—
87#13093947—42no—
88#13093945—22no—
89#13093945D414yescomplemented
90(C2×D47)⋊C8—24no—
91C4×C47⋊C8—22no—
92C47⋊#95—21no—
93C47⋊#95—22no—
94C22×D4×C47C2311yescharacteristic
95C23×C4×C47—21no—
96C22×C8×C47C2312yes—
97C4×D4×C47—22no—
98C4×D4×C47C2×C414yes—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2312yes—
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47—21no—
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106#102×C47—24no—
107C24×C47—21no—
108C23×D47C2×D412yes—
109C2×D4×C47—21no—
110C2×D4×C47—21no—
111C2×D4×C47—21no—
112C2×D4×C47C22×C412yes—
113C2×D4×C47C2×D412yes—
114C2×D4×C47#4112yes—
115C22×C4×C47—21no—
116C22×C4×C47—24no—
117C22×C4×C47—22no—
118C22×C4×C47—42no—
119C22×C4×C47—42no—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D411yescharacteristic
122C22×C4×C47C2411yescharacteristic
123C2×C22⋊D47—28no—
124C2×C22⋊D47—44no—
125C2×C22⋊D47—44no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47—28no—
129C22×Dic47—21no—
130C22×Dic47—44no—
131C22×Dic47—44no—
132C22×Dic47C2×D412yes—
133#10625352—216no—
134#10625352—48no—
135#10625352—48no—
136#10625352—48no—
137C2×D188—22no—
138C2×D188—22no—
139C2×D188C22⋊C414yes—
140C2×C4×D47—22no—
141C2×C4×D47—22no—
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47C22×C414yes—
145C2×C4×D47C2×D414yes—
146C2×Dic94—22no—
147C2×Dic94—22no—
148C2×Dic94—22no—
149C2×Dic94—22no—
150C2×Dic94—22no—
151C2×Dic94C22×C414yes—
152C2×C8×C47—22no—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—42no—
159C4⋊C4×C47—81no—
160C4⋊C4×C47C22×C412yes—
161C22⋊C4×C47—24no—
162C42×C47C22×C412yes—
163C42×C47C2×D412yes—
164(C2×D47)⋊C4—28no—
165(C2×D47)⋊C4—44no—
166(C2×D47)⋊C4—82no—
167(C2×D47)⋊C4C22⋊C418yes—
168C4⋊Dic47—24no—
169C4⋊Dic47—24no—
170C4⋊Dic47—21no—
171C4⋊Dic47—28no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47C22⋊C414yes—
175C4⋊Dic47C2×D412yes—
176C4×Dic47—44no—
177C4×Dic47—42no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8C2×D412yes—
184C8×D47—48no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—41no—
189C23×C47—41no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47C2×C22⋊C414yes—
200D4×C47—22no—
201D4×C47—22no—
202D4×C47—22no—
203D4×C47C4×D414yes—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—48no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47C2×C22⋊C412yes—
214C2×C4×C47C2×C22⋊C412yes—
215C2×C4×C47#13512yes—
216C2×C4×C47#13512yes—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×D411yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47H(5,𝔽2)11yescharacteristic
222C2×C4×C47H(5,𝔽2)11yescharacteristic
223C22⋊D47—416no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—24no—
227C2×Dic47—22no—
228C2×Dic47—416no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47C2×C22⋊C414yes—
237D188—28no—
238C4×D47—24no—
239C4×D47—24no—
240C4×D47—24no—
241C4×D47—24no—
242Dic94—24no—
243Dic94—24no—
244Dic94—42no—
245Dic94—42no—
246C8×C47—42no—
247C47⋊C8—22no—
248C47⋊C8—41no—
249#22417—471nomaximal, Sylow
250C22×C47—24no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47C22×C22⋊C411yescharacteristic
258C22×C47D4211yescharacteristic
259C22×C47#54611yescharacteristic
260C22×C47C2×#13511yescharacteristic
261C22×C47C2×#13511yescharacteristic
262C22×C47#57711yescharacteristic
263C22×C47#57811yescharacteristic
264C2×D47—24no—
265C2×D47—24no—
266C2×D47—24no—
267C2×D47—42no—
268C2×D47—42no—
269C2×D47—42no—
270C2×D47—42no—
271C4×C47—22no—
272C4×C47—42no—
273C4×C47—82no—
274C4×C47#41614yescomplemented
275C4×C47C2×C4×D411yescharacteristic
276C4×C47#51811yescharacteristic
277C4×C47D4211yescharacteristic
278C4×C47#54611yescharacteristic
279Dic47—24no—
280Dic47—44no—
281Dic47—42no—
282Dic47—81no—
283C2×#547—471no—
284C2×C8×D4—471no—
285C2×#416—472no—
286#2354—471no—
287#1981—471no—
288#1956—471no—
289#1892—472no—
290#1873—472no—
291#1802D4711yescharacteristic, complemented
292#1788—472no—
293C4×#102—471no—
294C2×C47—44no—
295C2×C47D4×C22⋊C411yescharacteristic
296C2×C47C2×#41611yescharacteristic
297C2×C47#289811yescharacteristic
298C2×C47#328311yescharacteristic
299C2×C47#328411yescharacteristic
300C2×C47#330211yescharacteristic
301C2×C47#330311yescharacteristic
302D47—44no—
303C22×#41—472no—
304C23×C8—472no—
305#547—478no—
306#547—944no—
307C2×C4⋊Q8—471no—
308C2×#123—942no—
309C2×C22⋊Q8—472no—
310C2×#121—941no—
311C2×C4×D4—471no—
312C2×C4×D4C2×D4711yescharacteristic
313C2×#117—472no—
314C22×C4⋊C4—941no—
315C8×D4—944no—
316C2×C4⋊C8C2×D4711yescharacteristic
317#416—948no—
318C2×#103—472no—
319C2×#103—472no—
320C2×#103—472no—
321C2×#103—472no—
322C2×#102—471no—
323C2×#102—471no—
324C2×#102C2×D4712yes—
325C2×C22⋊C8—472no—
326C2×C4×C8—471no—
327#390—941no—
328#388—942no—
329#384C2×D4711yescharacteristic
330#380—472no—
331C4×C4⋊C4—471no—
332C4×C4⋊C4—941no—
333#340—471no—
334#340—471no—
335#340C2×D4712yes—
336C47#2241711yescharacteristic, complemented, Sylow
337C2×#41—478no—
338C2×#41—944no—
339C2×#41—944no—
340C2×#41—944no—
341C2×#41—1882no—
342C2×#41—1882no—
343C22×Q8—472no—
344C22×D4—471no—
345C22×D4—942no—
346C22×D4C22×D4711yescharacteristic
347C23×C4—21no—
348C23×C4—472no—
349C23×C4—942no—
350C22×C8—471no—
351C22×C8—471no—
352C22×C8—944no—
353C22×C8—944no—
354C22×C8—1882no—
355C22×C8C22×D4712yes—
356C4⋊Q8—474no—
357C4⋊Q8—942no—
358#123—9416no—
359C22⋊Q8—478no—
360C22⋊Q8—944no—
361#121—948no—
362C4×D4—22no—
363C4×D4—474no—
364C4×D4—474no—
365C4×D4C4×D4714yes—
366#117—944no—
367#117—1882no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—948no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4C22×D4712yes—
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—472no—
381C2×C22⋊C4—942no—
382C2×C22⋊C4—942no—
383C2×C22⋊C4—942no—
384C2×C42—942no—
385C2×C42—941no—
386C2×C42C22×D4711yescharacteristic
387C4⋊C8—22no—
388#103—944no—
389#103—944no—
390#103—1882no—
391#103—1882no—
392#102—24no—
393#102—474no—
394#102—942no—
395C22⋊C8—944no—
396C4×C8—942no—
397#95—942no—
398#95—941no—
399C24—21no—
400C24—472no—
401#41—9416no—
402#41—1888no—
403#41—1888no—
404#41—1888no—
405C2×Q8—474no—
406C2×Q8—942no—
407C2×Q8—942no—
408C2×Q8—942no—
409C2×Q8—942no—
410C2×Q8—942no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—474no—
415C2×D4—948no—
416C2×D4—942no—
417C2×D4—942no—
418C2×D4—1884no—
419C2×D4—1884no—
420C2×D4C2×C4×D4712yes—
421C2×D4D4×D4712yes—
422C2×D4#1062535412yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—472no—
429C22×C4—474no—
430C22×C4—474no—
431C22×C4—948no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—941no—
437C22×C4—942no—
438C22×C4—941no—
439C22×C4—1884no—
440C22×C4—1884no—
441C22×C4D4×D4711yescharacteristic
442C22×C4D4×D4711yescharacteristic
443C22×C4C23×D4711yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—472no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—1888no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—81no—
459C4⋊C4—474no—
460C4⋊C4—472no—
461C4⋊C4—948no—
462C4⋊C4—944no—
463C4⋊C4—944no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4C2×C4×D4712yes—
468C22⋊C4—24no—
469C22⋊C4—478no—
470C22⋊C4—948no—
471C22⋊C4—3762no—
472C22⋊C4—1884no—
473C42—1884no—
474C42—1882no—
475C42C2×C4×D4712yes—
476C42D4×D4712yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—474no—
485C23—942no—
486C23—942no—
487C23—942no—
488C23—942no—
489C23—942no—
490C23—942no—
491C23C2×D4×D4711yescharacteristic
492Q8—944no—
493Q8—944no—
494Q8—1882no—
495Q8—1882no—
496D4—22no—
497D4—22no—
498D4—22no—
499D4—948no—
500D4—18816no—
501D4#1309398614yes—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—474no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—944no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—942no—
519C2×C4—944no—
520C2×C4—18816no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4C22⋊C4×D4712yes—
529C2×C4C22⋊C4×D4712yes—
530C2×C4#1309404312yes—
531C2×C4#1309404512yes—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4C2×#1062535411yescharacteristic
536C2×C4#1309407011yescharacteristic
537C2×C4#1309407011yescharacteristic
538C8—42no—
539C8—942no—
540C8—1881no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—944no—
549C22—944no—
550C22—944no—
551C22—1882no—
552C22—1882no—
553C22—1882no—
554C22—1882no—
555C22C2×C22⋊C4×D4711yescharacteristic
556C22#1568975711yescharacteristic
557C22#1568976411yescharacteristic
558C22C2×#1309404311yescharacteristic
559C22C2×#1309404511yescharacteristic
560C22#1568993311yescharacteristic
561C22#1568993911yescharacteristic
562C4—22no—
563C4—42no—
564C4—82no—
565C4—944no—
566C4—3761no—
567C4—1882no—
568C4—1884no—
569C4#1568911614yescomplemented
570C4C2×#1309398611yescharacteristic
571C4#1568959711yescharacteristic
572C4#1568975711yescharacteristic
573C4#1568976411yescharacteristic
574C2—44no—
575C2—1884no—
576C2#2024620811yescharacteristic
577C2C2×#1568911611yescharacteristic
578C2#2025191911yescharacteristic
579C2#2025510511yescharacteristic
580C2#2025512111yescharacteristic
581C2#2025530211yescharacteristic
582C2#2025531311yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1802C2
socleC23×C47C22×D4
prime radicalC2×#15689777C2
cosocleC22×C4×C47C24
hypercenter#1802D47
hypocenterC47#22417
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#15689188C2semidirect
3. C2×#15689116C2semidirect
4. #20246339C2semidirect
5. #20244310C2semidirect
6. #20244189C2semidirect
7. #20243810C2semidirect
8. #20243699C2semidirect
9. #20243325C2semidirect
10. C4×#13093945C2semidirect
11. C47×#1802C2semidirect
15. #15689777C4semidirect
89. #13093945D4semidirect
274. C4×C47#416semidirect
291. #1802D47semidirect
336. C47#22417semidirect
569. C4#15689116semidirect

Maximal subgroups

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

subgroupindex
C2×#156897772
C2×#156891882
C2×#156891162
#202463392
#202443102
#202441892
#202438102
#202436992
#202433252
C4×#130939452
C47×#18022
#2241747