Finite Group Databasebeta

← #26250211

Subgroups — #26250211

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689777C211yesmaximal, characteristic
2C2×#15689408C211yesmaximal, characteristic, complemented
3C2×C4×#10625336C211yesmaximal, characteristic, complemented
4#20244577C212yesmaximal, complemented
5#20244389C212yesmaximal, complemented
6#20244348C211yesmaximal, characteristic, complemented
7#20244183C211yesmaximal, characteristic, complemented
8#20243696C212yesmaximal, complemented
9#20243580C212yesmaximal, complemented
10#20243385C211yesmaximal, characteristic, complemented
11C47×#2538C211yesmaximal, characteristic, complemented
12C22×#10625352—21no—
13C22×#10625336C2211yescharacteristic
14#15689777—24no—
15#15689777—24no—
16C2×#13094024C2211yescharacteristic
17C2×#13094008—22no—
18C2×#13094004—21no—
19C2×#13093998C2212yes—
20C2×C4×D188C2211yescharacteristic
21C22×C4⋊Dic47C2212yes—
22#15689408—24no—
23C4×#10625336—24no—
24C2×#13093955C2212yes—
25C2×#13093955C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×#13093948C2212yes—
28C2×#13093948C2211yescharacteristic
29C2×#13093944C2211yescharacteristic
30C2×#13093944C2212yes—
31C2×C22⋊(C47⋊C8)C2212yes—
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×C4×C47⋊C8C2211yescharacteristic
34#15688885C2212yes—
35#15688885C2212yes—
36#15688885C2212yes—
37C2×C22⋊Q8×C47C2212yes—
38C2×C4×Q8×C47C2211yescharacteristic
39C2×C4×D4×C47C2211yescharacteristic
40C47×#396—22no—
41C47×#390—21no—
42C47×#389—21no—
43C47×#384C2211yescharacteristic
44C47×#380C2212yes—
45C2×#10625352—28no—
46C2×#10625352—44no—
47C2×#10625352—44no—
48C22×Dic94—21no—
49C22×D188C2311yescharacteristic
50C22×C22⋊D47—22no—
51C23×Dic47—22no—
52C22×C4×D47—21no—
53C2×#10625336—22no—
54C2×#10625336—22no—
55C2×#10625336—22no—
56C22×C47⋊C8C2311yescharacteristic
57C22×C47⋊C8C2312yes—
58C22×C47⋊C8C2311yescharacteristic
59#13094024—22no—
60#13094024D414yes—
61#13094008—216no—
62#13094004—28no—
63#13093998—24no—
64#13093998—24no—
65C4×D188—22no—
66C4×D188D414yes—
67C2×C4⋊Dic47—22no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—24no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2312yes—
73C2×(C2×D47)⋊C4—21no—
74C2×(C2×D47)⋊C4—22no—
75#13093955—24no—
76#13093955—22no—
77C4⋊(C47⋊C8)—22no—
78#13093948—24no—
79#13093948—22no—
80#13093944—22no—
81#13093944—42no—
82C22⋊(C47⋊C8)—24no—
83C22⋊(C47⋊C8)—42no—
84C4×C47⋊C8—22no—
85C22×Q8×C47C2311yescharacteristic
86C22×D4×C47D411yescharacteristic
87C23×C4×C47C2312yes—
88C22⋊Q8×C47—24no—
89C22⋊Q8×C47—42no—
90C4×Q8×C47—22no—
91C4×Q8×C47—22no—
92C4×D4×C47—22no—
93C4×D4×C47—22no—
94C2×C4⋊C4×C47—21no—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—21no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C4⋊C4×C47C2312yes—
101C2×C22⋊C4×C47—22no—
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47D412yes—
104C2×C42×C47—21no—
105C2×C42×C47C2311yescharacteristic
106#95×C47—22no—
107C24×C47C2×D412yes—
108C23×D47—21no—
109C2×Q8×C47—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—21no—
114C2×Q8×C47#4112yescomplemented
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47SD1614yescomplemented
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2411yescharacteristic
131C2×C22⋊D47—28no—
132C2×C22⋊D47—44no—
133C2×C22⋊D47—44no—
134C22×Dic47—28no—
135C22×Dic47—21no—
136C22×Dic47—22no—
137C22×Dic47—44no—
138C22×Dic47—44no—
139#10625352—432no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188C2×D412yes—
146C2×C4×D47—24no—
147C2×C4×D47—42no—
148C2×C4×D47—42no—
149C2×Dic94—24no—
150C2×Dic94—42no—
151C2×Dic94—42no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—22no—
156C4⋊C4×C47—22no—
157C4⋊C4×C47—44no—
158C4⋊C4×C47—42no—
159C4⋊C4×C47—42no—
160C4⋊C4×C47—81no—
161C22⋊C4×C47—24no—
162C22⋊C4×C47—44no—
163C22⋊C4×C47—82no—
164C42×C47—42no—
165C42×C47C2×D412yes—
166C42×C47C2×D412yes—
167#10625336—28no—
168(C2×D47)⋊C4—28no—
169(C2×D47)⋊C4—24no—
170C4⋊Dic47—28no—
171C4⋊Dic47—24no—
172C4⋊Dic47—21no—
173C4⋊Dic47—22no—
174C4⋊Dic47C2×D412yes—
175C4⋊Dic47C2×D414yes—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8#4112yescomplemented
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47C2×SD1614yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—22no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—41no—
199C22×D47—41no—
200Q8×C47—22no—
201Q8×C47—22no—
202Q8×C47—41no—
203Q8×C47—41no—
204D4×C47—28no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—41no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—42no—
222C2×C4×C47—41no—
223C2×C4×C47#12012yes—
224C2×C4×C47#12012yes—
225C2×C4×C47C2×SD1612yes—
226C2×C4×C47AGL(1,ℤ/8)12yes—
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47H(5,𝔽2)11yescharacteristic
232C2×C4×C472−1+411yescharacteristic
233C22⋊D47—416no—
234C2×Dic47—24no—
235C2×Dic47—22no—
236C2×Dic47—416no—
237C2×Dic47—42no—
238C2×Dic47—41no—
239C2×Dic47—42no—
240C2×Dic47—41no—
241D188—22no—
242D188—22no—
243D188—41no—
244D188—41no—
245C4×D47—48no—
246Dic94—48no—
247C47⋊C8—22no—
248C47⋊C8—41no—
249C47⋊C8—42no—
250#24120—471nomaximal, Sylow
251C22×C47—24no—
252C22×C47—24no—
253C22×C47—24no—
254C22×C47—42no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47C2×#12011yescharacteristic
259C22×C47#54611yescharacteristic
260C22×C47#54711yescharacteristic
261C22×C47C22×SD1611yescharacteristic
262C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C47#57711yescharacteristic
264C22×C47#57811yescharacteristic
265C2×D47—42no—
266C2×D47—42no—
267C2×D47—42no—
268C2×D47—81no—
269C2×D47—81no—
270C2×D47—81no—
271C2×D47—81no—
272C4×C47—22no—
273C4×C47—42no—
274C4×C47—42no—
275C4×C47—82no—
276C4×C47—81no—
277C4×C47#45014yes—
278C4×C47#52511yescharacteristic
279C4×C47#53711yescharacteristic
280C4×C47#54811yescharacteristic
281C4×C47#55011yescharacteristic
282Dic47—44no—
283Dic47—42no—
284C2×#547—471no—
285C2×#475—471no—
286C2×C4×SD16—471no—
287#2538D4711yescharacteristic, complemented
288#2034—472no—
289#2000—472no—
290#1990—471no—
291#1954—471no—
292#1872—472no—
293#1853—472no—
294#1802—471no—
295C2×C47—44no—
296C2×C47#240711yescharacteristic
297C2×C47C2×#45011yescharacteristic
298C2×C47#301311yescharacteristic
299C2×C47#329811yescharacteristic
300C2×C47#333211yescharacteristic
301C2×C47#333511yescharacteristic
302C2×C47#335311yescharacteristic
303D47—82no—
304C22×#41—941no—
305C22×SD16—471no—
306#547—944no—
307#547—944no—
308C2×C4⋊Q8—471no—
309C2×#123—942no—
310C2×C22⋊Q8—941no—
311C2×C22⋊Q8C2×D4712yes—
312C2×#121—472no—
313C2×C4×Q8C2×D4711yescharacteristic
314C2×C4×D4—471no—
315C2×C4×D4C2×D4711yescharacteristic
316C22×C4⋊C4—472no—
317#475—944no—
318C4×SD16—944no—
319C2×#106—472no—
320C2×#106—471no—
321C2×C4⋊C8—471no—
322C2×#103—472no—
323C2×#103—471no—
324C2×#102—471no—
325C2×#102—472no—
326C2×C22⋊C8—472no—
327C2×C22⋊C8—472no—
328C2×C4×C8—471no—
329#396—22no—
330#390—21no—
331#389—21no—
332#384C2×D4711yescharacteristic
333#380C2×D4712yes—
334#340—472no—
335#340—472no—
336#340—472no—
337C47#2412011yescharacteristic, complemented, Sylow
338C2×#41—948no—
339C2×#41—1884no—
340C2×#41—1884no—
341C22×Q8—941no—
342C22×Q8C22×D4711yescharacteristic
343C22×D4—471no—
344C22×D4—942no—
345C22×D4C22⋊D4711yescharacteristic
346C23×C4—942no—
347C23×C4—941no—
348C23×C4C22×D4712yes—
349C2×SD16—942no—
350C2×SD16—942no—
351C2×SD16—942no—
352C22×C8—471no—
353C22×C8—471no—
354C22×C8—472no—
355C4⋊Q8—474no—
356C4⋊Q8—942no—
357#123—9416no—
358C22⋊Q8—24no—
359C22⋊Q8—42no—
360C22⋊Q8—948no—
361#121—944no—
362#121—944no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—474no—
368C4×D4—942no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—472no—
375C2×C4⋊C4—944no—
376C2×C4⋊C4—944no—
377C2×C4⋊C4—942no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4C22⋊D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C4⋊C4C22×D4712yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—942no—
386C2×C22⋊C4C22⋊D4712yes—
387C2×C42—21no—
388C2×C42C22×D4711yescharacteristic
389#106—942no—
390#106—944no—
391C4⋊C8—942no—
392#103—942no—
393#103—944no—
394#102—942no—
395#102—1882no—
396C22⋊C8—944no—
397C22⋊C8—1882no—
398C4×C8—942no—
399#95—22no—
400C24—941no—
401C24C2×C22⋊D4712yes—
402#41—18832no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—944no—
409C2×Q8—1882no—
410C2×Q8—1882no—
411C2×Q8#1062535212yescomplemented
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—472no—
415C2×D4—948no—
416C2×D4—941no—
417C2×D4—941no—
418C2×D4—941no—
419C2×D4—941no—
420C2×D4—941no—
421C2×D4—1884no—
422C2×D4—1884no—
423C2×D4#1062533614yescomplemented
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—948no—
434C22×C4—941no—
435C22×C4—942no—
436C22×C4—944no—
437C22×C4—1884no—
438C22×C4—1884no—
439C22×C4—1882no—
440C22×C4—1882no—
441C22×C4C2×C22⋊D4712yes—
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C2×C22⋊D4711yescharacteristic
444C22×C4C23×D4711yescharacteristic
445SD16—948no—
446C2×C8—472no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—942no—
451C2×C8—942no—
452C2×C8—941no—
453C2×C8—942no—
454C2×C8—941no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—472no—
465C4⋊C4—474no—
466C4⋊C4—948no—
467C4⋊C4—944no—
468C4⋊C4—941no—
469C4⋊C4—942no—
470C22⋊C4—24no—
471C22⋊C4—44no—
472C22⋊C4—82no—
473C22⋊C4—948no—
474C22⋊C4—944no—
475C42—42no—
476C42D4×D4712yes—
477C42D4×D4712yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—942no—
485C23—1881no—
486C23—1881no—
487C23—1881no—
488C23—1881no—
489C23—1881no—
490C23—1881no—
491C23C2×#1062533614yes—
492C23C22×C22⋊D4711yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497Q8—1888no—
498D4—28no—
499D4—942no—
500D4—942no—
501D4—18816no—
502D4—1881no—
503D4—1881no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—942no—
523C2×C4—944no—
524C2×C4—18816no—
525C2×C4—1881no—
526C2×C4—1882no—
527C2×C4—1881no—
528C2×C4—1882no—
529C2×C4—1888no—
530C2×C4#1309399212yes—
531C2×C4#1309399212yes—
532C2×C4C2×#1062533612yes—
533C2×C4#1309404912yes—
534C2×C4C2×D4×D4711yescharacteristic
535C2×C4C2×D4×D4711yescharacteristic
536C2×C4#41×D4711yescharacteristic
537C2×C4C2×#1062535211yescharacteristic
538C2×C4#1309406911yescharacteristic
539C2×C4#1309407111yescharacteristic
540C8—942no—
541C8—1882no—
542C8—1881no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—3761no—
551C22—3761no—
552C22—3761no—
553C22—3761no—
554C22—1882no—
555C22—1882no—
556C22—1882no—
557C22C2×#1309399211yescharacteristic
558C22#1568976711yescharacteristic
559C22#1568977211yescharacteristic
560C22C22×#1062533611yescharacteristic
561C22C2×#1309404911yescharacteristic
562C22#1568993411yescharacteristic
563C22#1568993811yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—82no—
569C4—1882no—
570C4—1884no—
571C4#1568927314yes—
572C4#1568963311yescharacteristic
573C4#1568970211yescharacteristic
574C4#1568977911yescharacteristic
575C4#1568979111yescharacteristic
576C2—44no—
577C2—3762no—
578C2#2024672911yescharacteristic
579C2C2×#1568927311yescharacteristic
580C2#2025270011yescharacteristic
581C2#2025525311yescharacteristic
582C2#2025571511yescharacteristic
583C2#2025573911yescharacteristic
584C2#2025596611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2538C2
socleC23×C47C22×D4
prime radicalC2×#15689777C2
hypercenter#2538D47
hypocenterC47#24120
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#15689408C2semidirect
3. C2×C4×#10625336C2semidirect
4. #20244577C2semidirect
5. #20244389C2semidirect
6. #20244348C2semidirect
7. #20244183C2semidirect
8. #20243696C2semidirect
9. #20243580C2semidirect
10. #20243385C2semidirect
11. C47×#2538C2semidirect
114. C2×Q8×C47#41semidirect
117. C2×D4×C47SD16semidirect
184. C2×C47⋊C8#41semidirect
287. #2538D47semidirect
337. C47#24120semidirect
411. C2×Q8#10625352semidirect
423. C2×D4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156897772
C2×#156894082
C2×C4×#106253362
#202445772
#202443892
#202443482
#202441832
#202436962
#202435802
#202433852
C47×#25382
#2412047