Finite Group Databasebeta

← #26221346

Subgroups — #26221346

Order 12032.

1290 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×#15689659C211yesmaximal, characteristic
2C2×#15689173C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic, complemented
4#20244342C212yesmaximal, complemented
5#20244317C212yesmaximal, complemented
6#20243705C211yesmaximal, characteristic
7#20243703C211yesmaximal, characteristic
8#20243696C211yesmaximal, characteristic, complemented
9#103×Dic47C212yesmaximal, complemented
10#20243202C212yesmaximal, complemented
11C47×#1872C211yesmaximal, characteristic, complemented
12#15689659—24no—
13#15689659C418yescomplemented
14C2×#13094019—21no—
15C2×#13094015C2211yescharacteristic
16C2×#13094013C2211yescharacteristic
17C2×#13094008—21no—
18C2×#13093998C2211yescharacteristic
19C2×Q8×Dic47C2211yescharacteristic
20C2×#13093986C2211yescharacteristic
21C2×(C22⋊D47)⋊C4—21no—
22C2×#13093982—21no—
23C22×C4⋊Dic47C2211yescharacteristic
24C22×C4×Dic47—21no—
25C22×C4×Dic47C2211yescharacteristic
26#15689173—24no—
27C2×Dic47⋊C8C2212yes—
28C2×#13093948C2212yes—
29C2×#13093946C2212yes—
30C2×#13093944C2212yes—
31C2×C22⋊(C47⋊C8)C2211yescharacteristic
32C2×C8×Dic47C2212yes—
33C4⋊C4×Dic47—21no—
34C4⋊C4×Dic47—21no—
35C4⋊C4×Dic47C2212yes—
36C4⋊C4×Dic47C2212yes—
37C2×C47⋊#95—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C22⋊Q8×C47C2211yescharacteristic
42C22×C4⋊C4×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C2×C22⋊C8×C47C2211yescharacteristic
45C47×#340C2212yes—
46C22×D188C2311yescharacteristic
47C22×C22⋊D47—21no—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C23×Dic47D411yescharacteristic
51C22×C4×D47—21no—
52C22×C47⋊C8C2312yes—
53#13094019—28no—
54#13094015—22no—
55#13094015C2×C414yescomplemented
56#13094013—22no—
57#13094013C2×C414yescomplemented
58#13094008—28no—
59#13093998—22no—
60#13093998C2×C414yes—
61Q8×Dic47—22no—
62Q8×Dic47C2×C414yescomplemented
63#13093986—22no—
64#13093986C2×C414yescomplemented
65(C22⋊D47)⋊C4—28no—
66#13093982—28no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47—24no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2312yes—
73C2×Dic47⋊C4—21no—
74C2×(C2×D47)⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—22no—
80C2×C4×Dic47—22no—
81C2×C4×Dic47—44no—
82C2×C4×Dic47C2×C414yes—
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47D411yescharacteristic
85C2×C4×Dic47D414yes—
86C2×C4×Dic47C2311yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88Dic47⋊C8—24no—
89#13093948—24no—
90#13093946—24no—
91#13093944—24no—
92C22⋊(C47⋊C8)—22no—
93C8×Dic47—24no—
94C47⋊#95—22no—
95C47⋊#95—22no—
96C22×Q8×C47C2311yescharacteristic
97C23×C4×C47—21no—
98C23×C4×C47C2311yescharacteristic
99C22×C8×C47C2312yes—
100C22⋊Q8×C47—22no—
101C22⋊Q8×C47C2×C414yes—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47—21no—
104C2×C4⋊C4×C47—24no—
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2312yes—
107C2×C22⋊C4×C47—21no—
108#103×C47—24no—
109C22⋊C8×C47—22no—
110C24×C47C2×D411yescharacteristic
111C23×D47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47C22×C412yes—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—24no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
122C22×C4×C47—42no—
123C22×C4×C47—42no—
124C22×C4×C47C22×C412yes—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2411yescharacteristic
128C2×C22⋊D47—24no—
129C2×C22⋊D47—42no—
130C2×C22⋊D47—42no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—24no—
136C22×Dic47—22no—
137C22×Dic47—21no—
138C22×Dic47—42no—
139C22×Dic47—42no—
140C22×Dic47C22⋊C414yes—
141C22×Dic47C22⋊C414yes—
142C22×Dic47C2×D411yescharacteristic
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C2×D188—22no—
147C2×D188—22no—
148C2×D188—21no—
149C2×D188C22×C412yes—
150C2×C4×D47—24no—
151C2×C4×D47—42no—
152C2×C4×D47—42no—
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—22no—
160C4⋊C4×C47C22×C412yes—
161C4⋊C4×C47#4114yes—
162C22⋊C4×C47—24no—
163(C2×D47)⋊C4—24no—
164(C2×D47)⋊C4—24no—
165(C2×D47)⋊C4—24no—
166(C2×D47)⋊C4—24no—
167C4⋊Dic47—24no—
168C4⋊Dic47—21no—
169C4⋊Dic47—22no—
170C4⋊Dic47—24no—
171C4⋊Dic47—24no—
172C4⋊Dic47C22×C412yes—
173C4⋊Dic47#4114yes—
174Dic47⋊C4—24no—
175C4×Dic47—24no—
176C4×Dic47—44no—
177C4×Dic47—44no—
178C4×Dic47C22⋊C414yes—
179C4×Dic47C22⋊C414yes—
180C4×Dic47C22×C412yes—
181C4×Dic47C22×C412yes—
182C4×Dic47C2×D412yes—
183C4×Dic47C2×D412yes—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C23×C47—21no—
188C23×C47—21no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47C2×C22⋊C412yes—
192C23×C47#13512yes—
193C23×C47#13512yes—
194C23×C47C22×D411yescharacteristic
195C22×D47—22no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—41no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202Q8×C47—24no—
203Q8×C47—42no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—24no—
207C2×C4×C47—24no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—24no—
211C2×C4×C47—48no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47C2×C22⋊C412yes—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4112yes—
222C2×C4×C47C2×#4112yes—
223C22⋊D47—48no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—28no—
231C2×Dic47—28no—
232C2×Dic47—22no—
233C2×Dic47—24no—
234C2×Dic47—48no—
235C2×Dic47—42no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—42no—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47C2×C22⋊C412yes—
241C2×Dic47C2×C22⋊C412yes—
242C2×Dic47C2×C22⋊C412yes—
243C2×Dic47AGL(1,ℤ/8)12yes—
244C2×Dic47#13712yes—
245D188—24no—
246D188—42no—
247C4×D47—48no—
248C8×C47—42no—
249C47⋊C8—42no—
250#21388—471nomaximal, Sylow
251C22×C47—22no—
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—22no—
257C22×C47#41614yes—
258C22×C47C22×C22⋊C411yescharacteristic
259C22×C47#52511yescharacteristic
260C22×C47#52511yescharacteristic
261C22×C47C2×#13511yescharacteristic
262C22×C47C2×#13511yescharacteristic
263C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C47C2×#13711yescharacteristic
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—44no—
274C4×C47—42no—
275C4×C47—42no—
276C4×C47C4×#4112yes—
277C4×C47#53212yes—
278Dic47—22no—
279Dic47—22no—
280Dic47—22no—
281Dic47—44no—
282Dic47—42no—
283Dic47#41714yes—
284C2×#529—471no—
285C2×#431—471no—
286C2×C4×C4⋊C4—471no—
287#1989—472no—
288#1983—472no—
289#1874—471no—
290#1874—471no—
291#1872—471no—
292#1872D4711yescharacteristic, complemented
293C4×#103—472no—
294C4×#102—472no—
295C2×C47—24no—
296C2×C47#229711yescharacteristic
297C2×C47C2×#41611yescharacteristic
298C2×C47C2×#41711yescharacteristic
299C2×C47#310511yescharacteristic
300C2×C47#310511yescharacteristic
301C2×C47#310811yescharacteristic
302C2×C47#310811yescharacteristic
303D47—82no—
304#529—478no—
305#529—944no—
306C2×#126—941no—
307C2×#125—471no—
308C2×#124—471no—
309C2×#123—941no—
310C2×C22⋊Q8C2×D4711yescharacteristic
311C2×#121—471no—
312C2×C4×Q8—471no—
313C2×C4×D4—471no—
314C2×C4×D4—941no—
315C2×#117—941no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4C2×D4711yescharacteristic
318C22×C42—471no—
319C22×C42—941no—
320#431—944no—
321C2×C4⋊C8—472no—
322C2×#103—472no—
323C2×#103C2×D4712yes—
324C2×#102—472no—
325C2×#102—472no—
326C2×C22⋊C8—471no—
327C2×C22⋊C8C2×D4711yescharacteristic
328C2×C4×C8—472no—
329C4×C4⋊C4—472no—
330C4×C4⋊C4—472no—
331C4×C4⋊C4—941no—
332C4×C4⋊C4—941no—
333C2×#95—941no—
334#340—472no—
335#340—472no—
336#340—472no—
337#340C2×D4712yes—
338C47#2138811yescharacteristic, complemented, Sylow
339C22×Q8C22×D4711yescharacteristic
340C22×D4—471no—
341C22×D4—941no—
342C23×C4—21no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—941no—
346C23×C4—941no—
347C23×C4C22×D4711yescharacteristic
348C22×C8—472no—
349C22×C8C22×D4712yes—
350#126—948no—
351#125—474no—
352#125—942no—
353#124—474no—
354#124—942no—
355#123—948no—
356C22⋊Q8—22no—
357C22⋊Q8C4×D4714yes—
358#121—474no—
359#121—942no—
360C4×Q8—474no—
361C4×Q8—942no—
362C4×D4—474no—
363C4×D4—942no—
364C4×D4—948no—
365#117—948no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—472no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—944no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4712yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C42—474no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—474no—
387C2×C42—471no—
388C2×C42—471no—
389C2×C42—941no—
390C2×C42—942no—
391C2×C42—942no—
392C2×C42—1884no—
393C4⋊C8—944no—
394#103—24no—
395#103—944no—
396#102—944no—
397#102—944no—
398C22⋊C8—22no—
399C22⋊C8—942no—
400C4×C8—944no—
401#95—942no—
402#95—942no—
403C24—941no—
404C24D4×D4711yescharacteristic
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C2×C4×D4712yes—
409C2×D4—472no—
410C2×D4—944no—
411C2×D4—941no—
412C2×D4—942no—
413C2×D4—942no—
414C2×D4—1882no—
415C2×D4—1882no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—471no—
425C22×C4—474no—
426C22×C4—474no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—471no—
430C22×C4—944no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—944no—
436C22×C4—942no—
437C22×C4—941no—
438C22×C4—1882no—
439C22×C4—1882no—
440C22×C4—1882no—
441C22×C4—1882no—
442C22×C4C2×C4×D4712yes—
443C22×C4D4×D4711yescharacteristic
444C22×C4D4×D4712yes—
445C22×C4C23×D4711yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—942no—
450C2×C8—942no—
451C2×C8—942no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—472no—
457C4⋊C4—474no—
458C4⋊C4—944no—
459C4⋊C4—944no—
460C4⋊C4—944no—
461C4⋊C4—944no—
462C4⋊C4—941no—
463C4⋊C4—942no—
464C4⋊C4C2×C4×D4712yes—
465C4⋊C4#1062535214yes—
466C22⋊C4—24no—
467C22⋊C4—944no—
468C22⋊C4—944no—
469C22⋊C4—944no—
470C22⋊C4—944no—
471C42—474no—
472C42—474no—
473C42—472no—
474C42—472no—
475C42—472no—
476C42—472no—
477C42—944no—
478C42—1884no—
479C42—1884no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—942no—
485C23—1881no—
486C23—1881no—
487C23—1881no—
488C23—1881no—
489C23—1881no—
490C23—1881no—
491C23C22⋊C4×D4712yes—
492C23#1309404512yes—
493C23#1309404712yes—
494C23C2×D4×D4711yescharacteristic
495Q8—24no—
496Q8—42no—
497D4—944no—
498D4—1888no—
499D4—1882no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—472no—
515C2×C4—472no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—472no—
520C2×C4—948no—
521C2×C4—948no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—941no—
527C2×C4—941no—
528C2×C4—942no—
529C2×C4—944no—
530C2×C4—1888no—
531C2×C4—1888no—
532C2×C4—1881no—
533C2×C4—1882no—
534C2×C4—1881no—
535C2×C4—1882no—
536C2×C4C22⋊C4×D4712yes—
537C2×C4C22×C4×D4711yescharacteristic
538C2×C4C2×D4×D4711yescharacteristic
539C2×C4#41×D4712yes—
540C2×C4C2×#1062535212yes—
541C8—42no—
542C8—1882no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—3761no—
550C22—3761no—
551C22—3761no—
552C22—3761no—
553C22—1882no—
554C22—1882no—
555C22—1882no—
556C22#1568911814yes—
557C22C2×C22⋊C4×D4711yescharacteristic
558C22#1568962911yescharacteristic
559C22#1568963111yescharacteristic
560C22C2×#1309404511yescharacteristic
561C22C2×#1309404711yescharacteristic
562C22AGL(1,ℤ/8)×D4711yescharacteristic
563C22#137×D4711yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—474no—
569C4—942no—
570C4—942no—
571C4—942no—
572C4—1882no—
573C4—1884no—
574C4C4×#1062535212yes—
575C4#1568966812yes—
576C2—24no—
577C2—3762no—
578C2#2024587311yescharacteristic
579C2C2×#1568911811yescharacteristic
580C2#417×D4711yescharacteristic
581C2#2025343611yescharacteristic
582C2#2025343811yescharacteristic
583C2#2025346911yescharacteristic
584C2#2025347111yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1872C2
socleC23×C47C22×D4
prime radicalC2×#13093998C22
cosocleC22×C4×C47C24
hypercenter#1872D47
hypocenterC47#21388
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#15689173C2semidirect
3. C2×C4⋊C4×Dic47C2semidirect
4. #20244342C2semidirect
5. #20244317C2semidirect
8. #20243696C2semidirect
9. #103×Dic47C2semidirect
10. #20243202C2semidirect
11. C47×#1872C2semidirect
13. #15689659C4semidirect
55. #13094015C2×C4semidirect
57. #13094013C2×C4semidirect
62. Q8×Dic47C2×C4semidirect
64. #13093986C2×C4semidirect
292. #1872D47semidirect
338. C47#21388semidirect

Maximal subgroups

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

subgroupindex
C2×#156896592
C2×#156891732
C2×C4⋊C4×Dic472
#202443422
#202443172
#202437052
#202437032
#202436962
#103×Dic472
#202432022
C47×#18722
#2138847