Finite Group Databasebeta

← #26182215

Subgroups — #26182215

Order 12032.

1174 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×#15689602C211yesmaximal, characteristic
2#20252472C212yesmaximal, complemented
3#20241729C211yesmaximal, characteristic, complemented
4#20241727C212yesmaximal, complemented
5#20241720C212yesmaximal, complemented
6#20241719C212yesmaximal, complemented
7#20241707C212yesmaximal, complemented
8#20241699C212yesmaximal, complemented
9C47×#1466C211yesmaximal, characteristic, complemented
10C22×#10625352C411yescharacteristic
11#15689602—24no—
12#15689602—24no—
13C2×C4×Dic94C2211yescharacteristic
14C2×#13093989C411yescharacteristic
15C2×D4×Dic47C411yescharacteristic
16C2×C4×C22⋊D47C411yescharacteristic
17C2×C4×C22⋊D47C411yescharacteristic
18C2×(C22⋊D47)⋊C4C2211yescharacteristic
19C2×(C22⋊D47)⋊C4C2211yescharacteristic
20C2×#13093981C2211yescharacteristic
21C2×#13093980C2211yescharacteristic
22C2×#13093980C2211yescharacteristic
23C22×Dic47⋊C4C411yescharacteristic
24C22×Dic47⋊C4C411yescharacteristic
25C2×C4⋊C4×D47C411yescharacteristic
26#15689252—24no—
27#15689251—24no—
28#15689180C2212yes—
29#15689177C2214yes—
30#15689173C2214yes—
31#15688848C2214yes—
32#15688847C2212yes—
33#15688847C2212yes—
34#15688846C2214yes—
35C2×C4×D4×C47C2211yescharacteristic
36C47×#432C2212yes—
37C47×#323C2212yes—
38C47×#323C2212yes—
39C2×#10625352—24no—
40C2×#10625352—24no—
41C2×#10625352—24no—
42C22×Dic94C2×C411yescharacteristic
43C22×C22⋊D47C2×C411yescharacteristic
44C22×C22⋊D47C2×C411yescharacteristic
45C23×Dic47C2×C411yescharacteristic
46C23×Dic47C2×C411yescharacteristic
47C23×Dic47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47C2×C411yescharacteristic
50C22×C4×D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C4×Dic94D414yes—
53C4×Dic94D414yes—
54#13093989—24no—
55D4×Dic47—24no—
56C4×C22⋊D47—22no—
57C4×C22⋊D47—22no—
58C4×C22⋊D47—22no—
59C4×C22⋊D47—22no—
60(C22⋊D47)⋊C4—24no—
61(C22⋊D47)⋊C4—24no—
62#13093981—22no—
63#13093981—22no—
64#13093980—24no—
65#13093980—24no—
66C2×C4⋊Dic47C2×C411yescharacteristic
67C2×Dic47⋊C4—22no—
68C2×Dic47⋊C4—22no—
69C2×Dic47⋊C4—22no—
70C2×Dic47⋊C4—22no—
71C2×Dic47⋊C4C2×C411yescharacteristic
72C2×Dic47⋊C4C2×C411yescharacteristic
73C2×Dic47⋊C4C2×C411yescharacteristic
74C2×Dic47⋊C4C2×C411yescharacteristic
75C2×Dic47⋊C4C2×C411yescharacteristic
76C2×Dic47⋊C4C2×C411yescharacteristic
77C2×Dic47⋊C4C2×C411yescharacteristic
78C2×Dic47⋊C4C2311yescharacteristic
79C4⋊C4×D47—24no—
80C2×C22⋊Dic47C2×C411yescharacteristic
81C2×C22⋊Dic47C2×C411yescharacteristic
82C2×(C2×D47)⋊C4C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2311yescharacteristic
84C2×C4×Dic47C2×C411yescharacteristic
85C2×C4×Dic47C2×C411yescharacteristic
86C2×C4×Dic47C2×C411yescharacteristic
87C2×C4×Dic47C2311yescharacteristic
88C2×C4×Dic47C2311yescharacteristic
89C4⋊(C47⋊C8)—22no—
90Dic47⋊C8—24no—
91Dic47⋊C8—24no—
92C22⋊(C47⋊C8)C2314yes—
93#13093935—22no—
94#13093934—24no—
95C8×Dic47—24no—
96C22×D4×C47C2×C411yescharacteristic
97C23×C4×C47C2×C411yescharacteristic
98C23×C4×C47C2×C411yescharacteristic
99C4×D4×C47—22no—
100C4×D4×C47—22no—
101C2×C4⋊C4×C47C2×C411yescharacteristic
102C2×C22⋊C4×C47C2311yescharacteristic
103C2×C22⋊C4×C47C2311yescharacteristic
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106C22⋊C8×C47C2314yes—
107#97×C47—22no—
108C24×C47C22×C411yescharacteristic
109C24×C47C22×C411yescharacteristic
110C23×D47C22×C411yescharacteristic
111C2×D4×C47—22no—
112C2×D4×C47—22no—
113C2×D4×C47—22no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—21no—
118C22×C4×C47—81no—
119C22×C4×C47—81no—
120C22×C4×C47C22×C411yescharacteristic
121C22×C4×C47C22×C411yescharacteristic
122C22×C4×C47C22×C411yescharacteristic
123C22×C4×C47C22×C411yescharacteristic
124C22×C4×C47C2411yescharacteristic
125C2×C22⋊D47—22no—
126C2×C22⋊D47—22no—
127C2×C22⋊D47—22no—
128C2×C22⋊D47—22no—
129C2×C22⋊D47—22no—
130C2×C22⋊D47M1614yes—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—81no—
139C22×Dic47—81no—
140C22×Dic47M1614yes—
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143C22×Dic47C22×C411yescharacteristic
144C22×Dic47C22×C411yescharacteristic
145C22×Dic47C22×C411yescharacteristic
146C22×Dic47C22×C411yescharacteristic
147C22×Dic47C22×C411yescharacteristic
148#10625352—232no—
149C2×C4×D47—22no—
150C2×C4×D47—22no—
151C2×C4×D47—22no—
152C2×C4×D47—22no—
153C2×C4×D47—22no—
154C2×C4×D47—81no—
155C2×C4×D47—81no—
156C2×Dic94—22no—
157C2×Dic94—22no—
158C2×Dic94C22⋊C414yes—
159C2×C8×C47—24no—
160C4⋊C4×C47—22no—
161C22⋊C4×C47—22no—
162C22⋊C4×C47—22no—
163C42×C47C2×D412yes—
164C42×C47C2×D412yes—
165C22⋊Dic47—22no—
166C22⋊Dic47—22no—
167(C2×D47)⋊C4—21no—
168(C2×D47)⋊C4—21no—
169(C2×D47)⋊C4—22no—
170C4⋊Dic47C22⋊C414yes—
171Dic47⋊C4—21no—
172Dic47⋊C4—21no—
173Dic47⋊C4—22no—
174Dic47⋊C4—22no—
175Dic47⋊C4—22no—
176Dic47⋊C4—22no—
177Dic47⋊C4—22no—
178Dic47⋊C4C22⋊C412yes—
179Dic47⋊C4C22⋊C412yes—
180Dic47⋊C4C22⋊C412yes—
181Dic47⋊C4C22⋊C412yes—
182C4×Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47—22no—
185C4×Dic47C2×D414yes—
186C4×Dic47C2×D414yes—
187C2×C47⋊C8—24no—
188C23×C47—21no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47—41no—
196C23×C47C2×M1612yes—
197C23×C47C2×M1612yes—
198C23×C47C23×C411yescharacteristic
199C22×D47—21no—
200C22×D47—21no—
201C22×D47—41no—
202C22×D47—41no—
203C22×D47C2×M1612yes—
204D4×C47—28no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—81no—
214C2×C4×C47—81no—
215C2×C4×C47—81no—
216C2×C4×C47—81no—
217C2×C4×C47C2×C22⋊C412yes—
218C2×C4×C47C2×C22⋊C411yescharacteristic
219C2×C4×C47C2×C22⋊C411yescharacteristic
220C2×C4×C47C22×D412yes—
221C22⋊D47—28no—
222C22⋊D47—28no—
223C2×Dic47—28no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—28no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—41no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—82no—
236C2×Dic47—82no—
237C2×Dic47C2×C22⋊C412yes—
238C2×Dic47C2×C22⋊C412yes—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47C2×C22⋊C412yes—
241C2×Dic47C2×M1612yes—
242C2×Dic47C2×M1612yes—
243C2×Dic47C2×M1612yes—
244C4×D47—28no—
245C4×D47—82no—
246C4×D47—82no—
247Dic94—28no—
248C8×C47—44no—
249C47⋊C8—44no—
250#17372—471nomaximal, Sylow
251C22×C47—21no—
252C22×C47—21no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—42no—
256C22×C47—42no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47—41no—
263C22×C47C22×C22⋊C411yescharacteristic
264C22×C47C22×M1611yescharacteristic
265C22×C47C22×M1611yescharacteristic
266C2×D47—22no—
267C2×D47—42no—
268C2×D47—41no—
269C2×D47—41no—
270C2×D47—42no—
271C4×C47—22no—
272C4×C47—22no—
273C4×C47—21no—
274C4×C47—21no—
275C4×C47—41no—
276C4×C47—41no—
277Dic47—22no—
278Dic47—22no—
279Dic47—22no—
280Dic47—22no—
281Dic47—41no—
282Dic47—41no—
283Dic47#40714yes—
284C2×#520—471no—
285#2983—472no—
286#1466—472no—
287#1466—471no—
288#1466D4711yescharacteristic, complemented
289#1464—472no—
290#1464—472no—
291#1460—472no—
292#1458—472no—
293C2×C47—21no—
294C2×C47—21no—
295C2×C47—42no—
296C2×C47—42no—
297C2×C47C2×#40711yescharacteristic
298C2×C47#288411yescharacteristic
299C2×C47#288811yescharacteristic
300D47—42no—
301C22×#41—471no—
302#520—944no—
303#520—944no—
304C2×C4×Q8—471no—
305C2×C4×Q8—471no—
306C2×C4×D4—471no—
307C2×C4×D4—471no—
308C2×C4×D4—471no—
309C2×C4×D4—471no—
310C2×C4×D4—471no—
311C2×C4×D4C2×D4711yescharacteristic
312C2×#117—471no—
313C2×#117—471no—
314C2×#117—471no—
315C22×C4⋊C4—471no—
316C22×C4⋊C4—471no—
317C22×C4⋊C4—471no—
318#446—944no—
319#446—944no—
320#432—472no—
321#432—474no—
322#432C2×D4712yes—
323#431—474no—
324#324—474no—
325#323—472no—
326#323—472no—
327#323—474no—
328#323C2×D4712yes—
329#323C2×D4712yes—
330C47#1737211yescharacteristic, complemented, Sylow
331C2×#41—944no—
332C2×#41—944no—
333C2×#41—944no—
334C22×Q8—471no—
335C22×D4—471no—
336C22×D4—471no—
337C22×D4C4×D4711yescharacteristic
338C23×C4—471no—
339C23×C4—471no—
340C23×C4—471no—
341C23×C4—471no—
342C23×C4—471no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4C4×D4711yescharacteristic
346C23×C4C4×D4711yescharacteristic
347C4×Q8—474no—
348C4×Q8—474no—
349C4×Q8—944no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—944no—
353C4×D4—944no—
354C4×D4—942no—
355C4×D4—942no—
356C4×D4—942no—
357C4×D4—942no—
358C4×D4—944no—
359#117—944no—
360#117—944no—
361#117—942no—
362#117—942no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—471no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—944no—
373C2×C4⋊C4—942no—
374C2×C4⋊C4—942no—
375C2×C4⋊C4—942no—
376C2×C4⋊C4—942no—
377C2×C4⋊C4C4×D4711yescharacteristic
378C2×C22⋊C4—471no—
379C2×C22⋊C4—471no—
380C2×C22⋊C4—471no—
381C2×C22⋊C4—471no—
382C2×C22⋊C4C22×D4711yescharacteristic
383C2×C22⋊C4C22×D4711yescharacteristic
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—471no—
387C2×C42—471no—
388C2×C42—471no—
389C2×C42C22×D4711yescharacteristic
390C4⋊C8—22no—
391C4⋊C8—944no—
392C4⋊C8—944no—
393C4⋊C8—942no—
394C22⋊C8—474no—
395C22⋊C8C22×D4714yes—
396#97—22no—
397#97—944no—
398#97—942no—
399C4×C8—944no—
400C24—471no—
401C24C2×C4×D4711yescharacteristic
402C24C2×C4×D4711yescharacteristic
403#41—9432no—
404C2×Q8—474no—
405C2×Q8—942no—
406C2×Q8—942no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—474no—
411C2×D4—942no—
412C2×D4—942no—
413C2×D4—942no—
414C2×D4—942no—
415C2×D4—942no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—471no—
423C22×C4—474no—
424C22×C4—471no—
425C22×C4—471no—
426C22×C4—471no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—471no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—3761no—
443C22×C4—3761no—
444C22×C4—3761no—
445C22×C4—3761no—
446C22×C4C2×C4×D4711yescharacteristic
447C22×C4C2×C4×D4711yescharacteristic
448C22×C4C2×C4×D4711yescharacteristic
449C22×C4C2×C4×D4711yescharacteristic
450C22×C4C23×D4711yescharacteristic
451C2×C8—24no—
452C2×C8—944no—
453C4⋊C4—22no—
454C4⋊C4—472no—
455C4⋊C4—472no—
456C4⋊C4—472no—
457C4⋊C4—472no—
458C4⋊C4—474no—
459C4⋊C4—942no—
460C4⋊C4—942no—
461C4⋊C4—942no—
462C4⋊C4—942no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—942no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—942no—
469C22⋊C4—942no—
470C22⋊C4—941no—
471C22⋊C4—941no—
472C22⋊C4—942no—
473C42—474no—
474C42—474no—
475C42—942no—
476C42—942no—
477C42—942no—
478C42D4×D4712yes—
479C42D4×D4712yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—472no—
489C23—941no—
490C23—941no—
491C23—1881no—
492C23—1881no—
493C23M16×D4712yes—
494C23M16×D4712yes—
495C23C22×C4×D4711yescharacteristic
496Q8—948no—
497D4—28no—
498D4—948no—
499D4—948no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—472no—
513C2×C4—472no—
514C2×C4—472no—
515C2×C4—472no—
516C2×C4—472no—
517C2×C4—472no—
518C2×C4—472no—
519C2×C4—948no—
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—3762no—
529C2×C4—3762no—
530C2×C4—3762no—
531C2×C4—3762no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4C22⋊C4×D4712yes—
537C2×C4C22⋊C4×D4711yescharacteristic
538C2×C4C22⋊C4×D4711yescharacteristic
539C2×C4C2×D4×D4712yes—
540C8—44no—
541C8—1884no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—942no—
555C22—1882no—
556C22—1882no—
557C22—1881no—
558C22—1881no—
559C22C2×C22⋊C4×D4711yescharacteristic
560C22C2×M16×D4711yescharacteristic
561C22C2×M16×D4711yescharacteristic
562C4—22no—
563C4—22no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—474no—
569C4—942no—
570C4—942no—
571C4—942no—
572C4—942no—
573C4—1881no—
574C4—1881no—
575C2—21no—
576C2—21no—
577C2—42no—
578C2—42no—
579C2—1882no—
580C2#407×D4711yescharacteristic
581C2#2025185011yescharacteristic
582C2#2025186411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1466C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#10625352C4
cosocleC22×C4×C47C24
hypercenter#1466D47
hypocenterC47#17372
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #20252472C2semidirect
3. #20241729C2semidirect
4. #20241727C2semidirect
5. #20241720C2semidirect
6. #20241719C2semidirect
7. #20241707C2semidirect
8. #20241699C2semidirect
9. C47×#1466C2semidirect
288. #1466D47semidirect
330. C47#17372semidirect

Maximal subgroups

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

subgroupindex
C2×#156896022
#202524722
#202417292
#202417272
#202417202
#202417192
#202417072
#202416992
C47×#14662
#1737247