Finite Group Databasebeta

← #26217054

Subgroups — #26217054

Order 12032.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689603C211yesmaximal, characteristic
2C2×#15689165C211yesmaximal, characteristic, complemented
3C2×#15689159C211yesmaximal, characteristic, complemented
4C2×#15689091C211yesmaximal, characteristic
5#20245981C211yesmaximal, characteristic
6#20243590C212yesmaximal, complemented
7#20243588C212yesmaximal, complemented
8#20243582C212yesmaximal, complemented
9#20243577C212yesmaximal, complemented
10C2×#15688885C211yesmaximal, characteristic, complemented
11C47×#1736C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C23×C47⋊C8C2211yescharacteristic
14#15689603—24no—
15#15689603—24no—
16C2×C4×Dic94—21no—
17C2×C4×D188—21no—
18C2×(C22⋊D47)⋊C4—22no—
19C2×#13093983C2211yescharacteristic
20C2×#13093982C2212yes—
21C22×C4⋊Dic47C2211yescharacteristic
22C2×C4⋊C4×D47C2212yes—
23#15689165—24no—
24#15689159—24no—
25C2×#13093958C2212yes—
26C2×#13093957C2211yescharacteristic
27C2×#13093957C2211yescharacteristic
28C2×#13093956C2212yes—
29C2×#13093955C2211yescharacteristic
30C2×#13093955C2211yescharacteristic
31#15689091C418yescomplemented
32C2×(C2×D47)⋊C8C2212yes—
33C2×(C2×D47)⋊C8C2212yes—
34#15689028—21no—
35#15689024—21no—
36#15689002—22no—
37#15688980C2212yes—
38#15688980C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C22×M16×C47C2211yescharacteristic
42C2×#117×C47C2211yescharacteristic
43C22×C4⋊C4×C47C2211yescharacteristic
44C47×#340C2212yes—
45C47×#340C2212yes—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350C2×C414yes—
51C2×#10625350Q814yes—
52C22×Dic94D411yescharacteristic
53C22×D188D411yescharacteristic
54C22×C22⋊D47D412yes—
55C23×Dic47—21no—
56C22×C4×D47—22no—
57C22×C4×D47C2312yes—
58C22×C47⋊C8—22no—
59C22×C47⋊C8—22no—
60C22×C47⋊C8C2×C414yescomplemented
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63C4×Dic94—28no—
64C4×D188—28no—
65(C22⋊D47)⋊C4—216no—
66#13093983—22no—
67#13093983—22no—
68#13093982—24no—
69#13093982—24no—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—22no—
72C2×C4⋊Dic47D414yes—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2312yes—
76C2×Dic47⋊C4—22no—
77C4⋊C4×D47—24no—
78C4⋊C4×D47—24no—
79C2×C22⋊Dic47—21no—
80C2×(C2×D47)⋊C4—22no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×C4×Dic47—22no—
84C2×C4×Dic47—21no—
85#13093958—24no—
86#13093957—22no—
87#13093957—22no—
88#13093956—24no—
89#13093955—22no—
90#13093955—22no—
91(C2×D47)⋊C8C2×C418yescomplemented
92(C2×D47)⋊C8C2×C418yescomplemented
93C47⋊#95—22no—
94C47⋊#95—22no—
95C23×C4×C47—21no—
96C23×C4×C47C2311yescharacteristic
97C2×M16×C47—22no—
98C2×M16×C47—22no—
99C2×M16×C47C2×C414yescomplemented
100C22×C8×C47C2312yes—
101#117×C47—22no—
102#117×C47D414yes—
103C2×C4⋊C4×C47—22no—
104C2×C4⋊C4×C47—22no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47C2312yes—
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C42×C47—21no—
111C24×C47C2×D411yescharacteristic
112C23×D47C2×D412yes—
113C22×C4×C47—21no—
114C22×C4×C47—21no—
115C22×C4×C47—22no—
116C22×C4×C47—42no—
117C22×C4×C47—81no—
118C22×C4×C47—81no—
119C22×C4×C47C22×C412yes—
120C22×C4×C47C2×D411yescharacteristic
121C22×C4×C47C2×D412yes—
122C22×C4×C47C2×D412yes—
123C22×C4×C47C2×Q812yes—
124C22×C4×C47#4112yes—
125C22×C4×C47#4112yes—
126C22×C4×C47C2411yescharacteristic
127C2×C22⋊D47—24no—
128C2×C22⋊D47—24no—
129C2×C22⋊D47C4⋊C418yes—
130C22×Dic47—22no—
131C22×Dic47—24no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—81no—
135C22×Dic47—81no—
136C22×Dic47C2×D412yes—
137#10625350—28no—
138#10625350—28no—
139#10625350—28no—
140#10625350—28no—
141C2×D188—22no—
142C2×D188—22no—
143C2×D188C4⋊C414yes—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—28no—
149C2×C4×D47—82no—
150C2×C4×D47—82no—
151C2×C4×D47C22×C414yes—
152C2×C4×D47C2×Q814yes—
153C2×Dic94—22no—
154C2×Dic94—22no—
155C2×Dic94C4⋊C414yes—
156M16×C47—28no—
157C2×C8×C47—22no—
158C2×C8×C47—22no—
159C2×C8×C47C22×C414yes—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—42no—
165C4⋊C4×C47C2×D412yes—
166C4⋊C4×C47C2×D412yes—
167C22⋊C4×C47—24no—
168C42×C47—24no—
169C22⋊Dic47—42no—
170(C2×D47)⋊C4—28no—
171(C2×D47)⋊C4—82no—
172(C2×D47)⋊C4—82no—
173C4⋊Dic47—24no—
174C4⋊Dic47—22no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47C2×D414yes—
180Dic47⋊C4—28no—
181C4×Dic47—28no—
182C4×Dic47—42no—
183C2×C47⋊C8—28no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8C22×C412yes—
189C2×C47⋊C8C22×C412yes—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47C2×C4⋊C412yes—
195C23×C47#12012yes—
196C23×C47#12312yes—
197C23×C47C22×D411yescharacteristic
198C22×D47—22no—
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47C2×C4⋊C414yes—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—84no—
215C2×C4×C47C2×C4⋊C412yes—
216C2×C4×C47C4×D414yes—
217C2×C4×C47C4×D414yes—
218C2×C4×C47#12012yes—
219C2×C4×C47C22⋊Q814yes—
220C2×C4×C47C22⋊Q814yes—
221C2×C4×C47#12312yes—
222C2×C4×C47C23×C411yescharacteristic
223C2×C4×C47C22×D411yescharacteristic
224C2×C4×C47C22×D411yescharacteristic
225C2×C4×C47C22×Q811yescharacteristic
226C2×C4×C47C2×#4111yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C22⋊D47—216no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—24no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—84no—
241C2×Dic47C2×C4⋊C414yes—
242D188—28no—
243C4×D47—24no—
244C4×D47—24no—
245C4×D47—24no—
246C4×D47—24no—
247C4×D47—88no—
248Dic94—28no—
249C8×C47—24no—
250C47⋊C8—22no—
251C47⋊C8—22no—
252#20849—471nomaximal, Sylow
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47#38514yes—
260C22×C47C22×C4⋊C411yescharacteristic
261C22×C47C2×#12011yescharacteristic
262C22×C47C2×#12311yescharacteristic
263C22×C47#57611yescharacteristic
264C22×C47#57711yescharacteristic
265C22×C47#57711yescharacteristic
266C22×C47#57811yescharacteristic
267C2×D47—24no—
268C2×D47—24no—
269C2×D47—24no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—42no—
274C4×C47—42no—
275C4×C47—42no—
276C4×C47—82no—
277C4×C47#38514yes—
278C4×C47C2×C4×D411yescharacteristic
279C4×C47C2×C4×D411yescharacteristic
280C4×C47C2×C22⋊Q811yescharacteristic
281C4×C47C2×C22⋊Q811yescharacteristic
282Dic47—24no—
283Dic47—44no—
284Dic47—81no—
285Dic47—81no—
286C2×#520—471no—
287C2×#428—471no—
288C2×#427—471no—
289C2×#408—471no—
290#2311—471no—
291#1854—472no—
292#1854—472no—
293#1853—472no—
294#1852—472no—
295#1736D4711yescharacteristic, complemented
296C2×#340—471no—
297C2×C47—24no—
298C2×C47C2×#38511yescharacteristic
299C2×C47#291211yescharacteristic
300C2×C47#291311yescharacteristic
301C2×C47#301211yescharacteristic
302C2×C47#301311yescharacteristic
303C2×C47#309611yescharacteristic
304C2×C47#309711yescharacteristic
305D47—44no—
306C22×#41—471no—
307C22×M16C2×D4711yescharacteristic
308C23×C8—471no—
309#520—944no—
310#520—944no—
311C2×C4×Q8—941no—
312C2×C4×D4—941no—
313C2×C4×D4—942no—
314C2×#117—471no—
315C2×#117—472no—
316C2×#117C2×D4711yescharacteristic
317C22×C4⋊C4—471no—
318C22×C4⋊C4—472no—
319C22×C4⋊C4C2×D4711yescharacteristic
320#428—944no—
321#427—944no—
322C2×#107—472no—
323C2×#107—471no—
324C2×#107—471no—
325C2×#106—472no—
326C2×#106—471no—
327C2×#106—471no—
328#408—478no—
329C2×C22⋊C8—472no—
330C2×C22⋊C8—472no—
331#391—941no—
332#390—941no—
333#386—942no—
334#380—472no—
335#380—472no—
336#340—472no—
337#340—472no—
338#340C2×D4712yes—
339#340C2×D4712yes—
340C47#2084911yescharacteristic, complemented, Sylow
341C2×#41—474no—
342C2×#41—474no—
343C2×#41—942no—
344C2×#41—942no—
345C2×#41—942no—
346C2×#41—942no—
347C22×Q8—471no—
348C22×D4—472no—
349C22×D4—471no—
350C23×C4—21no—
351C23×C4—472no—
352C23×C4—942no—
353C23×C4—941no—
354C23×C4C22×D4711yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D4714yescomplemented
358C22×C8—474no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—942no—
362C22×C8—942no—
363C22×C8C22×D4712yes—
364C4×Q8—948no—
365C4×D4—948no—
366C4×D4—9416no—
367#117—22no—
368#117—944no—
369#117—944no—
370#117—942no—
371#117—942no—
372#117D18814yes—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—474no—
377C2×C4⋊C4—472no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—944no—
381C2×C4⋊C4—942no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—941no—
384C2×C4⋊C4—944no—
385C2×C4⋊C4C22×D4712yes—
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—942no—
390C2×C22⋊C4—942no—
391C2×C22⋊C4—942no—
392C2×C22⋊C4—941no—
393C2×C42—21no—
394C2×C42—942no—
395C2×C42—941no—
396#107—944no—
397#107—942no—
398#107—942no—
399#106—944no—
400#106—942no—
401#106—942no—
402C22⋊C8—478no—
403C22⋊C8—478no—
404#95—942no—
405#95—942no—
406C24—472no—
407C24D4×D4711yescharacteristic
408#41—948no—
409#41—948no—
410#41—948no—
411#41—948no—
412C2×Q8—474no—
413C2×Q8—942no—
414C2×Q8—942no—
415C2×D4—478no—
416C2×D4—474no—
417C2×D4—944no—
418C2×D4—944no—
419C2×D4—942no—
420C2×D4—942no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—472no—
428C22×C4—474no—
429C22×C4—474no—
430C22×C4—948no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—944no—
436C22×C4—942no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—3762no—
440C22×C4—3762no—
441C22×C4—3761no—
442C22×C4—3761no—
443C22×C4C2×C4×D4712yes—
444C22×C4C2×D18812yes—
445C22×C4#1062535012yes—
446C22×C4D4×D4711yescharacteristic
447C22×C4D4×D4712yes—
448C22×C4#1062535212yes—
449C22×C4Q8×D4712yes—
450C22×C4C23×D4711yescharacteristic
451M16—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—472no—
455C2×C8—472no—
456C2×C8—948no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8C2×C4×D4714yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—474no—
468C4⋊C4—948no—
469C4⋊C4—944no—
470C4⋊C4—942no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4C2×D18812yes—
476C4⋊C4C2×D18812yes—
477C22⋊C4—24no—
478C22⋊C4—948no—
479C22⋊C4—3762no—
480C22⋊C4—3762no—
481C22⋊C4—1882no—
482C42—24no—
483C42—948no—
484C42—1882no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—474no—
490C23—942no—
491C23—942no—
492C23—942no—
493C23—942no—
494C23—942no—
495C23—942no—
496C23C4⋊C4×D4712yes—
497C23#1309399412yes—
498C23#1309400612yes—
499C23C2×D4×D4711yescharacteristic
500Q8—948no—
501D4—9416no—
502D4—948no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4—474no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—944no—
520C2×C4—944no—
521C2×C4—3768no—
522C2×C4—3764no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4C4⋊C4×D4712yes—
532C2×C4(C22⋊D47)⋊C414yes—
533C2×C4C4×D18814yes—
534C2×C4#1309399412yes—
535C2×C4#1309400214yes—
536C2×C4#1309400414yes—
537C2×C4#1309400612yes—
538C2×C4C22×C4×D4711yescharacteristic
539C2×C4C2×D4×D4711yescharacteristic
540C2×C4C22×D18811yescharacteristic
541C2×C4C2×Q8×D4711yescharacteristic
542C2×C4C2×#1062535211yescharacteristic
543C2×C4C2×#1062535011yescharacteristic
544C8—24no—
545C8—942no—
546C8—942no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—944no—
554C22—944no—
555C22—944no—
556C22—1882no—
557C22—1882no—
558C22—1882no—
559C22—1882no—
560C22#1568899714yes—
561C22C2×C4⋊C4×D4711yescharacteristic
562C22C2×#1309399411yescharacteristic
563C22C2×#1309400611yescharacteristic
564C22#1568992711yescharacteristic
565C22#1568993211yescharacteristic
566C22#1568993311yescharacteristic
567C22#1568993911yescharacteristic
568C4—42no—
569C4—42no—
570C4—82no—
571C4—944no—
572C4—3761no—
573C4—3761no—
574C4—1884no—
575C4#1568899714yes—
576C4C2×(C22⋊D47)⋊C411yescharacteristic
577C4C2×C4×D18811yescharacteristic
578C4C2×#1309400211yescharacteristic
579C4C2×#1309400411yescharacteristic
580C2—24no—
581C2—1884no—
582C2C2×#1568899711yescharacteristic
583C2#2025198211yescharacteristic
584C2#2025198711yescharacteristic
585C2#2025269411yescharacteristic
586C2#2025270511yescharacteristic
587C2#2025334411yescharacteristic
588C2#2025335511yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20849
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#15689165C2semidirect
3. C2×#15689159C2semidirect
6. #20243590C2semidirect
7. #20243588C2semidirect
8. #20243582C2semidirect
9. #20243577C2semidirect
10. C2×#15688885C2semidirect
11. C47×#1736C2semidirect
31. #15689091C4semidirect
60. C22×C47⋊C8C2×C4semidirect
91. (C2×D47)⋊C8C2×C4semidirect
92. (C2×D47)⋊C8C2×C4semidirect
99. C2×M16×C47C2×C4semidirect
295. #1736D47semidirect
340. C47#20849semidirect
357. C2×M16C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156896032
C2×#156891652
C2×#156891592
C2×#156890912
#202459812
#202435902
#202435882
#202435822
#202435772
C2×#156888852
C47×#17362
#2084947