Finite Group Databasebeta

← #26182188

Subgroups — #26182188

Order 12032.

1210 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689603C211yesmaximal, characteristic
2#20252416C212yesmaximal, complemented
3#20241731C212yesmaximal, complemented
4#20241723C212yesmaximal, complemented
5#20241722C211yesmaximal, characteristic, complemented
6#20241716C212yesmaximal, complemented
7#20241703C212yesmaximal, complemented
8#324×D47C212yesmaximal, complemented
9C47×#1464C211yesmaximal, characteristic, complemented
10C22×#10625350C411yescharacteristic
11#15689603—24no—
12#15689603—24no—
13C2×C4×Dic94C411yescharacteristic
14C2×C4×Dic94C2211yescharacteristic
15C2×C4×D188C411yescharacteristic
16C2×C4×D188C2211yescharacteristic
17C2×(C22⋊D47)⋊C4C411yescharacteristic
18C2×(C22⋊D47)⋊C4C411yescharacteristic
19C2×(C22⋊D47)⋊C4C2211yescharacteristic
20C2×(C22⋊D47)⋊C4C2211yescharacteristic
21C2×#13093982C411yescharacteristic
22C2×#13093982C411yescharacteristic
23C22×C4⋊Dic47C411yescharacteristic
24C2×C4⋊C4×D47C2211yescharacteristic
25C2×C4⋊C4×D47C2211yescharacteristic
26#15689196—24no—
27#15689193—24no—
28#15689180C2212yes—
29#15689085C2214yes—
30C22⋊C8×D47C2214yes—
31#15688849C2212yes—
32#15688848C2214yes—
33#15688848C2214yes—
34#15688847C2212yes—
35C2×#117×C47C2211yescharacteristic
36C47×#432C2212yes—
37C47×#324C2212yes—
38C47×#323C2212yes—
39C2×#10625350—24no—
40C2×#10625350—22no—
41C2×#10625350—22no—
42C2×#10625350—24no—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47C2×C411yescharacteristic
46C22×C22⋊D47C2×C411yescharacteristic
47C23×Dic47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47C2×C411yescharacteristic
50C22×C4×D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2311yescharacteristic
54C22×C4×D47C2311yescharacteristic
55C2×#10625324—24no—
56C2×C8×D47—24no—
57C4×Dic94—22no—
58C4×Dic94—22no—
59C4×Dic94—22no—
60C4×Dic94—22no—
61C4×D188—22no—
62C4×D188—22no—
63C4×D188D414yes—
64C4×D188D414yes—
65(C22⋊D47)⋊C4—24no—
66(C22⋊D47)⋊C4—24no—
67(C22⋊D47)⋊C4—24no—
68(C22⋊D47)⋊C4—24no—
69#13093982—24no—
70#13093982—24no—
71C2×C4⋊Dic47—22no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47C2×C411yescharacteristic
74C2×C4⋊Dic47C2×C411yescharacteristic
75C2×C4⋊Dic47C2×C411yescharacteristic
76C2×C4⋊Dic47C2×C411yescharacteristic
77C2×Dic47⋊C4C2×C411yescharacteristic
78C2×Dic47⋊C4C2×C411yescharacteristic
79C2×Dic47⋊C4C2311yescharacteristic
80C2×Dic47⋊C4C2311yescharacteristic
81C4⋊C4×D47—24no—
82C4⋊C4×D47—24no—
83C2×(C2×D47)⋊C4C2×C411yescharacteristic
84C2×(C2×D47)⋊C4C2×C411yescharacteristic
85C2×(C2×D47)⋊C4C2×C411yescharacteristic
86C2×(C2×D47)⋊C4C2×C411yescharacteristic
87C2×C4×Dic47C2×C411yescharacteristic
88C2×C4×Dic47C2×C411yescharacteristic
89C2×C4×Dic47C2×C411yescharacteristic
90C2×C4×Dic47C2×C411yescharacteristic
91C4⋊(C47⋊C8)—22no—
92C22⋊(C47⋊C8)C2314yes—
93(C2×D47)⋊C8—24no—
94(C2×D47)⋊C8—24no—
95#13093935—22no—
96C23×C4×C47C2×C411yescharacteristic
97#117×C47—22no—
98#117×C47—22no—
99C2×C4⋊C4×C47C2×C411yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C22⋊C4×C47C2×C411yescharacteristic
102C2×C22⋊C4×C47C2311yescharacteristic
103C2×C42×C47C2×C411yescharacteristic
104C2×C42×C47C2311yescharacteristic
105C4⋊C8×C47—22no—
106C22⋊C8×C47C2314yes—
107#97×C47—22no—
108C24×C47C22×C411yescharacteristic
109C23×D47C22×C411yescharacteristic
110C23×D47C22×C411yescharacteristic
111C22×C4×C47—22no—
112C22×C4×C47—21no—
113C22×C4×C47—21no—
114C22×C4×C47—22no—
115C22×C4×C47C22×C411yescharacteristic
116C22×C4×C47C22×C411yescharacteristic
117C22×C4×C47C22×C411yescharacteristic
118C22×C4×C47C22×C411yescharacteristic
119C22×C4×C47C22×C411yescharacteristic
120C22×C4×C47C2411yescharacteristic
121C2×C22⋊D47—22no—
122C2×C22⋊D47—22no—
123C2×C22⋊D47—22no—
124C2×C22⋊D47—22no—
125C2×C22⋊D47—22no—
126C2×C22⋊D47M1614yescomplemented
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—81no—
130C22×Dic47—81no—
131C22×Dic47C22×C411yescharacteristic
132C22×Dic47C22×C411yescharacteristic
133C22×Dic47C22×C411yescharacteristic
134C22×Dic47C22×C411yescharacteristic
135C22×Dic47C22×C411yescharacteristic
136C22×Dic47C22×C411yescharacteristic
137#10625350—216no—
138#10625350—216no—
139C2×D188—22no—
140C2×D188—22no—
141C2×D188C22⋊C412yes—
142C2×D188C22⋊C412yes—
143C2×C4×D47—22no—
144C2×C4×D47—22no—
145C2×C4×D47—22no—
146C2×C4×D47—22no—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—22no—
150C2×C4×D47—81no—
151C2×C4×D47—81no—
152C2×C4×D47—81no—
153C2×C4×D47—81no—
154C2×C4×D47M1614yescomplemented
155C2×C4×D47C2×D414yes—
156C2×C4×D47C2×D414yes—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—22no—
160C2×Dic94—22no—
161C2×C8×C47—24no—
162C4⋊C4×C47—22no—
163C4⋊C4×C47—22no—
164C22⋊C4×C47—22no—
165C22⋊C4×C47—22no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47C2×D412yes—
169C42×C47C2×D412yes—
170(C2×D47)⋊C4—22no—
171(C2×D47)⋊C4—22no—
172(C2×D47)⋊C4C22⋊C414yes—
173(C2×D47)⋊C4C22⋊C414yes—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—22no—
177C4⋊Dic47—22no—
178C4⋊Dic47C22⋊C412yes—
179C4⋊Dic47C22⋊C412yes—
180Dic47⋊C4—22no—
181Dic47⋊C4—22no—
182Dic47⋊C4—22no—
183Dic47⋊C4—22no—
184C4×Dic47—22no—
185C4×Dic47—22no—
186C4×Dic47—22no—
187C4×Dic47—22no—
188C2×C47⋊C8—24no—
189#10625324—84no—
190C8×D47—84no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—41no—
194C23×C47—41no—
195C23×C47C2×M1612yes—
196C23×C47C23×C411yescharacteristic
197C22×D47—21no—
198C22×D47—21no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202C22×D47—41no—
203C22×D47C2×C22⋊C412yes—
204C22×D47C2×C22⋊C412yes—
205C22×D47C2×M1612yes—
206C22×D47C2×M1612yes—
207C2×C4×C47—24no—
208C2×C4×C47—24no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—41no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47C2×C22⋊C412yes—
220C2×C4×C47C2×C22⋊C411yescharacteristic
221C2×C4×C47C2×C22⋊C411yescharacteristic
222C2×C4×C47C2×M1612yes—
223C2×C4×C47C22×D412yes—
224C22⋊D47—28no—
225C22⋊D47—28no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—41no—
233C2×Dic47—41no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—82no—
237C2×Dic47—82no—
238C2×Dic47C2×C22⋊C412yes—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47C2×M1612yes—
241C2×Dic47C2×M1612yes—
242D188—24no—
243D188—24no—
244C4×D47—28no—
245C4×D47—28no—
246C4×D47—44no—
247C4×D47—44no—
248C4×D47—82no—
249C4×D47—82no—
250Dic94—24no—
251Dic94—24no—
252C8×C47—44no—
253C47⋊C8—44no—
254#17368—471nomaximal, Sylow
255C22×C47—21no—
256C22×C47—21no—
257C22×C47—22no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47C22×C22⋊C411yescharacteristic
263C22×C47C22×M1611yescharacteristic
264C22×C47C22×M1611yescharacteristic
265C2×D47—22no—
266C2×D47—22no—
267C2×D47—22no—
268C2×D47—22no—
269C2×D47—22no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47#40714yes—
275C4×C47—21no—
276C4×C47—21no—
277C4×C47—22no—
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—41no—
281C4×C47—41no—
282Dic47—22no—
283Dic47—22no—
284Dic47—22no—
285Dic47—22no—
286Dic47—41no—
287Dic47—41no—
288C2×#520—471no—
289#2973—472no—
290#1467—472no—
291#1465—472no—
292#1464—471no—
293#1464D4711yescharacteristic, complemented
294#1463—472no—
295#1459—472no—
296C2×#324—472no—
297C2×C47—21no—
298C2×C47—21no—
299C2×C47—42no—
300C2×C47C2×#40711yescharacteristic
301C2×C47#288311yescharacteristic
302C2×C47#288911yescharacteristic
303D47—24no—
304D47—42no—
305C22×#41—471no—
306#520—944no—
307#520—944no—
308C2×C4×Q8—471no—
309C2×C4×Q8—471no—
310C2×C4×D4—471no—
311C2×C4×D4—471no—
312C2×C4×D4—471no—
313C2×C4×D4—471no—
314C2×C4×D4—471no—
315C2×C4×D4—471no—
316C2×#117—471no—
317C2×#117—471no—
318C2×#117C2×D4711yescharacteristic
319C22×C4⋊C4—471no—
320C22×C4⋊C4—471no—
321C22×C4⋊C4—471no—
322#435—944no—
323#435—944no—
324#432—472no—
325#432C2×D4712yes—
326#407—474no—
327C2×C22⋊C8—474no—
328#324—472no—
329#324—474no—
330#324—474no—
331#324C2×D4712yes—
332#323—472no—
333#323C2×D4712yes—
334C47#1736811yescharacteristic, complemented, Sylow
335C2×#41—944no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—944no—
339C22×Q8—471no—
340C22×D4—471no—
341C22×D4—471no—
342C22×D4—471no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—471no—
347C23×C4—471no—
348C23×C4—471no—
349C23×C4—471no—
350C23×C4—471no—
351C23×C4C4×D4711yescharacteristic
352C2×M16—944no—
353C22×C8—944no—
354C4×Q8—942no—
355C4×Q8—942no—
356C4×Q8—942no—
357C4×Q8—942no—
358C4×D4—474no—
359C4×D4—474no—
360C4×D4—942no—
361C4×D4—942no—
362C4×D4—944no—
363C4×D4—944no—
364C4×D4—944no—
365C4×D4—944no—
366#117—22no—
367#117—22no—
368#117—944no—
369#117—944no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—944no—
379C2×C4⋊C4—944no—
380C2×C4⋊C4—942no—
381C2×C4⋊C4—942no—
382C2×C4⋊C4C4×D4711yescharacteristic
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C22⋊C4—471no—
385C2×C22⋊C4—471no—
386C2×C22⋊C4—471no—
387C2×C22⋊C4—471no—
388C2×C22⋊C4C4×D4711yescharacteristic
389C2×C22⋊C4C22×D4711yescharacteristic
390C2×C42—471no—
391C2×C42—471no—
392C2×C42—471no—
393C2×C42—471no—
394C2×C42C4×D4711yescharacteristic
395C2×C42C22×D4711yescharacteristic
396C4⋊C8—22no—
397C4⋊C8—942no—
398C22⋊C8—474no—
399C22⋊C8—944no—
400C22⋊C8—944no—
401C22⋊C8C22×D4714yes—
402#97—22no—
403#97—942no—
404C24—471no—
405C24—471no—
406C24C2×C4×D4711yescharacteristic
407#41—9416no—
408#41—9416no—
409C2×Q8—942no—
410C2×Q8—942no—
411C2×Q8—941no—
412C2×Q8—941no—
413C2×D4—474no—
414C2×D4—472no—
415C2×D4—472no—
416C2×D4—942no—
417C2×D4—942no—
418C2×D4—942no—
419C2×D4—942no—
420C2×D4—942no—
421C2×D4—942no—
422C2×D4—942no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—471no—
428C22×C4—471no—
429C22×C4—471no—
430C22×C4—471no—
431C22×C4—474no—
432C22×C4—474no—
433C22×C4—474no—
434C22×C4—471no—
435C22×C4—471no—
436C22×C4—942no—
437C22×C4—942no—
438C22×C4—942no—
439C22×C4—942no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—942no—
443C22×C4—942no—
444C22×C4—942no—
445C22×C4—3761no—
446C22×C4—3761no—
447C22×C4—3761no—
448C22×C4—3761no—
449C22×C4—3761no—
450C22×C4—3761no—
451C22×C4C2×C4×D4711yescharacteristic
452C22×C4C2×C4×D4711yescharacteristic
453C22×C4C2×C4×D4711yescharacteristic
454C22×C4C2×C4×D4711yescharacteristic
455C22×C4C2×C4×D4711yescharacteristic
456C22×C4C23×D4711yescharacteristic
457M16—3764no—
458C2×C8—24no—
459C2×C8—944no—
460C2×C8—3764no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—472no—
464C4⋊C4—472no—
465C4⋊C4—942no—
466C4⋊C4—942no—
467C4⋊C4—942no—
468C4⋊C4—942no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—474no—
476C22⋊C4—474no—
477C22⋊C4—942no—
478C22⋊C4—942no—
479C42—21no—
480C42—21no—
481C42—942no—
482C42—942no—
483C42—942no—
484C42—942no—
485C42D4×D4712yes—
486C42D4×D4712yes—
487C23—21no—
488C23—21no—
489C23—41no—
490C23—41no—
491C23—472no—
492C23—472no—
493C23—472no—
494C23—472no—
495C23—941no—
496C23—941no—
497C23—1881no—
498C23—1881no—
499C23—1881no—
500C23—1881no—
501C23M16×D4712yes—
502C23C22×C4×D4711yescharacteristic
503Q8—944no—
504Q8—944no—
505D4—944no—
506D4—944no—
507D4—948no—
508D4—948no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—472no—
522C2×C4—472no—
523C2×C4—472no—
524C2×C4—472no—
525C2×C4—948no—
526C2×C4—948no—
527C2×C4—941no—
528C2×C4—941no—
529C2×C4—941no—
530C2×C4—941no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—3762no—
534C2×C4—3762no—
535C2×C4—3762no—
536C2×C4—3762no—
537C2×C4—1884no—
538C2×C4—1884no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4C22⋊C4×D4711yescharacteristic
544C2×C4C22⋊C4×D4711yescharacteristic
545C2×C4C22⋊C4×D4712yes—
546C2×C4M16×D4712yes—
547C2×C4C2×D4×D4712yes—
548C8—44no—
549C8—1884no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—474no—
558C22—942no—
559C22—942no—
560C22—942no—
561C22—942no—
562C22—942no—
563C22—1882no—
564C22—1882no—
565C22—1882no—
566C22—1882no—
567C22C2×C22⋊C4×D4711yescharacteristic
568C22C2×M16×D4711yescharacteristic
569C22C2×M16×D4711yescharacteristic
570C4—22no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—942no—
578C4—942no—
579C4—942no—
580C4—942no—
581C4—1881no—
582C4—1881no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—944no—
587C2—1882no—
588C2#407×D4711yescharacteristic
589C2#2025184711yescharacteristic
590C2#2025186911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1464C2
socleC22×C47C22×C22⋊C4
prime radicalC22×#10625350C4
cosocleC22×C4×C47C24
hypercenter#1464D47
hypocenterC47#17368
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20252416C2semidirect
3. #20241731C2semidirect
4. #20241723C2semidirect
5. #20241722C2semidirect
6. #20241716C2semidirect
7. #20241703C2semidirect
8. #324×D47C2semidirect
9. C47×#1464C2semidirect
126. C2×C22⋊D47M16semidirect
154. C2×C4×D47M16semidirect
293. #1464D47semidirect
334. C47#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#156896032
#202524162
#202417312
#202417232
#202417222
#202417162
#202417032
#324×D472
C47×#14642
#1736847