Finite Group Databasebeta

← #26182181

Subgroups — #26182181

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689595C211yesmaximal, characteristic
2#20252417C211yesmaximal, characteristic, complemented
3#20241728C212yesmaximal, complemented
4#20241724C212yesmaximal, complemented
5#20241720C212yesmaximal, complemented
6#20241714C212yesmaximal, complemented
7#20241704C212yesmaximal, complemented
8C2×#15688846C212yesmaximal, complemented
9C47×#2973C211yesmaximal, characteristic, complemented
10C22×D4×D47C411yescharacteristic
11#15689595—24no—
12#15689595—24no—
13C2×C4×D188C2211yescharacteristic
14C2×D4×Dic47C411yescharacteristic
15C2×#13093986C411yescharacteristic
16C2×C4×C22⋊D47C2211yescharacteristic
17C2×C4×C22⋊D47C2211yescharacteristic
18C2×(C22⋊D47)⋊C4C411yescharacteristic
19C2×(C22⋊D47)⋊C4C411yescharacteristic
20C2×#13093982C411yescharacteristic
21C2×#13093981C2211yescharacteristic
22C22×(C2×D47)⋊C4C2211yescharacteristic
23C22×(C2×D47)⋊C4C2211yescharacteristic
24C2×C22⋊C4×D47C411yescharacteristic
25C2×C22⋊C4×D47C411yescharacteristic
26#15689195—24no—
27#15689180C2212yescomplemented
28#15689087C2212yescomplemented
29C2×C22⋊(C47⋊C8)C2212yescomplemented
30#15688848C2214yescomplemented
31#15688846C2214yes—
32#15688846C2214yescomplemented
33#15688846C2214yescomplemented
34C2×C4×D4×C47C2211yescharacteristic
35C47×#435—24no—
36C47×#432C2212yescomplemented
37C47×#407C2212yescomplemented
38C2×C22⋊C8×C47C2212yescomplemented
39C24×D47C2×C411yescharacteristic
40C24×D47C2×C411yescharacteristic
41C22×D188C2×C411yescharacteristic
42C22×C22⋊D47C2×C411yescharacteristic
43C22×C22⋊D47C2×C411yescharacteristic
44C2×D4×D47—24no—
45C2×D4×D47—24no—
46C2×D4×D47—24no—
47C23×Dic47C2×C411yescharacteristic
48C23×Dic47C2×C411yescharacteristic
49C22×C4×D47C2×C411yescharacteristic
50C22×C4×D47C2×C411yescharacteristic
51C22×C4×D47C2×C411yescharacteristic
52C22×C4×D47C2×C411yescharacteristic
53C22×C4×D47C2×C411yescharacteristic
54C2×#10625329—22no—
55C22×C47⋊C8—22no—
56C4×D188—22no—
57C4×D188—22no—
58D4×Dic47—24no—
59#13093986—24no—
60C4×C22⋊D47—22no—
61C4×C22⋊D47—22no—
62C4×C22⋊D47—22no—
63C4×C22⋊D47—22no—
64(C22⋊D47)⋊C4—24no—
65(C22⋊D47)⋊C4—24no—
66#13093982—24no—
67#13093981—22no—
68#13093981—22no—
69C2×C4⋊Dic47C2×C411yescharacteristic
70C2×Dic47⋊C4C2×C411yescharacteristic
71C2×Dic47⋊C4C2311yescharacteristic
72C2×C22⋊Dic47C2×C411yescharacteristic
73C2×C22⋊Dic47C2×C411yescharacteristic
74C2×(C2×D47)⋊C4—22no—
75C2×(C2×D47)⋊C4—22no—
76C2×(C2×D47)⋊C4—22no—
77C2×(C2×D47)⋊C4—22no—
78C2×(C2×D47)⋊C4C2×C411yescharacteristic
79C2×(C2×D47)⋊C4C2×C411yescharacteristic
80C2×(C2×D47)⋊C4C2×C411yescharacteristic
81C2×(C2×D47)⋊C4C2×C411yescharacteristic
82C2×(C2×D47)⋊C4C2×C411yescharacteristic
83C2×(C2×D47)⋊C4C2311yescharacteristic
84C2×(C2×D47)⋊C4C2311yescharacteristic
85C2×(C2×D47)⋊C4C2311yescharacteristic
86C22⋊C4×D47—24no—
87C22⋊C4×D47—24no—
88C2×C4×Dic47C2×C411yescharacteristic
89C2×C4×Dic47C2×C411yescharacteristic
90C2×C4×Dic47C2×C411yescharacteristic
91C4⋊(C47⋊C8)—22no—
92C22⋊(C47⋊C8)—22no—
93C22⋊(C47⋊C8)—22no—
94C22⋊(C47⋊C8)C2314yes—
95#13093935—22no—
96C22×D4×C47C2×C411yescharacteristic
97C23×C4×C47C2311yescharacteristic
98C23×C4×C47C2311yescharacteristic
99C2×M16×C47—22no—
100C22×C8×C47—22no—
101C4×D4×C47D414yes—
102C4×D4×C47D414yes—
103C2×C4⋊C4×C47C2×C411yescharacteristic
104C2×C22⋊C4×C47C2×C411yescharacteristic
105C2×C22⋊C4×C47C2×C411yescharacteristic
106C2×C42×C47C2311yescharacteristic
107C4⋊C8×C47—22no—
108C22⋊C8×C47—22no—
109C22⋊C8×C47—22no—
110C22⋊C8×C47C2314yes—
111#97×C47—22no—
112C24×C47C22×C411yescharacteristic
113C24×C47C22×C411yescharacteristic
114C23×D47—22no—
115C23×D47—22no—
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—22no—
119C23×D47—81no—
120C23×D47—81no—
121C23×D47—81no—
122C23×D47—81no—
123C23×D47M1614yescomplemented
124C23×D47C22×C411yescharacteristic
125C23×D47C22×C411yescharacteristic
126C23×D47C22×C411yescharacteristic
127C2×D4×C47—22no—
128C2×D4×C47—22no—
129C2×D4×C47C22⋊C414yes—
130C22×C4×C47—81no—
131C22×C4×C47—81no—
132C22×C4×C47C22×C411yescharacteristic
133C22×C4×C47C22×C411yescharacteristic
134C22×C4×C47C22×C411yescharacteristic
135C22×C4×C47C22×C411yescharacteristic
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2411yescharacteristic
141C2×C22⋊D47—22no—
142C2×C22⋊D47—22no—
143C2×C22⋊D47—22no—
144C2×C22⋊D47—22no—
145C2×C22⋊D47—22no—
146C2×C22⋊D47M1614yescomplemented
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—81no—
150C22×Dic47—81no—
151C22×Dic47C22×C411yescharacteristic
152C22×Dic47C22×C411yescharacteristic
153C22×Dic47C22×C411yescharacteristic
154C22×Dic47C22×C411yescharacteristic
155C22×Dic47C22×C411yescharacteristic
156D4×D47—232no—
157C2×D188—22no—
158C2×D188—22no—
159C2×D188—22no—
160C2×C4×D47—22no—
161C2×C4×D47—22no—
162C2×C4×D47—22no—
163C2×C4×D47—22no—
164C2×C4×D47—22no—
165C2×C4×D47—22no—
166C2×C4×D47—22no—
167C2×C4×D47—81no—
168C2×C4×D47—81no—
169C2×C4×D47—81no—
170C2×C4×D47—81no—
171M16×C47—82no—
172C2×C8×C47—24no—
173C2×C8×C47—82no—
174C4⋊C4×C47C22⋊C414yes—
175C22⋊C4×C47C22⋊C414yes—
176C22⋊C4×C47C22⋊C414yes—
177C42×C47C2×D412yes—
178C42×C47C2×D412yes—
179C22⋊Dic47—22no—
180C22⋊Dic47—22no—
181(C2×D47)⋊C4—22no—
182(C2×D47)⋊C4—22no—
183(C2×D47)⋊C4—22no—
184(C2×D47)⋊C4—22no—
185(C2×D47)⋊C4—22no—
186(C2×D47)⋊C4—21no—
187(C2×D47)⋊C4—21no—
188(C2×D47)⋊C4—21no—
189(C2×D47)⋊C4—21no—
190(C2×D47)⋊C4—21no—
191(C2×D47)⋊C4—21no—
192C4⋊Dic47—22no—
193Dic47⋊C4—22no—
194Dic47⋊C4—21no—
195Dic47⋊C4—21no—
196C4×Dic47—22no—
197C4×Dic47—22no—
198C4×Dic47—22no—
199#10625329—82no—
200C2×C47⋊C8—24no—
201C2×C47⋊C8—82no—
202C23×C47—21no—
203C23×C47—21no—
204C23×C47—41no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—41no—
208C23×C47C2×C22⋊C412yes—
209C23×C47C2×C22⋊C412yes—
210C23×C47C2×M1612yes—
211C23×C47C2×M1612yes—
212C23×C47C23×C411yescharacteristic
213C22×D47—21no—
214C22×D47—21no—
215C22×D47—21no—
216C22×D47—21no—
217C22×D47—21no—
218C22×D47—21no—
219C22×D47—28no—
220C22×D47—28no—
221C22×D47—41no—
222C22×D47—41no—
223C22×D47—41no—
224C22×D47—41no—
225C22×D47—41no—
226C22×D47—41no—
227C22×D47—82no—
228C22×D47—82no—
229C22×D47—82no—
230C22×D47—82no—
231C22×D47—84no—
232C22×D47—84no—
233C22×D47—84no—
234C22×D47—84no—
235C22×D47C2×M1612yes—
236C22×D47C2×M1612yes—
237C22×D47C2×M1612yes—
238D4×C47—28no—
239C2×C4×C47—21no—
240C2×C4×C47—21no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47C2×C22⋊C412yes—
250C2×C4×C47C2×C22⋊C412yes—
251C2×C4×C47C2×C22⋊C412yes—
252C2×C4×C47C2×C22⋊C412yes—
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C22⋊D47—28no—
256C22⋊D47—28no—
257C2×Dic47—21no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—21no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—82no—
268C2×Dic47—82no—
269C2×Dic47C2×M1612yes—
270D188—28no—
271C4×D47—28no—
272C4×D47—82no—
273C4×D47—82no—
274C4×D47—82no—
275C4×D47—82no—
276C8×C47—44no—
277C47⋊C8—44no—
278#17367—471nomaximal, Sylow
279C22×C47—21no—
280C22×C47—22no—
281C22×C47—21no—
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—22no—
286C22×C47—42no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47#40714yes—
291C22×C47C22×C22⋊C411yescharacteristic
292C22×C47C22×M1611yescharacteristic
293C22×C47C22×M1611yescharacteristic
294C2×D47—22no—
295C2×D47—22no—
296C2×D47—22no—
297C2×D47—42no—
298C2×D47—42no—
299C2×D47—42no—
300C2×D47—41no—
301C2×D47—41no—
302C2×D47—41no—
303C2×D47—41no—
304C2×D47—41no—
305C2×D47—41no—
306C2×D47—42no—
307C2×D47—42no—
308C2×D47—42no—
309C2×D47—88no—
310C2×D47—88no—
311C4×C47—22no—
312C4×C47—21no—
313C4×C47—21no—
314C4×C47—22no—
315C4×C47—22no—
316C4×C47—22no—
317Dic47—22no—
318Dic47—41no—
319Dic47—41no—
320Dic47—41no—
321Dic47—41no—
322C2×#518—471no—
323#2973—471no—
324#2973D4711yescharacteristic, complemented
325#1466—472no—
326#1465—472no—
327#1464—472no—
328#1462—472no—
329#1459—472no—
330C2×#323—472no—
331C2×C47—21no—
332C2×C47—21no—
333C2×C47—24no—
334C2×C47—42no—
335C2×C47C2×#40711yescharacteristic
336C2×C47#288311yescharacteristic
337C2×C47#288811yescharacteristic
338D47—42no—
339D47—42no—
340D47—42no—
341C23×D4—471no—
342#518—944no—
343#518—944no—
344C2×C4×D4—471no—
345C2×C4×D4—471no—
346C2×C4×D4—471no—
347C2×C4×D4—471no—
348C2×C4×D4—471no—
349C2×C4×D4—471no—
350C2×C4×D4—471no—
351C2×C4×D4C2×D4711yescharacteristic
352C2×#117—471no—
353C2×#117—471no—
354C22×C22⋊C4—471no—
355C22×C22⋊C4—471no—
356C22×C22⋊C4—471no—
357C22×C22⋊C4—471no—
358#435—24no—
359#435—944no—
360#432—472no—
361#432C2×D4712yescomplemented
362#407—472no—
363#407C2×D4712yescomplemented
364C2×C22⋊C8—472no—
365C2×C22⋊C8C2×D4712yescomplemented
366#324—474no—
367#323—474no—
368#323—474no—
369#323—474no—
370C47#1736711yescharacteristic, complemented, Sylow
371C25—471no—
372C25—471no—
373C22×D4—471no—
374C22×D4—471no—
375C22×D4—471no—
376C22×D4—944no—
377C22×D4—944no—
378C22×D4—944no—
379C22×D4C4×D4711yescharacteristic
380C23×C4—471no—
381C23×C4—471no—
382C23×C4—471no—
383C23×C4—471no—
384C23×C4—471no—
385C23×C4—471no—
386C23×C4—471no—
387C23×C4C22×D4711yescharacteristic
388C23×C4C22×D4711yescharacteristic
389C2×M16—22no—
390C2×M16—942no—
391C22×C8—22no—
392C22×C8—942no—
393C4×D4—944no—
394C4×D4—942no—
395C4×D4—942no—
396C4×D4—942no—
397C4×D4—942no—
398C4×D4—944no—
399C4×D4—944no—
400C4×D4—944no—
401C4×D4—942no—
402C4×D4—942no—
403C4×D4D18814yes—
404C4×D4C22⋊D4714yes—
405#117—944no—
406#117—942no—
407#117—942no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—471no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4C4×D4711yescharacteristic
412C2×C22⋊C4—471no—
413C2×C22⋊C4—471no—
414C2×C22⋊C4—471no—
415C2×C22⋊C4—471no—
416C2×C22⋊C4—471no—
417C2×C22⋊C4—471no—
418C2×C22⋊C4—471no—
419C2×C22⋊C4—471no—
420C2×C22⋊C4—471no—
421C2×C22⋊C4—471no—
422C2×C22⋊C4—942no—
423C2×C22⋊C4—942no—
424C2×C22⋊C4—942no—
425C2×C22⋊C4—942no—
426C2×C22⋊C4—944no—
427C2×C22⋊C4—944no—
428C2×C22⋊C4C4×D4711yescharacteristic
429C2×C22⋊C4C4×D4711yescharacteristic
430C2×C42—471no—
431C2×C42—471no—
432C2×C42—471no—
433C2×C42C22×D4711yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—942no—
436C22⋊C8—22no—
437C22⋊C8—22no—
438C22⋊C8—474no—
439C22⋊C8—942no—
440C22⋊C8—942no—
441C22⋊C8C22×D4714yes—
442#97—22no—
443#97—942no—
444C24—474no—
445C24—471no—
446C24—471no—
447C24—471no—
448C24—942no—
449C24—942no—
450C24—942no—
451C24—942no—
452C24—942no—
453C24—3761no—
454C24—3761no—
455C24—3761no—
456C24—3761no—
457C24C2×C4×D4711yescharacteristic
458C24C2×C4×D4711yescharacteristic
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—474no—
462C2×D4—942no—
463C2×D4—942no—
464C2×D4—942no—
465C2×D4—9432no—
466C2×D4—942no—
467C2×D4—942no—
468C2×D4—942no—
469C2×D4—942no—
470C2×D4—942no—
471C2×D4(C2×D47)⋊C414yes—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—471no—
475C22×C4—471no—
476C22×C4—471no—
477C22×C4—471no—
478C22×C4—471no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—942no—
488C22×C4—3761no—
489C22×C4—3761no—
490C22×C4—3761no—
491C22×C4—3761no—
492C22×C4—3761no—
493C22×C4—3761no—
494C22×C4C2×C4×D4711yescharacteristic
495C22×C4C2×C4×D4711yescharacteristic
496C22×C4C2×C4×D4711yescharacteristic
497C22×C4C2×C4×D4711yescharacteristic
498C22×C4C2×D18812yes—
499C22×C4C2×D18812yes—
500C22×C4C2×C22⋊D4712yes—
501C22×C4C2×C22⋊D4712yes—
502C22×C4C23×D4711yescharacteristic
503M16—82no—
504M16—3762no—
505C2×C8—24no—
506C2×C8—82no—
507C2×C8—944no—
508C2×C8—3762no—
509C4⋊C4—942no—
510C4⋊C4—942no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4(C2×D47)⋊C414yes—
514C22⋊C4—942no—
515C22⋊C4—942no—
516C22⋊C4—942no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C22⋊C4—942no—
520C22⋊C4—942no—
521C22⋊C4—941no—
522C22⋊C4—941no—
523C22⋊C4—941no—
524C22⋊C4—941no—
525C22⋊C4—941no—
526C22⋊C4—941no—
527C22⋊C4(C2×D47)⋊C414yes—
528C22⋊C4(C2×D47)⋊C414yes—
529C42—942no—
530C42—942no—
531C42—942no—
532C42C2×D18812yes—
533C42C2×C22⋊D4712yes—
534C23—21no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—472no—
541C23—472no—
542C23—472no—
543C23—948no—
544C23—948no—
545C23—941no—
546C23—941no—
547C23—941no—
548C23—941no—
549C23—941no—
550C23—941no—
551C23—3762no—
552C23—3762no—
553C23—3762no—
554C23—3762no—
555C23—3764no—
556C23—3764no—
557C23—3764no—
558C23—3764no—
559C23—1881no—
560C23—1881no—
561C23—1881no—
562C23—1881no—
563C23—1881no—
564C23—1881no—
565C23C2×(C2×D47)⋊C412yes—
566C23C2×(C2×D47)⋊C412yes—
567C23M16×D4712yes—
568C23M16×D4712yes—
569C23C22×C4×D4711yescharacteristic
570D4—28no—
571D4—948no—
572D4—948no—
573D4—948no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—472no—
585C2×C4—948no—
586C2×C4—941no—
587C2×C4—941no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—3762no—
593C2×C4—3762no—
594C2×C4—3762no—
595C2×C4—3762no—
596C2×C4—3762no—
597C2×C4—3762no—
598C2×C4—1881no—
599C2×C4—1881no—
600C2×C4—1881no—
601C2×C4—1881no—
602C2×C4C2×(C2×D47)⋊C412yes—
603C2×C4C2×(C2×D47)⋊C412yes—
604C2×C4C2×(C2×D47)⋊C412yes—
605C2×C4C2×(C2×D47)⋊C412yes—
606C2×C4C22×C22⋊D4711yescharacteristic
607C2×C4C22×D18811yescharacteristic
608C8—44no—
609C8—1884no—
610C22—21no—
611C22—22no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—942no—
622C22—942no—
623C22—942no—
624C22—3768no—
625C22—3768no—
626C22—1882no—
627C22—1882no—
628C22—1882no—
629C22—1882no—
630C22—1882no—
631C22—1882no—
632C22—1881no—
633C22—1881no—
634C22—1881no—
635C22—1881no—
636C22—1881no—
637C22—1881no—
638C22#1568908614yes—
639C22C22×(C2×D47)⋊C411yescharacteristic
640C22C2×M16×D4711yescharacteristic
641C22C2×M16×D4711yescharacteristic
642C4—22no—
643C4—21no—
644C4—21no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—942no—
649C4—1881no—
650C4—1881no—
651C4—1881no—
652C4—1881no—
653C2—21no—
654C2—21no—
655C2—24no—
656C2—42no—
657C2—1882no—
658C2—1882no—
659C2—1882no—
660C2C2×#1568908611yescharacteristic
661C2#2025184811yescharacteristic
662C2#2025186611yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2973C2
socleC22×C47C22×C22⋊C4
prime radicalC22×D4×D47C4
cosocleC22×C4×C47C24
hypercenter#2973D47
hypocenterC47#17367
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20252417C2semidirect
3. #20241728C2semidirect
4. #20241724C2semidirect
5. #20241720C2semidirect
6. #20241714C2semidirect
7. #20241704C2semidirect
8. C2×#15688846C2semidirect
9. C47×#2973C2semidirect
27. #15689180C22semidirect
28. #15689087C22semidirect
29. C2×C22⋊(C47⋊C8)C22semidirect
30. #15688848C22semidirect
32. #15688846C22semidirect
33. #15688846C22semidirect
36. C47×#432C22semidirect
37. C47×#407C22semidirect
38. C2×C22⋊C8×C47C22semidirect
123. C23×D47M16semidirect
146. C2×C22⋊D47M16semidirect
324. #2973D47semidirect
361. #432C2×D47semidirect
363. #407C2×D47semidirect
365. C2×C22⋊C8C2×D47semidirect
370. C47#17367semidirect

Maximal subgroups

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

subgroupindex
C2×#156895952
#202524172
#202417282
#202417242
#202417202
#202417142
#202417042
C2×#156888462
C47×#29732
#1736747