Finite Group Databasebeta

← #26217978

Subgroups — #26217978

Order 12032.

1986 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C211yesmaximal, characteristic, complemented
2C2×#15689634C211yesmaximal, characteristic
3C2×#15689122C212yesmaximal, complemented
4#20243824C212yesmaximal, complemented
5#20243819C212yesmaximal, complemented
6#20243814C212yesmaximal, complemented
7#20243809C212yesmaximal, complemented
8#20243107C212yesmaximal, complemented
9C2×C47×#380C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C22×#10625329—28no—
12C22×#10625329—24no—
13C22×#10625329C2212yes—
14C22×#10625329C2212yes—
15C23×C47⋊C8C2212yes—
16#15689634—24no—
17#15689634C418yescomplemented
18C2×#13094007—21no—
19C2×#13094003C2211yescharacteristic
20C2×#13094003C2211yescharacteristic
21C2×#13093998C2211yescharacteristic
22C2×#13093998C2211yescharacteristic
23C2×#13093993—21no—
24C2×C4×C22⋊D47—21no—
25C2×C4×C22⋊D47—21no—
26C2×#13093983C2211yescharacteristic
27#15689122—28no—
28C2×#13093947C2214yescomplemented
29C2×#13093944C2214yescomplemented
30#15688885C2214yescomplemented
31#15688885C2214yescomplemented
32C24×C4×C47C2211yescharacteristic
33C22×C4⋊C4×C47C2212yes—
34C22×C22⋊C4×C47—21no—
35C47×#380—21no—
36C47×#380—21no—
37C47×#380C2212yes—
38C47×#380C2212yes—
39C2×#95×C47—21no—
40C2×#10625350—22no—
41C2×#10625350—24no—
42C2×#10625350—24no—
43C2×#10625350C2×C414yes—
44C22×Dic94C2311yescharacteristic
45C22×D188C2311yescharacteristic
46C22×C22⋊D47—21no—
47C22×C22⋊D47—21no—
48C22×C22⋊D47—21no—
49C22×C4×D47—21no—
50C2×#10625329—232no—
51C2×#10625329—24no—
52C2×#10625329—24no—
53C2×#10625329—24no—
54C2×#10625329—28no—
55C2×#10625329D418yescomplemented
56C22×C47⋊C8—24no—
57C22×C47⋊C8—28no—
58C22×C47⋊C8C2314yes—
59#13094007—28no—
60#13094003—22no—
61#13094003—22no—
62#13094003C2×C414yescomplemented
63#13094003C2×C414yescomplemented
64#13093998—22no—
65#13093998—22no—
66#13093998C2×C414yescomplemented
67#13093998C2×C414yescomplemented
68#13093993—28no—
69C4×C22⋊D47—28no—
70C4×C22⋊D47—28no—
71#13093983—22no—
72#13093983C2×C414yes—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C22⋊Dic47—21no—
80C2×(C2×D47)⋊C4—21no—
81C2×(C2×D47)⋊C4—21no—
82C2×C4×Dic47—21no—
83#13093947—28no—
84#13093944—28no—
85C25×C47D411yescharacteristic
86C23×C4×C47—24no—
87C23×C4×C47—24no—
88C23×C4×C47—22no—
89C23×C4×C47C2×C414yes—
90C23×C4×C47D411yescharacteristic
91C23×C4×C47D411yescharacteristic
92C23×C4×C47D411yescharacteristic
93C23×C4×C47D414yes—
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2311yescharacteristic
96C23×C4×C47C2311yescharacteristic
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47—24no—
99C2×C4⋊C4×C47—22no—
100C2×C4⋊C4×C47C2314yes—
101C2×C22⋊C4×C47—22no—
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47—44no—
104#95×C47—22no—
105#95×C47—22no—
106C24×C47—24no—
107C24×C47—22no—
108C24×C47—22no—
109C24×C47—42no—
110C24×C47C22⋊C414yes—
111C24×C47C2×D411yescharacteristic
112C24×C47C2×D411yescharacteristic
113C24×C47C2×D411yescharacteristic
114C23×D47—21no—
115C22×C4×C47—24no—
116C22×C4×C47—216no—
117C22×C4×C47—28no—
118C22×C4×C47—24no—
119C22×C4×C47—24no—
120C22×C4×C47—28no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—24no—
126C22×C4×C47—44no—
127C22×C4×C47—44no—
128C22×C4×C47—44no—
129C22×C4×C47C22⋊C414yes—
130C22×C4×C47C22⋊C414yes—
131C22×C4×C47C22⋊C414yes—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C22×C412yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D414yes—
142C22×C4×C47#4114yes—
143C22×C4×C47C2411yescharacteristic
144C2×C22⋊D47—24no—
145C2×C22⋊D47—24no—
146C2×C22⋊D47—24no—
147C2×C22⋊D47—44no—
148C2×C22⋊D47—44no—
149C2×C22⋊D47—42no—
150C2×C22⋊D47—42no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—21no—
154#10625350—216no—
155#10625350—44no—
156#10625350—44no—
157C2×D188—22no—
158C2×D188—22no—
159C2×D188—21no—
160C2×D188C22×C412yes—
161C2×C4×D47—24no—
162C2×C4×D47—44no—
163C2×Dic94—22no—
164C2×Dic94—22no—
165C2×Dic94—21no—
166C2×Dic94C22×C412yes—
167C4⋊C4×C47—24no—
168C4⋊C4×C47—24no—
169C4⋊C4×C47—44no—
170C22⋊C4×C47—44no—
171C22⋊C4×C47—44no—
172C22⋊Dic47—24no—
173C22⋊Dic47—24no—
174C22⋊Dic47—24no—
175(C2×D47)⋊C4—24no—
176(C2×D47)⋊C4—24no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47C22×C412yes—
180C4⋊Dic47C22×C412yes—
181Dic47⋊C4—24no—
182Dic47⋊C4—24no—
183C4×Dic47—24no—
184#10625329—216no—
185#10625329—216no—
186C2×C47⋊C8—216no—
187C2×C47⋊C8—24no—
188C2×C47⋊C8—24no—
189C2×C47⋊C8C2×D418yes—
190C23×C47—22no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47—22no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—22no—
199C23×C47—28no—
200C23×C47—28no—
201C23×C47—44no—
202C23×C47—48no—
203C23×C47—48no—
204C23×C47C2×C22⋊C412yes—
205C23×C47C2×C22⋊C412yes—
206C23×C47C2×C22⋊C412yes—
207C23×C47#12114yes—
208C23×C47AGL(1,ℤ/8)12yes—
209C23×C47#13712yes—
210C23×C47C22×D411yescharacteristic
211C22×D47—22no—
212C22×D47—42no—
213C22×D47—41no—
214C22×D47—41no—
215C22×D47—42no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—28no—
219C2×C4×C47—24no—
220C2×C4×C47—24no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—28no—
226C2×C4×C47—28no—
227C2×C4×C47—44no—
228C2×C4×C47—44no—
229C2×C4×C47—44no—
230C2×C4×C47—88no—
231C2×C4×C47C2×C22⋊C412yes—
232C2×C4×C47C2×C22⋊C412yes—
233C2×C4×C47C2×C22⋊C412yes—
234C2×C4×C47C4×D418yes—
235C2×C4×C47#12014yes—
236C2×C4×C47#12114yes—
237C2×C4×C47#12314yes—
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D412yes—
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C2×#4112yes—
242C22⋊D47—48no—
243C22⋊D47—48no—
244C22⋊D47—48no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—42no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—42no—
252D188—24no—
253D188—41no—
254D188—41no—
255C4×D47—48no—
256Dic94—24no—
257Dic94—41no—
258Dic94—41no—
259C47⋊C8—28no—
260#20971—471nomaximal, Sylow
261C22×C47—22no—
262C22×C47—22no—
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—24no—
266C22×C47—24no—
267C22×C47—24no—
268C22×C47—24no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—416no—
273C22×C47—42no—
274C22×C47—42no—
275C22×C47#38614yes—
276C22×C47#41714yescomplemented
277C22×C47C22×C22⋊C411yescharacteristic
278C22×C47C2×#12112yes—
279C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C47C2×#13711yescharacteristic
281C22×C47#57512yes—
282C2×D47—42no—
283C2×D47—42no—
284C2×D47—42no—
285C2×D47—82no—
286C2×D47—82no—
287C4×C47—22no—
288C4×C47—22no—
289C4×C47—84no—
290C4×C47#38614yes—
291C4×C47C2×C4×D412yes—
292C4×C47C2×#12011yescharacteristic
293C4×C47C2×#12311yescharacteristic
294Dic47—42no—
295Dic47—42no—
296Dic47—42no—
297C23×M16—471no—
298C2×#525—471no—
299C2×#417—472no—
300C2×#380D4711yescharacteristic, complemented
301#1895—472no—
302#1894—472no—
303#1893—472no—
304#1892—472no—
305#1736—472no—
306C2×C47—24no—
307C2×C47—24no—
308C2×C47—42no—
309C2×C47C2×#38611yescharacteristic
310C2×C47C2×#41711yescharacteristic
311C2×C47#289611yescharacteristic
312C2×C47#305512yes—
313C2×C47#305612yes—
314D47—82no—
315C22×#41—471no—
316C24×C4C2×D4711yescharacteristic
317C22×M16—472no—
318C22×M16—472no—
319C22×M16—948no—
320C22×M16—944no—
321C23×C8—472no—
322#525—478no—
323#525—944no—
324C2×#123—941no—
325C2×C22⋊Q8—471no—
326C2×C22⋊Q8—471no—
327C2×#121—471no—
328C2×#121—471no—
329C2×#120—941no—
330C2×C4×D4—941no—
331C2×C4×D4—941no—
332C2×#117—471no—
333C22×C4⋊C4C2×D4712yes—
334C22×C22⋊C4—21no—
335#417—948no—
336C2×#103—474no—
337C2×#102—474no—
338#380—21no—
339#380—21no—
340#380C2×D4712yes—
341#380C2×D4712yes—
342C2×#95—21no—
343#340—474no—
344#340—474no—
345C47#2097111yescharacteristic, complemented, Sylow
346C25C22⋊D4711yescharacteristic
347C2×#41—474no—
348C2×#41—942no—
349C2×#41—944no—
350C2×#41—944no—
351C22×Q8—471no—
352C22×D4—471no—
353C22×D4—941no—
354C22×D4—941no—
355C22×D4—941no—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—22no—
359C23×C4—941no—
360C23×C4C4×D4714yes—
361C23×C4D18814yes—
362C23×C4C22⋊D4711yescharacteristic
363C23×C4C22⋊D4711yescharacteristic
364C23×C4C22⋊D4711yescharacteristic
365C23×C4C22×D4711yescharacteristic
366C23×C4C22×D4711yescharacteristic
367C23×C4C22×D4711yescharacteristic
368C2×M16—478no—
369C2×M16—948no—
370C2×M16—9432no—
371C2×M16—944no—
372C2×M16—944no—
373C2×M16—944no—
374C22×C8—474no—
375C22×C8—948no—
376C22×C8—944no—
377#123—948no—
378C22⋊Q8—474no—
379C22⋊Q8—474no—
380C22⋊Q8—942no—
381C22⋊Q8—942no—
382#121—474no—
383#121—474no—
384#121—942no—
385#121—942no—
386#120—948no—
387C4×D4—948no—
388C4×D4—948no—
389#117—474no—
390#117—942no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—471no—
395C2×C4⋊C4—471no—
396C2×C4⋊C4—941no—
397C2×C4⋊C4—941no—
398C2×C4⋊C4C22×D4714yes—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—44no—
402C2×C22⋊C4—941no—
403C2×C22⋊C4—941no—
404C2×C22⋊C4—941no—
405C2×C22⋊C4—941no—
406C2×C22⋊C4—941no—
407C2×C42—941no—
408#103—948no—
409#102—948no—
410#95—22no—
411#95—22no—
412C24—24no—
413C24—22no—
414C24—22no—
415C24—42no—
416C24—941no—
417C24(C2×D47)⋊C414yes—
418C24C2×C22⋊D4711yescharacteristic
419C24C2×C22⋊D4711yescharacteristic
420C24C2×C22⋊D4711yescharacteristic
421#41—9416no—
422#41—1884no—
423#41—1884no—
424C2×Q8—472no—
425C2×Q8—941no—
426C2×Q8—942no—
427C2×Q8—942no—
428C2×D4—472no—
429C2×D4—944no—
430C2×D4—944no—
431C2×D4—944no—
432C2×D4—942no—
433C2×D4—941no—
434C2×D4—942no—
435C2×D4—1884no—
436C2×D4—1884no—
437C2×D4—1882no—
438C2×D4—1882no—
439C22×C4—24no—
440C22×C4—216no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—941no—
454C22×C4—941no—
455C22×C4—941no—
456C22×C4—944no—
457C22×C4—1884no—
458C22×C4(C2×D47)⋊C414yes—
459C22×C4(C2×D47)⋊C414yes—
460C22×C4(C2×D47)⋊C414yes—
461C22×C4C2×C4×D4712yes—
462C22×C4C2×C4×D4712yes—
463C22×C4C2×C4×D4712yes—
464C22×C4C2×D18812yes—
465C22×C4C2×D18812yes—
466C22×C4C2×D18812yes—
467C22×C4D4×D4714yes—
468C22×C4#1062535214yes—
469C22×C4C2×C22⋊D4711yescharacteristic
470C22×C4C2×C22⋊D4711yescharacteristic
471C22×C4C2×C22⋊D4711yescharacteristic
472C22×C4C23×D4711yescharacteristic
473M16—9416no—
474M16—9416no—
475C2×C8—478no—
476C2×C8—9416no—
477C2×C8—944no—
478C2×C8—944no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—44no—
482C4⋊C4—472no—
483C4⋊C4—472no—
484C4⋊C4—944no—
485C4⋊C4—944no—
486C4⋊C4—941no—
487C4⋊C4—941no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—944no—
491C22⋊C4—944no—
492C22⋊C4—944no—
493C22⋊C4—944no—
494C22⋊C4—944no—
495C42—944no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—22no—
503C23—28no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—942no—
511C23—1882no—
512C23—1881no—
513C23—1881no—
514C23—1882no—
515C23C2×(C2×D47)⋊C412yes—
516C23C2×(C2×D47)⋊C412yes—
517C23C2×(C2×D47)⋊C412yes—
518C23#1309399614yes—
519C23#1309405012yes—
520C23#1309405512yes—
521C23C22×C22⋊D4711yescharacteristic
522Q8—944no—
523Q8—1881no—
524Q8—1881no—
525D4—944no—
526D4—1888no—
527D4—1888no—
528D4—1888no—
529D4—1881no—
530D4—1881no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—88no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—942no—
549C2×C4—1882no—
550C2×C4—1881no—
551C2×C4—1881no—
552C2×C4—1882no—
553C2×C4—1888no—
554C2×C4C2×(C2×D47)⋊C412yes—
555C2×C4C2×(C2×D47)⋊C412yes—
556C2×C4C2×(C2×D47)⋊C412yes—
557C2×C4(C22⋊D47)⋊C418yes—
558C2×C4#1309399414yes—
559C2×C4#1309399614yes—
560C2×C4#1309400814yes—
561C2×C4C22×C4×D4711yescharacteristic
562C2×C4C2×D4×D4712yes—
563C2×C4C22×D18811yescharacteristic
564C2×C4C2×#1062535212yes—
565C8—948no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—42no—
579C22—42no—
580C22—3762no—
581C22—3762no—
582C22—1882no—
583C22—1882no—
584C22—1882no—
585C22#1568900014yes—
586C22#1568912214yescomplemented
587C22C22×(C2×D47)⋊C411yescharacteristic
588C22C2×#1309399612yes—
589C22C2×#1309405011yescharacteristic
590C22C2×#1309405511yescharacteristic
591C22#1568992112yes—
592C4—22no—
593C4—22no—
594C4—84no—
595C4—1882no—
596C4—1882no—
597C4—1882no—
598C4#1568900014yes—
599C4C2×(C22⋊D47)⋊C412yes—
600C4C2×#1309399411yescharacteristic
601C4C2×#1309400811yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—3762no—
606C2C2×#1568900011yescharacteristic
607C2C2×#1568912211yescharacteristic
608C2#2025190511yescharacteristic
609C2#2025300312yes—
610C2#2025301412yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#380C2
socleC23×C47C22×D4
prime radicalC2×#15689634C2
cosocleC22×C4×C47C24
hypercenterC2×#380D47
hypocenterC47#20971
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C23×#10625329C2semidirect
3. C2×#15689122C2semidirect
4. #20243824C2semidirect
5. #20243819C2semidirect
6. #20243814C2semidirect
7. #20243809C2semidirect
8. #20243107C2semidirect
9. C2×C47×#380C2semidirect
17. #15689634C4semidirect
28. C2×#13093947C22semidirect
29. C2×#13093944C22semidirect
30. #15688885C22semidirect
31. #15688885C22semidirect
55. C2×#10625329D4semidirect
62. #13094003C2×C4semidirect
63. #13094003C2×C4semidirect
66. #13093998C2×C4semidirect
67. #13093998C2×C4semidirect
276. C22×C47#417semidirect
300. C2×#380D47semidirect
345. C47#20971semidirect
586. C22#15689122semidirect

Maximal subgroups

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

subgroupindex
C23×#106253292
C2×#156896342
C2×#156891222
#202438242
#202438192
#202438142
#202438092
#202431072
C2×C47×#3802
#2097147