Finite Group Databasebeta

← #26231924

Subgroups — #26231924

Order 12032.

1530 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×#15689778C211yesmaximal, characteristic
2C2×#15689201C211yesmaximal, characteristic, complemented
3C2×#15689122C212yesmaximal, complemented
4#20244311C211yesmaximal, characteristic, complemented
5#20244189C211yesmaximal, characteristic, complemented
6#20243819C212yesmaximal, complemented
7#20243695C212yesmaximal, complemented
8#20243385C211yesmaximal, characteristic, complemented
9#20243326C212yesmaximal, complemented
10C4×#13093944C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C47C211yesmaximal, characteristic, complemented
12C22×#10625350C2212yes—
13C22×#10625329C2212yes—
14#15689778—24no—
15#15689778C418yescomplemented
16C2×#13094024C2211yescharacteristic
17C2×#13094006—22no—
18C2×#13094003C2212yes—
19C2×#13093995—21no—
20C2×C4×D188C2211yescharacteristic
21C2×C4⋊C4×D47—21no—
22#15689201—24no—
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24#15689122—28no—
25C2×#13093947C2212yes—
26C2×#13093947C2212yes—
27C2×#13093947C2212yes—
28C2×#13093947C2212yes—
29C2×#13093944C2211yescharacteristic
30C2×#13093944C2212yes—
31C2×#13093944C2211yescharacteristic
32C2×C22⋊(C47⋊C8)C2212yes—
33C2×C4×C47⋊C8C2211yescharacteristic
34#15688885C2211yescharacteristic
35#15688885C2211yescharacteristic
36#15688885C2212yes—
37C2×C4×D4×C47—21no—
38C2×C4×D4×C47C2211yescharacteristic
39C22×C4⋊C4×C47C2212yes—
40C47×#389—21no—
41C47×#385—22no—
42C47×#384C2211yescharacteristic
43C47×#380C2212yes—
44C4×C4⋊C4×C47C2211yescharacteristic
45C2×#10625350—24no—
46C2×#10625350—24no—
47C2×#10625350—24no—
48C2×#10625350—42no—
49C2×#10625350—42no—
50C2×#10625350C2×C418yescomplemented
51C22×Dic94C2312yes—
52C22×D188C2311yescharacteristic
53C22×C22⋊D47—22no—
54C22×C4×D47—21no—
55C22×C4×D47—22no—
56C2×#10625329—24no—
57C2×#10625329—24no—
58C2×#10625329—42no—
59C22×C47⋊C8C2311yescharacteristic
60C22×C47⋊C8C2312yes—
61C22×C47⋊C8C2311yescharacteristic
62#13094024—22no—
63#13094024C2×C414yescomplemented
64#13094006—216no—
65#13094003—24no—
66#13094003C2×C418yescomplemented
67#13093995—28no—
68C4×D188—22no—
69C4×D188C2×C414yescomplemented
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—22no—
74C4⋊C4×D47—28no—
75C2×C22⋊Dic47—22no—
76C2×(C2×D47)⋊C4—21no—
77C4⋊(C47⋊C8)—22no—
78#13093947—24no—
79#13093947—24no—
80#13093947—42no—
81#13093947—42no—
82#13093944—24no—
83#13093944—22no—
84#13093944D414yescomplemented
85C22⋊(C47⋊C8)—24no—
86C4×C47⋊C8—22no—
87C22×D4×C47D411yescharacteristic
88C23×C4×C47—22no—
89C23×C4×C47C2312yes—
90C4×D4×C47—44no—
91C4×D4×C47C2×C414yes—
92C4×D4×C47D414yes—
93C2×C4⋊C4×C47—24no—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—42no—
98C2×C4⋊C4×C47D411yescharacteristic
99C2×C4⋊C4×C47C2312yes—
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C22⋊C4×C47—22no—
103C2×C22⋊C4×C47D412yes—
104C2×C42×C47—21no—
105C2×C42×C47C2311yescharacteristic
106#95×C47—21no—
107#95×C47—22no—
108C24×C47C2×D412yes—
109C23×D47—21no—
110C2×D4×C47—22no—
111C2×D4×C47—22no—
112C2×D4×C47C22⋊C414yes—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—44no—
121C22×C4×C47—44no—
122C22×C4×C47—82no—
123C22×C4×C47C22×C414yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D414yes—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2411yescharacteristic
129C2×C22⋊D47—28no—
130C2×C22⋊D47—44no—
131C2×C22⋊D47—44no—
132C22×Dic47—21no—
133C22×Dic47—22no—
134#10625350—216no—
135#10625350—48no—
136#10625350—48no—
137#10625350—48no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188C22×C412yes—
142C2×D188C2×D412yes—
143C2×D188#4112yes—
144C2×C4×D47—24no—
145C2×C4×D47—28no—
146C2×C4×D47—44no—
147C2×C4×D47—44no—
148C2×C4×D47—42no—
149C2×C4×D47—42no—
150C2×Dic94—22no—
151C2×Dic94—22no—
152C2×Dic94—22no—
153C2×Dic94—22no—
154C2×Dic94—22no—
155C2×Dic94C22×C414yes—
156C4⋊C4×C47—21no—
157C4⋊C4×C47—22no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—44no—
162C4⋊C4×C47—42no—
163C4⋊C4×C47—81no—
164C4⋊C4×C47C22⋊C414yes—
165C4⋊C4×C47C2×D412yes—
166C22⋊C4×C47—44no—
167C22⋊C4×C47C22⋊C418yes—
168C42×C47—42no—
169C42×C47C22×C412yes—
170C42×C47C2×D412yes—
171C22⋊Dic47—28no—
172(C2×D47)⋊C4—24no—
173C4⋊Dic47—24no—
174C4⋊Dic47—24no—
175C4⋊Dic47—21no—
176C4⋊Dic47C22×C412yes—
177Dic47⋊C4—28no—
178#10625329—48no—
179C2×C47⋊C8—21no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—22no—
183C2×C47⋊C8—22no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8C2×D412yes—
188C23×C47—22no—
189C23×C47—22no—
190C23×C47—22no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—22no—
194C23×C47C2×C22⋊C414yes—
195C23×C47C22×D411yescharacteristic
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—22no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202C22×D47—41no—
203D4×C47—28no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—24no—
209C2×C4×C47—24no—
210C2×C4×C47—24no—
211C2×C4×C47—24no—
212C2×C4×C47—22no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47—42no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—42no—
219C2×C4×C47—42no—
220C2×C4×C47—41no—
221C2×C4×C47—88no—
222C2×C4×C47C2×C22⋊C414yes—
223C2×C4×C47C2×C22⋊C412yes—
224C2×C4×C47C2×C22⋊C412yes—
225C2×C4×C47AGL(1,ℤ/8)12yes—
226C2×C4×C47#13712yes—
227C2×C4×C47C23×C411yescharacteristic
228C2×C4×C47C22×D411yescharacteristic
229C2×C4×C47C22×D411yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C47H(5,𝔽2)11yescharacteristic
232C2×C4×C47H(5,𝔽2)11yescharacteristic
233C22⋊D47—416no—
234C2×Dic47—22no—
235C2×Dic47—24no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240D188—22no—
241D188—22no—
242D188—22no—
243D188C4×D414yescomplemented
244C4×D47—416no—
245C4×D47—48no—
246Dic94—24no—
247Dic94—24no—
248Dic94—42no—
249Dic94—42no—
250C47⋊C8—22no—
251C47⋊C8—42no—
252C47⋊C8—41no—
253#22431—471nomaximal, Sylow
254C22×C47—24no—
255C22×C47—24no—
256C22×C47—24no—
257C22×C47—42no—
258C22×C47—42no—
259C22×C47—42no—
260C22×C47—42no—
261C22×C47C22×C22⋊C411yescharacteristic
262C22×C47D4211yescharacteristic
263C22×C47#54611yescharacteristic
264C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C47C2×#13711yescharacteristic
266C22×C47#57711yescharacteristic
267C22×C47#57811yescharacteristic
268C2×D47—24no—
269C2×D47—42no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—42no—
275C4×C47—24no—
276C4×C47—22no—
277C4×C47—42no—
278C4×C47—81no—
279C4×C47—82no—
280C4×C47#41714yescomplemented
281C4×C47C2×C4×D411yescharacteristic
282C4×C47#51811yescharacteristic
283C4×C47D4211yescharacteristic
284C4×C47#54611yescharacteristic
285Dic47—44no—
286Dic47—42no—
287C2×#547—471no—
288C2×#436—471no—
289C2×#417—472no—
290D4×C4⋊C4D4711yescharacteristic, complemented
291#1981—471no—
292#1956—471no—
293#1894—472no—
294#1872—472no—
295#1802—471no—
296#1788—472no—
297C4×#102—471no—
298C2×C47—44no—
299C2×C47D4×C22⋊C411yescharacteristic
300C2×C47C2×#41711yescharacteristic
301C2×C47#289811yescharacteristic
302C2×C47#327211yescharacteristic
303C2×C47#327611yescharacteristic
304C2×C47#333011yescharacteristic
305C2×C47#333111yescharacteristic
306D47—44no—
307C22×#41—472no—
308C22×M16—472no—
309#547—478no—
310#547—944no—
311C2×C4⋊Q8—471no—
312C2×#123—942no—
313C2×C22⋊Q8—472no—
314C2×#121—941no—
315C2×C4×D4—21no—
316C2×C4×D4—471no—
317C2×C4×D4C2×D4711yescharacteristic
318C22×C4⋊C4—941no—
319C22×C4⋊C4C2×D4712yes—
320#436—944no—
321C2×C4⋊C8—471no—
322#417—948no—
323C2×#103—472no—
324C2×#103—472no—
325C2×#103—472no—
326C2×#103—472no—
327C2×#102—471no—
328C2×#102—472no—
329C2×#102—471no—
330C2×C22⋊C8—472no—
331C2×C4×C8—471no—
332#389—21no—
333#385—22no—
334#384C2×D4711yescharacteristic
335#380C2×D4712yes—
336C4×C4⋊C4C2×D4711yescharacteristic
337#340—472no—
338#340—471no—
339#340—471no—
340C47#2243111yescharacteristic, complemented, Sylow
341C2×#41—478no—
342C2×#41—944no—
343C2×#41—944no—
344C2×#41—944no—
345C2×#41—1882no—
346C2×#41—1882no—
347C22×Q8—472no—
348C22×D4—471no—
349C22×D4—942no—
350C22×D4C22⋊D4711yescharacteristic
351C23×C4—22no—
352C23×C4—941no—
353C23×C4—942no—
354C23×C4C22×D4712yes—
355C2×M16—944no—
356C2×M16—944no—
357C2×M16—1882no—
358C22×C8—471no—
359C22×C8—472no—
360C22×C8—471no—
361C4⋊Q8—474no—
362C4⋊Q8—942no—
363#123—9416no—
364C22⋊Q8—478no—
365C22⋊Q8—944no—
366#121—948no—
367C4×D4—44no—
368C4×D4—474no—
369C4×D4—942no—
370C4×D4C4×D4714yes—
371C4×D4D18814yes—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4—948no—
382C2×C4⋊C4C22⋊D4711yescharacteristic
383C2×C4⋊C4C22×D4712yes—
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—942no—
388C2×C22⋊C4—941no—
389C2×C22⋊C4C22⋊D4712yes—
390C2×C42—21no—
391C2×C42C22×D4711yescharacteristic
392C4⋊C8—942no—
393#103—944no—
394#103—944no—
395#103—1882no—
396#103—1882no—
397#102—474no—
398#102—942no—
399#102—944no—
400C22⋊C8—944no—
401C4×C8—942no—
402#95—21no—
403#95—22no—
404C24—941no—
405C24C2×C22⋊D4712yes—
406#41—9416no—
407#41—1888no—
408#41—1888no—
409#41—1888no—
410C2×Q8—474no—
411C2×Q8—942no—
412C2×Q8—942no—
413C2×Q8—942no—
414C2×Q8—942no—
415C2×Q8—942no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—472no—
419C2×D4—472no—
420C2×D4—472no—
421C2×D4—948no—
422C2×D4—941no—
423C2×D4—941no—
424C2×D4—941no—
425C2×D4—1884no—
426C2×D4—1884no—
427C2×D4(C2×D47)⋊C414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—941no—
439C22×C4—942no—
440C22×C4—944no—
441C22×C4—948no—
442C22×C4—1884no—
443C22×C4—1884no—
444C22×C4—1882no—
445C22×C4—1882no—
446C22×C4C2×C4×D4714yes—
447C22×C4C2×D18814yes—
448C22×C4C2×C22⋊D4711yescharacteristic
449C22×C4C2×C22⋊D4712yes—
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C23×D4711yescharacteristic
452M16—1888no—
453C2×C8—472no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—942no—
457C2×C8—941no—
458C2×C8—942no—
459C2×C8—941no—
460C2×C8—941no—
461C2×C8—942no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—472no—
471C4⋊C4—948no—
472C4⋊C4—944no—
473C4⋊C4—944no—
474C4⋊C4—941no—
475C4⋊C4(C2×D47)⋊C414yes—
476C4⋊C4D4×D4712yes—
477C22⋊C4—44no—
478C22⋊C4—948no—
479C22⋊C4—944no—
480C22⋊C4(C2×D47)⋊C418yes—
481C42—42no—
482C42C2×C4×D4712yes—
483C42C2×D18812yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—942no—
491C23—942no—
492C23—942no—
493C23—1881no—
494C23—1881no—
495C23—1881no—
496C23—1881no—
497C23C2×(C2×D47)⋊C414yes—
498C23C22×C22⋊D4711yescharacteristic
499Q8—944no—
500Q8—944no—
501Q8—1882no—
502Q8—1882no—
503D4—28no—
504D4—474no—
505D4—942no—
506D4—942no—
507D4—942no—
508D4—18816no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—88no—
527C2×C4—942no—
528C2×C4—944no—
529C2×C4—18816no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1888no—
535C2×C4C2×(C2×D47)⋊C412yes—
536C2×C4C2×(C2×D47)⋊C414yes—
537C2×C4C2×(C2×D47)⋊C412yes—
538C2×C4#1309405012yes—
539C2×C4#1309405512yes—
540C2×C4C22×C4×D4711yescharacteristic
541C2×C4C2×D4×D4711yescharacteristic
542C2×C4C22×D18811yescharacteristic
543C2×C4#41×D4711yescharacteristic
544C2×C4#1309406911yescharacteristic
545C2×C4#1309407011yescharacteristic
546C8—942no—
547C8—1882no—
548C8—1881no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—944no—
557C22—1882no—
558C22—1882no—
559C22—1882no—
560C22—1882no—
561C22—1882no—
562C22—1882no—
563C22C22×(C2×D47)⋊C411yescharacteristic
564C22D4×C22⋊D4711yescharacteristic
565C22#1568976111yescharacteristic
566C22C2×#1309405011yescharacteristic
567C22C2×#1309405511yescharacteristic
568C22#1568993411yescharacteristic
569C22#1568993811yescharacteristic
570C4—22no—
571C4—24no—
572C4—42no—
573C4—82no—
574C4—81no—
575C4—1884no—
576C4—1882no—
577C4#1568912214yescomplemented
578C4C4×D4×D4711yescharacteristic
579C4#1568959511yescharacteristic
580C4D4×D18811yescharacteristic
581C4#1568976811yescharacteristic
582C2—44no—
583C2—1884no—
584C2D4×(C2×D47)⋊C411yescharacteristic
585C2C2×#1568912211yescharacteristic
586C2#2025191811yescharacteristic
587C2#2025498411yescharacteristic
588C2#2025503411yescharacteristic
589C2#2025568711yescharacteristic
590C2#2025569811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingD4×C4⋊C4×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689778C2
cosocleC22×C4×C47C24
hypercenterD4×C4⋊C4D47
hypocenterC47#22431
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15689201C2semidirect
3. C2×#15689122C2semidirect
4. #20244311C2semidirect
5. #20244189C2semidirect
6. #20243819C2semidirect
7. #20243695C2semidirect
8. #20243385C2semidirect
9. #20243326C2semidirect
10. C4×#13093944C2semidirect
11. D4×C4⋊C4×C47C2semidirect
15. #15689778C4semidirect
50. C2×#10625350C2×C4semidirect
63. #13094024C2×C4semidirect
66. #13094003C2×C4semidirect
69. C4×D188C2×C4semidirect
84. #13093944D4semidirect
243. D188C4×D4semidirect
280. C4×C47#417semidirect
290. D4×C4⋊C4D47semidirect
340. C47#22431semidirect
577. C4#15689122semidirect

Maximal subgroups

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

subgroupindex
C2×#156897782
C2×#156892012
C2×#156891222
#202443112
#202441892
#202438192
#202436952
#202433852
#202433262
C4×#130939442
D4×C4⋊C4×C472
#2243147