Finite Group Databasebeta

← #26244340

Subgroups — #26244340

Order 12032.

1202 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689676C211yesmaximal, characteristic
2C2×#15689166C211yesmaximal, characteristic
3C2×#15689165C211yesmaximal, characteristic
4C2×#15689022C211yesmaximal, characteristic
5#20244729C212yesmaximal, complemented
6#20244725C212yesmaximal, complemented
7#20244720C212yesmaximal, complemented
8#20244715C212yesmaximal, complemented
9#20243107C211yesmaximal, characteristic
10#20243081C211yesmaximal, characteristic
11C47×#1736C211yesmaximal, characteristic
12C22×#10625329C2211yescharacteristic
13#15689676—24no—
14#15689676—24no—
15C2×#13094023C2211yescharacteristic
16C2×#13094023C2211yescharacteristic
17C2×#13094015C2211yescharacteristic
18C2×#13094015C2211yescharacteristic
19C2×#13094001—21no—
20C2×#13094001—21no—
21C2×C4×Dic94—21no—
22C2×C4×Dic94—21no—
23C2×#13093983C2211yescharacteristic
24C22×C4⋊Dic47—21no—
25C22×C4⋊Dic47C2211yescharacteristic
26C22×C4×Dic47C2211yescharacteristic
27#15689166—24no—
28#15689165—24no—
29C2×#13093958C2212yes—
30C2×#13093957C2212yes—
31C2×#13093956C2212yes—
32C2×#13093955C2212yes—
33#15689022—21no—
34#15689022—21no—
35#15689022C2212yes—
36#15689022C2212yes—
37C2×C47⋊#95—21no—
38#15688885C2212yes—
39#15688885C2212yes—
40#15688877C2212yes—
41#15688877C2212yes—
42C22×M16×C47C2211yescharacteristic
43C2×#117×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2211yescharacteristic
45C47×#340C2212yes—
46C47×#340C2212yes—
47C22×Dic94—21no—
48C22×Dic94—21no—
49C23×Dic47—21no—
50C23×Dic47D411yescharacteristic
51C23×Dic47D411yescharacteristic
52C2×#10625329—22no—
53C2×#10625329—22no—
54C2×#10625329Q814yescomplemented
55C22×C47⋊C8C2312yes—
56#13094023—22no—
57#13094023—22no—
58#13094023—22no—
59#13094023—22no—
60#13094015—22no—
61#13094015—22no—
62#13094015—22no—
63#13094015—22no—
64#13094001—28no—
65#13094001—28no—
66C4×Dic94—28no—
67C4×Dic94—28no—
68#13093983—22no—
69#13093983D414yes—
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47—22no—
73C2×C4⋊Dic47—22no—
74C2×C4⋊Dic47—21no—
75C2×C4⋊Dic47—22no—
76C2×C4⋊Dic47—22no—
77C2×C4⋊Dic47—44no—
78C2×C4⋊Dic47C2312yes—
79C2×C4⋊Dic47C2311yescharacteristic
80C2×C4⋊Dic47C2311yescharacteristic
81C2×Dic47⋊C4—21no—
82C2×Dic47⋊C4—21no—
83C2×Dic47⋊C4—21no—
84C2×Dic47⋊C4—21no—
85C2×C22⋊Dic47—21no—
86C2×C4×Dic47—22no—
87C2×C4×Dic47—22no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47D411yescharacteristic
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47C2311yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93#13093958—24no—
94#13093957—24no—
95#13093956—24no—
96#13093955—24no—
97C47⋊#95—22no—
98C47⋊#95—22no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—22no—
102C2×M16×C47—22no—
103C2×M16×C47Q814yescomplemented
104C22×C8×C47C2312yes—
105#117×C47—22no—
106#117×C47D414yes—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—22no—
109C2×C4⋊C4×C47Q814yescomplemented
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2312yes—
113C2×C22⋊C4×C47—21no—
114C2×C42×C47—21no—
115C24×C47C2×D411yescharacteristic
116C22×C4×C47—24no—
117C22×C4×C47—21no—
118C22×C4×C47—21no—
119C22×C4×C47—22no—
120C22×C4×C47—42no—
121C22×C4×C47—42no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×Q812yes—
126C22×C4×C47C2×Q812yes—
127C22×C4×C47C2×Q812yes—
128C22×C4×C47#4112yes—
129C22×C4×C47C2411yescharacteristic
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—41no—
138C22×Dic47—41no—
139C22×Dic47—42no—
140C22×Dic47—42no—
141C22×Dic47—42no—
142C22×Dic47C2×D411yescharacteristic
143C22×Dic47C2×D411yescharacteristic
144C22×Dic47C2×D411yescharacteristic
145C22×Dic47C2×D411yescharacteristic
146C2×Dic94—24no—
147C2×Dic94—24no—
148C2×Dic94—42no—
149C2×Dic94—42no—
150C2×Dic94—42no—
151C2×Dic94—42no—
152M16×C47—28no—
153C2×C8×C47—22no—
154C2×C8×C47—22no—
155C2×C8×C47C2×Q814yes—
156C4⋊C4×C47—24no—
157C4⋊C4×C47—21no—
158C4⋊C4×C47—21no—
159C4⋊C4×C47—22no—
160C4⋊C4×C47C2×D412yes—
161C4⋊C4×C47C2×D412yes—
162C4⋊C4×C47C2×Q814yes—
163C22⋊C4×C47—24no—
164C42×C47—24no—
165C22⋊Dic47—24no—
166C4⋊Dic47—24no—
167C4⋊Dic47—24no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—22no—
171C4⋊Dic47—22no—
172C4⋊Dic47—44no—
173C4⋊Dic47—42no—
174C4⋊Dic47—44no—
175C4⋊Dic47C2×D412yes—
176C4⋊Dic47C2×D412yes—
177Dic47⋊C4—24no—
178Dic47⋊C4—24no—
179Dic47⋊C4—24no—
180Dic47⋊C4—24no—
181C4×Dic47—22no—
182C4×Dic47—22no—
183C4×Dic47—24no—
184C4×Dic47—21no—
185C4×Dic47—21no—
186C4×Dic47—21no—
187C4×Dic47—21no—
188#10625329—28no—
189C2×C47⋊C8—22no—
190C2×C47⋊C8—22no—
191C2×C47⋊C8C2×Q814yes—
192C23×C47—21no—
193C23×C47—21no—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47#12012yes—
197C23×C47C22⋊Q812yes—
198C23×C47C4⋊Q812yes—
199C23×C47C22×D411yescharacteristic
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—24no—
203C2×C4×C47—48no—
204C2×C4×C47—42no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—42no—
210C2×C4×C47#12012yes—
211C2×C4×C47C22⋊Q814yes—
212C2×C4×C47C22⋊Q812yes—
213C2×C4×C47C22⋊Q814yes—
214C2×C4×C47C4⋊Q814yes—
215C2×C4×C47C4⋊Q812yes—
216C2×C4×C47C4⋊Q814yes—
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×Q811yescharacteristic
220C2×C4×C47C22×Q811yescharacteristic
221C2×C4×C47C22×Q811yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—22no—
230C2×Dic47—22no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—44no—
236C2×Dic47—44no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—42no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—84no—
245C2×Dic47AGL(1,ℤ/8)12yes—
246C2×Dic47#13712yes—
247Dic94—48no—
248Dic94—48no—
249C8×C47—24no—
250C47⋊C8—24no—
251#23554—471nomaximal, Sylow
252C22×C47—22no—
253C22×C47—22no—
254C22×C47—22no—
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47#39514yes—
259C22×C47C2×#12011yescharacteristic
260C22×C47C2×C22⋊Q811yescharacteristic
261C22×C47C2×C4⋊Q811yescharacteristic
262C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C47C2×#13711yescharacteristic
264C22×C47#57511yescharacteristic
265C22×C47#57511yescharacteristic
266C4×C47—44no—
267C4×C47—42no—
268C4×C47—42no—
269C4×C47#39514yes—
270C4×C47C2×C22⋊Q811yescharacteristic
271C4×C47C2×C22⋊Q811yescharacteristic
272C4×C47C2×C4⋊Q811yescharacteristic
273C4×C47C2×C4⋊Q811yescharacteristic
274Dic47—22no—
275Dic47—22no—
276Dic47—42no—
277Dic47—41no—
278Dic47—42no—
279Dic47—41no—
280Dic47—82no—
281C2×#533—471no—
282C2×#428—471no—
283C2×#428—471no—
284C2×#389—471no—
285#2059—472no—
286#2059—472no—
287#2058—472no—
288#2057—472no—
289#1736—471no—
290#1736D4711yescharacteristic
291#1728—471no—
292C2×C47—24no—
293C2×C47C2×#39511yescharacteristic
294C2×C47#300611yescharacteristic
295C2×C47#300711yescharacteristic
296C2×C47#307711yescharacteristic
297C2×C47#307811yescharacteristic
298C2×C47#316511yescharacteristic
299C2×C47#316611yescharacteristic
300C22×M16—471no—
301C22×M16C2×D4711yescharacteristic
302#533—944no—
303#533—944no—
304C2×C4⋊Q8—471no—
305C2×C4⋊Q8—471no—
306C2×#125—471no—
307C2×#125—471no—
308C2×C22⋊Q8—941no—
309C2×C22⋊Q8—941no—
310C2×C4×Q8—941no—
311C2×C4×Q8—941no—
312C2×#117—471no—
313C2×#117C2×D4711yescharacteristic
314C22×C4⋊C4—471no—
315C22×C4⋊C4—941no—
316C22×C4⋊C4C2×D4711yescharacteristic
317C22×C42—471no—
318#428—944no—
319#428—944no—
320C2×#107—472no—
321C2×#107—472no—
322C2×#106—472no—
323C2×#106—472no—
324#389—472no—
325#389—472no—
326#389—941no—
327#389—941no—
328C2×#95—941no—
329#340—472no—
330#340—472no—
331#340C2×D4712yes—
332#340C2×D4712yes—
333#336—472no—
334#336—472no—
335C47#2355411yescharacteristic, complemented, Sylow
336C22×Q8—941no—
337C22×Q8—941no—
338C23×C4—21no—
339C23×C4—471no—
340C23×C4—471no—
341C23×C4—941no—
342C23×C4C22×D4711yescharacteristic
343C2×M16—22no—
344C2×M16—22no—
345C2×M16—474no—
346C2×M16—942no—
347C2×M16—942no—
348C2×M16Dic9414yescomplemented
349C22×C8—472no—
350C22×C8C22×D4712yes—
351C4⋊Q8—942no—
352C4⋊Q8—942no—
353C4⋊Q8—942no—
354C4⋊Q8—942no—
355#125—942no—
356#125—942no—
357#125—942no—
358#125—942no—
359C22⋊Q8—948no—
360C22⋊Q8—948no—
361C4×Q8—948no—
362C4×Q8—948no—
363#117—22no—
364#117—474no—
365#117—942no—
366#117D18814yes—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—471no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—472no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4—941no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—942no—
379C2×C4⋊C4—942no—
380C2×C4⋊C4—942no—
381C2×C4⋊C4—942no—
382C2×C4⋊C4—941no—
383C2×C4⋊C4—1884no—
384C2×C4⋊C4Dic9414yescomplemented
385C2×C4⋊C4C22×D4712yes—
386C2×C4⋊C4C22×D4711yescharacteristic
387C2×C4⋊C4C22×D4711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—941no—
390C2×C42—21no—
391C2×C42—471no—
392C2×C42—471no—
393C2×C42—471no—
394C2×C42—471no—
395C2×C42—942no—
396C2×C42—941no—
397C2×C42—942no—
398#107—944no—
399#107—944no—
400#106—944no—
401#106—944no—
402#95—942no—
403#95—942no—
404C24D4×D4711yescharacteristic
405C2×Q8—944no—
406C2×Q8—944no—
407C2×Q8—1882no—
408C2×Q8—1882no—
409C2×Q8—1882no—
410C2×Q8—1882no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—471no—
418C22×C4—471no—
419C22×C4—471no—
420C22×C4—471no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4—942no—
425C22×C4—941no—
426C22×C4—941no—
427C22×C4—942no—
428C22×C4—1881no—
429C22×C4—1881no—
430C22×C4—1882no—
431C22×C4—1882no—
432C22×C4—1882no—
433C22×C4C2×Dic9412yes—
434C22×C4C2×Dic9412yes—
435C22×C4C2×D18812yes—
436C22×C4D4×D4711yescharacteristic
437C22×C4D4×D4712yes—
438C22×C4#1062535212yes—
439C22×C4Q8×D4712yes—
440C22×C4C23×D4711yescharacteristic
441M16—28no—
442M16—948no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—474no—
446C2×C8—942no—
447C2×C8—942no—
448C2×C8C2×Dic9414yes—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—472no—
454C4⋊C4—472no—
455C4⋊C4—944no—
456C4⋊C4—944no—
457C4⋊C4—944no—
458C4⋊C4—944no—
459C4⋊C4—944no—
460C4⋊C4—944no—
461C4⋊C4—941no—
462C4⋊C4—941no—
463C4⋊C4—942no—
464C4⋊C4—942no—
465C4⋊C4—1884no—
466C4⋊C4—1884no—
467C4⋊C4—1882no—
468C4⋊C4C2×Dic9414yes—
469C4⋊C4C2×D18812yes—
470C4⋊C4C2×D18812yes—
471C22⋊C4—24no—
472C22⋊C4—944no—
473C42—24no—
474C42—942no—
475C42—942no—
476C42—944no—
477C42—941no—
478C42—941no—
479C42—941no—
480C42—941no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23#1309399412yes—
486C23#1309400112yes—
487C23#1309402312yes—
488C23C2×D4×D4711yescharacteristic
489Q8—1888no—
490Q8—1888no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—472no—
502C2×C4—472no—
503C2×C4—941no—
504C2×C4—941no—
505C2×C4—941no—
506C2×C4—941no—
507C2×C4—941no—
508C2×C4—941no—
509C2×C4—942no—
510C2×C4—942no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—3764no—
516C2×C4—1884no—
517C2×C4—1884no—
518C2×C4—1882no—
519C2×C4—1882no—
520C2×C4—1882no—
521C2×C4—1881no—
522C2×C4—1881no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4#1309399412yes—
526C2×C4#1309400112yes—
527C2×C4#1309400114yes—
528C2×C4#1309400414yes—
529C2×C4#1309402312yes—
530C2×C4#1309402314yes—
531C2×C4#1309402414yes—
532C2×C4C2×D4×D4711yescharacteristic
533C2×C4C22×D18811yescharacteristic
534C2×C4C2×Q8×D4711yescharacteristic
535C2×C4C22×Dic9411yescharacteristic
536C2×C4C22×Dic9411yescharacteristic
537C2×C4C2×#1062535211yescharacteristic
538C8—24no—
539C8—944no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22#1568904414yes—
547C22C2×#1309399411yescharacteristic
548C22C2×#1309400111yescharacteristic
549C22C2×#1309402311yescharacteristic
550C22AGL(1,ℤ/8)×D4711yescharacteristic
551C22#137×D4711yescharacteristic
552C22#1568991911yescharacteristic
553C22#1568992211yescharacteristic
554C4—42no—
555C4—42no—
556C4—44no—
557C4—942no—
558C4—942no—
559C4—3762no—
560C4—1881no—
561C4—1882no—
562C4—1881no—
563C4—1882no—
564C4#1568904414yes—
565C4C2×#1309400111yescharacteristic
566C4C2×#1309400411yescharacteristic
567C4C2×#1309402311yescharacteristic
568C4C2×#1309402411yescharacteristic
569C2—24no—
570C2C2×#1568904411yescharacteristic
571C2#2025264111yescharacteristic
572C2#2025264911yescharacteristic
573C2#2025318211yescharacteristic
574C2#2025319011yescharacteristic
575C2#2025393211yescharacteristic
576C2#2025393811yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#1736D47
hypocenterC47#23554
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
5. #20244729C2semidirect
6. #20244725C2semidirect
7. #20244720C2semidirect
8. #20244715C2semidirect
54. C2×#10625329Q8semidirect
103. C2×M16×C47Q8semidirect
109. C2×C4⋊C4×C47Q8semidirect
335. C47#23554semidirect
348. C2×M16Dic94semidirect
384. C2×C4⋊C4Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156896762
C2×#156891662
C2×#156891652
C2×#156890222
#202447292
#202447252
#202447202
#202447152
#202431072
#202430812
C47×#17362
#2355447