Finite Group Databasebeta

← #26138685

Subgroups — #26138685

Order 12032.

1710 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093989C211yesmaximal, characteristic
2C2×#15688987C211yesmaximal, characteristic
3C2×C4×Dic47⋊C4C211yesmaximal, characteristic
4#20241658C212yesmaximal
5#20241654C212yesmaximal
6#20241623C211yesmaximal, characteristic
7#20241621C211yesmaximal, characteristic
8#20241620C211yesmaximal, characteristic
9Dic47⋊#95C212yesmaximal
10Dic47⋊#95C212yesmaximal
11C47×#1436C211yesmaximal, characteristic
12C23×Dic94C2211yescharacteristic
13C2×#13093989—24no—
14C2×#13093989—24no—
15C2×#13093989C418yescomplemented
16C22×Dic47⋊C4C2212yes—
17C22×Dic47⋊C4C2211yescharacteristic
18C22×Dic47⋊C4C2211yescharacteristic
19C22×Dic47⋊C4C2212yes—
20C22×C4×Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23C22×C4×Dic47C2211yescharacteristic
24#15688987—24no—
25C4×Dic47⋊C4—24no—
26C2×C47⋊#95C2211yescharacteristic
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C22×C4⋊C4×C47C2211yescharacteristic
36C22×C42×C47C2211yescharacteristic
37C2×#95×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2212yes—
40C22×Dic94—24no—
41C22×Dic94—22no—
42C22×Dic94—22no—
43C22×Dic94—24no—
44C22×Dic94C2×C414yes—
45C23×Dic47C2312yes—
46C23×Dic47C2311yescharacteristic
47C23×Dic47C2312yes—
48C23×Dic47C2311yescharacteristic
49C23×Dic47C2311yescharacteristic
50C23×Dic47C2311yescharacteristic
51#13093989—232no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—24no—
55C2×Dic47⋊C4—22no—
56C2×Dic47⋊C4—22no—
57C2×Dic47⋊C4—24no—
58C2×Dic47⋊C4—24no—
59C2×Dic47⋊C4—24no—
60C2×Dic47⋊C4—24no—
61C2×Dic47⋊C4C2×C414yes—
62C2×Dic47⋊C4C2×C414yes—
63C2×Dic47⋊C4Q818yescomplemented
64C2×C4×Dic47—22no—
65C2×C4×Dic47—22no—
66C2×C4×Dic47—22no—
67C2×C4×Dic47—22no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47—42no—
70C2×C4×Dic47C2×C414yes—
71C2×C4×Dic47C2×C414yes—
72C2×C4×Dic47C2×C414yes—
73C2×C4×Dic47D414yes—
74C2×C4×Dic47D414yes—
75C47⋊#95—22no—
76C47⋊#95—24no—
77C47⋊#95—24no—
78C47⋊#95—24no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—42no—
82C47⋊#95—42no—
83C47⋊#95—42no—
84C23×C4×C47C2312yes—
85C23×C4×C47C2311yescharacteristic
86C23×C4×C47C2312yes—
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—22no—
91C2×C4⋊C4×C47C2×C414yes—
92C2×C42×C47—22no—
93C2×C42×C47—42no—
94#95×C47—22no—
95#95×C47—24no—
96#95×C47—42no—
97C24×C47C2411yescharacteristic
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—21no—
105C22×C4×C47—21no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—22no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47C22×C412yes—
121C22×C4×C47C22×C412yes—
122C22×C4×C47C22×C412yes—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×Q814yes—
126C22×C4×C47#4114yes—
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—21no—
134C22×Dic47—21no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47C22×C412yes—
150C22×Dic47C22×C412yes—
151C22×Dic47C22×C412yes—
152C22×Dic47C22×C412yes—
153C22×Dic47C2×D412yes—
154C22×Dic47C2×D412yes—
155C22×Dic47C2×D412yes—
156C22×Dic47C2×D412yes—
157C22×Dic47C2×Q814yes—
158C22×Dic47C2×Q814yes—
159C22×Dic47#4114yes—
160C22×Dic47#4114yes—
161C2×Dic94—216no—
162C2×Dic94—216no—
163C2×Dic94—28no—
164C2×Dic94—28no—
165C2×Dic94—28no—
166C4⋊C4×C47—28no—
167C42×C47—44no—
168Dic47⋊C4—28no—
169Dic47⋊C4—28no—
170Dic47⋊C4—216no—
171Dic47⋊C4—48no—
172C4×Dic47—28no—
173C4×Dic47—28no—
174C4×Dic47—44no—
175C4×Dic47C22⋊C4116yes—
176C23×C47C23×C411yescharacteristic
177C23×C47C22×D411yescharacteristic
178C23×C47C22×D411yescharacteristic
179C23×C47C22×Q812yes—
180C23×C47C2×#4112yes—
181C23×C47C2×#4112yes—
182C23×C47C2×#4112yes—
183C23×C47H(5,𝔽2)12yes—
184C23×C472−1+412yes—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—24no—
188C2×C4×C47—24no—
189C2×C4×C47—24no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—24no—
193C2×C4×C47—22no—
194C2×C4×C47—24no—
195C2×C4×C47—24no—
196C2×C4×C47—24no—
197C2×C4×C47—24no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47—42no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47C2×C22⋊C414yes—
213C2×C4×C47#11714yes—
214C2×Dic47—22no—
215C2×Dic47—22no—
216C2×Dic47—22no—
217C2×Dic47—22no—
218C2×Dic47—24no—
219C2×Dic47—24no—
220C2×Dic47—24no—
221C2×Dic47—24no—
222C2×Dic47—22no—
223C2×Dic47—24no—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—24no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—42no—
240C2×Dic47—42no—
241C2×Dic47C2×C22⋊C414yes—
242C2×Dic47C2×C22⋊C414yes—
243C2×Dic47#11714yes—
244C2×Dic47C4×Q818yes—
245C2×Dic47C22⋊Q818yes—
246C2×Dic47#12414yes—
247C2×Dic47C4⋊Q814yes—
248Dic94—232no—
249#13925—471nomaximal, Sylow
250C22×C47C22×C22⋊C411yescharacteristic
251C22×C47C2×#11711yescharacteristic
252C22×C47C2×#11711yescharacteristic
253C22×C47C2×C4×Q812yes—
254C22×C47#52012yes—
255C22×C47C2×C22⋊Q812yes—
256C22×C47#52512yes—
257C22×C47#52512yes—
258C22×C47C2×#12411yescharacteristic
259C22×C47C2×C4⋊Q811yescharacteristic
260C22×C47#53511yescharacteristic
261C22×C47#53611yescharacteristic
262C22×C47#53712yes—
263C22×C47#55412yes—
264C22×C47#55412yes—
265C22×C47#55612yes—
266C22×C47#55612yes—
267C22×C47#55712yes—
268C22×C47#55712yes—
269C22×C47#55912yes—
270C22×C47#55912yes—
271C4×C47—24no—
272C4×C47—24no—
273C4×C47—42no—
274C4×C47—44no—
275C4×C47—82no—
276Dic47—24no—
277Dic47—24no—
278Dic47—42no—
279Dic47—44no—
280Dic47—82no—
281Dic47#39118yes—
282C22×C4×Q8—471no—
283C2×#382—471no—
284C2×C4×C4⋊C4—471no—
285#1446—472no—
286#1445—472no—
287#1436—471no—
288#1436—471no—
289#1436—471no—
290#1436D4711yescharacteristic
291C4⋊#95—472no—
292C4⋊#95—472no—
293C2×C47#228011yescharacteristic
294C2×C47C2×#39111yescharacteristic
295C2×C47#231111yescharacteristic
296C2×C47#235512yes—
297C2×C47#235512yes—
298C2×C47#258012yes—
299C2×C47#258112yes—
300C2×C47#258412yes—
301C2×C47#259212yes—
302C23×Q8—471no—
303C2×C4×Q8—478no—
304C2×C4×Q8—944no—
305C2×C4×Q8—944no—
306C22×C4⋊C4—471no—
307C22×C4⋊C4—471no—
308C22×C4⋊C4—472no—
309C22×C4⋊C4—472no—
310C22×C4⋊C4C2×D4711yescharacteristic
311C22×C42—471no—
312C22×C42—471no—
313C22×C42—471no—
314C22×C42—471no—
315C22×C42C2×D4711yescharacteristic
316#382—944no—
317C4×C4⋊C4—944no—
318C2×#95—472no—
319C2×#95—472no—
320C2×#95—472no—
321C2×#95—471no—
322C2×#95—472no—
323C2×#95—472no—
324C2×#95—472no—
325C2×#95—472no—
326C2×#95—472no—
327C2×#95C2×D4712yes—
328C2×#95C2×D4711yescharacteristic
329C2×#95C2×D4712yes—
330C47#1392511yescharacteristic, complemented, Sylow
331C22×Q8—474no—
332C22×Q8—942no—
333C22×Q8—942no—
334C22×Q8—944no—
335C22×Q8—944no—
336C23×C4—471no—
337C23×C4—471no—
338C23×C4—471no—
339C23×C4—471no—
340C23×C4—472no—
341C23×C4—472no—
342C23×C4C22×D4712yes—
343C23×C4C22×D4711yescharacteristic
344C23×C4C22×D4712yes—
345C23×C4C22×D4711yescharacteristic
346C23×C4C22×D4711yescharacteristic
347C4×Q8—9432no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—474no—
351C2×C4⋊C4—474no—
352C2×C4⋊C4—478no—
353C2×C4⋊C4—942no—
354C2×C4⋊C4—942no—
355C2×C4⋊C4—944no—
356C2×C4⋊C4—944no—
357C2×C4⋊C4—942no—
358C2×C4⋊C4—942no—
359C2×C4⋊C4—944no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—944no—
362C2×C4⋊C4C4×D4714yes—
363C2×C42—22no—
364C2×C42—42no—
365C2×C42—474no—
366C2×C42—474no—
367C2×C42—474no—
368C2×C42—474no—
369C2×C42—474no—
370C2×C42—942no—
371C2×C42—942no—
372C2×C42—942no—
373C2×C42—942no—
374C2×C42—942no—
375C2×C42—1882no—
376#95—24no—
377#95—22no—
378#95—42no—
379#95—944no—
380#95—944no—
381#95—944no—
382#95—944no—
383#95—944no—
384#95—942no—
385#95—1882no—
386#95—1882no—
387#95—1882no—
388C24C23×D4711yescharacteristic
389C2×Q8—948no—
390C2×Q8—948no—
391C2×Q8—9416no—
392C2×Q8—9416no—
393C2×Q8—948no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—472no—
417C22×C4—472no—
418C22×C4—472no—
419C22×C4—474no—
420C22×C4—474no—
421C22×C4—472no—
422C22×C4—472no—
423C22×C4—472no—
424C22×C4—472no—
425C22×C4—474no—
426C22×C4—472no—
427C22×C4—474no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—942no—
431C22×C4—942no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—941no—
435C22×C4—941no—
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—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4C2×C4×D4712yes—
451C22×C4C2×C4×D4712yes—
452C22×C4C2×C4×D4712yes—
453C22×C4#1062535014yes—
454C22×C4D4×D4712yes—
455C22×C4D4×D4712yes—
456C22×C4Q8×D4714yes—
457C4⋊C4—28no—
458C4⋊C4—9416no—
459C4⋊C4—948no—
460C4⋊C4—948no—
461C4⋊C4—1888no—
462C42—44no—
463C42—4716no—
464C42—948no—
465C42—948no—
466C42—1884no—
467C23C22×C4×D4711yescharacteristic
468C23C2×D4×D4711yescharacteristic
469C23C2×D4×D4711yescharacteristic
470C23C2×Q8×D4712yes—
471C23#41×D4712yes—
472C23#41×D4712yes—
473C23C2×#1062535012yes—
474C23#1309406912yes—
475C23#1309407112yes—
476Q8—9432no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—474no—
505C2×C4—478no—
506C2×C4—478no—
507C2×C4—474no—
508C2×C4—474no—
509C2×C4—474no—
510C2×C4—474no—
511C2×C4—942no—
512C2×C4—942no—
513C2×C4—942no—
514C2×C4—942no—
515C2×C4—944no—
516C2×C4—944no—
517C2×C4—944no—
518C2×C4—944no—
519C2×C4—942no—
520C2×C4—944no—
521C2×C4—944no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4—1882no—
530C2×C4—1882no—
531C2×C4—1882no—
532C2×C4—1882no—
533C2×C4—1882no—
534C2×C4—1882no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1882no—
538C2×C4C22⋊C4×D4714yes—
539C2×C4#1309398114yes—
540C22C2×C22⋊C4×D4711yescharacteristic
541C22#117×D4711yescharacteristic
542C22C2×#1309398111yescharacteristic
543C22C4×Q8×D4712yes—
544C22#1568960212yes—
545C22C22⋊Q8×D4712yes—
546C22#1568963212yes—
547C22#1568963212yes—
548C22#124×D4711yescharacteristic
549C22C4⋊Q8×D4711yescharacteristic
550C22#1568968811yescharacteristic
551C22#1568969511yescharacteristic
552C22#1568970012yes—
553C22#1568982712yes—
554C22#1568982812yes—
555C22#1568984012yes—
556C22#1568984112yes—
557C22#1568985012yes—
558C22#1568985112yes—
559C22#1568985612yes—
560C22#1568985712yes—
561C4—24no—
562C4—24no—
563C4—44no—
564C4—42no—
565C4—82no—
566C4—478no—
567C4—944no—
568C4—944no—
569C4—3762no—
570C4—1884no—
571C4—1882no—
572C2#2024576911yescharacteristic
573C2#391×D4711yescharacteristic
574C2#2024597511yescharacteristic
575C2#2024634112yes—
576C2#2024634212yes—
577C2#2024874512yes—
578C2#2024875312yes—
579C2#2024877412yes—
580C2#2024887412yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1436C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1436D47
hypocenterC47#13925
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
15. C2×#13093989C4semidirect
63. C2×Dic47⋊C4Q8semidirect
330. C47#13925semidirect

Maximal subgroups

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

subgroupindex
C22×#130939892
C2×#156889872
C2×C4×Dic47⋊C42
#202416582
#202416542
#202416232
#202416212
#202416202
Dic47⋊#952
C47×#14362
#1392547