Finite Group Databasebeta

← #26421943

Subgroups — #26421943

Order 12032.

1776 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251735C211yesmaximal, characteristic
2#20251735C212yesmaximal
3#20248788C214yesmaximal, complemented
4#20248606C214yesmaximal
5#20248419C211yesmaximal, characteristic, complemented
6#20247965C212yesmaximal, complemented
7#20247939C212yesmaximal, complemented
8#20247160C212yesmaximal, complemented
9#20247110C212yesmaximal
10#20246489C212yesmaximal, complemented
11#20246448C211yesmaximal, characteristic, complemented
12#20246345C211yesmaximal, characteristic, complemented
13#20246264C212yesmaximal
14#20245926C212yesmaximal
15C4×#13094000C211yesmaximal, characteristic
16C4×C4⋊C4×D47C211yesmaximal, characteristic
17C47×#2546C211yesmaximal, characteristic, complemented
18C2×#13094015C2212yes—
19C2×#13094000C2211yescharacteristic
20C2×#13094000C2212yes—
21C2×Q8×Dic47C2212yes—
22C2×C4⋊C4×D47C2211yescharacteristic
23C2×C4⋊C4×D47C2211yescharacteristic
24C2×C42×D47C2211yescharacteristic
25C2×C42×D47C2212yes—
26#15689065C2214yes—
27#15689063C2214yes—
28#15689062C2214yes—
29#15689055C2214yes—
30#15689053C2214yes—
31#15689030C2212yes—
32#15689030C2212yes—
33#15689030C2212yes—
34#15689030C2211yescharacteristic
35#15689022C2212yes—
36#15689022C2212yes—
37#15689021C2212yes—
38#15689019C2214yes—
39#15689019C2212yes—
40#15689019C2212yes—
41#15689019C2214yes—
42#15689019C2211yescharacteristic
43#15689010C2214yes—
44#15689010C2214yes—
45#15689009C2214yes—
46#15689009C2214yes—
47#15689008C2214yes—
48#15689004C2214yes—
49#15689004C2214yes—
50#15689004C2212yes—
51#15688997C2214yes—
52#15688994C2212yes—
53#15688993C2211yescharacteristic
54#15688993C2212yes—
55#15688989C2212yes—
56#15688980C2212yes—
57#15688980C2214yes—
58#15688980C2214yes—
59#15688980C2212yes—
60#15688980C2211yescharacteristic
61C4⋊C4×Dic47C2212yes—
62C4⋊C4×Dic47C2211yescharacteristic
63C4⋊C4×Dic47C2212yes—
64C4⋊C4×Dic47C2211yescharacteristic
65C4×Dic47⋊C4C2212yes—
66C4×Dic47⋊C4C2212yes—
67C4×C4⋊Dic47C2211yescharacteristic
68C4×C4⋊Dic47C2212yes—
69C4×C4⋊Dic47C2211yescharacteristic
70#15688974C2214yes—
71#15688974C2212yes—
72C4×(C2×D47)⋊C4C2212yes—
73#15688970C2214yes—
74#95×D47C2212yes—
75C2×C4⋊Q8×C47C2211yescharacteristic
76C2×C4⋊Q8×C47C2212yes—
77C2×C4×Q8×C47C2211yescharacteristic
78C47×#398C2214yes—
79C47×#391C2212yes—
80C47×#389C2212yes—
81C47×#389C2212yes—
82C4×C4⋊C4×C47C2211yescharacteristic
83C22×C4×D47C2311yescharacteristic
84C22×C4×D47C2312yes—
85C22×C4×D47C2311yescharacteristic
86C22×C4×D47C2311yescharacteristic
87C22×C4×D47C2312yes—
88#13094015—28no—
89#13094000—24no—
90#13094000—28no—
91Q8×Dic47—44no—
92C2×C4⋊Dic47C2311yescharacteristic
93C2×C4⋊Dic47C2312yes—
94C2×C4⋊Dic47C2311yescharacteristic
95C2×C4⋊Dic47C2311yescharacteristic
96C2×C4⋊Dic47C2312yes—
97C2×C4⋊Dic47C2314yes—
98C2×C4⋊Dic47C2312yes—
99C2×C4⋊Dic47C2314yes—
100C2×C4⋊Dic47C2314yes—
101C2×C4⋊Dic47C2311yescharacteristic
102C2×Dic47⋊C4C2312yes—
103C2×Dic47⋊C4C2314yes—
104C2×Dic47⋊C4C2314yes—
105C2×Dic47⋊C4C2312yes—
106C2×Dic47⋊C4C2312yes—
107C2×Dic47⋊C4C2312yes—
108C4⋊C4×D47—24no—
109C4⋊C4×D47Q818yescomplemented
110C2×(C2×D47)⋊C4C2312yes—
111C2×(C2×D47)⋊C4C2314yes—
112C2×(C2×D47)⋊C4C2312yes—
113C2×C4×Dic47C2312yes—
114C2×C4×Dic47C2311yescharacteristic
115C2×C4×Dic47C2312yes—
116C2×C4×Dic47C2311yescharacteristic
117C2×C4×Dic47C2311yescharacteristic
118C2×C4×Dic47C2314yes—
119C2×C4×Dic47C2314yes—
120C2×C4×Dic47C2312yes—
121C2×C4×Dic47C2312yes—
122C42×D47—24no—
123C42×D47—44no—
124C47⋊#95C2312yes—
125C47⋊#95C2312yes—
126C47⋊#95C2312yes—
127C47⋊#95C2314yes—
128C47⋊#95C2312yes—
129C47⋊#95C2312yes—
130C47⋊#95C2312yes—
131C47⋊#95C2314yes—
132C47⋊#95C2312yes—
133C47⋊#95C2312yes—
134C47⋊#95C2314yes—
135C47⋊#95C2314yes—
136C47⋊#95C2314yes—
137C47⋊#95C2314yes—
138C47⋊#95C2314yes—
139C47⋊#95C2314yes—
140C22×Q8×C47C2311yescharacteristic
141C22×Q8×C47C2312yes—
142C4⋊Q8×C47—24no—
143C4⋊Q8×C47—44no—
144C4×Q8×C47—24no—
145C2×C4⋊C4×C47C2311yescharacteristic
146C2×C4⋊C4×C47C2312yes—
147C2×C4⋊C4×C47C2312yes—
148C2×C4⋊C4×C47C2312yes—
149C2×C4⋊C4×C47C2314yes—
150C2×C4⋊C4×C47C2314yes—
151C2×C4⋊C4×C47C2311yescharacteristic
152C2×C4⋊C4×C47C2311yescharacteristic
153C2×C42×C47C2312yes—
154C2×C42×C47C2312yes—
155C2×C42×C47C2311yescharacteristic
156#95×C47C2312yes—
157#95×C47C2314yes—
158#95×C47C2314yes—
159C23×D47C2411yescharacteristic
160C2×Q8×C47—22no—
161C2×Q8×C47—24no—
162C2×Q8×C47—22no—
163C2×Q8×C47—24no—
164C2×Q8×C47—22no—
165C2×Q8×C47—24no—
166C22×C4×C47C2412yes—
167C22×C4×C47C2412yes—
168C22×C4×C47C2414yes—
169C22×C4×C47C2412yes—
170C22×C4×C47C2411yescharacteristic
171C22×C4×C47C2412yes—
172C22×C4×C47C2411yescharacteristic
173C22×C4×C47C2411yescharacteristic
174C22×Dic47C2411yescharacteristic
175C22×Dic47C2412yes—
176C22×Dic47C2412yes—
177C22×Dic47C2412yes—
178C22×Dic47C2414yes—
179C22×Dic47C2411yescharacteristic
180C22×Dic47C2412yes—
181C22×Dic47C2411yescharacteristic
182C2×C4×D47—22no—
183C2×C4×D47—22no—
184C2×C4×D47—22no—
185C2×C4×D47—22no—
186C2×C4×D47—22no—
187C2×C4×D47—24no—
188C2×C4×D47—22no—
189C2×C4×D47—24no—
190C2×C4×D47—42no—
191C2×C4×D47—42no—
192C2×C4×D47—42no—
193C2×C4×D47—41no—
194C2×C4×D47C2×Q814yes—
195C2×C4×D47C2×Q818yes—
196C2×C4×D47#4114yes—
197C4⋊C4×C47—22no—
198C4⋊C4×C47—24no—
199C4⋊C4×C47—24no—
200C4⋊C4×C47—24no—
201C4⋊C4×C47—44no—
202C4⋊C4×C47—44no—
203C4⋊C4×C47C2×Q814yes—
204C4⋊C4×C47#4114yes—
205C42×C47—24no—
206C42×C47—24no—
207C42×C47#4114yes—
208(C2×D47)⋊C4—28no—
209(C2×D47)⋊C4—24no—
210(C2×D47)⋊C4—24no—
211C4⋊Dic47—22no—
212C4⋊Dic47—22no—
213C4⋊Dic47—24no—
214C4⋊Dic47—28no—
215C4⋊Dic47—24no—
216C4⋊Dic47—22no—
217C4⋊Dic47—42no—
218C4⋊Dic47—44no—
219C4⋊Dic47—44no—
220C4⋊Dic47C2×Q814yes—
221Dic47⋊C4—24no—
222Dic47⋊C4—28no—
223Dic47⋊C4—28no—
224Dic47⋊C4—24no—
225Dic47⋊C4—42no—
226Dic47⋊C4C2×Q818yes—
227C4×Dic47—22no—
228C4×Dic47—22no—
229C4×Dic47—22no—
230C4×Dic47—24no—
231C4×Dic47—24no—
232C4×Dic47—42no—
233C4×Dic47—42no—
234C4×Dic47—44no—
235C4×Dic47—44no—
236C23×C47C2511yescharacteristic
237C22×D47—21no—
238C22×D47—21no—
239C22×D47—21no—
240C22×D47—21no—
241C22×D47C22×Q812yes—
242C22×D47C2×#4112yes—
243C22×D472−1+412yes—
244Q8×C47—44no—
245Q8×C47—48no—
246C2×C4×C47—22no—
247C2×C4×C47—22no—
248C2×C4×C47—21no—
249C2×C4×C47—22no—
250C2×C4×C47—21no—
251C2×C4×C47—22no—
252C2×C4×C47—22no—
253C2×C4×C47—24no—
254C2×C4×C47—22no—
255C2×C4×C47—22no—
256C2×C4×C47—24no—
257C2×C4×C47—24no—
258C2×C4×C47—22no—
259C2×C4×C47—22no—
260C2×C4×C47—22no—
261C2×C4×C47C22×Q814yes—
262C2×C4×C47C22×Q812yes—
263C2×C4×C47C2×#4112yes—
264C2×C4×C47C2×#4112yes—
265C2×C4×C47C2×#4112yes—
266C2×C4×C47C2×#4112yes—
267C2×C4×C47C2×#4114yes—
268C2×C4×C47C2×#4112yes—
269C2×C4×C47H(5,𝔽2)12yes—
270C2×Dic47—21no—
271C2×Dic47—21no—
272C2×Dic47—21no—
273C2×Dic47—22no—
274C2×Dic47—21no—
275C2×Dic47—22no—
276C2×Dic47—21no—
277C2×Dic47—22no—
278C2×Dic47—22no—
279C2×Dic47—24no—
280C2×Dic47—22no—
281C2×Dic47—24no—
282C2×Dic47—22no—
283C2×Dic47—22no—
284C2×Dic47—22no—
285C2×Dic47—22no—
286C2×Dic47—21no—
287C2×Dic47—22no—
288C2×Dic47—24no—
289C2×Dic47—21no—
290C2×Dic47—22no—
291C2×Dic47C22×Q812yes—
292C2×Dic47C22×Q814yes—
293C2×Dic47C2×#4112yes—
294C4×D47—28no—
295C4×D47—28no—
296C4×D47—48no—
297C4×D47—44no—
298C4×D47—48no—
299#36599—471nomaximal, Sylow
300C22×C47C23×Q811yescharacteristic
301C22×C47C22×#4111yescharacteristic
302C22×C47C22×#4111yescharacteristic
303C22×C47C22×#4111yescharacteristic
304C22×C47C2×H(5,𝔽2)11yescharacteristic
305C22×C47C2×2−1+411yescharacteristic
306C22×C47#58511yescharacteristic
307C2×D47—22no—
308C2×D47—22no—
309C2×D47—22no—
310C2×D47—22no—
311C2×D47—22no—
312C2×D47—22no—
313C2×D47#55714yes—
314C4×C47—22no—
315C4×C47—24no—
316C4×C47—24no—
317C4×C47—42no—
318C4×C47—44no—
319C4×C47—42no—
320C4×C47#53314yes—
321C4×C47#55314yes—
322Dic47—22no—
323Dic47—22no—
324Dic47—24no—
325Dic47—42no—
326Dic47—44no—
327Dic47—41no—
328Dic47—42no—
329Dic47—42no—
330#2857—471no—
331#2857—472no—
332#2585—474no—
333#2569—474no—
334#2552—471no—
335#2546D4711yescharacteristic, complemented
336#2511—472no—
337#2509—472no—
338#2446—472no—
339#2442—472no—
340#2370—472no—
341#2365—471no—
342#2355—471no—
343#2345—472no—
344C4×C22⋊Q8—472no—
345C4×C22⋊Q8—471no—
346C2×C4×C4⋊C4—471no—
347C2×C47C2×#53311yescharacteristic
348C2×C47C2×#55311yescharacteristic
349C2×C47C2×#55711yescharacteristic
350C2×C47#348011yescharacteristic
351C2×C47#349511yescharacteristic
352C2×C47#350111yescharacteristic
353C2×C47#352011yescharacteristic
354D47—24no—
355C2×C4⋊Q8C2×D4711yescharacteristic
356C2×C4⋊Q8C2×D4712yes—
357C2×#125—472no—
358C2×C22⋊Q8—472no—
359C2×C22⋊Q8—471no—
360C2×C4×Q8—472no—
361C2×C4×Q8C2×D4711yescharacteristic
362C22×C4⋊C4—471no—
363C22×C4⋊C4—471no—
364C22×C42—471no—
365C22×C42—472no—
366#400—474no—
367#400—474no—
368#400—474no—
369#398—474no—
370#398—474no—
371#398C2×D4714yes—
372#391—472no—
373#391—472no—
374#391—472no—
375#391—471no—
376#391C2×D4712yes—
377#389—472no—
378#389—474no—
379#389—474no—
380#389—472no—
381#389—472no—
382#389—471no—
383#389—472no—
384#389—472no—
385#389C2×D4712yes—
386#389C2×D4712yes—
387#387—474no—
388#387—474no—
389#387—474no—
390#387—472no—
391#387—474no—
392#387—474no—
393#387—474no—
394#387—474no—
395#385—472no—
396#385—474no—
397#384—471no—
398#384—472no—
399#382—472no—
400#380—472no—
401#380—474no—
402#380—474no—
403#380—472no—
404#380—471no—
405C4×C4⋊C4—472no—
406C4×C4⋊C4—472no—
407C4×C4⋊C4—471no—
408C4×C4⋊C4—471no—
409C4×C4⋊C4—472no—
410C4×C4⋊C4—471no—
411C4×C4⋊C4—471no—
412C4×C4⋊C4—472no—
413C4×C4⋊C4—472no—
414C4×C4⋊C4—472no—
415C4×C4⋊C4—474no—
416C4×C4⋊C4C2×D4711yescharacteristic
417C4×C22⋊C4—472no—
418C4×C22⋊C4—474no—
419C2×#95—472no—
420C47#3659911yescharacteristic, complemented, Sylow
421C22×Q8C22×D4712yes—
422C22×Q8C22×D4711yescharacteristic
423C23×C4—471no—
424C23×C4—471no—
425C23×C4—472no—
426C23×C4—472no—
427C23×C4—471no—
428C4⋊Q8—24no—
429C4⋊Q8—44no—
430#125—948no—
431C22⋊Q8—948no—
432C22⋊Q8—944no—
433C4×Q8—24no—
434C4×Q8—1884no—
435C2×C4⋊C4—472no—
436C2×C4⋊C4—471no—
437C2×C4⋊C4—478no—
438C2×C4⋊C4—472no—
439C2×C4⋊C4—471no—
440C2×C4⋊C4—471no—
441C2×C4⋊C4—472no—
442C2×C4⋊C4—474no—
443C2×C4⋊C4—471no—
444C2×C4⋊C4—472no—
445C2×C4⋊C4—474no—
446C2×C4⋊C4—472no—
447C2×C4⋊C4—474no—
448C2×C4⋊C4—474no—
449C2×C4⋊C4—472no—
450C2×C4⋊C4—472no—
451C2×C4⋊C4—474no—
452C2×C4⋊C4—944no—
453C2×C4⋊C4C22×D4711yescharacteristic
454C2×C4⋊C4C22×D4712yes—
455C2×C4⋊C4C22×D4712yes—
456C2×C4⋊C4C22×D4712yes—
457C2×C4⋊C4C22×D4714yes—
458C2×C4⋊C4C22×D4714yes—
459C2×C4⋊C4C22×D4711yescharacteristic
460C2×C4⋊C4C22×D4711yescharacteristic
461C2×C22⋊C4—472no—
462C2×C22⋊C4—474no—
463C2×C22⋊C4—472no—
464C2×C42—472no—
465C2×C42—472no—
466C2×C42—471no—
467C2×C42—471no—
468C2×C42—471no—
469C2×C42—474no—
470C2×C42—474no—
471C2×C42—472no—
472C2×C42—472no—
473C2×C42—944no—
474C2×C42—1884no—
475C2×C42C22×D4712yes—
476C2×C42C22×D4712yes—
477C2×C42C22×D4711yescharacteristic
478#95—472no—
479#95—472no—
480#95—472no—
481#95—472no—
482#95—472no—
483#95—474no—
484#95—472no—
485#95—474no—
486#95—474no—
487#95—474no—
488#95—472no—
489#95—472no—
490#95—474no—
491#95—474no—
492#95—474no—
493#95—474no—
494#95C22×D4712yes—
495#95C22×D4714yes—
496#95C22×D4714yes—
497C24—471no—
498C2×Q8—24no—
499C2×Q8—22no—
500C2×Q8—22no—
501C2×Q8—24no—
502C2×Q8—24no—
503C2×Q8—22no—
504C22×C4—471no—
505C22×C4—472no—
506C22×C4—471no—
507C22×C4—474no—
508C22×C4—474no—
509C22×C4—478no—
510C22×C4—471no—
511C22×C4—472no—
512C22×C4—472no—
513C22×C4—472no—
514C22×C4—474no—
515C22×C4—944no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4—942no—
519C22×C4—942no—
520C22×C4—942no—
521C22×C4—942no—
522C22×C4—944no—
523C22×C4—1882no—
524C22×C4—1882no—
525C22×C4—1882no—
526C22×C4—1881no—
527C22×C4C23×D4712yes—
528C22×C4C23×D4712yes—
529C22×C4C23×D4714yes—
530C22×C4C23×D4712yes—
531C22×C4C23×D4711yescharacteristic
532C22×C4C23×D4712yes—
533C22×C4C23×D4711yescharacteristic
534C22×C4C23×D4711yescharacteristic
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—478no—
542C4⋊C4—474no—
543C4⋊C4—944no—
544C4⋊C4—948no—
545C4⋊C4—942no—
546C4⋊C4—944no—
547C4⋊C4—944no—
548C4⋊C4—948no—
549C4⋊C4—942no—
550C4⋊C4—948no—
551C4⋊C4—944no—
552C4⋊C4—942no—
553C4⋊C4—1882no—
554C4⋊C4—1882no—
555C4⋊C4—1884no—
556C4⋊C4—1884no—
557C4⋊C4Q8×D4714yes—
558C4⋊C4#1062535414yes—
559C22⋊C4—944no—
560C22⋊C4—944no—
561C22⋊C4—948no—
562C42—24no—
563C42—24no—
564C42—942no—
565C42—942no—
566C42—942no—
567C42—944no—
568C42—944no—
569C42—1884no—
570C42—1882no—
571C42—1882no—
572C42—1884no—
573C42#1062535214yes—
574C23—472no—
575C23—472no—
576C23—472no—
577C23—941no—
578C23—941no—
579C23—941no—
580C23—941no—
581C23C24×D4711yescharacteristic
582Q8—44no—
583Q8—48no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—474no—
600C2×C4—472no—
601C2×C4—472no—
602C2×C4—941no—
603C2×C4—941no—
604C2×C4—941no—
605C2×C4—942no—
606C2×C4—941no—
607C2×C4—942no—
608C2×C4—948no—
609C2×C4—948no—
610C2×C4—942no—
611C2×C4—944no—
612C2×C4—941no—
613C2×C4—942no—
614C2×C4—942no—
615C2×C4—944no—
616C2×C4—942no—
617C2×C4—942no—
618C2×C4—942no—
619C2×C4—942no—
620C2×C4—941no—
621C2×C4—942no—
622C2×C4—944no—
623C2×C4—942no—
624C2×C4—941no—
625C2×C4—1888no—
626C2×C4—1884no—
627C2×C4—1888no—
628C2×C4C2×Q8×D4714yes—
629C2×C4C2×Q8×D4712yes—
630C2×C4#41×D4712yes—
631C2×C4C2×#1062535212yes—
632C2×C4C2×#1062535212yes—
633C2×C4C2×#1062535212yes—
634C2×C4C2×#1062535414yes—
635C2×C4C2×#1062535412yes—
636C2×C4#1309406912yes—
637C22—474no—
638C22—942no—
639C22—942no—
640C22—942no—
641C22—942no—
642C22—942no—
643C22—942no—
644C22C22×Q8×D4711yescharacteristic
645C22C2×#41×D4711yescharacteristic
646C22C22×#1062535211yescharacteristic
647C22C22×#1062535411yescharacteristic
648C22C2×#1309406911yescharacteristic
649C222−1+4×D4711yescharacteristic
650C22#1568996011yescharacteristic
651C4—22no—
652C4—24no—
653C4—24no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—942no—
658C4—944no—
659C4—942no—
660C4—1881no—
661C4—1882no—
662C4—1882no—
663C4—1882no—
664C4—1884no—
665C4#1568967814yes—
666C4#1568982014yes—
667C2—944no—
668C2C2×#1568967811yescharacteristic
669C2C2×#1568982011yescharacteristic
670C2#557×D4711yescharacteristic
671C2#2025698811yescharacteristic
672C2#2025713711yescharacteristic
673C2#2025718311yescharacteristic
674C2#2025743811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC47×#2546C2
prime radicalC23×D47C24
hypercenter#2546D47
hypocenterC47#36599
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2546

Complements

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

NH ≅ G/Ntype
3. #20248788C2semidirect
5. #20248419C2semidirect
6. #20247965C2semidirect
7. #20247939C2semidirect
8. #20247160C2semidirect
10. #20246489C2semidirect
11. #20246448C2semidirect
12. #20246345C2semidirect
17. C47×#2546C2semidirect
109. C4⋊C4×D47Q8semidirect
335. #2546D47semidirect
420. C47#36599semidirect

Maximal subgroups

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

subgroupindex
#202517352
#202487882
#202486062
#202484192
#202479652
#202479392
#202471602
#202471102
#202464892
#202464482
#202463452
#202462642
#202459262
C4×#130940002
C4×C4⋊C4×D472
C47×#25462
#3659947