Finite Group Databasebeta

← #26141573

Subgroups — #26141573

Order 12032.

2062 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689034C211yesmaximal, characteristic
2C2×#15688994C212yesmaximal
3C2×C4×(C2×D47)⋊C4C211yesmaximal, characteristic
4#20241659C211yesmaximal, characteristic, complemented
5#20241651C211yesmaximal, characteristic, complemented
6#20241630C211yesmaximal, characteristic
7#20241626C212yesmaximal
8#20241621C212yesmaximal, complemented
9#20241620C212yesmaximal, complemented
10#20241615C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C47C211yesmaximal, characteristic, complemented
12C23×C4×D47C2211yescharacteristic
13C22×Dic47⋊C4C2212yes—
14C22×(C2×D47)⋊C4C2212yes—
15C22×(C2×D47)⋊C4C2212yes—
16C22×(C2×D47)⋊C4C2211yescharacteristic
17C22×(C2×D47)⋊C4C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19#15689034C418yescomplemented
20#15688994—28no—
21C4×(C2×D47)⋊C4—24no—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2211yescharacteristic
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2211yescharacteristic
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C23×Q8×C47C2211yescharacteristic
34C2×C4×Q8×C47—28no—
35C2×C4×Q8×C47—24no—
36C22×C4⋊C4×C47C2211yescharacteristic
37C22×C4⋊C4×C47C2212yes—
38C22×C42×C47C2212yes—
39C22×C42×C47C2211yescharacteristic
40C24×D47C2311yescharacteristic
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2312yes—
44C23×Dic47C2311yescharacteristic
45C23×Dic47C2311yescharacteristic
46C22×C4×D47—22no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—24no—
50C22×C4×D47—42no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—24no—
53C2×Dic47⋊C4—42no—
54C2×(C2×D47)⋊C4—22no—
55C2×(C2×D47)⋊C4—22no—
56C2×(C2×D47)⋊C4—24no—
57C2×(C2×D47)⋊C4—24no—
58C2×(C2×D47)⋊C4—22no—
59C2×(C2×D47)⋊C4—22no—
60C2×(C2×D47)⋊C4—24no—
61C2×(C2×D47)⋊C4—24no—
62C2×(C2×D47)⋊C4—42no—
63C2×(C2×D47)⋊C4C2×C418yes—
64C2×(C2×D47)⋊C4C2×C414yes—
65C2×(C2×D47)⋊C4Q814yescomplemented
66C2×C4×Dic47—22no—
67C2×C4×Dic47—42no—
68C47⋊#95—22no—
69C47⋊#95—22no—
70C47⋊#95—24no—
71C47⋊#95—24no—
72C47⋊#95—24no—
73C47⋊#95—24no—
74C47⋊#95—22no—
75C47⋊#95—42no—
76C47⋊#95—42no—
77C47⋊#95C2×C414yescomplemented
78C47⋊#95C2×C418yescomplemented
79C22×Q8×C47—24no—
80C22×Q8×C47—22no—
81C22×Q8×C47—22no—
82C22×Q8×C47—24no—
83C22×Q8×C47C2×C414yescomplemented
84C23×C4×C47C2312yes—
85C23×C4×C47C2312yes—
86C23×C4×C47C2311yescharacteristic
87C23×C4×C47C2311yescharacteristic
88C23×C4×C47C2311yescharacteristic
89C4×Q8×C47—232no—
90C2×C4⋊C4×C47—24no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47—24no—
93C2×C4⋊C4×C47—24no—
94C2×C4⋊C4×C47D418yes—
95C2×C42×C47—24no—
96C2×C42×C47—24no—
97C2×C42×C47—24no—
98C2×C42×C47—24no—
99C2×C42×C47D414yes—
100C24×C47C2411yescharacteristic
101C23×D47—21no—
102C23×D47—22no—
103C23×D47—21no—
104C23×D47—22no—
105C23×D47—21no—
106C23×D47—21no—
107C23×D47—22no—
108C23×D47—22no—
109C23×D47C22×C412yes—
110C23×D47C2×Q812yes—
111C23×D47#4112yes—
112C2×Q8×C47—28no—
113C2×Q8×C47—28no—
114C2×Q8×C47—216no—
115C2×Q8×C47—28no—
116C2×Q8×C47—216no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47C22×C414yes—
135C22×C4×C47C22×C412yes—
136C22×C4×C47C2×D414yes—
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D414yes—
140C22×C4×C47C2×D412yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×Q812yes—
143C22×C4×C47#4112yes—
144C22×C4×C47#4114yes—
145C22×C4×C47#4112yes—
146C22×Dic47—21no—
147C22×Dic47—21no—
148C22×Dic47—21no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—21no—
154C22×Dic47—22no—
155C22×Dic47—21no—
156C22×Dic47—22no—
157C22×Dic47—21no—
158C22×Dic47—22no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—22no—
164C22×Dic47—22no—
165C22×Dic47—22no—
166C22×Dic47—22no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47C22×C414yes—
172C22×Dic47C22×C412yes—
173C22×Dic47C2×Q812yes—
174C22×Dic47#4112yes—
175C2×C4×D47—28no—
176C2×C4×D47—48no—
177C2×C4×D47—48no—
178C2×C4×D47—44no—
179C2×C4×D47—44no—
180C4⋊C4×C47—28no—
181C4⋊C4×C47—216no—
182C42×C47—216no—
183C42×C47—28no—
184(C2×D47)⋊C4—28no—
185(C2×D47)⋊C4—216no—
186(C2×D47)⋊C4—28no—
187(C2×D47)⋊C4—48no—
188Dic47⋊C4—48no—
189C4×Dic47—44no—
190C23×C47C23×C411yescharacteristic
191C23×C47C22×D411yescharacteristic
192C23×C47C22×D412yes—
193C23×C47C22×Q811yescharacteristic
194C23×C47C2×#4111yescharacteristic
195C23×C47C2×#4111yescharacteristic
196C23×C47C2×#4111yescharacteristic
197C23×C47C2×#4112yes—
198C23×C47C2×#4111yescharacteristic
199C23×C47H(5,𝔽2)12yes—
200C23×C472−1+412yes—
201C22×D47—24no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—24no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—24no—
210C22×D47—24no—
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—41no—
214C22×D47—41no—
215C22×D47—42no—
216C22×D47—41no—
217C22×D47—41no—
218C22×D47—42no—
219C22×D47—42no—
220C22×D47—42no—
221C22×D47—42no—
222C22×D47—42no—
223C22×D47C4×Q814yescomplemented
224Q8×C47—232no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—22no—
238C2×C4×C47—22no—
239C2×C4×C47—24no—
240C2×C4×C47—24no—
241C2×C4×C47—24no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47—24no—
245C2×C4×C47—24no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47C4×D414yes—
251C2×C4×C47C4×D418yes—
252C2×C4×C47#12014yes—
253C2×C4×C47C22⋊Q814yes—
254C2×Dic47—22no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—22no—
260C2×Dic47—22no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271C2×Dic47—42no—
272C2×Dic47—42no—
273C2×Dic47—42no—
274C2×Dic47—42no—
275C2×Dic47—42no—
276C2×Dic47—42no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282C2×Dic47—42no—
283C4×D47—416no—
284#14172—471nomaximal, Sylow
285C22×C47C2×C4×D411yescharacteristic
286C22×C47C2×C4×D412yes—
287C22×C47C2×C4×Q811yescharacteristic
288C22×C47C4×#4111yescharacteristic
289C22×C47#52012yes—
290C22×C47C2×#12011yescharacteristic
291C22×C47C2×C22⋊Q811yescharacteristic
292C22×C47#52511yescharacteristic
293C22×C47#52511yescharacteristic
294C22×C47#52512yes—
295C22×C47#52911yescharacteristic
296C22×C47#53311yescharacteristic
297C22×C47#53511yescharacteristic
298C22×C47#53611yescharacteristic
299C22×C47#54212yes—
300C22×C47#54612yes—
301C22×C47#54612yes—
302C22×C47#54812yes—
303C22×C47D4×Q812yes—
304C22×C47#55212yes—
305C22×C47#55412yes—
306C22×C47#55612yes—
307C22×C47#55912yes—
308C2×D47—28no—
309C2×D47—24no—
310C2×D47—42no—
311C2×D47—42no—
312C2×D47—42no—
313C2×D47—44no—
314C2×D47—42no—
315C2×D47—42no—
316C2×D47—44no—
317C2×D47—44no—
318C2×D47—42no—
319C4×C47—24no—
320C4×C47—24no—
321C4×C47—28no—
322C4×C47—24no—
323C4×C47—44no—
324Dic47—42no—
325Dic47—42no—
326Dic47—44no—
327Dic47—42no—
328Dic47—82no—
329C22×C4×Q8D4711yescharacteristic, complemented
330C2×#393—471no—
331C2×#385—472no—
332C2×C4×C22⋊C4—471no—
333#1446—471no—
334#1444—471no—
335#1439—471no—
336#1437—472no—
337#1436—472no—
338#1436—472no—
339#1434—471no—
340C2×C47C4×#12011yescharacteristic
341C2×C47C4×C22⋊Q811yescharacteristic
342C2×C47#232611yescharacteristic
343C2×C47#236012yes—
344C2×C47#236712yes—
345C2×C47#240411yescharacteristic
346C2×C47#240911yescharacteristic
347C2×C47#245812yes—
348C2×C47#246512yes—
349C2×C47#258011yescharacteristic
350C2×C47#259911yescharacteristic
351D47—44no—
352C23×Q8C2×D4711yescharacteristic
353C24×C4—471no—
354C2×C4×Q8—28no—
355C2×C4×Q8—24no—
356C22×C4⋊C4—472no—
357C22×C4⋊C4C2×D4711yescharacteristic
358C22×C4⋊C4C2×D4712yes—
359C22×C22⋊C4—472no—
360C22×C22⋊C4—471no—
361C22×C22⋊C4—471no—
362C22×C22⋊C4—472no—
363C22×C42—471no—
364C22×C42C2×D4712yes—
365C22×C42C2×D4711yescharacteristic
366#393—478no—
367#385—948no—
368C4×C22⋊C4—944no—
369C2×#95—471no—
370C2×#95—472no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95—471no—
374C2×#95—471no—
375C2×#95—472no—
376C2×#95—472no—
377C2×#95—471no—
378C2×#95—472no—
379C2×#95—472no—
380C47#1417211yescharacteristic, complemented, Sylow
381C25—471no—
382C22×Q8—22no—
383C22×Q8—24no—
384C22×Q8—24no—
385C22×Q8—22no—
386C22×Q8C4×D4714yescomplemented
387C23×C4—472no—
388C23×C4—471no—
389C23×C4—471no—
390C23×C4—471no—
391C23×C4—472no—
392C23×C4—942no—
393C23×C4—942no—
394C23×C4—942no—
395C23×C4—944no—
396C23×C4—1882no—
397C23×C4C22×D4712yes—
398C23×C4C22×D4712yes—
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401C23×C4C22×D4711yescharacteristic
402C4×Q8—232no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—24no—
407C2×C4⋊C4—944no—
408C2×C4⋊C4—944no—
409C2×C4⋊C4—1882no—
410C2×C4⋊C4C22⋊D4718yes—
411C2×C22⋊C4—478no—
412C2×C22⋊C4—474no—
413C2×C22⋊C4—474no—
414C2×C22⋊C4—944no—
415C2×C22⋊C4—942no—
416C2×C22⋊C4—942no—
417C2×C22⋊C4—942no—
418C2×C22⋊C4—942no—
419C2×C22⋊C4—944no—
420C2×C22⋊C4—944no—
421C2×C22⋊C4—944no—
422C2×C22⋊C4—1882no—
423C2×C42—24no—
424C2×C42—24no—
425C2×C42—24no—
426C2×C42—24no—
427C2×C42—942no—
428C2×C42—1882no—
429C2×C42C22⋊D4714yes—
430#95—474no—
431#95—478no—
432#95—944no—
433#95—942no—
434#95—944no—
435#95—942no—
436#95—942no—
437#95—944no—
438#95—944no—
439#95—1882no—
440#95—1882no—
441C24—472no—
442C24—472no—
443C24—472no—
444C24—942no—
445C24—941no—
446C24—941no—
447C24—942no—
448C24—941no—
449C24—941no—
450C24—942no—
451C24—942no—
452C24C23×D4711yescharacteristic
453C2×Q8—28no—
454C2×Q8—216no—
455C2×Q8—28no—
456C2×Q8—28no—
457C2×Q8—216no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—474no—
476C22×C4—472no—
477C22×C4—472no—
478C22×C4—472no—
479C22×C4—948no—
480C22×C4—941no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—942no—
486C22×C4—941no—
487C22×C4—942no—
488C22×C4—941no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—941no—
492C22×C4—942no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—1888no—
506C22×C4—1888no—
507C22×C4—1884no—
508C22×C4—1884no—
509C22×C4C2×C4×D4714yes—
510C22×C4C2×C4×D4712yes—
511C22×C4#1062535012yes—
512C22×C4#1062535014yes—
513C22×C4Q8×D4712yes—
514C22×C4#1062535412yes—
515C22×C4C2×C22⋊D4714yes—
516C22×C4C2×C22⋊D4714yes—
517C22×C4C2×C22⋊D4712yes—
518C22×C4C2×C22⋊D4714yes—
519C22×C4C2×C22⋊D4712yes—
520C22×C4C2×C22⋊D4712yes—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—1888no—
524C22⋊C4—948no—
525C22⋊C4—9416no—
526C22⋊C4—948no—
527C22⋊C4—1888no—
528C42—216no—
529C42—28no—
530C42—1884no—
531C23—474no—
532C23—944no—
533C23—944no—
534C23—944no—
535C23—944no—
536C23—944no—
537C23—942no—
538C23—942no—
539C23—944no—
540C23—942no—
541C23—942no—
542C23—942no—
543C23—942no—
544C23—1882no—
545C23—1882no—
546C23—1882no—
547C23—1882no—
548C23—1882no—
549C23—1881no—
550C23—1882no—
551C23—1881no—
552C23—1881no—
553C23—1881no—
554C23C22×C4×D4711yescharacteristic
555C23C22×C22⋊D4712yes—
556C23C22×C22⋊D4711yescharacteristic
557C23C2×Q8×D4711yescharacteristic
558C23#41×D4711yescharacteristic
559C23#41×D4711yescharacteristic
560C23C2×#1062535011yescharacteristic
561C23C2×#1062535012yes—
562C23C2×#1062535411yescharacteristic
563C23#1309407012yes—
564C23#1309407212yes—
565Q8—232no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—944no—
592C2×C4—944no—
593C2×C4—942no—
594C2×C4—944no—
595C2×C4—944no—
596C2×C4—944no—
597C2×C4—944no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—18816no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1882no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4—1882no—
610C2×C4—1882no—
611C2×C4—1882no—
612C2×C4—1882no—
613C2×C4—1882no—
614C2×C4—1882no—
615C2×C4—1881no—
616C2×C4—1881no—
617C2×C4—1881no—
618C2×C4—1881no—
619C2×C4—1881no—
620C2×C4—1881no—
621C2×C4C4×C22⋊D4714yes—
622C2×C4C4×C22⋊D4718yes—
623C2×C4#1309399314yes—
624C2×C4#1309400014yes—
625C22—948no—
626C22—944no—
627C22—1884no—
628C22—1882no—
629C22—1884no—
630C22—1882no—
631C22—1882no—
632C22—1882no—
633C22—1882no—
634C22—1884no—
635C22—1882no—
636C22C2×C4×C22⋊D4711yescharacteristic
637C22C2×C4×C22⋊D4712yes—
638C22C4×Q8×D4711yescharacteristic
639C22C4×#1062535411yescharacteristic
640C22#1568960512yes—
641C22C2×#1309399311yescharacteristic
642C22C2×#1309400011yescharacteristic
643C22#1568963011yescharacteristic
644C22#1568963412yes—
645C22#1568963411yescharacteristic
646C22#1568966011yescharacteristic
647C22#1568967711yescharacteristic
648C22#1568968711yescharacteristic
649C22#1568969711yescharacteristic
650C22#1568974512yes—
651C22#1568976112yes—
652C22#1568976112yes—
653C22#1568978412yes—
654C22Q8×C22⋊D4712yes—
655C22#1568981412yes—
656C22#1568983312yes—
657C22#1568984512yes—
658C22#1568985812yes—
659C4—28no—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—44no—
664C4—3762no—
665C4—1882no—
666C4—1882no—
667C4—1882no—
668C4—1884no—
669C2—1884no—
670C2C4×#1309399311yescharacteristic
671C2C4×#1309400011yescharacteristic
672C2#2024609511yescharacteristic
673C2#2024638912yes—
674C2#2024645512yes—
675C2#2024670311yescharacteristic
676C2#2024674711yescharacteristic
677C2#2024730412yes—
678C2#2024740312yes—
679C2#2024874611yescharacteristic
680C2#2024895911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC22×C4×Q8×C47C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC22×C4×Q8D47
hypocenterC47#14172
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #20241659C2semidirect
5. #20241651C2semidirect
8. #20241621C2semidirect
9. #20241620C2semidirect
10. #20241615C2semidirect
11. C22×C4×Q8×C47C2semidirect
19. #15689034C4semidirect
65. C2×(C2×D47)⋊C4Q8semidirect
77. C47⋊#95C2×C4semidirect
78. C47⋊#95C2×C4semidirect
83. C22×Q8×C47C2×C4semidirect
223. C22×D47C4×Q8semidirect
329. C22×C4×Q8D47semidirect
380. C47#14172semidirect
386. C22×Q8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156890342
C2×#156889942
C2×C4×(C2×D47)⋊C42
#202416592
#202416512
#202416302
#202416262
#202416212
#202416202
#202416152
C22×C4×Q8×C472
#1417247