Finite Group Databasebeta

← #26147567

Subgroups — #26147567

Order 12032.

2734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094000C211yesmaximal, characteristic
2C22×C42×D47C211yesmaximal, characteristic
3C2×#15689030C211yesmaximal, characteristic, complemented
4C2×#15688980C211yesmaximal, characteristic
5#20241619C214yesmaximal, complemented
6#20241618C211yesmaximal, characteristic, complemented
7#20241617C211yesmaximal, characteristic, complemented
8(C2×D47)⋊#95C214yesmaximal
9C2×C47×#391C211yesmaximal, characteristic, complemented
10C23×C4×D47C2212yes—
11C23×C4×D47C2211yescharacteristic
12C2×#13094000—24no—
13C2×#13094000—24no—
14C2×#13094000C418yescomplemented
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C4⋊Dic47C2211yescharacteristic
17C22×Dic47⋊C4C2212yes—
18C22×(C2×D47)⋊C4C2212yes—
19C22×(C2×D47)⋊C4C2212yes—
20C22×C4×Dic47C2212yes—
21C22×C4×Dic47C2211yescharacteristic
22C2×C42×D47—24no—
23C2×C42×D47—24no—
24C2×C42×D47—24no—
25#15689030—24no—
26#15688980—24no—
27C2×C47⋊#95C2214yes—
28C2×C47⋊#95C2214yes—
29C2×C47⋊#95C2214yes—
30C2×C47⋊#95C2212yes—
31C23×Q8×C47C2211yescharacteristic
32C22×C4⋊C4×C47C2211yescharacteristic
33C22×C42×C47C2211yescharacteristic
34C47×#391—24no—
35C2×#95×C47C2214yes—
36C24×D47C2311yescharacteristic
37C23×Dic47C2312yes—
38C23×Dic47C2312yes—
39C23×Dic47C2311yescharacteristic
40C23×Dic47C2312yes—
41C22×C4×D47—24no—
42C22×C4×D47—24no—
43C22×C4×D47—24no—
44C22×C4×D47—22no—
45C22×C4×D47—22no—
46C22×C4×D47—24no—
47C22×C4×D47—22no—
48C22×C4×D47—22no—
49C22×C4×D47—24no—
50C22×C4×D47—24no—
51C22×C4×D47C2×C414yes—
52C22×C4×D47D414yes—
53C22×C4×D47D414yes—
54C22×C4×D47Q818yescomplemented
55#13094000—232no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47—22no—
59C2×C4⋊Dic47—22no—
60C2×C4⋊Dic47—22no—
61C2×C4⋊Dic47C2×C414yescomplemented
62C2×Dic47⋊C4—24no—
63C2×Dic47⋊C4—24no—
64C2×Dic47⋊C4C2×C418yescomplemented
65C2×(C2×D47)⋊C4—24no—
66C2×(C2×D47)⋊C4—24no—
67C2×(C2×D47)⋊C4—24no—
68C2×(C2×D47)⋊C4—28no—
69C2×(C2×D47)⋊C4C2×C418yes—
70C2×C4×Dic47—24no—
71C2×C4×Dic47—22no—
72C2×C4×Dic47—22no—
73C2×C4×Dic47—24no—
74C2×C4×Dic47—24no—
75C2×C4×Dic47D414yes—
76C42×D47—232no—
77C47⋊#95—28no—
78C47⋊#95—24no—
79C47⋊#95—28no—
80C47⋊#95—28no—
81C22×Q8×C47—22no—
82C22×Q8×C47—22no—
83C22×Q8×C47—22no—
84C22×Q8×C47—22no—
85C22×Q8×C47—22no—
86C22×Q8×C47—22no—
87C22×Q8×C47C2×C414yescomplemented
88C23×C4×C47C2312yes—
89C23×C4×C47C2312yes—
90C23×C4×C47C2312yes—
91C23×C4×C47C2311yescharacteristic
92C2×C4⋊C4×C47—22no—
93C2×C4⋊C4×C47—22no—
94C2×C4⋊C4×C47—22no—
95C2×C42×C47—22no—
96C2×C42×C47—22no—
97C2×C42×C47D414yes—
98#95×C47—28no—
99C24×C47C2411yescharacteristic
100C23×D47—21no—
101C23×D47—21no—
102C23×D47—21no—
103C23×D47—21no—
104C23×D47—22no—
105C23×D47—22no—
106C23×D47C22×C412yes—
107C23×D47C2×D412yes—
108C23×D47C2×D412yes—
109C23×D47C2×Q814yes—
110C23×D47#4114yes—
111C2×Q8×C47—28no—
112C2×Q8×C47—28no—
113C2×Q8×C47—28no—
114C2×Q8×C47—44no—
115C2×Q8×C47—44no—
116C2×Q8×C47—44no—
117C2×Q8×C47—44no—
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—22no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—24no—
132C22×C4×C47—24no—
133C22×C4×C47C22×C414yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C2×D414yes—
136C22×C4×C47C2×D412yes—
137C22×C4×C47C2×D412yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47C2×D412yes—
140C22×C4×C47C2×Q814yes—
141C22×C4×C47#4114yes—
142C22×C4×C47#4112yes—
143C22×C4×C47#4112yes—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—21no—
149C22×Dic47—21no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—24no—
155C22×Dic47—24no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—24no—
159C22×Dic47—24no—
160C22×Dic47—22no—
161C22×Dic47C22×C414yes—
162C22×Dic47C22×C412yes—
163C22×Dic47C2×D412yes—
164C22×Dic47C2×D412yes—
165C22×Dic47C2×D414yes—
166C22×Dic47C2×Q814yes—
167C22×Dic47#4114yes—
168C2×C4×D47—216no—
169C2×C4×D47—216no—
170C2×C4×D47—216no—
171C2×C4×D47—216no—
172C2×C4×D47—216no—
173C2×C4×D47—28no—
174C2×C4×D47—28no—
175C2×C4×D47—28no—
176C2×C4×D47—28no—
177C2×C4×D47—28no—
178C2×C4×D47—216no—
179C2×C4×D47—28no—
180C2×C4×D47—216no—
181C2×C4×D47C22⋊C4116yes—
182C4⋊C4×C47—44no—
183C42×C47—28no—
184(C2×D47)⋊C4—216no—
185(C2×D47)⋊C4—48no—
186C4⋊Dic47—28no—
187C4⋊Dic47—44no—
188Dic47⋊C4—216no—
189C4×Dic47—216no—
190C4×Dic47—28no—
191C23×C47C23×C411yescharacteristic
192C23×C47C22×D411yescharacteristic
193C23×C47C22×D411yescharacteristic
194C23×C47C22×D411yescharacteristic
195C23×C47C22×D411yescharacteristic
196C23×C47C22×Q812yes—
197C23×C47C2×#4112yes—
198C23×C47C2×#4112yes—
199C23×C47C2×#4111yescharacteristic
200C23×C47C2×#4111yescharacteristic
201C23×C47C2×#4112yes—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—24no—
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—24no—
209C22×D47—24no—
210C22×D47—22no—
211C22×D47—24no—
212C22×D47—24no—
213C22×D47—24no—
214C22×D47—24no—
215C22×D47—24no—
216C22×D47—22no—
217C22×D47—22no—
218C22×D47—24no—
219C22×D47—24no—
220C22×D47C2×C22⋊C414yes—
221C22×D47C4×Q818yes—
222C22×D47C22⋊Q818yes—
223C22×D47#12414yes—
224C22×D47C4⋊Q814yes—
225Q8×C47—416no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—44no—
241C2×C4×C47—42no—
242C2×C4×C47—44no—
243C2×C4×C47—42no—
244C2×C4×C47C2×C22⋊C414yes—
245C2×C4×C47C4×D414yes—
246C2×C4×C47C4×D414yes—
247C2×C4×C47#12014yes—
248C2×C4×C47#12114yes—
249C2×C4×C47#12114yes—
250C2×C4×C47C22⋊Q818yescomplemented
251C2×C4×C47#12314yes—
252C2×Dic47—22no—
253C2×Dic47—22no—
254C2×Dic47—24no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—22no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—22no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—24no—
265C2×Dic47—28no—
266C2×Dic47—44no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—44no—
271C2×Dic47—42no—
272C2×Dic47—44no—
273C2×Dic47—42no—
274C2×Dic47C2×C22⋊C414yes—
275C2×Dic47C22⋊Q818yescomplemented
276C4×D47—264no—
277C4×D47—232no—
278#14702—471nomaximal, Sylow
279C22×C47C22×C22⋊C411yescharacteristic
280C22×C47C2×C4×D411yescharacteristic
281C22×C47C2×C4×D411yescharacteristic
282C22×C47C2×C4×Q812yes—
283C22×C47C4×#4112yes—
284C22×C47C2×#12011yescharacteristic
285C22×C47C2×#12111yescharacteristic
286C22×C47C2×#12111yescharacteristic
287C22×C47C2×C22⋊Q812yes—
288C22×C47C2×C22⋊Q812yes—
289C22×C47C2×C22⋊Q812yes—
290C22×C47C2×#12311yescharacteristic
291C22×C47#52512yes—
292C22×C47#52512yes—
293C22×C47#52512yes—
294C22×C47C2×#12411yescharacteristic
295C22×C47#52912yes—
296C22×C47#52912yes—
297C22×C47C2×C4⋊Q811yescharacteristic
298C22×C47#53211yescharacteristic
299C22×C47#53211yescharacteristic
300C22×C47#53312yes—
301C22×C47#53312yes—
302C2×D47—28no—
303C2×D47—24no—
304C2×D47—24no—
305C2×D47—28no—
306C2×D47—24no—
307C2×D47—24no—
308C2×D47—24no—
309C2×D47—24no—
310C2×D47—28no—
311C2×D47—28no—
312C2×D47#39118yescomplemented
313C4×C47—28no—
314C4×C47—44no—
315C4×C47—82no—
316C4×C47#38618yes—
317Dic47—28no—
318Dic47—24no—
319Dic47—44no—
320Dic47—82no—
321C22×C22⋊Q8—471no—
322C23×C42—471no—
323C2×#391—471no—
324C2×#391D4711yescharacteristic, complemented
325C2×#380—471no—
326#1436—474no—
327C4⋊#95—471no—
328C4⋊#95—471no—
329C22⋊#95—474no—
330C2×C47C2×#38611yescharacteristic
331C2×C47C2×#39111yescharacteristic
332C2×C47#229711yescharacteristic
333C2×C47C4×C22⋊Q812yes—
334C2×C47C4×C22⋊Q812yes—
335C2×C47#284812yes—
336C2×C47#285011yescharacteristic
337C2×C47#285011yescharacteristic
338C2×C47#285512yes—
339C2×C47#285711yescharacteristic
340C2×C47#285711yescharacteristic
341D47—28no—
342C23×Q8C2×D4711yescharacteristic
343C24×C4—471no—
344C24×C4—472no—
345C2×C22⋊Q8—478no—
346C2×C22⋊Q8—944no—
347C2×C22⋊Q8—944no—
348C22×C4⋊C4—472no—
349C22×C4⋊C4—471no—
350C22×C4⋊C4—471no—
351C22×C4⋊C4C2×D4711yescharacteristic
352C22×C22⋊C4—472no—
353C22×C22⋊C4—472no—
354C22×C42—471no—
355C22×C42—472no—
356C22×C42—944no—
357C22×C42—944no—
358C22×C42—944no—
359C22×C42C2×D4711yescharacteristic
360#391—24no—
361#391—944no—
362#380—944no—
363C2×#95—474no—
364C2×#95—474no—
365C2×#95—474no—
366C2×#95—472no—
367C2×#95C2×D4714yes—
368C47#1470211yescharacteristic, complemented, Sylow
369C25—471no—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8C4×D4714yescomplemented
377C23×C4—471no—
378C23×C4—472no—
379C23×C4—474no—
380C23×C4—474no—
381C23×C4—474no—
382C23×C4—472no—
383C23×C4—478no—
384C23×C4—472no—
385C23×C4—944no—
386C23×C4—944no—
387C23×C4—944no—
388C23×C4—944no—
389C23×C4—942no—
390C23×C4—942no—
391C23×C4—942no—
392C23×C4—944no—
393C23×C4—944no—
394C23×C4—942no—
395C23×C4C22×D4712yes—
396C23×C4C22×D4712yes—
397C23×C4C22×D4712yes—
398C23×C4C22×D4711yescharacteristic
399C22⋊Q8—9432no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—478no—
404C2×C4⋊C4—474no—
405C2×C4⋊C4—944no—
406C2×C4⋊C4—944no—
407C2×C4⋊C4—942no—
408C2×C4⋊C4—942no—
409C2×C4⋊C4—942no—
410C2×C4⋊C4—942no—
411C2×C4⋊C4—942no—
412C2×C22⋊C4—478no—
413C2×C22⋊C4—944no—
414C2×C22⋊C4—944no—
415C2×C22⋊C4—948no—
416C2×C22⋊C4—944no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—474no—
420C2×C42—9432no—
421C2×C42—944no—
422C2×C42—944no—
423C2×C42—942no—
424C2×C42—944no—
425C2×C42—942no—
426C2×C42C22⋊D4714yes—
427#95—28no—
428#95—948no—
429#95—944no—
430#95—948no—
431#95—948no—
432C24—472no—
433C24—472no—
434C24—472no—
435C24—474no—
436C24—474no—
437C24—941no—
438C24—941no—
439C24—942no—
440C24—941no—
441C24—941no—
442C24—942no—
443C24C23×D4711yescharacteristic
444C2×Q8—28no—
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—472no—
467C22×C4—474no—
468C22×C4—472no—
469C22×C4—472no—
470C22×C4—474no—
471C22×C4—474no—
472C22×C4—4716no—
473C22×C4—474no—
474C22×C4—9416no—
475C22×C4—9416no—
476C22×C4—9416no—
477C22×C4—9416no—
478C22×C4—9416no—
479C22×C4—9416no—
480C22×C4—941no—
481C22×C4—942no—
482C22×C4—941no—
483C22×C4—942no—
484C22×C4—942no—
485C22×C4—941no—
486C22×C4—941no—
487C22×C4—942no—
488C22×C4—948no—
489C22×C4—948no—
490C22×C4—948no—
491C22×C4—948no—
492C22×C4—948no—
493C22×C4—9416no—
494C22×C4—948no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—944no—
498C22×C4—944no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—944no—
502C22×C4—942no—
503C22×C4—944no—
504C22×C4C2×C4×D4714yes—
505C22×C4C2×C4×D4712yes—
506C22×C4#1062535012yes—
507C22×C4D4×D4712yes—
508C22×C4D4×D4712yes—
509C22×C4D4×D4712yes—
510C22×C4#1062535212yes—
511C22×C4Q8×D4714yes—
512C22×C4#1062535414yes—
513C22×C4C2×C22⋊D4714yes—
514C22×C4C2×C22⋊D4712yes—
515C4⋊C4—44no—
516C4⋊C4—9416no—
517C4⋊C4—948no—
518C4⋊C4—1884no—
519C22⋊C4—9416no—
520C22⋊C4—1888no—
521C42—28no—
522C42—9416no—
523C42—948no—
524C23—474no—
525C23—478no—
526C23—478no—
527C23—474no—
528C23—474no—
529C23—942no—
530C23—942no—
531C23—944no—
532C23—942no—
533C23—942no—
534C23—942no—
535C23—944no—
536C23—944no—
537C23—942no—
538C23—944no—
539C23—944no—
540C23—944no—
541C23—944no—
542C23—944no—
543C23—942no—
544C23—942no—
545C23—944no—
546C23—944no—
547C23C22×C4×D4711yescharacteristic
548C23C2×D4×D4711yescharacteristic
549C23C2×D4×D4711yescharacteristic
550C23C2×D4×D4711yescharacteristic
551C23C22×C22⋊D4711yescharacteristic
552C23C2×Q8×D4712yes—
553C23#41×D4712yes—
554C23#41×D4712yes—
555C23C2×#1062535211yescharacteristic
556C23C2×#1062535011yescharacteristic
557C23C2×#1062535412yes—
558Q8—416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—474no—
578C2×C4—478no—
579C2×C4—9464no—
580C2×C4—942no—
581C2×C4—942no—
582C2×C4—944no—
583C2×C4—942no—
584C2×C4—942no—
585C2×C4—942no—
586C2×C4—944no—
587C2×C4—944no—
588C2×C4—942no—
589C2×C4—944no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—9432no—
593C2×C4—944no—
594C2×C4—948no—
595C2×C4—1882no—
596C2×C4—1884no—
597C2×C4—1882no—
598C2×C4—1884no—
599C2×C4—1882no—
600C2×C4—1884no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4C22⋊C4×D4714yes—
604C2×C4(C22⋊D47)⋊C414yes—
605C2×C4C4×C22⋊D4714yes—
606C2×C4#1309399214yes—
607C2×C4#1309399514yes—
608C2×C4#1309399714yes—
609C2×C4#1309400018yescomplemented
610C2×C4#1309400614yes—
611C22—478no—
612C22—948no—
613C22—944no—
614C22—944no—
615C22—944no—
616C22—948no—
617C22—944no—
618C22—944no—
619C22—944no—
620C22—948no—
621C22—948no—
622C22C2×C22⋊C4×D4711yescharacteristic
623C22C2×(C22⋊D47)⋊C411yescharacteristic
624C22C2×C4×C22⋊D4711yescharacteristic
625C22C4×Q8×D4712yes—
626C22C4×#1062535412yes—
627C22C2×#1309399211yescharacteristic
628C22C2×#1309399511yescharacteristic
629C22C2×#1309399711yescharacteristic
630C22C22⋊Q8×D4712yes—
631C22C22⋊Q8×D4712yes—
632C22C2×#1309400012yes—
633C22C2×#1309400611yescharacteristic
634C22#1568963012yes—
635C22#1568963612yes—
636C22#1568963612yes—
637C22#124×D4711yescharacteristic
638C22#1568965212yes—
639C22#1568966012yes—
640C22C4⋊Q8×D4711yescharacteristic
641C22#1568967211yescharacteristic
642C22#1568967411yescharacteristic
643C22#1568967712yes—
644C22#1568967812yes—
645C4—28no—
646C4—44no—
647C4—82no—
648C4—948no—
649C4—944no—
650C4—3762no—
651C4—1884no—
652C4#1568899918yes—
653C2—948no—
654C2C2×#1568899911yescharacteristic
655C2#391×D4711yescharacteristic
656C2#2024587711yescharacteristic
657C2C4×#1309400012yes—
658C2#2024592612yes—
659C2#2025167212yes—
660C2#2025168411yescharacteristic
661C2#2025169211yescharacteristic
662C2#2025172912yes—
663C2#2025173511yescharacteristic
664C2#2025173711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#391C2
socle, cosocleC24×C47C24
prime radicalC24×D47C23
hypercenterC2×#391D47
hypocenterC47#14702
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×#15689030C2semidirect
5. #20241619C2semidirect
6. #20241618C2semidirect
7. #20241617C2semidirect
9. C2×C47×#391C2semidirect
14. C2×#13094000C4semidirect
54. C22×C4×D47Q8semidirect
61. C2×C4⋊Dic47C2×C4semidirect
64. C2×Dic47⋊C4C2×C4semidirect
87. C22×Q8×C47C2×C4semidirect
250. C2×C4×C47C22⋊Q8semidirect
275. C2×Dic47C22⋊Q8semidirect
312. C2×D47#391semidirect
324. C2×#391D47semidirect
368. C47#14702semidirect
376. C22×Q8C4×D47semidirect
609. C2×C4#13094000semidirect

Maximal subgroups

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

subgroupindex
C22×#130940002
C22×C42×D472
C2×#156890302
C2×#156889802
#202416192
#202416182
#202416172
(C2×D47)⋊#952
C2×C47×#3912
#1470247