Finite Group Databasebeta

← #26142652

Subgroups — #26142652

Order 12032.

2142 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094001C211yesmaximal, characteristic
2C22×#13093980C211yesmaximal, characteristic
3C2×#15689027C212yesmaximal, complemented
4C2×#15688981C211yesmaximal, characteristic
5#20241640C212yesmaximal, complemented
6#20241639C212yesmaximal, complemented
7#20241637C211yesmaximal, characteristic
8#20241635C211yesmaximal, characteristic
9#20241635C211yesmaximal, characteristic
10Dic47⋊#95C212yesmaximal, complemented
11C47×#1440C211yesmaximal, characteristic
12C23×Dic94C2211yescharacteristic
13C24×Dic47C2211yescharacteristic
14C2×#13094001—24no—
15C2×#13094001—24no—
16C2×#13094001C418yescomplemented
17C2×#13093980—24no—
18C2×#13093980—28no—
19C22×C4⋊Dic47C2211yescharacteristic
20C22×Dic47⋊C4C2212yes—
21C22×Dic47⋊C4C2212yes—
22C22×Dic47⋊C4C2211yescharacteristic
23C22×Dic47⋊C4C2211yescharacteristic
24C22×C22⋊Dic47C2211yescharacteristic
25C22×C22⋊Dic47C2211yescharacteristic
26C22×C22⋊Dic47C2211yescharacteristic
27C22×C4×Dic47C2212yes—
28#15689027—28no—
29#15688981—24no—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C2×C47⋊#95C2212yes—
36C2×C47⋊#95C2212yes—
37C22×C22⋊C4×C47C2211yescharacteristic
38C22×C22⋊C4×C47C2211yescharacteristic
39C22×C22⋊C4×C47C2211yescharacteristic
40C2×#95×C47C2212yes—
41C2×#95×C47C2212yes—
42C22×Dic94—24no—
43C22×Dic94—22no—
44C22×Dic94—22no—
45C22×Dic94—24no—
46C22×Dic94C2×C414yes—
47C23×Dic47—24no—
48C23×Dic47—24no—
49C23×Dic47C2×C414yes—
50C23×Dic47D414yes—
51C23×Dic47D414yes—
52C23×Dic47C2312yes—
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2311yescharacteristic
55C23×Dic47C2312yes—
56C23×Dic47C2311yescharacteristic
57C23×Dic47C2311yescharacteristic
58#13094001—232no—
59#13093980—232no—
60C2×C4⋊Dic47—22no—
61C2×C4⋊Dic47—22no—
62C2×C4⋊Dic47C2×C414yes—
63C2×Dic47⋊C4—24no—
64C2×Dic47⋊C4—24no—
65C2×Dic47⋊C4—24no—
66C2×Dic47⋊C4—22no—
67C2×Dic47⋊C4—22no—
68C2×Dic47⋊C4—22no—
69C2×Dic47⋊C4—22no—
70C2×Dic47⋊C4—24no—
71C2×Dic47⋊C4—24no—
72C2×Dic47⋊C4—24no—
73C2×Dic47⋊C4C2×C414yes—
74C2×Dic47⋊C4C2×C414yes—
75C2×C22⋊Dic47—22no—
76C2×C22⋊Dic47—22no—
77C2×C22⋊Dic47—22no—
78C2×C22⋊Dic47—24no—
79C2×C22⋊Dic47—22no—
80C2×C22⋊Dic47—42no—
81C2×C22⋊Dic47C2×C414yes—
82C2×C4×Dic47—24no—
83C2×C4×Dic47—24no—
84C2×C4×Dic47D418yes—
85C47⋊#95—24no—
86C47⋊#95—24no—
87C47⋊#95—24no—
88C47⋊#95—24no—
89C47⋊#95—42no—
90C47⋊#95—42no—
91C47⋊#95—42no—
92C25×C47C2311yescharacteristic
93C23×C4×C47C2312yes—
94C23×C4×C47C2311yescharacteristic
95C23×C4×C47C2311yescharacteristic
96C23×C4×C47C2312yes—
97C2×C22⋊C4×C47—22no—
98C2×C22⋊C4×C47—22no—
99C2×C22⋊C4×C47—22no—
100C2×C22⋊C4×C47—24no—
101C2×C22⋊C4×C47—22no—
102C2×C22⋊C4×C47—42no—
103C2×C22⋊C4×C47C2×C414yes—
104#95×C47—24no—
105#95×C47—42no—
106C24×C47—22no—
107C24×C47—22no—
108C24×C47—22no—
109C24×C47—22no—
110C24×C47—22no—
111C24×C47—22no—
112C24×C47C22×C412yes—
113C24×C47C2×D412yes—
114C24×C47C2×D412yes—
115C24×C47C2411yescharacteristic
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—21no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
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—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47C22×C412yes—
138C22×C4×C47C22×C412yes—
139C22×C4×C47C2×D414yes—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—216no—
143C22×Dic47—28no—
144C22×Dic47—216no—
145C22×Dic47—28no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
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×Dic47C22⋊C4116yes—
168C22×Dic47C22×C412yes—
169C22×Dic47C22×C412yes—
170C22×Dic47C22×C412yes—
171C22×Dic47C22×C412yes—
172C22×Dic47C2×D414yes—
173C22×Dic47C2×D412yes—
174C22×Dic47C2×D412yes—
175C22×Dic47C2×D412yes—
176C22×Dic47C2×D412yes—
177C22×Dic47C2×D414yes—
178C22×Dic47#4114yes—
179C2×Dic94—28no—
180C2×Dic94—216no—
181C2×Dic94—44no—
182C2×Dic94—48no—
183C2×Dic94—44no—
184C22⋊C4×C47—28no—
185C22⋊C4×C47—28no—
186C22⋊C4×C47—44no—
187C22⋊Dic47—28no—
188C22⋊Dic47—28no—
189C22⋊Dic47—44no—
190C4⋊Dic47—28no—
191Dic47⋊C4—216no—
192Dic47⋊C4—28no—
193Dic47⋊C4—28no—
194Dic47⋊C4—48no—
195C4×Dic47—216no—
196C23×C47—24no—
197C23×C47—22no—
198C23×C47—22no—
199C23×C47—24no—
200C23×C47—22no—
201C23×C47—24no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—22no—
207C23×C47—24no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47—42no—
212C23×C47—42no—
213C23×C47—42no—
214C23×C47—42no—
215C23×C47—42no—
216C23×C47C2×C22⋊C414yes—
217C23×C47C23×C411yescharacteristic
218C23×C47C22×D411yescharacteristic
219C23×C47C22×D411yescharacteristic
220C23×C47C22×D412yes—
221C23×C47C2×#4112yes—
222C23×C47H(5,𝔽2)12yes—
223C23×C472−1+412yes—
224C23×C472−1+412yes—
225C23×C472−1+412yes—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—24no—
229C2×C4×C47—24no—
230C2×C4×C47—24no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—41no—
234C2×C4×C47—42no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—42no—
238C2×C4×C47—41no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47—42no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×Dic47—24no—
251C2×Dic47—22no—
252C2×Dic47—24no—
253C2×Dic47—22no—
254C2×Dic47—232no—
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—41no—
263C2×Dic47—42no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—42no—
268C2×Dic47—42no—
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×Dic47C2×C22⋊C414yes—
280C2×Dic47C2×C22⋊C414yes—
281C2×Dic47C4×D418yes—
282C2×Dic47#12014yes—
283C2×Dic47#12118yes—
284C2×Dic47#12314yes—
285Dic94—416no—
286#14258—471nomaximal, Sylow
287C22×C47—24no—
288C22×C47—24no—
289C22×C47—24no—
290C22×C47—44no—
291C22×C47—44no—
292C22×C47—44no—
293C22×C47—44no—
294C22×C47—44no—
295C22×C47—44no—
296C22×C47C22×C22⋊C411yescharacteristic
297C22×C47C2×C4×D412yes—
298C22×C47#51911yescharacteristic
299C22×C47#51911yescharacteristic
300C22×C47#52012yes—
301C22×C47C2×#12011yescharacteristic
302C22×C47C2×#12112yes—
303C22×C47C2×#12311yescharacteristic
304C22×C47#53511yescharacteristic
305C22×C47#53612yes—
306C22×C47#53612yes—
307C22×C47#53612yes—
308C22×C47#53611yescharacteristic
309C22×C47#53612yes—
310C22×C47#53712yes—
311C22×C47#53712yes—
312C22×C47#53712yes—
313C22×C47#54112yes—
314C22×C47#54112yes—
315C22×C47#54212yes—
316C22×C47#54212yes—
317C4×C47—42no—
318C4×C47—44no—
319C4×C47—42no—
320C4×C47—82no—
321Dic47—24no—
322Dic47—42no—
323Dic47—44no—
324Dic47—42no—
325Dic47—82no—
326Dic47#38618yes—
327C22×C22⋊Q8—471no—
328C22×#117—471no—
329C2×#391—472no—
330C2×#380—471no—
331#1441—472no—
332#1441—472no—
333#1440—471no—
334#1440—471no—
335#1440—471no—
336#1440D4711yescharacteristic
337C4⋊#95—472no—
338C2×C47—44no—
339C2×C47C2×#38611yescharacteristic
340C2×C47#231011yescharacteristic
341C2×C47#231111yescharacteristic
342C2×C47#232312yes—
343C2×C47#232312yes—
344C2×C47#261612yes—
345C2×C47#262012yes—
346C2×C47#262112yes—
347C2×C47#262512yes—
348C23×Q8—471no—
349C24×C4—471no—
350C2×C22⋊Q8—478no—
351C2×C22⋊Q8—944no—
352C2×C22⋊Q8—944no—
353C2×#117—948no—
354C2×#117—944no—
355C22×C4⋊C4—471no—
356C22×C4⋊C4—471no—
357C22×C4⋊C4—471no—
358C22×C4⋊C4—472no—
359C22×C4⋊C4—472no—
360C22×C22⋊C4—471no—
361C22×C22⋊C4—471no—
362C22×C22⋊C4—471no—
363C22×C22⋊C4C2×D4711yescharacteristic
364C22×C22⋊C4C2×D4711yescharacteristic
365C22×C22⋊C4C2×D4711yescharacteristic
366C22×C42—472no—
367#391—948no—
368#380—944no—
369C2×#95—472no—
370C2×#95—472no—
371C2×#95—472no—
372C2×#95—472no—
373C2×#95—472no—
374C2×#95—472no—
375C2×#95—472no—
376C2×#95C2×D4712yes—
377C2×#95C2×D4712yes—
378C47#1425811yescharacteristic, complemented, Sylow
379C25C22×D4711yescharacteristic
380C22×Q8—474no—
381C22×Q8—942no—
382C22×Q8—942no—
383C22×Q8—944no—
384C22×Q8—944no—
385C23×C4—471no—
386C23×C4—474no—
387C23×C4—474no—
388C23×C4—474no—
389C23×C4—471no—
390C23×C4—471no—
391C23×C4—471no—
392C23×C4—472no—
393C23×C4—472no—
394C23×C4—944no—
395C23×C4—944no—
396C23×C4C22×D4712yes—
397C23×C4C22×D4711yescharacteristic
398C23×C4C22×D4711yescharacteristic
399C23×C4C22×D4712yes—
400C22⋊Q8—9432no—
401#117—9432no—
402C2×C4⋊C4—474no—
403C2×C4⋊C4—474no—
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×C4⋊C4—942no—
413C2×C4⋊C4—944no—
414C2×C4⋊C4—944no—
415C2×C4⋊C4—944no—
416C2×C4⋊C4—944no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—42no—
423C2×C22⋊C4—474no—
424C2×C22⋊C4—944no—
425C2×C22⋊C4—942no—
426C2×C22⋊C4—942no—
427C2×C22⋊C4—942no—
428C2×C22⋊C4—942no—
429C2×C22⋊C4—1882no—
430C2×C22⋊C4C4×D4714yes—
431C2×C42—478no—
432C2×C42—944no—
433C2×C42—944no—
434#95—24no—
435#95—42no—
436#95—944no—
437#95—944no—
438#95—944no—
439#95—944no—
440#95—1882no—
441#95—1882no—
442#95—1882no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24C2×C4×D4712yes—
450C24D4×D4712yes—
451C24D4×D4712yes—
452C24C23×D4711yescharacteristic
453C2×Q8—948no—
454C2×Q8—9416no—
455C2×Q8—1884no—
456C2×Q8—1888no—
457C2×Q8—1884no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
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—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—472no—
480C22×C4—474no—
481C22×C4—472no—
482C22×C4—472no—
483C22×C4—474no—
484C22×C4—4716no—
485C22×C4—472no—
486C22×C4—472no—
487C22×C4—472no—
488C22×C4—472no—
489C22×C4—472no—
490C22×C4—474no—
491C22×C4—948no—
492C22×C4—9416no—
493C22×C4—9416no—
494C22×C4—948no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—942no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4—942no—
512C22×C4—942no—
513C22×C4—942no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—942no—
517C22×C4—942no—
518C22×C4C2×C4×D4712yes—
519C22×C4C2×C4×D4712yes—
520C22×C4D4×D4714yes—
521C4⋊C4—948no—
522C4⋊C4—948no—
523C4⋊C4—948no—
524C4⋊C4—9416no—
525C4⋊C4—1888no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—948no—
530C22⋊C4—948no—
531C22⋊C4—1884no—
532C42—9416no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C22⋊C4×D4714yes—
554C23C22×C4×D4711yescharacteristic
555C23C2×D4×D4711yescharacteristic
556C23C2×D4×D4711yescharacteristic
557C23C2×D4×D4712yes—
558C23#41×D4712yes—
559C23#1309406912yes—
560C23#1309407112yes—
561C23#1309407112yes—
562C23#1309407112yes—
563Q8—18816no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—474no—
589C2×C4—478no—
590C2×C4—474no—
591C2×C4—478no—
592C2×C4—474no—
593C2×C4—474no—
594C2×C4—9432no—
595C2×C4—944no—
596C2×C4—942no—
597C2×C4—944no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—944no—
604C2×C4—944no—
605C2×C4—944no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4—1882no—
610C2×C4—1882no—
611C2×C4—1882no—
612C2×C4—1882no—
613C2×C4—1881no—
614C2×C4—1882no—
615C2×C4—1881no—
616C2×C4—1881no—
617C2×C4—1882no—
618C2×C4—1882no—
619C2×C4—1881no—
620C2×C4—1882no—
621C2×C4—1882no—
622C2×C4—1882no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22C2×C22⋊C4×D4711yescharacteristic
633C22C4×D4×D4712yes—
634C22#1568960011yescharacteristic
635C22#1568960011yescharacteristic
636C22#1568960212yes—
637C22#120×D4711yescharacteristic
638C22#121×D4712yes—
639C22#123×D4711yescharacteristic
640C22#1568968811yescharacteristic
641C22#1568969312yes—
642C22#1568969512yes—
643C22#1568969512yes—
644C22#1568969512yes—
645C22#1568969511yescharacteristic
646C22#1568970012yes—
647C22#1568970012yes—
648C22#1568970212yes—
649C22#1568973012yes—
650C22#1568973012yes—
651C22#1568973712yes—
652C22#1568974112yes—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—478no—
658C4—944no—
659C4—3762no—
660C4—1884no—
661C4—1882no—
662C4—1882no—
663C2—44no—
664C2#386×D4711yescharacteristic
665C2#2024597311yescharacteristic
666C2#2024597511yescharacteristic
667C2#2024606312yes—
668C2#2024606712yes—
669C2#2024911612yes—
670C2#2024914412yes—
671C2#2024915312yes—
672C2#2024918412yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1440C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenter#1440D47
hypocenterC47#14258
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
3. C2×#15689027C2semidirect
5. #20241640C2semidirect
6. #20241639C2semidirect
10. Dic47⋊#95C2semidirect
16. C2×#13094001C4semidirect
378. C47#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#130940012
C22×#130939802
C2×#156890272
C2×#156889812
#202416402
#202416392
#202416372
#202416352
Dic47⋊#952
C47×#14402
#1425847