Finite Group Databasebeta

← #26141580

Subgroups — #26141580

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
1C22×Q8×Dic47C211yesmaximal, characteristic, complemented
2C2×#15688998C212yesmaximal
3C2×C22⋊C4×Dic47C211yesmaximal, characteristic
4#20241659C211yesmaximal, characteristic, complemented
5#20241653C211yesmaximal, characteristic, complemented
6#20241633C211yesmaximal, characteristic
7#20241625C212yesmaximal
8#20241622C212yesmaximal, complemented
9#20241622C212yesmaximal, complemented
10#20241613C211yesmaximal, characteristic, complemented
11C2×C47×#393C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C2×Q8×Dic47—28no—
14C2×Q8×Dic47—24no—
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C4⋊Dic47C2212yes—
17C22×C4⋊Dic47C2212yes—
18C22×C22⋊Dic47C2212yes—
19C22×C22⋊Dic47C2211yescharacteristic
20C22×C4×Dic47C2212yes—
21C22×C4×Dic47C2211yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23#15688998—28no—
24C22⋊C4×Dic47—24no—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2211yescharacteristic
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C23×Q8×C47C2211yescharacteristic
35C22×C22⋊C4×C47C2212yes—
36C22×C22⋊C4×C47C2211yescharacteristic
37C47×#393C418yescomplemented
38C2×#95×C47C2211yescharacteristic
39C2×#95×C47C2212yes—
40C23×Dic47—22no—
41C23×Dic47—22no—
42C23×Dic47—22no—
43C23×Dic47—24no—
44C23×Dic47—42no—
45C23×Dic47C2312yes—
46C23×Dic47C2312yes—
47C23×Dic47C2311yescharacteristic
48C23×Dic47C2311yescharacteristic
49C23×Dic47C2311yescharacteristic
50C23×Dic47C2311yescharacteristic
51Q8×Dic47—232no—
52C2×C4⋊Dic47—22no—
53C2×C4⋊Dic47—24no—
54C2×C4⋊Dic47—24no—
55C2×C4⋊Dic47—24no—
56C2×C4⋊Dic47—24no—
57C2×C4⋊Dic47—24no—
58C2×C4⋊Dic47—42no—
59C2×C4⋊Dic47D418yes—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C22⋊Dic47—24no—
63C2×C22⋊Dic47—24no—
64C2×C22⋊Dic47—42no—
65C2×C22⋊Dic47Q814yescomplemented
66C2×C4×Dic47—24no—
67C2×C4×Dic47—24no—
68C2×C4×Dic47—24no—
69C2×C4×Dic47—24no—
70C2×C4×Dic47—22no—
71C2×C4×Dic47—42no—
72C2×C4×Dic47D414yes—
73C47⋊#95—22no—
74C47⋊#95—24no—
75C47⋊#95—24no—
76C47⋊#95—22no—
77C47⋊#95—22no—
78C47⋊#95—24no—
79C47⋊#95—24no—
80C47⋊#95—42no—
81C47⋊#95—42no—
82C25×C47C2311yescharacteristic
83C22×Q8×C47—24no—
84C22×Q8×C47—22no—
85C22×Q8×C47—22no—
86C22×Q8×C47—24no—
87C22×Q8×C47C2×C414yescomplemented
88C23×C4×C47C2311yescharacteristic
89C23×C4×C47C2312yes—
90C23×C4×C47C2312yes—
91C23×C4×C47C2311yescharacteristic
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47—24no—
94C2×C22⋊C4×C47—24no—
95C2×C22⋊C4×C47—22no—
96C2×C22⋊C4×C47C2×C418yes—
97C2×C22⋊C4×C47C2×C414yes—
98#95×C47C2×C414yescomplemented
99#95×C47C2×C418yescomplemented
100C24×C47—21no—
101C24×C47—22no—
102C24×C47—21no—
103C24×C47—22no—
104C24×C47—21no—
105C24×C47—21no—
106C24×C47—22no—
107C24×C47—22no—
108C24×C47C22×C412yes—
109C24×C47C2×Q812yes—
110C24×C47#4112yes—
111C24×C47C2411yescharacteristic
112C2×Q8×C47—28no—
113C2×Q8×C47—216no—
114C2×Q8×C47—28no—
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—21no—
124C22×C4×C47—22no—
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×C47C22×C414yes—
134C22×C4×C47C22×C412yes—
135C22×C4×C47C22×C414yes—
136C22×C4×C47C22×C412yes—
137C22×C4×C47C2×D414yes—
138C22×C4×C47C2×D412yes—
139C22×C4×C47#4112yes—
140C22×C4×C47#4114yes—
141C22×Dic47—22no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—28no—
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—22no—
156C22×Dic47—21no—
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×Dic47—22no—
168C22×Dic47—48no—
169C22×Dic47—48no—
170C22×Dic47—44no—
171C22×Dic47—44no—
172C22×Dic47C2×D414yes—
173C22×Dic47C2×D414yes—
174C22×Dic47C2×D412yes—
175C22×Dic47C2×D412yes—
176C22×Dic47C2×Q812yes—
177C22×Dic47C2×Q812yes—
178C22×Dic47#4112yes—
179C22×Dic47#4112yes—
180C22⋊C4×C47—28no—
181C22⋊C4×C47—216no—
182C22⋊Dic47—28no—
183C22⋊Dic47—48no—
184C4⋊Dic47—28no—
185C4⋊Dic47—216no—
186C4⋊Dic47—48no—
187C4×Dic47—216no—
188C4×Dic47—28no—
189C4×Dic47—44no—
190C23×C47—24no—
191C23×C47—22no—
192C23×C47—22no—
193C23×C47—24no—
194C23×C47—22no—
195C23×C47—22no—
196C23×C47—22no—
197C23×C47—22no—
198C23×C47—24no—
199C23×C47—24no—
200C23×C47—24no—
201C23×C47—24no—
202C23×C47—41no—
203C23×C47—41no—
204C23×C47—42no—
205C23×C47—41no—
206C23×C47—41no—
207C23×C47—42no—
208C23×C47—42no—
209C23×C47—42no—
210C23×C47—42no—
211C23×C47—42no—
212C23×C47C4×Q814yescomplemented
213C23×C47C23×C411yescharacteristic
214C23×C47C22×D411yescharacteristic
215C23×C47C22×D412yes—
216C23×C47C22×Q811yescharacteristic
217C23×C47C2×#4111yescharacteristic
218C23×C47C2×#4111yescharacteristic
219C23×C47C2×#4111yescharacteristic
220C23×C47C2×#4112yes—
221C23×C47C2×#4111yescharacteristic
222C23×C47H(5,𝔽2)12yes—
223C23×C472−1+412yes—
224Q8×C47—232no—
225C2×C4×C47—24no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—24no—
229C2×C4×C47—22no—
230C2×C4×C47—22no—
231C2×C4×C47—24no—
232C2×C4×C47—24no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—24no—
239C2×C4×C47—24no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47C4×D414yes—
248C2×C4×C47C4×D418yes—
249C2×Dic47—22no—
250C2×Dic47—22no—
251C2×Dic47—22no—
252C2×Dic47—22no—
253C2×Dic47—24no—
254C2×Dic47—24no—
255C2×Dic47—24no—
256C2×Dic47—24no—
257C2×Dic47—24no—
258C2×Dic47—22no—
259C2×Dic47—22no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—416no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—42no—
268C2×Dic47—41no—
269C2×Dic47—42no—
270C2×Dic47—41no—
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#12014yes—
283C2×Dic47C22⋊Q814yes—
284#14172—471nomaximal, Sylow
285C22×C47—28no—
286C22×C47—24no—
287C22×C47—42no—
288C22×C47—42no—
289C22×C47—42no—
290C22×C47—44no—
291C22×C47—42no—
292C22×C47—42no—
293C22×C47—44no—
294C22×C47—44no—
295C22×C47—42no—
296C22×C47C2×C4×D411yescharacteristic
297C22×C47C2×C4×D412yes—
298C22×C47C2×C4×Q811yescharacteristic
299C22×C47C4×#4111yescharacteristic
300C22×C47#52012yes—
301C22×C47C2×#12011yescharacteristic
302C22×C47C2×C22⋊Q811yescharacteristic
303C22×C47#52511yescharacteristic
304C22×C47#52512yes—
305C22×C47#52511yescharacteristic
306C22×C47#52911yescharacteristic
307C22×C47#53311yescharacteristic
308C22×C47#53511yescharacteristic
309C22×C47#53611yescharacteristic
310C22×C47#54212yes—
311C22×C47#54612yes—
312C22×C47#54612yes—
313C22×C47#54812yes—
314C22×C47D4×Q812yes—
315C22×C47#55212yes—
316C22×C47#55412yes—
317C22×C47#55612yes—
318C22×C47#55912yes—
319C4×C47—28no—
320C4×C47—24no—
321C4×C47—42no—
322C4×C47—44no—
323Dic47—24no—
324Dic47—24no—
325Dic47—42no—
326Dic47—44no—
327Dic47—42no—
328Dic47—82no—
329C22×C4×Q8—471no—
330C2×#393D4711yescharacteristic
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×C47—44no—
341C2×C47C4×#12011yescharacteristic
342C2×C47C4×C22⋊Q811yescharacteristic
343C2×C47#232611yescharacteristic
344C2×C47#236012yes—
345C2×C47#236712yes—
346C2×C47#240411yescharacteristic
347C2×C47#240911yescharacteristic
348C2×C47#245812yes—
349C2×C47#246512yes—
350C2×C47#258011yescharacteristic
351C2×C47#259911yescharacteristic
352C23×Q8C2×D4711yescharacteristic
353C24×C4—471no—
354C2×C4×Q8—944no—
355C2×C4×Q8—948no—
356C22×C4⋊C4—472no—
357C22×C4⋊C4—471no—
358C22×C4⋊C4—472no—
359C22×C22⋊C4—471no—
360C22×C22⋊C4—472no—
361C22×C22⋊C4C2×D4712yes—
362C22×C22⋊C4C2×D4711yescharacteristic
363C22×C42—471no—
364C22×C42—472no—
365C22×C42—471no—
366#393Dic4718yescomplemented
367#385—948no—
368C4×C22⋊C4—944no—
369C2×#95—472no—
370C2×#95—472no—
371C2×#95—471no—
372C2×#95—471no—
373C2×#95—472no—
374C2×#95—472no—
375C2×#95—471no—
376C2×#95—472no—
377C2×#95—472no—
378C2×#95C2×D4711yescharacteristic
379C2×#95C2×D4712yes—
380C47#1417211yescharacteristic, complemented, Sylow
381C25C22×D4711yescharacteristic
382C22×Q8—22no—
383C22×Q8—24no—
384C22×Q8—22no—
385C22×Q8—24no—
386C22×Q8C2×Dic4714yescomplemented
387C23×C4—471no—
388C23×C4—471no—
389C23×C4—472no—
390C23×C4—471no—
391C23×C4—471no—
392C23×C4—472no—
393C23×C4—942no—
394C23×C4—942no—
395C23×C4—944no—
396C23×C4—942no—
397C23×C4—1882no—
398C23×C4C22×D4712yes—
399C23×C4C22×D4711yescharacteristic
400C23×C4C22×D4711yescharacteristic
401C23×C4C22×D4712yes—
402C4×Q8—9432no—
403C2×C4⋊C4—478no—
404C2×C4⋊C4—944no—
405C2×C4⋊C4—944no—
406C2×C4⋊C4—944no—
407C2×C4⋊C4—942no—
408C2×C4⋊C4—944no—
409C2×C4⋊C4—944no—
410C2×C4⋊C4—1882no—
411C2×C22⋊C4—22no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—474no—
416C2×C22⋊C4—942no—
417C2×C22⋊C4—942no—
418C2×C22⋊C4—944no—
419C2×C22⋊C4—944no—
420C2×C22⋊C4—1882no—
421C2×C22⋊C4C2×Dic4718yes—
422C2×C22⋊C4C2×Dic4714yes—
423C2×C42—474no—
424C2×C42—944no—
425C2×C42—944no—
426C2×C42—944no—
427C2×C42—944no—
428C2×C42—942no—
429C2×C42—1882no—
430#95—944no—
431#95—942no—
432#95—944no—
433#95—942no—
434#95—944no—
435#95—944no—
436#95—942no—
437#95—1882no—
438#95—1882no—
439#95C2×Dic4714yescomplemented
440#95C2×Dic4718yescomplemented
441C24—22no—
442C24—21no—
443C24—21no—
444C24—21no—
445C24—22no—
446C24—21no—
447C24—22no—
448C24—22no—
449C24Q8×D4712yes—
450C24#1062535412yes—
451C24C22×Dic4712yes—
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—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—472no—
475C22×C4—474no—
476C22×C4—472no—
477C22×C4—472no—
478C22×C4—472no—
479C22×C4—474no—
480C22×C4—472no—
481C22×C4—472no—
482C22×C4—942no—
483C22×C4—941no—
484C22×C4—941no—
485C22×C4—942no—
486C22×C4—942no—
487C22×C4—948no—
488C22×C4—941no—
489C22×C4—941no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—941no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4—942no—
497C22×C4—941no—
498C22×C4—942no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—942no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—1884no—
510C22×C4—1888no—
511C22×C4—1888no—
512C22×C4—1884no—
513C22×C4D4×D4712yes—
514C22×C4D4×D4714yes—
515C22×C4#1062535212yes—
516C22×C4#1062535214yes—
517C22×C4C22×Dic4714yes—
518C22×C4C22×Dic4712yes—
519C22×C4C22×Dic4714yes—
520C22×C4C22×Dic4712yes—
521C4⋊C4—9416no—
522C4⋊C4—948no—
523C4⋊C4—1888no—
524C22⋊C4—28no—
525C22⋊C4—216no—
526C22⋊C4—948no—
527C22⋊C4—1888no—
528C42—948no—
529C42—9416no—
530C42—1884no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23Q8×Dic4714yescomplemented
554C23C23×Dic4711yescharacteristic
555C23C2×D4×D4712yes—
556C23C2×D4×D4711yescharacteristic
557C23C2×Q8×D4711yescharacteristic
558C23#41×D4711yescharacteristic
559C23#41×D4711yescharacteristic
560C23C2×#1062535211yescharacteristic
561C23C2×#1062535212yes—
562C23C2×#1062535411yescharacteristic
563C23#1309407012yes—
564C23#1309407112yes—
565Q8—232no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—474no—
589C2×C4—474no—
590C2×C4—944no—
591C2×C4—944no—
592C2×C4—944no—
593C2×C4—944no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—944no—
599C2×C4—944no—
600C2×C4—944no—
601C2×C4—944no—
602C2×C4—944no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—18816no—
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—1882no—
616C2×C4—1882no—
617C2×C4—1881no—
618C2×C4—1881no—
619C2×C4—1882no—
620C2×C4—1881no—
621C2×C4—1882no—
622C2×C4—1881no—
623C2×C4D4×Dic4714yes—
624C2×C4D4×Dic4718yes—
625C22—28no—
626C22—24no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—42no—
636C22C2×D4×Dic4711yescharacteristic
637C22C2×D4×Dic4712yes—
638C22C2×Q8×Dic4711yescharacteristic
639C22#41×Dic4711yescharacteristic
640C22#1568960612yes—
641C22#120×D4711yescharacteristic
642C22C22⋊Q8×D4711yescharacteristic
643C22#1568962912yes—
644C22#1568962911yescharacteristic
645C22#1568963611yescharacteristic
646C22#1568965911yescharacteristic
647C22#1568967511yescharacteristic
648C22#1568969111yescharacteristic
649C22#1568969311yescharacteristic
650C22#1568974212yes—
651C22#1568976412yes—
652C22#1568976612yes—
653C22#1568978112yes—
654C22#1568978612yes—
655C22#1568981512yes—
656C22#1568983112yes—
657C22#1568984212yes—
658C22#1568985512yes—
659C4—28no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—944no—
664C4—944no—
665C4—3762no—
666C4—1882no—
667C4—1882no—
668C4—1884no—
669C2—44no—
670C2#120×Dic4711yescharacteristic
671C2C22⋊Q8×Dic4711yescharacteristic
672C2#2024610111yescharacteristic
673C2#2024639712yes—
674C2#2024646312yes—
675C2#2024670811yescharacteristic
676C2#2024674611yescharacteristic
677C2#2024730312yes—
678C2#2024740212yes—
679C2#2024874111yescharacteristic
680C2#2024895811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#393C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC2×#393D47
hypocenterC47#14172
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic47C2semidirect
4. #20241659C2semidirect
5. #20241653C2semidirect
8. #20241622C2semidirect
9. #20241622C2semidirect
10. #20241613C2semidirect
37. C47×#393C4semidirect
65. C2×C22⋊Dic47Q8semidirect
87. C22×Q8×C47C2×C4semidirect
98. #95×C47C2×C4semidirect
99. #95×C47C2×C4semidirect
212. C23×C47C4×Q8semidirect
366. #393Dic47semidirect
380. C47#14172semidirect
386. C22×Q8C2×Dic47semidirect
439. #95C2×Dic47semidirect
440. #95C2×Dic47semidirect
553. C23Q8×Dic47semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic472
C2×#156889982
C2×C22⋊C4×Dic472
#202416592
#202416532
#202416332
#202416252
#202416222
#202416132
C2×C47×#3932
#1417247