Finite Group Databasebeta

← #26247376

Subgroups — #26247376

Order 12032.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625337C211yesmaximal, characteristic, complemented
2C2×#15689279C212yesmaximal, complemented
3C2×#15689278C211yesmaximal, characteristic, complemented
4#20246753C211yesmaximal, characteristic
5#20244488C212yesmaximal, complemented
6#20244396C212yesmaximal, complemented
7#20243812C211yesmaximal, characteristic, complemented
8#20243729C212yesmaximal, complemented
9#20243681C211yesmaximal, characteristic
10#20243323C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C47C211yesmaximal, characteristic
12C23×Dic94C2211yescharacteristic
13C22×#10625337—28no—
14C22×#10625337—24no—
15C22×#10625337C2212yes—
16C22×#10625337C2212yes—
17C22×#10625337C2212yes—
18C23×C47⋊C8C2211yescharacteristic
19C2×#13094003C2212yes—
20#15689279—28no—
21#15689278—24no—
22C2×#13093948C2212yes—
23C2×#13093948C2212yes—
24C2×#13093948C2212yes—
25C2×#13093947C2212yes—
26C2×C22⋊(C47⋊C8)C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2212yes—
28#15689043—21no—
29#15689035—21no—
30#15689029C2212yes—
31#15689003—21no—
32#15688995—21no—
33#15688983C2211yescharacteristic
34#15688885C2212yes—
35#15688878C2212yes—
36C23×Q8×C47C2211yescharacteristic
37C23×Q8×C47C2211yescharacteristic
38D4×Q8×C47—24no—
39D4×Q8×C47—24no—
40C2×C4⋊Q8×C47—21no—
41C2×C4⋊Q8×C47C2211yescharacteristic
42C2×C22⋊Q8×C47—21no—
43C2×C22⋊Q8×C47—21no—
44C2×C22⋊Q8×C47C2211yescharacteristic
45C2×C22⋊Q8×C47C2211yescharacteristic
46C2×C4×Q8×C47C2211yescharacteristic
47C2×C4×D4×C47—21no—
48C2×C4×D4×C47C2211yescharacteristic
49C22×Dic94—22no—
50C22×Dic94—24no—
51C22×Dic94—22no—
52C22×Dic94—21no—
53C22×Dic94—42no—
54C22×Dic94C2312yes—
55C23×Dic47—21no—
56C2×#10625337—232no—
57C2×#10625337—24no—
58C2×#10625337—24no—
59C2×#10625337—24no—
60C2×#10625337—24no—
61C2×#10625337—24no—
62C2×#10625337—24no—
63C2×#10625337—24no—
64C2×#10625337—42no—
65C2×#10625337D418yescomplemented
66C22×C47⋊C8—22no—
67C22×C47⋊C8—24no—
68C22×C47⋊C8C2312yes—
69C22×C47⋊C8C2312yes—
70#13094003—24no—
71#13094003—42no—
72C2×C4⋊Dic47C2312yes—
73C2×Dic47⋊C4—22no—
74C2×C22⋊Dic47—21no—
75C2×C22⋊Dic47—22no—
76#13093948—24no—
77#13093948—24no—
78#13093948—42no—
79#13093947—24no—
80C22⋊(C47⋊C8)—22no—
81C22⋊(C47⋊C8)—24no—
82C47⋊#95—21no—
83C47⋊#95—22no—
84C47⋊#95—22no—
85C22×Q8×C47—22no—
86C22×Q8×C47—22no—
87C22×Q8×C47—21no—
88C22×Q8×C47—21no—
89C22×Q8×C47—22no—
90C22×Q8×C47—24no—
91C22×Q8×C47—24no—
92C22×Q8×C47—42no—
93C22×Q8×C47—42no—
94C22×Q8×C47D414yes—
95C22×Q8×C47C2311yescharacteristic
96C22×Q8×C47C2311yescharacteristic
97C22×Q8×C47C2311yescharacteristic
98C22×D4×C47D411yescharacteristic
99C23×C4×C47—21no—
100C23×C4×C47—21no—
101C23×C4×C47C2311yescharacteristic
102C23×C4×C47C2311yescharacteristic
103C4⋊Q8×C47—28no—
104C4⋊Q8×C47—22no—
105C4⋊Q8×C47—22no—
106C22⋊Q8×C47—28no—
107C22⋊Q8×C47—28no—
108C22⋊Q8×C47—22no—
109C22⋊Q8×C47—22no—
110C22⋊Q8×C47—22no—
111C22⋊Q8×C47D414yes—
112C4×Q8×C47—22no—
113C4×Q8×C47—22no—
114C4×D4×C47—28no—
115C4×D4×C47—22no—
116C4×D4×C47—22no—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47—21no—
119C2×C4⋊C4×C47—21no—
120C2×C4⋊C4×C47—21no—
121C2×C4⋊C4×C47—21no—
122C2×C4⋊C4×C47D411yescharacteristic
123C2×C4⋊C4×C47C2311yescharacteristic
124C2×C22⋊C4×C47—21no—
125C2×C22⋊C4×C47—21no—
126C2×C22⋊C4×C47D411yescharacteristic
127C2×C22⋊C4×C47D411yescharacteristic
128C2×C42×C47—21no—
129C2×C42×C47C2311yescharacteristic
130C24×C47C2×D411yescharacteristic
131C24×C47C2×D411yescharacteristic
132C2×Q8×C47—24no—
133C2×Q8×C47—24no—
134C2×Q8×C47—21no—
135C2×Q8×C47—21no—
136C2×Q8×C47—21no—
137C2×Q8×C47—22no—
138C2×Q8×C47—28no—
139C2×Q8×C47—22no—
140C2×Q8×C47—22no—
141C2×Q8×C47—22no—
142C2×Q8×C47—22no—
143C2×Q8×C47—44no—
144C2×Q8×C47—44no—
145C2×Q8×C47—44no—
146C2×Q8×C47—48no—
147C2×Q8×C47—48no—
148C2×Q8×C47—44no—
149C2×Q8×C47—44no—
150C2×Q8×C47C2×D412yes—
151C2×Q8×C47C2×D412yes—
152C2×Q8×C47C2×D414yes—
153C2×Q8×C47C2×D412yes—
154C2×D4×C47—22no—
155C2×D4×C47—22no—
156C2×D4×C47Q1614yescomplemented
157C22×C4×C47—21no—
158C22×C4×C47—22no—
159C22×C4×C47—22no—
160C22×C4×C47—24no—
161C22×C4×C47—24no—
162C22×C4×C47—21no—
163C22×C4×C47—21no—
164C22×C4×C47—21no—
165C22×C4×C47—21no—
166C22×C4×C47—21no—
167C22×C4×C47—22no—
168C22×C4×C47—22no—
169C22×C4×C47—44no—
170C22×C4×C47—44no—
171C22×C4×C47C2×D411yescharacteristic
172C22×C4×C47C2×D411yescharacteristic
173C22×C4×C47C2×D411yescharacteristic
174C22×C4×C47C2×D412yes—
175C22×C4×C47C2×D412yes—
176C22×C4×C47C2×D411yescharacteristic
177C22×C4×C47C2411yescharacteristic
178C22×Dic47—22no—
179C22×Dic47—22no—
180C22×Dic47—44no—
181C22×Dic47—42no—
182C2×Dic94—28no—
183C2×Dic94—22no—
184C2×Dic94—22no—
185C2×Dic94—22no—
186C2×Dic94—22no—
187C2×Dic94—22no—
188C2×Dic94—48no—
189C2×Dic94—44no—
190C2×Dic94—44no—
191C2×Dic94—42no—
192C2×Dic94C2×D414yes—
193C4⋊C4×C47—24no—
194C4⋊C4×C47—24no—
195C4⋊C4×C47—24no—
196C4⋊C4×C47—24no—
197C4⋊C4×C47—24no—
198C4⋊C4×C47—22no—
199C4⋊C4×C47—21no—
200C4⋊C4×C47C2×D412yes—
201C22⋊C4×C47—24no—
202C22⋊C4×C47—24no—
203C22⋊C4×C47—22no—
204C22⋊C4×C47—22no—
205C42×C47—24no—
206C42×C47—21no—
207C42×C47—21no—
208C22⋊Dic47—44no—
209C22⋊Dic47—42no—
210#10625337—216no—
211#10625337—216no—
212#10625337—48no—
213C4⋊Dic47—22no—
214C4⋊Dic47—22no—
215Dic47⋊C4—44no—
216C2×C47⋊C8—28no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—22no—
220C2×C47⋊C8—22no—
221C2×C47⋊C8—22no—
222C2×C47⋊C8C2×D414yes—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47—21no—
226C23×C47—22no—
227C23×C47—21no—
228C23×C47—22no—
229C23×C47#12012yes—
230C23×C47C2×Q1612yes—
231C23×C47C2×Q1612yescomplemented
232C23×C47#13712yes—
233C23×C47C22×D411yescharacteristic
234Q8×C47—22no—
235Q8×C47—24no—
236Q8×C47—22no—
237Q8×C47—22no—
238Q8×C47—24no—
239Q8×C47—48no—
240Q8×C47—48no—
241Q8×C47—41no—
242Q8×C47—41no—
243Q8×C47—42no—
244Q8×C47#12014yes—
245D4×C47—28no—
246C2×C4×C47—21no—
247C2×C4×C47—22no—
248C2×C4×C47—22no—
249C2×C4×C47—24no—
250C2×C4×C47—21no—
251C2×C4×C47—22no—
252C2×C4×C47—22no—
253C2×C4×C47—24no—
254C2×C4×C47—21no—
255C2×C4×C47—22no—
256C2×C4×C47—22no—
257C2×C4×C47—22no—
258C2×C4×C47—21no—
259C2×C4×C47—22no—
260C2×C4×C47—22no—
261C2×C4×C47—48no—
262C2×C4×C47—48no—
263C2×C4×C47—42no—
264C2×C4×C47—42no—
265C2×C4×C47—42no—
266C2×C4×C47—41no—
267C2×C4×C47—41no—
268C2×C4×C47—42no—
269C2×C4×C47#12012yes—
270C2×C4×C47C2×Q1612yescomplemented
271C2×C4×C47C22×D412yes—
272C2×C4×C47C22×D411yescharacteristic
273C2×C4×C47C22×D411yescharacteristic
274C2×C4×C47H(5,𝔽2)12yes—
275C2×Dic47—24no—
276C2×Dic47—48no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—42no—
280C2×Dic47—42no—
281C2×Dic47—42no—
282Dic94—24no—
283Dic94—24no—
284Dic94—48no—
285Dic94—42no—
286Dic94—42no—
287C47⋊C8—24no—
288C47⋊C8—42no—
289#23856—471nomaximal, Sylow
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—24no—
293C22×C47—24no—
294C22×C47—22no—
295C22×C47—24no—
296C22×C47—41no—
297C22×C47—41no—
298C22×C47—42no—
299C22×C47#45114yescomplemented
300C22×C47C2×#12011yescharacteristic
301C22×C47D4212yes—
302C22×C47C22×Q1611yescharacteristic
303C22×C47C2×#13711yescharacteristic
304C22×C47#57812yes—
305C4×C47—22no—
306C4×C47—24no—
307C4×C47—22no—
308C4×C47—41no—
309C4×C47—42no—
310C4×C47—42no—
311C4×C47—41no—
312C4×C47—42no—
313C4×C47C2×#12011yescharacteristic
314C4×C47#53411yescharacteristic
315C4×C47D4212yes—
316Dic47—44no—
317Dic47—82no—
318C23×Q16—471no—
319C2×D4×Q8D4711yescharacteristic
320C2×#451—472no—
321C2×#451—471no—
322#2409—471no—
323#2016—472no—
324#2001—472no—
325#1892—471no—
326#1878—472no—
327#1869—471no—
328#1787—471no—
329C2×C47—24no—
330C2×C47—42no—
331C2×C47#240311yescharacteristic
332C2×C47C2×#45111yescharacteristic
333C2×C47#301111yescharacteristic
334C2×C47#327012yes—
335C2×C47D4×Q1612yes—
336C23×Q8—471no—
337C23×Q8C2×D4711yescharacteristic
338C23×Q8C2×D4711yescharacteristic
339C22×Q16—472no—
340C22×Q16—472no—
341C22×Q16—472no—
342C22×Q16—948no—
343C22×Q16—944no—
344C23×C8—471no—
345D4×Q8—24no—
346D4×Q8—24no—
347C2×C4⋊Q8—21no—
348C2×C4⋊Q8C2×D4711yescharacteristic
349C2×C22⋊Q8—21no—
350C2×C22⋊Q8—21no—
351C2×C22⋊Q8—472no—
352C2×C22⋊Q8C2×D4711yescharacteristic
353C2×C22⋊Q8C2×D4711yescharacteristic
354C2×C4×Q8C2×D4711yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4C2×D4711yescharacteristic
357#451—948no—
358#451—944no—
359C2×#103—472no—
360C2×#103—472no—
361C2×#103—472no—
362C2×#103—472no—
363C2×C22⋊C8—472no—
364C2×C22⋊C8—471no—
365#395—941no—
366#393—941no—
367#391—472no—
368#386—941no—
369#385—941no—
370#380—471no—
371#340—472no—
372#336—472no—
373C47#2385611yescharacteristic, complemented, Sylow
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8—21no—
377C22×Q8—21no—
378C22×Q8—22no—
379C22×Q8—24no—
380C22×Q8—24no—
381C22×Q8—42no—
382C22×Q8—42no—
383C22×Q8—472no—
384C22×Q8—944no—
385C22×Q8—941no—
386C22×Q8—942no—
387C22×Q8—942no—
388C22×Q8—1882no—
389C22×Q8C22⋊D4714yes—
390C22×Q8C22×D4711yescharacteristic
391C22×Q8C22×D4711yescharacteristic
392C22×Q8C22×D4711yescharacteristic
393C22×D4C22⋊D4711yescharacteristic
394C23×C4—21no—
395C23×C4—21no—
396C23×C4—941no—
397C23×C4C22×D4711yescharacteristic
398C23×C4C22×D4711yescharacteristic
399C2×Q16—478no—
400C2×Q16—944no—
401C2×Q16—944no—
402C2×Q16—9432no—
403C2×Q16—944no—
404C2×Q16—944no—
405C2×Q16—944no—
406C2×Q16—944no—
407C2×Q16—944no—
408C2×Q16—1882no—
409C22×C8—472no—
410C22×C8—472no—
411C22×C8—944no—
412C22×C8—942no—
413C4⋊Q8—28no—
414C4⋊Q8—22no—
415C4⋊Q8—22no—
416C22⋊Q8—28no—
417C22⋊Q8—28no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421C22⋊Q8—944no—
422C22⋊Q8—1882no—
423C22⋊Q8C22⋊D4714yes—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×D4—28no—
427C4×D4—22no—
428C4×D4—22no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—472no—
435C2×C4⋊C4—942no—
436C2×C4⋊C4C22⋊D4711yescharacteristic
437C2×C4⋊C4C22×D4711yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—942no—
441C2×C22⋊C4—941no—
442C2×C22⋊C4C22⋊D4711yescharacteristic
443C2×C22⋊C4C22⋊D4711yescharacteristic
444C2×C42—21no—
445C2×C42C22×D4711yescharacteristic
446#103—944no—
447#103—944no—
448#103—944no—
449#103—1882no—
450C22⋊C8—944no—
451C22⋊C8—942no—
452#95—941no—
453#95—942no—
454#95—942no—
455C24C2×C22⋊D4711yescharacteristic
456C24C2×C22⋊D4711yescharacteristic
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—28no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—44no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—48no—
472C2×Q8—48no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8—474no—
476C2×Q8—948no—
477C2×Q8—942no—
478C2×Q8—942no—
479C2×Q8—942no—
480C2×Q8—942no—
481C2×Q8—942no—
482C2×Q8—1884no—
483C2×Q8—1888no—
484C2×Q8—1884no—
485C2×Q8—1882no—
486C2×Q8D4×D4714yes—
487C2×Q8C2×C22⋊D4712yes—
488C2×Q8C2×C22⋊D4712yes—
489C2×Q8C2×C22⋊D4712yes—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4#1062533714yescomplemented
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—1884no—
510C22×C4—1882no—
511C22×C4C2×C22⋊D4711yescharacteristic
512C22×C4C2×C22⋊D4711yescharacteristic
513C22×C4C2×C22⋊D4711yescharacteristic
514C22×C4C2×C22⋊D4712yes—
515C22×C4C2×C22⋊D4712yes—
516C22×C4C2×C22⋊D4711yescharacteristic
517C22×C4C23×D4711yescharacteristic
518Q16—9416no—
519Q16—9416no—
520Q16—1888no—
521C2×C8—474no—
522C2×C8—948no—
523C2×C8—942no—
524C2×C8—942no—
525C2×C8—942no—
526C2×C8—942no—
527C2×C8—942no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—942no—
536C4⋊C4—942no—
537C4⋊C4—1884no—
538C4⋊C4C2×C22⋊D4712yes—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—1884no—
544C22⋊C4—1882no—
545C42—24no—
546C42—21no—
547C42—21no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23#1309399312yes—
555C23C2×#1062533712yes—
556C23C2×#1062533712yescomplemented
557C23#1309405512yes—
558C23C22×C22⋊D4711yescharacteristic
559Q8—24no—
560Q8—22no—
561Q8—24no—
562Q8—22no—
563Q8—22no—
564Q8—48no—
565Q8—48no—
566Q8—42no—
567Q8—41no—
568Q8—41no—
569Q8—944no—
570Q8—944no—
571Q8—1888no—
572Q8—1882no—
573Q8—1882no—
574Q8#1309399214yes—
575D4—28no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—944no—
600C2×C4—1882no—
601C2×C4—1882no—
602C2×C4—1882no—
603C2×C4—1888no—
604C2×C4—1882no—
605C2×C4—1882no—
606C2×C4#1309399312yes—
607C2×C4C2×#1062533712yescomplemented
608C2×C4C2×D4×D4712yes—
609C2×C4C22×C22⋊D4711yescharacteristic
610C2×C4C22×C22⋊D4711yescharacteristic
611C2×C4#1309406912yes—
612C8—944no—
613C8—1882no—
614C22—22no—
615C22—24no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22#1568927914yescomplemented
624C22C2×#1309399311yescharacteristic
625C22D4×C22⋊D4712yes—
626C22C22×#1062533711yescharacteristic
627C22C2×#1309405511yescharacteristic
628C22#1568993812yes—
629C4—22no—
630C4—22no—
631C4—24no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—3762no—
638C4—1884no—
639C4C2×#1309399211yescharacteristic
640C4#1568968211yescharacteristic
641C4D4×C22⋊D4712yes—
642C2—24no—
643C2—42no—
644C2#2024669811yescharacteristic
645C2C2×#1568927911yescharacteristic
646C2#2025268311yescharacteristic
647C2#2025495912yes—
648C2D4×#1062533712yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×D4×Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×D4×C47D4
hypercenterC2×D4×Q8D47
hypocenterC47#23856
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#10625337C2semidirect
2. C2×#15689279C2semidirect
3. C2×#15689278C2semidirect
5. #20244488C2semidirect
6. #20244396C2semidirect
7. #20243812C2semidirect
8. #20243729C2semidirect
10. #20243323C2semidirect
65. C2×#10625337D4semidirect
156. C2×D4×C47Q16semidirect
231. C23×C47C2×Q16semidirect
270. C2×C4×C47C2×Q16semidirect
299. C22×C47#451semidirect
373. C47#23856semidirect
492. C2×D4#10625337semidirect
556. C23C2×#10625337semidirect
607. C2×C4C2×#10625337semidirect
623. C22#15689279semidirect

Maximal subgroups

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

subgroupindex
C23×#106253372
C2×#156892792
C2×#156892782
#202467532
#202444882
#202443962
#202438122
#202437292
#202436812
#202433232
C2×D4×Q8×C472
#2385647