Finite Group Databasebeta

← #26217420

Subgroups — #26217420

Order 12032.

1938 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D47C211yesmaximal, characteristic, complemented
2C2×#15689635C211yesmaximal, characteristic
3C2×#15689091C211yesmaximal, characteristic, complemented
4C2×#15689089C211yesmaximal, characteristic, complemented
5C22×(C2×D47)⋊C8C211yesmaximal, characteristic, complemented
6#20243684C214yesmaximal, complemented
7#20243679C214yesmaximal, complemented
8#20243085C211yesmaximal, characteristic, complemented
9C47×#1729C211yesmaximal, characteristic, complemented
10C22×#10625350C2211yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C22×#10625324C2212yes—
14C2×M16×D47—24no—
15C2×M16×D47—24no—
16C2×M16×D47—24no—
17C23×C47⋊C8C2211yescharacteristic
18C22×C8×D47C2212yes—
19#15689635—24no—
20#15689635—24no—
21C2×#13094008C412yes—
22C2×#13094004C411yescharacteristic
23C2×#13094004C411yescharacteristic
24C2×#13093999C411yescharacteristic
25C2×#13093999C411yescharacteristic
26C2×#13093994C412yes—
27C2×C4×D188C2212yes—
28C2×C4×D188C2212yes—
29#15689091—24no—
30#15689089—24no—
31C2×(C2×D47)⋊C8—24no—
32C2×(C2×D47)⋊C8C2212yes—
33C2×(C2×D47)⋊C8C2212yes—
34C2×(C2×D47)⋊C8C2214yescomplemented
35C2×(C2×D47)⋊C8C2214yescomplemented
36#15688878C2214yescomplemented
37C22×M16×C47C2211yescharacteristic
38C23×C8×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C47×#336C2214yescomplemented
41C24×D47C2×C411yescharacteristic
42C2×#10625350—22no—
43C2×#10625350—22no—
44C2×#10625350—22no—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C22×Dic94C2×C411yescharacteristic
49C22×D188C2×C412yes—
50C22×D188C2×C411yescharacteristic
51C22×D188C2×C412yes—
52C22×C22⋊D47C2×C412yes—
53C23×Dic47C2×C411yescharacteristic
54C22×C4×D47—22no—
55C22×C4×D47—22no—
56C22×C4×D47—22no—
57C22×C4×D47—22no—
58C22×C4×D47C2×C411yescharacteristic
59C22×C4×D47C2×C411yescharacteristic
60C22×C4×D47D414yes—
61C22×C4×D47D414yes—
62C22×C4×D47C2312yes—
63C22×C4×D47C2311yescharacteristic
64C22×C4×D47C2311yescharacteristic
65C2×#10625329—22no—
66C2×#10625329—22no—
67C2×#10625329D414yescomplemented
68C2×#10625324—24no—
69C2×#10625324—24no—
70C2×#10625324—24no—
71M16×D47—232no—
72C22×C47⋊C8—22no—
73C22×C47⋊C8—22no—
74C22×C47⋊C8—22no—
75C22×C47⋊C8C2312yes—
76C22×C47⋊C8C2312yes—
77C2×C8×D47—24no—
78C2×C8×D47—24no—
79C2×C8×D47—24no—
80#13094008—28no—
81#13094004—24no—
82#13094004—24no—
83#13093999—24no—
84#13093999—24no—
85#13093994—28no—
86C4×D188—24no—
87C4×D188—24no—
88C4×D188—24no—
89C4×D188—24no—
90C2×C4⋊Dic47C2×C412yes—
91C2×C4⋊Dic47C2×C412yes—
92C2×(C2×D47)⋊C4C2×C412yes—
93C2×(C2×D47)⋊C4C2×C412yes—
94C2×(C2×D47)⋊C4C2×C412yes—
95C2×(C2×D47)⋊C4C2×C412yes—
96(C2×D47)⋊C8—24no—
97(C2×D47)⋊C8—24no—
98(C2×D47)⋊C8—28no—
99(C2×D47)⋊C8—28no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—22no—
102C2×M16×C47—22no—
103C2×M16×C47D414yescomplemented
104C22×C8×C47—22no—
105C22×C8×C47—22no—
106C22×C8×C47—22no—
107C22×C8×C47C2312yes—
108C22×C8×C47C2312yes—
109#117×C47—22no—
110#117×C47—22no—
111C2×C4⋊C4×C47C2×C411yescharacteristic
112C2×C4⋊C4×C47C2×C411yescharacteristic
113C2×C22⋊C4×C47C2×C412yes—
114C2×C42×C47C2312yes—
115C24×C47C22×C411yescharacteristic
116C23×D47—22no—
117C23×D47—22no—
118C23×D47—24no—
119C23×D47—42no—
120C23×D47C22⋊C414yes—
121C23×D47C22×C412yes—
122C23×D47C22×C411yescharacteristic
123C23×D47C22×C411yescharacteristic
124C22×C4×C47C22×C412yes—
125C22×C4×C47C22×C412yes—
126C22×C4×C47C22×C411yescharacteristic
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47#4112yes—
132C22×C4×C47#4112yes—
133C22×C4×C47C2411yescharacteristic
134C2×C22⋊D47—24no—
135C2×C22⋊D47—24no—
136C2×C22⋊D47—24no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47C22⋊C414yes—
140C22×Dic47C22×C412yes—
141C22×Dic47C22×C411yescharacteristic
142C22×Dic47C22×C411yescharacteristic
143#10625350—44no—
144#10625350—44no—
145#10625350—44no—
146#10625350—44no—
147C2×D188—24no—
148C2×D188—24no—
149C2×D188—28no—
150C2×D188—22no—
151C2×D188—22no—
152C2×D188—22no—
153C2×D188—44no—
154C2×C4×D47—22no—
155C2×C4×D47—22no—
156C2×C4×D47—22no—
157C2×C4×D47—22no—
158C2×C4×D47—28no—
159C2×C4×D47—28no—
160C2×C4×D47—28no—
161C2×C4×D47—28no—
162C2×C4×D47—21no—
163C2×C4×D47—21no—
164C2×C4×D47—21no—
165C2×C4×D47—21no—
166C2×C4×D47—21no—
167C2×C4×D47—21no—
168C2×C4×D47—21no—
169C2×C4×D47—21no—
170C2×C4×D47—22no—
171C2×C4×D47—22no—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—22no—
175C2×C4×D47—22no—
176C2×C4×D47C22⋊C414yes—
177C2×C4×D47C22⋊C414yes—
178C2×C4×D47C2×D412yes—
179C2×C4×D47C2×D412yes—
180C2×C4×D47C2×D412yes—
181C2×C4×D47C2×D412yes—
182C2×Dic94—22no—
183C2×Dic94—22no—
184C2×Dic94—22no—
185M16×C47—28no—
186C2×C8×C47—22no—
187C2×C8×C47—22no—
188C2×C8×C47—22no—
189C2×C8×C47—22no—
190C2×C8×C47—22no—
191C2×C8×C47—44no—
192C2×C8×C47C2×D414yes—
193C4⋊C4×C47—22no—
194C4⋊C4×C47—22no—
195C22⋊C4×C47—24no—
196C42×C47—22no—
197C42×C47—22no—
198(C2×D47)⋊C4—24no—
199(C2×D47)⋊C4—24no—
200(C2×D47)⋊C4—24no—
201(C2×D47)⋊C4—24no—
202C4⋊Dic47—24no—
203C4⋊Dic47—24no—
204#10625329—28no—
205C2×C47⋊C8—22no—
206C2×C47⋊C8—22no—
207C2×C47⋊C8—22no—
208C2×C47⋊C8—22no—
209C2×C47⋊C8—22no—
210C2×C47⋊C8—44no—
211C2×C47⋊C8C2×D414yes—
212#10625324—216no—
213C8×D47—216no—
214C23×C47—22no—
215C23×C47—22no—
216C23×C47C2×C22⋊C412yes—
217C23×C47C4×D412yes—
218C23×C47C4×D412yes—
219C23×C47C23×C411yescharacteristic
220C22×D47—21no—
221C22×D47—21no—
222C22×D47—22no—
223C22×D47—21no—
224C22×D47—21no—
225C22×D47—22no—
226C22×D47—22no—
227C22×D47—28no—
228C22×D47—28no—
229C22×D47—22no—
230C22×D47—22no—
231C22×D47—22no—
232C22×D47—24no—
233C22×D47—24no—
234C22×D47—44no—
235C22×D47—48no—
236C22×D47—48no—
237C22×D47C2×C22⋊C412yes—
238C22×D47C2×C22⋊C412yes—
239C22×D47#13114yes—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—24no—
243C2×C4×C47—24no—
244C2×C4×C47C2×C22⋊C412yes—
245C2×C4×C47C4×D412yes—
246C2×C4×C47C4×D412yes—
247C2×C4×C47#12014yes—
248C2×C4×C47#12114yes—
249C2×C4×C47#12114yes—
250C2×C4×C47#12314yes—
251C2×C4×C47C22×D411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C2×C4×C47C22×D411yescharacteristic
254C2×C4×C47C22×D411yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C22⋊D47—48no—
258C2×Dic47—21no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261C2×Dic47—21no—
262C2×Dic47—28no—
263C2×Dic47—22no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47C2×C22⋊C412yes—
267C2×Dic47C2×C22⋊C412yes—
268D188—48no—
269D188—48no—
270D188—44no—
271C4×D47—22no—
272C4×D47—22no—
273C4×D47—22no—
274C4×D47—22no—
275C4×D47—22no—
276C4×D47—22no—
277C4×D47—22no—
278C4×D47—22no—
279C4×D47—28no—
280C4×D47—28no—
281C4×D47—42no—
282C4×D47—42no—
283C4×D47—42no—
284C4×D47—42no—
285Dic94—44no—
286C8×C47—24no—
287C8×C47—42no—
288C47⋊C8—24no—
289C47⋊C8—42no—
290#20897—471nomaximal, Sylow
291C22×C47—22no—
292C22×C47—22no—
293C22×C47—24no—
294C22×C47—24no—
295C22×C47#38614yes—
296C22×C47C22×C22⋊C411yescharacteristic
297C22×C47C2×C4×D411yescharacteristic
298C22×C47C2×C4×D411yescharacteristic
299C22×C47C2×#13112yes—
300C22×C47#56812yes—
301C2×D47—22no—
302C2×D47—24no—
303C2×D47—24no—
304C2×D47—22no—
305C2×D47—22no—
306C2×D47—22no—
307C2×D47—22no—
308C2×D47—416no—
309C2×D47—42no—
310C2×D47—42no—
311C2×D47—42no—
312C2×D47—42no—
313C2×D47—44no—
314C2×D47—44no—
315C2×D47—42no—
316C2×D47#40814yescomplemented
317C4×C47—42no—
318C4×C47—42no—
319C4×C47#38614yes—
320C4×C47C2×#12011yescharacteristic
321C4×C47C2×#12111yescharacteristic
322C4×C47C2×#12111yescharacteristic
323C4×C47C2×#12311yescharacteristic
324Dic47—22no—
325Dic47—22no—
326Dic47—42no—
327C23×M16—471no—
328C2×#525—471no—
329C2×#408—471no—
330C2×#408—471no—
331C22×C22⋊C8—471no—
332#1870—474no—
333#1869—474no—
334#1729—471no—
335#1729D4711yescharacteristic, complemented
336C2×C47—24no—
337C2×C47C2×#38611yescharacteristic
338C2×C47C2×#40811yescharacteristic
339C2×C47#287911yescharacteristic
340C2×C47#293312yes—
341C2×C47#293312yes—
342D47—24no—
343D47—42no—
344D47—82no—
345C22×#41—471no—
346C24×C4—471no—
347C22×M16—471no—
348C22×M16—472no—
349C22×M16—944no—
350C22×M16—944no—
351C22×M16—944no—
352C22×M16C2×D4711yescharacteristic
353C23×C8—471no—
354C23×C8—472no—
355C23×C8C2×D4711yescharacteristic
356#525—944no—
357#525—944no—
358C2×#123—472no—
359C2×C22⋊Q8—471no—
360C2×C22⋊Q8—471no—
361C2×#121—471no—
362C2×#121—471no—
363C2×#120—472no—
364C2×C4×D4—472no—
365C2×C4×D4—472no—
366C2×#117C2×D4711yescharacteristic
367#408—944no—
368#408—944no—
369C2×C22⋊C8—472no—
370C2×C22⋊C8—472no—
371C2×C22⋊C8—474no—
372C2×C22⋊C8—474no—
373C2×C22⋊C8—944no—
374#336—474no—
375#336C2×D4714yescomplemented
376C47#2089711yescharacteristic, complemented, Sylow
377C25—471no—
378C2×#41—942no—
379C2×#41—942no—
380C2×#41—942no—
381C2×#41—942no—
382C2×#41—942no—
383C2×#41—942no—
384C22×Q8—471no—
385C22×D4—472no—
386C22×D4—472no—
387C22×D4—471no—
388C22×D4—472no—
389C23×C4—471no—
390C23×C4—471no—
391C23×C4—471no—
392C23×C4—474no—
393C23×C4—474no—
394C23×C4—471no—
395C23×C4—471no—
396C23×C4—472no—
397C23×C4—942no—
398C23×C4—942no—
399C23×C4—942no—
400C23×C4—942no—
401C23×C4C22×D4711yescharacteristic
402C2×M16—22no—
403C2×M16—22no—
404C2×M16—474no—
405C2×M16—944no—
406C2×M16—944no—
407C2×M16—9432no—
408C2×M16—942no—
409C2×M16—942no—
410C2×M16—944no—
411C2×M16D18814yescomplemented
412C22×C8—22no—
413C22×C8—22no—
414C22×C8—22no—
415C22×C8—472no—
416C22×C8—472no—
417C22×C8—944no—
418C22×C8—944no—
419C22×C8—944no—
420C22×C8—942no—
421C22×C8—942no—
422C22×C8—942no—
423C22×C8C22×D4712yes—
424C22×C8C22×D4712yes—
425#123—948no—
426C22⋊Q8—944no—
427C22⋊Q8—944no—
428#121—944no—
429#121—944no—
430#120—948no—
431C4×D4—944no—
432C4×D4—944no—
433C4×D4—944no—
434C4×D4—944no—
435#117—22no—
436#117—22no—
437C2×C4⋊C4—472no—
438C2×C4⋊C4—472no—
439C2×C4⋊C4C4×D4711yescharacteristic
440C2×C4⋊C4C4×D4711yescharacteristic
441C2×C22⋊C4—472no—
442C2×C22⋊C4—472no—
443C2×C22⋊C4—472no—
444C2×C22⋊C4—472no—
445C2×C22⋊C4C4×D4712yes—
446C2×C42C22×D4712yes—
447C22⋊C8—948no—
448C22⋊C8—944no—
449C22⋊C8—944no—
450C22⋊C8—948no—
451C24—471no—
452C24—471no—
453C24—474no—
454C24—472no—
455C24—944no—
456C24—942no—
457C24—942no—
458C24—1882no—
459C24C2×C4×D4711yescharacteristic
460#41—1884no—
461#41—1884no—
462#41—1884no—
463#41—1884no—
464C2×Q8—942no—
465C2×Q8—942no—
466C2×Q8—942no—
467C2×D4—944no—
468C2×D4—944no—
469C2×D4—948no—
470C2×D4—944no—
471C2×D4—944no—
472C2×D4—944no—
473C2×D4—942no—
474C2×D4—942no—
475C2×D4—942no—
476C2×D4—1884no—
477C22×C4—474no—
478C22×C4—471no—
479C22×C4—471no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—472no—
483C22×C4—472no—
484C22×C4—472no—
485C22×C4—472no—
486C22×C4—472no—
487C22×C4—942no—
488C22×C4—942no—
489C22×C4—942no—
490C22×C4—942no—
491C22×C4—942no—
492C22×C4—942no—
493C22×C4—948no—
494C22×C4—948no—
495C22×C4—948no—
496C22×C4—948no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4—941no—
500C22×C4—941no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—941no—
505C22×C4—942no—
506C22×C4—942no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—942no—
511C22×C4C2×C4×D4712yes—
512C22×C4C2×C4×D4712yes—
513C22×C4C2×C4×D4711yescharacteristic
514C22×C4C2×D18812yes—
515C22×C4#1062535012yes—
516C22×C4D4×D4712yes—
517C22×C4D4×D4712yes—
518C22×C4D4×D4712yes—
519C22×C4#1062535412yes—
520C22×C4C23×D4711yescharacteristic
521M16—28no—
522M16—9416no—
523M16—948no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—44no—
530C2×C8—474no—
531C2×C8—9416no—
532C2×C8—942no—
533C2×C8—942no—
534C2×C8—942no—
535C2×C8—942no—
536C2×C8—942no—
537C2×C8—1884no—
538C2×C8C2×D18814yes—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—944no—
542C4⋊C4—944no—
543C22⋊C4—24no—
544C22⋊C4—944no—
545C22⋊C4—944no—
546C22⋊C4—944no—
547C22⋊C4—944no—
548C42—22no—
549C42—22no—
550C23—22no—
551C23—22no—
552C23—472no—
553C23—474no—
554C23—472no—
555C23—948no—
556C23—941no—
557C23—941no—
558C23—942no—
559C23—948no—
560C23—941no—
561C23—941no—
562C23—942no—
563C23—942no—
564C23—942no—
565C23—942no—
566C23—942no—
567C23—944no—
568C23—944no—
569C23—1884no—
570C23—1888no—
571C23—1888no—
572C23C22⋊C4×D4712yes—
573C23#1309398612yes—
574C23C4×D18812yes—
575C23C22×C4×D4711yescharacteristic
576Q8—1884no—
577D4—1888no—
578D4—1888no—
579D4—1888no—
580D4—1884no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—472no—
586C2×C4—472no—
587C2×C4—948no—
588C2×C4—941no—
589C2×C4—941no—
590C2×C4—941no—
591C2×C4—941no—
592C2×C4—948no—
593C2×C4—948no—
594C2×C4—942no—
595C2×C4—942no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—942no—
603C2×C4—942no—
604C2×C4—942no—
605C2×C4—1882no—
606C2×C4—1882no—
607C2×C4—1882no—
608C2×C4—1882no—
609C2×C4C22⋊C4×D4712yes—
610C2×C4#1309398612yes—
611C2×C4C4×D18812yes—
612C2×C4#1309399414yes—
613C2×C4#1309399714yes—
614C2×C4#1309399914yes—
615C2×C4#1309400914yes—
616C2×C4C2×D4×D4711yescharacteristic
617C2×C4C2×D4×D4711yescharacteristic
618C2×C4C2×D4×D4711yescharacteristic
619C2×C4C22×D18811yescharacteristic
620C2×C4C2×#1062535011yescharacteristic
621C2×C4C2×#1062535411yescharacteristic
622C8—24no—
623C8—42no—
624C8—944no—
625C8—1882no—
626C22—22no—
627C22—22no—
628C22—24no—
629C22—24no—
630C22—474no—
631C22—942no—
632C22—942no—
633C22—944no—
634C22—944no—
635C22—942no—
636C22—942no—
637C22—942no—
638C22—18816no—
639C22—1882no—
640C22—1882no—
641C22—1882no—
642C22—1884no—
643C22—1884no—
644C22—1882no—
645C22—1882no—
646C22#1568900114yes—
647C22C2×C22⋊C4×D4711yescharacteristic
648C22C2×#1309398611yescharacteristic
649C22C2×C4×D18811yescharacteristic
650C22#131×D4712yes—
651C22#1568988812yes—
652C4—42no—
653C4—42no—
654C4—942no—
655C4—942no—
656C4—1882no—
657C4#1568900114yes—
658C4C2×#1309399411yescharacteristic
659C4C2×#1309399711yescharacteristic
660C4C2×#1309399911yescharacteristic
661C4C2×#1309400911yescharacteristic
662C2—24no—
663C2—944no—
664C2—3762no—
665C2—1882no—
666C2C2×#1568900111yescharacteristic
667C2#408×D4711yescharacteristic
668C2#2025183011yescharacteristic
669C2#2025211212yes—
670C2#2025211612yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1729C2
prime radicalC2×#15689635C2
cosocleC22×C4×C47C24
hypercenter#1729D47
hypocenterC47#20897
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
1. C22×M16×D47C2semidirect
3. C2×#15689091C2semidirect
4. C2×#15689089C2semidirect
5. C22×(C2×D47)⋊C8C2semidirect
6. #20243684C2semidirect
7. #20243679C2semidirect
8. #20243085C2semidirect
9. C47×#1729C2semidirect
34. C2×(C2×D47)⋊C8C22semidirect
35. C2×(C2×D47)⋊C8C22semidirect
36. #15688878C22semidirect
40. C47×#336C22semidirect
67. C2×#10625329D4semidirect
103. C2×M16×C47D4semidirect
316. C2×D47#408semidirect
335. #1729D47semidirect
375. #336C2×D47semidirect
376. C47#20897semidirect
411. C2×M16D188semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D472
C2×#156896352
C2×#156890912
C2×#156890892
C22×(C2×D47)⋊C82
#202436842
#202436792
#202430852
C47×#17292
#2089747