Finite Group Databasebeta

← #26213684

Subgroups — #26213684

Order 12032.

2270 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D47C211yesmaximal, characteristic
2C22×#13093982C211yesmaximal, characteristic
3C2×#117×D47C211yesmaximal, characteristic
4#20243109C211yesmaximal, characteristic, complemented
5#20243108C211yesmaximal, characteristic, complemented
6#20243107C211yesmaximal, characteristic, complemented
7#20243106C214yesmaximal, complemented
8#20243103C214yesmaximal, complemented
9C47×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D47C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C22×#10625324C2212yes—
13C2×M16×D47—24no—
14C2×M16×D47—24no—
15C2×M16×D47C418yescomplemented
16C22×C8×D47C2212yes—
17C2×#13093983C2211yescharacteristic
18C2×#13093982—24no—
19C2×#13093982—21no—
20C2×#13093982—21no—
21C2×#13093982C418yescomplemented
22C2×#13093982C2212yes—
23C2×#13093982C2212yes—
24C2×#13093982C2212yes—
25C2×#13093981—21no—
26C2×#13093980—21no—
27#117×D47—24no—
28#117×D47C418yescomplemented
29C22×C4⋊Dic47C2211yescharacteristic
30C2×C4⋊C4×D47C2212yes—
31C22×(C2×D47)⋊C4—21no—
32C2×C22⋊C4×D47—21no—
33C22×C4×Dic47—21no—
34C2×C42×D47—21no—
35#15688886C2214yescomplemented
36#15688885C2214yescomplemented
37#15688885C2214yescomplemented
38C22×M16×C47C2211yescharacteristic
39C2×#117×C47C2211yescharacteristic
40C22×C4⋊C4×C47C2211yescharacteristic
41C47×#340C2214yescomplemented
42C24×D47D411yescharacteristic
43C23×Dic47—21no—
44C23×Dic47D411yescharacteristic
45C22×C4×D47—22no—
46C22×C4×D47C2×C414yes—
47C22×C4×D47C2×C414yes—
48C22×C4×D47C2×C414yes—
49C22×C4×D47D411yescharacteristic
50C22×C4×D47D411yescharacteristic
51C22×C4×D47D414yes—
52C22×C4×D47D414yes—
53C22×C4×D47Q814yes—
54C22×C4×D47C2311yescharacteristic
55C22×C4×D47C2311yescharacteristic
56C2×#10625329—22no—
57C2×#10625329—22no—
58C2×#10625329C2×C414yescomplemented
59C2×#10625324—24no—
60C2×#10625324—24no—
61C2×#10625324C2×C418yescomplemented
62M16×D47—232no—
63C22×C47⋊C8C2312yes—
64C2×C8×D47—24no—
65C2×C8×D47—24no—
66C2×C8×D47C2×C418yescomplemented
67#13093983—22no—
68#13093983C2×C414yes—
69#13093982—28no—
70#13093982—28no—
71#13093982—24no—
72#13093982—24no—
73#13093982—24no—
74#13093982C2×C418yes—
75#13093982C2×C418yescomplemented
76#13093982C2×C418yescomplemented
77#13093981—28no—
78#13093980—28no—
79C2×C4⋊Dic47—21no—
80C2×C4⋊Dic47—22no—
81C2×C4⋊Dic47C2×C414yescomplemented
82C2×C4⋊Dic47C2312yes—
83C2×C4⋊Dic47C2312yes—
84C2×Dic47⋊C4—22no—
85C4⋊C4×D47—24no—
86C4⋊C4×D47C2×C418yescomplemented
87C2×C22⋊Dic47—21no—
88C2×(C2×D47)⋊C4—28no—
89C2×(C2×D47)⋊C4—22no—
90C2×(C2×D47)⋊C4—22no—
91C2×(C2×D47)⋊C4—22no—
92C22⋊C4×D47—28no—
93C2×C4×Dic47—21no—
94C2×C4×Dic47—28no—
95C2×C4×Dic47—22no—
96C2×C4×Dic47—22no—
97C2×C4×Dic47—22no—
98C42×D47—28no—
99C23×C4×C47—21no—
100C23×C4×C47C2311yescharacteristic
101C2×M16×C47—22no—
102C2×M16×C47—22no—
103C2×M16×C47C2×C414yescomplemented
104C22×C8×C47C2312yes—
105#117×C47—22no—
106#117×C47C2×C414yes—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—22no—
109C2×C4⋊C4×C47C2×C414yescomplemented
110C2×C4⋊C4×C47C2312yes—
111C2×C4⋊C4×C47C2312yes—
112C2×C22⋊C4×C47—21no—
113C2×C42×C47—21no—
114C24×C47C2×D411yescharacteristic
115C23×D47—42no—
116C23×D47—42no—
117C23×D47C22⋊C414yes—
118C23×D47C22⋊C414yes—
119C23×D47C4⋊C414yes—
120C23×D47C2×D411yescharacteristic
121C23×D47C2×D411yescharacteristic
122C22×C4×C47—24no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—44no—
126C22×C4×C47C22×C412yes—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×Q812yes—
133C22×C4×C47C2411yescharacteristic
134C22×Dic47—24no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—44no—
138C22×Dic47C22⋊C414yes—
139C22×Dic47C22⋊C414yes—
140C22×Dic47C4⋊C414yes—
141C22×Dic47C2×D411yescharacteristic
142C22×Dic47C2×D411yescharacteristic
143C2×C4×D47—28no—
144C2×C4×D47—44no—
145C2×C4×D47—44no—
146C2×C4×D47C42116yes—
147C2×C4×D47C22⋊C414yes—
148C2×C4×D47C22⋊C414yes—
149C2×C4×D47C22⋊C414yes—
150C2×C4×D47C22⋊C414yes—
151C2×C4×D47C22⋊C4116yes—
152C2×C4×D47C4⋊C414yes—
153C2×C4×D47C4⋊C414yes—
154C2×C4×D47C4⋊C4116yes—
155C2×C4×D47C4⋊C4116yes—
156C2×C4×D47C22×C412yes—
157C2×C4×D47C22×C412yes—
158C2×C4×D47C22×C412yes—
159C2×C4×D47C22×C412yes—
160C2×C4×D47C22×C412yes—
161C2×C4×D47C22×C412yes—
162C2×C4×D47C2×D412yes—
163C2×C4×D47C2×D412yes—
164C2×C4×D47C2×D412yes—
165C2×C4×D47C2×D412yes—
166C2×C4×D47C2×Q812yes—
167C2×C4×D47C2×Q812yes—
168M16×C47—28no—
169C2×C8×C47—22no—
170C2×C8×C47—22no—
171C2×C8×C47C22×C414yes—
172C4⋊C4×C47—24no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47C22×C414yes—
176C4⋊C4×C47C22×C414yes—
177C22⋊C4×C47—24no—
178C42×C47—24no—
179C22⋊Dic47—24no—
180(C2×D47)⋊C4—28no—
181(C2×D47)⋊C4—28no—
182(C2×D47)⋊C4—28no—
183C4⋊Dic47—24no—
184C4⋊Dic47—22no—
185C4⋊Dic47—22no—
186C4⋊Dic47C22×C414yes—
187C4⋊Dic47C22×C414yes—
188Dic47⋊C4—28no—
189C4×Dic47—24no—
190C4×Dic47—28no—
191C4×Dic47—28no—
192C4×Dic47—28no—
193#10625329—28no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8C22×C414yes—
197#10625324—216no—
198C8×D47—216no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47C2×C22⋊C412yes—
202C23×C47C2×C22⋊C412yes—
203C23×C47C2×C4⋊C412yes—
204C23×C47C22×D411yescharacteristic
205C22×D47—22no—
206C22×D47—22no—
207C22×D47—22no—
208C22×D47—22no—
209C22×D47—48no—
210C22×D47—44no—
211C22×D47—44no—
212C22×D47—48no—
213C22×D47#95116yes—
214C22×D47C2×C22⋊C412yes—
215C22×D47C2×C22⋊C412yes—
216C22×D47C2×C22⋊C412yes—
217C22×D47C2×C22⋊C412yes—
218C22×D47C2×C4⋊C412yes—
219C22×D47C2×C4⋊C412yes—
220C2×C4×C47—24no—
221C2×C4×C47—24no—
222C2×C4×C47—48no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—44no—
226C2×C4×C47C2×C4214yes—
227C2×C4×C47C2×C22⋊C414yes—
228C2×C4×C47C2×C22⋊C412yes—
229C2×C4×C47C2×C22⋊C412yes—
230C2×C4×C47C2×C4⋊C414yes—
231C2×C4×C47C2×C4⋊C414yes—
232C2×C4×C47C2×C4⋊C412yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C23×C411yescharacteristic
235C2×C4×C47C23×C411yescharacteristic
236C2×C4×C47C22×D411yescharacteristic
237C2×C4×C47C22×D411yescharacteristic
238C2×C4×C47C22×Q811yescharacteristic
239C2×Dic47—24no—
240C2×Dic47—24no—
241C2×Dic47—48no—
242C2×Dic47—42no—
243C2×Dic47—42no—
244C2×Dic47—44no—
245C2×Dic47#95116yes—
246C2×Dic47C2×C22⋊C412yes—
247C2×Dic47C2×C22⋊C412yes—
248C2×Dic47C2×C22⋊C412yes—
249C2×Dic47C2×C22⋊C412yes—
250C2×Dic47C2×C4⋊C412yes—
251C2×Dic47C2×C4⋊C412yes—
252C4×D47—416no—
253C4×D47#95116yes—
254C4×D47#95116yes—
255C4×D47C2×C4214yes—
256C4×D47C2×C4214yes—
257C4×D47C2×C22⋊C414yes—
258C4×D47C2×C22⋊C414yes—
259C4×D47C2×C4⋊C414yes—
260C4×D47C2×C4⋊C414yes—
261C4×D47C2×C4⋊C414yes—
262C4×D47C2×C4⋊C414yes—
263C8×C47—24no—
264C47⋊C8—24no—
265#20347—471nomaximal, Sylow
266C22×C47—22no—
267C22×C47—22no—
268C22×C47—24no—
269C22×C47—24no—
270C22×C47C2×#9514yes—
271C22×C47C22×C22⋊C411yescharacteristic
272C22×C47C22×C22⋊C411yescharacteristic
273C22×C47C22×C4⋊C411yescharacteristic
274C22×C47#57512yes—
275C22×C47#57512yes—
276C2×D47—24no—
277C2×D47—24no—
278C2×D47—22no—
279C2×D47—22no—
280C2×D47—22no—
281C2×D47—22no—
282C2×D47—24no—
283C2×D47—24no—
284C2×D47—416no—
285C2×D47C2×#9514yes—
286C2×D47C2×#9514yes—
287C4×C47—44no—
288C4×C47—44no—
289C4×C47C2×#9514yes—
290C4×C47C22×C4211yescharacteristic
291C4×C47C22×C22⋊C411yescharacteristic
292C4×C47C22×C4⋊C411yescharacteristic
293C4×C47C22×C4⋊C411yescharacteristic
294Dic47—44no—
295Dic47—44no—
296Dic47C2×#9514yes—
297Dic47C2×#9514yes—
298C23×M16—471no—
299C22×#117—471no—
300C22×#117—471no—
301#1736—474no—
302#1736—471no—
303#1736—471no—
304#1736—471no—
305#1736D4711yescharacteristic, complemented
306#1735—474no—
307C2×C47—24no—
308C2×C47C22×#9511yescharacteristic
309C2×C47#289612yes—
310C2×C47#289611yescharacteristic
311C2×C47#289611yescharacteristic
312C2×C47#291112yes—
313D47—24no—
314D47—24no—
315C24×C4—471no—
316C22×M16—471no—
317C22×M16—478no—
318C22×M16—472no—
319C22×M16—944no—
320C22×M16—944no—
321C22×M16C2×D4711yescharacteristic
322C23×C8—472no—
323C2×#117—471no—
324C2×#117—478no—
325C2×#117—478no—
326C2×#117—472no—
327C2×#117—472no—
328C2×#117—472no—
329C2×#117—941no—
330C2×#117—941no—
331C2×#117—941no—
332C2×#117—944no—
333C2×#117—941no—
334C2×#117—944no—
335C2×#117C2×D4711yescharacteristic
336C22×C4⋊C4—471no—
337C22×C4⋊C4—472no—
338C22×C4⋊C4C2×D4711yescharacteristic
339C22×C22⋊C4—941no—
340C22×C22⋊C4—941no—
341C22×C42—941no—
342C22×C42—941no—
343#340—474no—
344#340—474no—
345#340—474no—
346#340C2×D4714yescomplemented
347C47#2034711yescharacteristic, complemented, Sylow
348C25—471no—
349C23×C4—21no—
350C23×C4—471no—
351C23×C4—474no—
352C23×C4—474no—
353C23×C4—474no—
354C23×C4—471no—
355C23×C4—474no—
356C23×C4—474no—
357C23×C4—471no—
358C23×C4—474no—
359C23×C4—471no—
360C23×C4—471no—
361C23×C4—942no—
362C23×C4—941no—
363C23×C4C22×D4711yescharacteristic
364C2×M16—22no—
365C2×M16—22no—
366C2×M16—474no—
367C2×M16—478no—
368C2×M16—944no—
369C2×M16—944no—
370C2×M16—9432no—
371C2×M16—942no—
372C2×M16—942no—
373C2×M16C4×D4714yescomplemented
374C22×C8—472no—
375C22×C8—478no—
376C22×C8—944no—
377C22×C8—944no—
378C22×C8C22×D4712yes—
379#117—22no—
380#117—474no—
381#117—478no—
382#117—478no—
383#117—478no—
384#117—948no—
385#117—948no—
386#117—948no—
387#117—948no—
388#117—944no—
389#117—944no—
390#117—944no—
391#117—942no—
392#117C4×D4714yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—474no—
396C2×C4⋊C4—472no—
397C2×C4⋊C4—472no—
398C2×C4⋊C4—478no—
399C2×C4⋊C4—944no—
400C2×C4⋊C4—942no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—942no—
403C2×C4⋊C4C4×D4714yescomplemented
404C2×C4⋊C4C22×D4712yes—
405C2×C4⋊C4C22×D4712yes—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—948no—
408C2×C22⋊C4—948no—
409C2×C22⋊C4—942no—
410C2×C22⋊C4—942no—
411C2×C22⋊C4—942no—
412C2×C22⋊C4—941no—
413C2×C42—21no—
414C2×C42—948no—
415C2×C42—948no—
416C2×C42—942no—
417C2×C42—942no—
418C2×C42—942no—
419C2×C42—941no—
420C24—471no—
421C24—471no—
422C24—474no—
423C24—474no—
424C24—474no—
425C24—1882no—
426C24—1882no—
427C24D4×D4711yescharacteristic
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—474no—
433C22×C4—474no—
434C22×C4—474no—
435C22×C4—471no—
436C22×C4—471no—
437C22×C4—4716no—
438C22×C4—474no—
439C22×C4—474no—
440C22×C4—474no—
441C22×C4—474no—
442C22×C4—4716no—
443C22×C4—4716no—
444C22×C4—4716no—
445C22×C4—474no—
446C22×C4—474no—
447C22×C4—472no—
448C22×C4—472no—
449C22×C4—472no—
450C22×C4—472no—
451C22×C4—472no—
452C22×C4—472no—
453C22×C4—472no—
454C22×C4—472no—
455C22×C4—472no—
456C22×C4—472no—
457C22×C4—472no—
458C22×C4—472no—
459C22×C4—948no—
460C22×C4—944no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—1884no—
464C22×C4—1884no—
465C22×C4—1884no—
466C22×C4C2×C4×D4712yes—
467C22×C4C2×C4×D4712yes—
468C22×C4C2×C4×D4712yes—
469C22×C4D4×D4711yescharacteristic
470C22×C4D4×D4712yes—
471C22×C4D4×D4712yes—
472C22×C4Q8×D4712yes—
473C22×C4C23×D4711yescharacteristic
474M16—28no—
475M16—9416no—
476M16—948no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—474no—
480C2×C8—9416no—
481C2×C8—942no—
482C2×C8—942no—
483C2×C8C2×C4×D4714yes—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—474no—
488C4⋊C4—474no—
489C4⋊C4—948no—
490C4⋊C4—944no—
491C4⋊C4—942no—
492C4⋊C4—942no—
493C4⋊C4C2×C4×D4714yes—
494C4⋊C4C2×C4×D4714yes—
495C22⋊C4—24no—
496C22⋊C4—948no—
497C22⋊C4—948no—
498C22⋊C4—948no—
499C22⋊C4—944no—
500C42—24no—
501C42—948no—
502C42—948no—
503C42—948no—
504C42—944no—
505C23—22no—
506C23—22no—
507C23—472no—
508C23—472no—
509C23—472no—
510C23—472no—
511C23—472no—
512C23—472no—
513C23—4716no—
514C23—942no—
515C23—942no—
516C23—942no—
517C23—942no—
518C23—1888no—
519C23—1884no—
520C23—1888no—
521C23—1884no—
522C23C22⋊C4×D4712yes—
523C23C22⋊C4×D4712yes—
524C23C4⋊C4×D4712yes—
525C23C2×D4×D4711yescharacteristic
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—4716no—
533C2×C4—472no—
534C2×C4—472no—
535C2×C4—472no—
536C2×C4—472no—
537C2×C4—472no—
538C2×C4—472no—
539C2×C4—4716no—
540C2×C4—4716no—
541C2×C4—474no—
542C2×C4—474no—
543C2×C4—474no—
544C2×C4—474no—
545C2×C4—474no—
546C2×C4—474no—
547C2×C4—474no—
548C2×C4—474no—
549C2×C4—944no—
550C2×C4—944no—
551C2×C4—18816no—
552C2×C4—1888no—
553C2×C4—1882no—
554C2×C4—1882no—
555C2×C4—1884no—
556C2×C4C42×D4714yes—
557C2×C4C22⋊C4×D4712yes—
558C2×C4C22⋊C4×D4712yes—
559C2×C4C22⋊C4×D4714yes—
560C2×C4C4⋊C4×D4712yes—
561C2×C4C4⋊C4×D4714yes—
562C2×C4C4⋊C4×D4714yes—
563C2×C4C22×C4×D4711yescharacteristic
564C2×C4C22×C4×D4711yescharacteristic
565C2×C4C22×C4×D4711yescharacteristic
566C2×C4C2×D4×D4711yescharacteristic
567C2×C4C2×D4×D4711yescharacteristic
568C2×C4C2×Q8×D4711yescharacteristic
569C8—24no—
570C8—944no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—474no—
576C22—474no—
577C22—944no—
578C22—944no—
579C22—942no—
580C22—942no—
581C22—944no—
582C22—944no—
583C22—942no—
584C22—942no—
585C22—18816no—
586C22#95×D4714yes—
587C22C2×C22⋊C4×D4711yescharacteristic
588C22C2×C22⋊C4×D4711yescharacteristic
589C22C2×C4⋊C4×D4711yescharacteristic
590C22#1568991912yes—
591C22#1568992212yes—
592C4—44no—
593C4—44no—
594C4—474no—
595C4—474no—
596C4—1884no—
597C4—1884no—
598C4#95×D4714yes—
599C4C2×C42×D4711yescharacteristic
600C4C2×C22⋊C4×D4711yescharacteristic
601C4C2×C4⋊C4×D4711yescharacteristic
602C4C2×C4⋊C4×D4711yescharacteristic
603C2—24no—
604C2—944no—
605C2—944no—
606C2C2×#95×D4711yescharacteristic
607C2#2025190211yescharacteristic
608C2#2025190412yes—
609C2#2025190611yescharacteristic
610C2#2025197512yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC24×D47D4
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20347
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #20243109C2semidirect
5. #20243108C2semidirect
6. #20243107C2semidirect
7. #20243106C2semidirect
8. #20243103C2semidirect
9. C47×#1736C2semidirect
15. C2×M16×D47C4semidirect
21. C2×#13093982C4semidirect
28. #117×D47C4semidirect
35. #15688886C22semidirect
36. #15688885C22semidirect
37. #15688885C22semidirect
41. C47×#340C22semidirect
58. C2×#10625329C2×C4semidirect
61. C2×#10625324C2×C4semidirect
66. C2×C8×D47C2×C4semidirect
75. #13093982C2×C4semidirect
76. #13093982C2×C4semidirect
81. C2×C4⋊Dic47C2×C4semidirect
86. C4⋊C4×D47C2×C4semidirect
103. C2×M16×C47C2×C4semidirect
109. C2×C4⋊C4×C47C2×C4semidirect
305. #1736D47semidirect
346. #340C2×D47semidirect
347. C47#20347semidirect
373. C2×M16C4×D47semidirect
403. C2×C4⋊C4C4×D47semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D472
C22×#130939822
C2×#117×D472
#202431092
#202431082
#202431072
#202431062
#202431032
C47×#17362
#2034747