Finite Group Databasebeta

← #26213691

Subgroups — #26213691

Order 12032.

2158 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094032C211yesmaximal, characteristic
2C2×#15689604C211yesmaximal, characteristic
3C2×#15689603C211yesmaximal, characteristic
4#20243108C211yesmaximal, characteristic, complemented
5#20243106C212yesmaximal, complemented
6#20243106C212yesmaximal, complemented
7#20243104C211yesmaximal, characteristic, complemented
8#20243103C212yesmaximal, complemented
9C2×#15688885C211yesmaximal, characteristic, complemented
10#340×D47C212yesmaximal, complemented
11C47×#1736C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13C2×#13094032—24no—
14C2×#13094032—24no—
15C2×#13094032C418yescomplemented
16C22×#10625324C2212yes—
17C23×C47⋊C8C2211yescharacteristic
18C22×C8×D47C2212yes—
19#15689604—24no—
20#15689604C418yescomplemented
21#15689603—24no—
22#15689603C418yescomplemented
23C2×C4×Dic94—21no—
24C2×#13093989—21no—
25C2×C4×D188—21no—
26C2×#13093986—21no—
27C2×C4×C22⋊D47—22no—
28C2×(C22⋊D47)⋊C4—22no—
29C2×#13093983C2211yescharacteristic
30C2×#13093982C2212yes—
31C2×#13093982C2212yes—
32C22×C4⋊Dic47C2211yescharacteristic
33C2×C4⋊C4×D47C2212yes—
34C2×C4⋊C4×D47C2212yes—
35#15688886C2212yes—
36#15688886C2212yes—
37#15688885C2211yescharacteristic
38#15688885C2212yes—
39#15688885C2212yes—
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43C22×M16×C47C2211yescharacteristic
44C2×#117×C47C2211yescharacteristic
45C22×C4⋊C4×C47C2211yescharacteristic
46C47×#340C2212yes—
47C47×#340C2212yes—
48C2×#10625350C2×C414yes—
49C2×#10625350C2×C414yes—
50C2×#10625350C2×C414yes—
51C2×#10625350D414yes—
52C2×#10625350D414yes—
53C2×#10625350Q814yes—
54C22×Dic94D411yescharacteristic
55C22×D188D411yescharacteristic
56C22×C22⋊D47D412yes—
57C23×Dic47—21no—
58C22×C4×D47—22no—
59C22×C4×D47—22no—
60C22×C4×D47C2312yes—
61#13094032—232no—
62C2×#10625324—24no—
63C2×#10625324—24no—
64C2×#10625324C2×C418yescomplemented
65C22×C47⋊C8—22no—
66C22×C47⋊C8—22no—
67C22×C47⋊C8C2×C414yescomplemented
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C2×C8×D47—24no—
71C2×C8×D47—24no—
72C2×C8×D47C2×C418yescomplemented
73C4×Dic94—28no—
74#13093989—28no—
75C4×D188—28no—
76#13093986—28no—
77C4×C22⋊D47—216no—
78(C22⋊D47)⋊C4—216no—
79#13093983—22no—
80#13093983C2×C414yescomplemented
81#13093982—24no—
82#13093982—24no—
83#13093982C2×C418yescomplemented
84#13093982C2×C418yescomplemented
85C2×C4⋊Dic47—21no—
86C2×C4⋊Dic47—22no—
87C2×C4⋊Dic47C2×C414yescomplemented
88C2×C4⋊Dic47C2311yescharacteristic
89C2×C4⋊Dic47C2312yes—
90C2×C4⋊Dic47C2311yescharacteristic
91C2×Dic47⋊C4—22no—
92C2×Dic47⋊C4—22no—
93C4⋊C4×D47—24no—
94C4⋊C4×D47—24no—
95C4⋊C4×D47C2×C418yescomplemented
96C4⋊C4×D47C2×C418yescomplemented
97C2×C22⋊Dic47—21no—
98C2×(C2×D47)⋊C4—22no—
99C2×(C2×D47)⋊C4—22no—
100C2×C4×Dic47—21no—
101C2×C4×Dic47—22no—
102C2×C4×Dic47—22no—
103C23×C4×C47—21no—
104C23×C4×C47C2311yescharacteristic
105C2×M16×C47—22no—
106C2×M16×C47—22no—
107C2×M16×C47C2×C414yescomplemented
108C22×C8×C47C2312yes—
109#117×C47—22no—
110#117×C47C2×C414yescomplemented
111C2×C4⋊C4×C47—21no—
112C2×C4⋊C4×C47—22no—
113C2×C4⋊C4×C47C2×C414yescomplemented
114C2×C4⋊C4×C47C2312yes—
115C2×C4⋊C4×C47C2311yescharacteristic
116C2×C4⋊C4×C47C2311yescharacteristic
117C2×C22⋊C4×C47—21no—
118C2×C42×C47—21no—
119C24×C47C2×D411yescharacteristic
120C23×D47C2×D412yes—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—24no—
125C22×C4×C47—81no—
126C22×C4×C47—81no—
127C22×C4×C47C22×C412yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D412yes—
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×Q812yes—
134C22×C4×C47C2411yescharacteristic
135C2×C22⋊D47C22⋊C418yes—
136C2×C22⋊D47C22⋊C418yes—
137C2×C22⋊D47C4⋊C418yes—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C22×Dic47—21no—
141C22×Dic47—24no—
142C22×Dic47—81no—
143C22×Dic47—81no—
144C22×Dic47C2×D412yes—
145#10625350C42116yes—
146#10625350C22⋊C4116yes—
147#10625350C4⋊C4116yes—
148#10625350C4⋊C4116yes—
149C2×D188C22⋊C414yes—
150C2×D188C22⋊C414yes—
151C2×D188C4⋊C414yes—
152C2×C4×D47—28no—
153C2×C4×D47—28no—
154C2×C4×D47—82no—
155C2×C4×D47—82no—
156C2×C4×D47—82no—
157C2×C4×D47—82no—
158C2×C4×D47C22×C414yes—
159C2×C4×D47C22×C414yes—
160C2×C4×D47C22×C414yes—
161C2×C4×D47C2×D414yes—
162C2×C4×D47C2×D414yes—
163C2×C4×D47C2×Q814yes—
164C2×Dic94C22⋊C414yes—
165C2×Dic94C22⋊C414yes—
166C2×Dic94C4⋊C414yes—
167M16×C47—28no—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47C22×C414yes—
171C4⋊C4×C47—24no—
172C4⋊C4×C47—22no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47—21no—
175C4⋊C4×C47C22×C414yes—
176C4⋊C4×C47C22×C412yes—
177C4⋊C4×C47C22×C412yes—
178C22⋊C4×C47—24no—
179C42×C47—24no—
180C22⋊Dic47—24no—
181(C2×D47)⋊C4—28no—
182(C2×D47)⋊C4—28no—
183C4⋊Dic47—24no—
184C4⋊Dic47—21no—
185C4⋊Dic47—22no—
186C4⋊Dic47—21no—
187C4⋊Dic47C22×C412yes—
188C4⋊Dic47C22×C414yes—
189C4⋊Dic47C22×C412yes—
190Dic47⋊C4—28no—
191Dic47⋊C4—28no—
192C4×Dic47—24no—
193C4×Dic47—28no—
194C4×Dic47—28no—
195C2×C47⋊C8—28no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8C22×C412yes—
201C2×C47⋊C8C22×C412yes—
202#10625324—216no—
203C8×D47—216no—
204C23×C47—21no—
205C23×C47—21no—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47C2×C22⋊C412yes—
209C23×C47C2×C22⋊C412yes—
210C23×C47C2×C4⋊C412yes—
211C23×C47C22×D411yescharacteristic
212C22×D47—22no—
213C22×D47—22no—
214C22×D47—22no—
215C22×D47—22no—
216C22×D47C2×C22⋊C414yes—
217C22×D47C2×C22⋊C414yes—
218C22×D47C2×C4⋊C414yes—
219C2×C4×C47—22no—
220C2×C4×C47—24no—
221C2×C4×C47—22no—
222C2×C4×C47—41no—
223C2×C4×C47—42no—
224C2×C4×C47—41no—
225C2×C4×C47—42no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—84no—
229C2×C4×C47C2×C4214yes—
230C2×C4×C47C2×C22⋊C414yes—
231C2×C4×C47C2×C22⋊C412yes—
232C2×C4×C47C2×C22⋊C412yes—
233C2×C4×C47C2×C4⋊C414yes—
234C2×C4×C47C2×C4⋊C414yes—
235C2×C4×C47C2×C4⋊C412yes—
236C2×C4×C47C23×C411yescharacteristic
237C2×C4×C47C23×C411yescharacteristic
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C2×C4×C47C22×Q811yescharacteristic
242C22⋊D47#95132yes—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—24no—
246C2×Dic47—41no—
247C2×Dic47—42no—
248C2×Dic47—41no—
249C2×Dic47—42no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—84no—
253C2×Dic47C2×C22⋊C414yes—
254C2×Dic47C2×C22⋊C414yes—
255C2×Dic47C2×C4⋊C414yes—
256D188#95116yes—
257C4×D47—88no—
258C4×D47—88no—
259C4×D47C2×C4218yes—
260C4×D47C2×C22⋊C418yes—
261C4×D47C2×C4⋊C418yes—
262C4×D47C2×C4⋊C418yes—
263Dic94#95116yes—
264C8×C47—24no—
265C47⋊C8—22no—
266C47⋊C8—22no—
267#20348—471nomaximal, Sylow
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—22no—
274C22×C47C2×#9514yes—
275C22×C47C22×C22⋊C411yescharacteristic
276C22×C47C22×C22⋊C411yescharacteristic
277C22×C47C22×C4⋊C411yescharacteristic
278C22×C47#57611yescharacteristic
279C22×C47#57711yescharacteristic
280C22×C47#57711yescharacteristic
281C22×C47#57811yescharacteristic
282C2×D47—24no—
283C2×D47—24no—
284C2×D47—24no—
285C2×D47—24no—
286C2×D47—24no—
287C2×D47—24no—
288C2×D47C2×#9518yes—
289C4×C47—42no—
290C4×C47—44no—
291C4×C47—42no—
292C4×C47C2×#9514yes—
293C4×C47C22×C4211yescharacteristic
294C4×C47C22×C22⋊C411yescharacteristic
295C4×C47C22×C4⋊C411yescharacteristic
296C4×C47C22×C4⋊C411yescharacteristic
297Dic47—42no—
298Dic47—44no—
299Dic47—42no—
300Dic47C2×#9518yes—
301C22×#131—471no—
302C2×#520—471no—
303C2×#520—471no—
304#1736—472no—
305#1736—472no—
306#1736—471no—
307#1736D4711yescharacteristic, complemented
308#1735—472no—
309#1735—471no—
310C2×#340—472no—
311C2×#340—471no—
312C2×C47—24no—
313C2×C47C22×#9511yescharacteristic
314C2×C47#289711yescharacteristic
315C2×C47#289711yescharacteristic
316C2×C47#289811yescharacteristic
317C2×C47#289811yescharacteristic
318C2×C47#291211yescharacteristic
319C2×C47#291311yescharacteristic
320D47—28no—
321C22×#41—471no—
322C2×#131—478no—
323C2×#131—944no—
324C2×#131—944no—
325C22×M16—472no—
326C22×M16C2×D4711yescharacteristic
327C23×C8—471no—
328C23×C8—472no—
329#520—478no—
330#520—478no—
331#520—944no—
332#520—944no—
333C2×C4×Q8—941no—
334C2×C4×Q8—941no—
335C2×C4×D4—941no—
336C2×C4×D4—942no—
337C2×C4×D4—941no—
338C2×C4×D4—942no—
339C2×#117—471no—
340C2×#117—472no—
341C2×#117—472no—
342C2×#117C2×D4711yescharacteristic
343C22×C4⋊C4—471no—
344C22×C4⋊C4—472no—
345C22×C4⋊C4—472no—
346C22×C4⋊C4C2×D4711yescharacteristic
347#340—472no—
348#340—472no—
349#340—471no—
350#340—472no—
351#340—472no—
352#340—471no—
353#340—471no—
354#340—471no—
355#340C2×D4712yes—
356#340C2×D4712yes—
357C47#2034811yescharacteristic, complemented, Sylow
358C2×#41—474no—
359C2×#41—474no—
360C2×#41—474no—
361C2×#41—474no—
362C2×#41—474no—
363C2×#41—474no—
364C22×Q8—471no—
365C22×D4—472no—
366C22×D4—471no—
367C23×C4—21no—
368C23×C4—472no—
369C23×C4—942no—
370C23×C4—942no—
371C23×C4—941no—
372C23×C4C22×D4711yescharacteristic
373#131—9432no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—478no—
377C2×M16—944no—
378C2×M16—944no—
379C2×M16C4×D4714yescomplemented
380C22×C8—474no—
381C22×C8—471no—
382C22×C8—471no—
383C22×C8—478no—
384C22×C8—944no—
385C22×C8—944no—
386C22×C8—942no—
387C22×C8—942no—
388C22×C8C22×D4712yes—
389C4×Q8—948no—
390C4×Q8—948no—
391C4×D4—948no—
392C4×D4—9416no—
393C4×D4—948no—
394C4×D4—9416no—
395#117—22no—
396#117—474no—
397#117—478no—
398#117—478no—
399#117—944no—
400#117—944no—
401#117—942no—
402#117C4×D4714yescomplemented
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—474no—
406C2×C4⋊C4—472no—
407C2×C4⋊C4—471no—
408C2×C4⋊C4—471no—
409C2×C4⋊C4—478no—
410C2×C4⋊C4—478no—
411C2×C4⋊C4—944no—
412C2×C4⋊C4—944no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—942no—
417C2×C4⋊C4C4×D4714yescomplemented
418C2×C4⋊C4C22×D4711yescharacteristic
419C2×C4⋊C4C22×D4712yes—
420C2×C4⋊C4C22×D4711yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—942no—
423C2×C22⋊C4—942no—
424C2×C22⋊C4—941no—
425C2×C42—21no—
426C2×C42—942no—
427C2×C42—942no—
428C2×C42—941no—
429C24—472no—
430C24D4×D4711yescharacteristic
431#41—4716no—
432#41—4716no—
433#41—4716no—
434#41—4716no—
435C2×Q8—474no—
436C2×Q8—474no—
437C2×Q8—474no—
438C2×D4—478no—
439C2×D4—478no—
440C2×D4—478no—
441C2×D4—474no—
442C2×D4—474no—
443C2×D4—474no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—472no—
451C22×C4—474no—
452C22×C4—474no—
453C22×C4—474no—
454C22×C4—474no—
455C22×C4—474no—
456C22×C4—474no—
457C22×C4—948no—
458C22×C4—948no—
459C22×C4—944no—
460C22×C4—941no—
461C22×C4—942no—
462C22×C4—941no—
463C22×C4—3762no—
464C22×C4—3762no—
465C22×C4—3762no—
466C22×C4—3762no—
467C22×C4—3761no—
468C22×C4—3761no—
469C22×C4C2×C4×D4712yes—
470C22×C4C2×C4×D4712yes—
471C22×C4C2×C4×D4712yes—
472C22×C4D4×D4711yescharacteristic
473C22×C4D4×D4712yes—
474C22×C4D4×D4712yes—
475C22×C4Q8×D4712yes—
476C22×C4C23×D4711yescharacteristic
477M16—28no—
478M16—9416no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—472no—
482C2×C8—472no—
483C2×C8—9416no—
484C2×C8—948no—
485C2×C8—941no—
486C2×C8—941no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8C2×C4×D4714yes—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—472no—
495C4⋊C4—474no—
496C4⋊C4—472no—
497C4⋊C4—948no—
498C4⋊C4—948no—
499C4⋊C4—944no—
500C4⋊C4—941no—
501C4⋊C4—942no—
502C4⋊C4—941no—
503C4⋊C4C2×C4×D4712yes—
504C4⋊C4C2×C4×D4714yes—
505C4⋊C4C2×C4×D4712yes—
506C22⋊C4—24no—
507C22⋊C4—948no—
508C22⋊C4—948no—
509C22⋊C4—944no—
510C42—24no—
511C42—948no—
512C42—948no—
513C42—944no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—474no—
519C23—474no—
520C23—474no—
521C23—942no—
522C23—942no—
523C23—942no—
524C23—942no—
525C23C22⋊C4×D4712yes—
526C23C22⋊C4×D4712yes—
527C23C4⋊C4×D4712yes—
528C23C2×D4×D4711yescharacteristic
529Q8—4716no—
530D4—4732no—
531D4—4716no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—474no—
543C2×C4—474no—
544C2×C4—474no—
545C2×C4—478no—
546C2×C4—478no—
547C2×C4—478no—
548C2×C4—478no—
549C2×C4—942no—
550C2×C4—944no—
551C2×C4—942no—
552C2×C4—3768no—
553C2×C4—3768no—
554C2×C4—3764no—
555C2×C4—1882no—
556C2×C4—1881no—
557C2×C4—1881no—
558C2×C4—1882no—
559C2×C4—1881no—
560C2×C4—1881no—
561C2×C4C42×D4714yes—
562C2×C4C22⋊C4×D4712yes—
563C2×C4C22⋊C4×D4712yes—
564C2×C4C22⋊C4×D4714yes—
565C2×C4C4⋊C4×D4712yes—
566C2×C4C4⋊C4×D4714yes—
567C2×C4C4⋊C4×D4714yes—
568C2×C4C22×C4×D4711yescharacteristic
569C2×C4C22×C4×D4711yescharacteristic
570C2×C4C22×C4×D4711yescharacteristic
571C2×C4C2×D4×D4711yescharacteristic
572C2×C4C2×D4×D4711yescharacteristic
573C2×C4C2×Q8×D4711yescharacteristic
574C8—24no—
575C8—942no—
576C8—942no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—478no—
584C22—944no—
585C22—944no—
586C22—944no—
587C22—944no—
588C22—944no—
589C22—944no—
590C22#95×D4714yes—
591C22C2×C22⋊C4×D4711yescharacteristic
592C22C2×C22⋊C4×D4711yescharacteristic
593C22C2×C4⋊C4×D4711yescharacteristic
594C22#1568992711yescharacteristic
595C22#1568993211yescharacteristic
596C22#1568993311yescharacteristic
597C22#1568993911yescharacteristic
598C4—42no—
599C4—44no—
600C4—42no—
601C4—478no—
602C4—1884no—
603C4—1882no—
604C4—1882no—
605C4#95×D4714yes—
606C4C2×C42×D4711yescharacteristic
607C4C2×C22⋊C4×D4711yescharacteristic
608C4C2×C4⋊C4×D4711yescharacteristic
609C4C2×C4⋊C4×D4711yescharacteristic
610C2—24no—
611C2—948no—
612C2C2×#95×D4711yescharacteristic
613C2#2025191211yescharacteristic
614C2#2025191311yescharacteristic
615C2#2025191911yescharacteristic
616C2#2025192011yescharacteristic
617C2#2025198211yescharacteristic
618C2#2025198711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC47×#1736C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
cosocleC22×C4×C47C24
hypercenter#1736D47
hypocenterC47#20348
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. #20243108C2semidirect
5. #20243106C2semidirect
6. #20243106C2semidirect
7. #20243104C2semidirect
8. #20243103C2semidirect
9. C2×#15688885C2semidirect
10. #340×D47C2semidirect
11. C47×#1736C2semidirect
15. C2×#13094032C4semidirect
20. #15689604C4semidirect
22. #15689603C4semidirect
64. C2×#10625324C2×C4semidirect
67. C22×C47⋊C8C2×C4semidirect
72. C2×C8×D47C2×C4semidirect
80. #13093983C2×C4semidirect
83. #13093982C2×C4semidirect
84. #13093982C2×C4semidirect
87. C2×C4⋊Dic47C2×C4semidirect
95. C4⋊C4×D47C2×C4semidirect
96. C4⋊C4×D47C2×C4semidirect
107. C2×M16×C47C2×C4semidirect
110. #117×C47C2×C4semidirect
113. C2×C4⋊C4×C47C2×C4semidirect
307. #1736D47semidirect
357. C47#20348semidirect
379. C2×M16C4×D47semidirect
402. #117C4×D47semidirect
417. 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×#130940322
C2×#156896042
C2×#156896032
#202431082
#202431062
#202431042
#202431032
C2×#156888852
#340×D472
C47×#17362
#2034847