Finite Group Databasebeta

← #26243780

Subgroups — #26243780

Order 12032.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094003C211yesmaximal, characteristic
2C2×#15689165C211yesmaximal, characteristic, complemented
3C2×#15689133C211yesmaximal, characteristic, complemented
4C2×#15689133C211yesmaximal, characteristic, complemented
5#20245771C211yesmaximal, characteristic
6#20244592C214yesmaximal, complemented
7#20244585C214yesmaximal, complemented
8#20243108C211yesmaximal, characteristic, complemented
9C2×C47×#407C211yesmaximal, characteristic
10C23×Dic94C2211yescharacteristic
11C2×#13094003—24no—
12C2×#13094003—24no—
13C2×#13094003—22no—
14C2×#13094003C2214yes—
15C2×#13093983C2211yescharacteristic
16C2×#13093983C2211yescharacteristic
17C22×C4⋊Dic47C2211yescharacteristic
18C22×Dic47⋊C4—21no—
19C22×C22⋊Dic47—21no—
20#15689165—24no—
21C2×#13093958C2212yescomplemented
22C2×#13093956C2212yescomplemented
23#15689133—24no—
24#15689133—24no—
25C2×#13093949C2212yescomplemented
26C2×#13093949C2214yescomplemented
27C2×#13093949C2212yescomplemented
28#15688986—21no—
29#15688983C2212yes—
30#15688983C2212yes—
31#15688979—21no—
32C4×C22⋊Dic47—22no—
33#15688886C2214yescomplemented
34C24×C4×C47C2211yescharacteristic
35C22×M16×C47C2211yescharacteristic
36C22×M16×C47C2211yescharacteristic
37C47×#407—24no—
38C2×C22⋊C8×C47C2214yes—
39C22×Dic94—22no—
40C22×Dic94—22no—
41C22×Dic94—22no—
42C22×Dic94—22no—
43C22×Dic94—21no—
44C22×Dic94—22no—
45C22×Dic94D414yes—
46C22×Dic94C2312yes—
47C23×Dic47—21no—
48C23×Dic47—21no—
49#13094003—216no—
50#13094003—28no—
51#13094003—28no—
52#13093983—22no—
53#13093983—22no—
54#13093983—22no—
55#13093983—22no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—22no—
58C2×C4⋊Dic47D414yes—
59C2×C4⋊Dic47C2312yes—
60C2×C4⋊Dic47C2311yescharacteristic
61C2×C4⋊Dic47C2311yescharacteristic
62C2×C4⋊Dic47C2312yes—
63C2×Dic47⋊C4—28no—
64C2×Dic47⋊C4—24no—
65C2×C22⋊Dic47—28no—
66C2×C22⋊Dic47—24no—
67C2×C22⋊Dic47—22no—
68C2×C22⋊Dic47—22no—
69C2×C22⋊Dic47—21no—
70C2×C22⋊Dic47—21no—
71C2×C4×Dic47—21no—
72C2×C4×Dic47—21no—
73#13093958—24no—
74#13093956—24no—
75#13093949—24no—
76#13093949—24no—
77#13093949—28no—
78C47⋊#95—22no—
79C47⋊#95—22no—
80C25×C47D411yescharacteristic
81C23×C4×C47—22no—
82C23×C4×C47—22no—
83C23×C4×C47—22no—
84C23×C4×C47—22no—
85C23×C4×C47—22no—
86C23×C4×C47—22no—
87C23×C4×C47D412yes—
88C23×C4×C47D411yescharacteristic
89C23×C4×C47C2312yes—
90C23×C4×C47C2311yescharacteristic
91C2×M16×C47—22no—
92C2×M16×C47—22no—
93C2×M16×C47—22no—
94C2×M16×C47—22no—
95C2×M16×C47—22no—
96C2×M16×C47D414yes—
97C22×C8×C47C2312yes—
98C22×C8×C47C2312yes—
99C22⋊C8×C47—28no—
100C24×C47—22no—
101C24×C47—24no—
102C24×C47—22no—
103C24×C47—22no—
104C24×C47—42no—
105C24×C47C2×D412yes—
106C24×C47C2×D411yescharacteristic
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—24no—
111C22×C4×C47—24no—
112C22×C4×C47—24no—
113C22×C4×C47—22no—
114C22×C4×C47—22no—
115C22×C4×C47—22no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—44no—
120C22×C4×C47—44no—
121C22×C4×C47—44no—
122C22×C4×C47—44no—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D412yes—
126C22×C4×C47C2×D412yes—
127C22×C4×C47C2×D412yes—
128C22×C4×C47#4112yes—
129C22×C4×C47#4112yes—
130C22×C4×C47#4112yes—
131C22×C4×C47C2411yescharacteristic
132C22×Dic47—24no—
133C22×Dic47—24no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—22no—
139C22×Dic47—44no—
140C22×Dic47—42no—
141C22×Dic47—42no—
142C2×Dic94—28no—
143C2×Dic94—28no—
144C2×Dic94—22no—
145C2×Dic94—22no—
146C2×Dic94—22no—
147C2×Dic94—22no—
148C2×Dic94—24no—
149C2×Dic94—22no—
150C2×Dic94—44no—
151C2×Dic94—42no—
152C2×Dic94—42no—
153C2×Dic94—44no—
154C2×Dic94C2×D414yes—
155M16×C47—28no—
156M16×C47—44no—
157C2×C8×C47—22no—
158C2×C8×C47—22no—
159C2×C8×C47—22no—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47C2×D414yes—
163C22⋊Dic47—216no—
164C22⋊Dic47—44no—
165C22⋊Dic47—42no—
166C22⋊Dic47—42no—
167C22⋊Dic47—44no—
168C4⋊Dic47—24no—
169C4⋊Dic47—22no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—22no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—22no—
176C4⋊Dic47C2×D414yes—
177Dic47⋊C4—216no—
178C4×Dic47—42no—
179C4×Dic47—42no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—28no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—22no—
189C23×C47—48no—
190C23×C47—48no—
191C23×C47—44no—
192C23×C47—44no—
193C23×C47#12012yes—
194C23×C47#12112yes—
195C23×C47#12312yes—
196C23×C47#13714yes—
197C23×C47C22×D411yescharacteristic
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—44no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—48no—
207C2×C4×C47#12012yes—
208C2×C4×C47#12114yes—
209C2×C4×C47#12114yes—
210C2×C4×C47#12112yes—
211C2×C4×C47#12314yes—
212C2×C4×C47#12312yes—
213C2×C4×C47#12414yes—
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C22×D411yescharacteristic
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×Dic47—24no—
221C2×Dic47—24no—
222C2×Dic47—48no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—44no—
226C2×Dic47—48no—
227C2×Dic47—41no—
228C2×Dic47—41no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234Dic94—24no—
235Dic94—24no—
236Dic94—42no—
237Dic94—42no—
238Dic94—48no—
239C8×C47—24no—
240C8×C47—42no—
241#23481—471nomaximal, Sylow
242C22×C47—24no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—24no—
246C22×C47—24no—
247C22×C47—24no—
248C22×C47—24no—
249C22×C47—416no—
250C22×C47—42no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47#39414yes—
255C22×C47C2×#12011yescharacteristic
256C22×C47C2×#12111yescharacteristic
257C22×C47C2×#12311yescharacteristic
258C22×C47C2×#13712yes—
259C22×C47#57512yes—
260C4×C47—42no—
261C4×C47#39414yes—
262C4×C47C2×#12111yescharacteristic
263C4×C47C2×#12111yescharacteristic
264C4×C47C2×#12311yescharacteristic
265C4×C47C2×#12411yescharacteristic
266Dic47—44no—
267Dic47—44no—
268Dic47—81no—
269Dic47—81no—
270Dic47—82no—
271C22×C22⋊Q8—471no—
272C2×#428—471no—
273C2×#419—471no—
274C2×#419—471no—
275C2×#407D4711yescharacteristic
276#2280—471no—
277#2036—474no—
278#2035—474no—
279#1736—471no—
280C2×C47—24no—
281C2×C47—44no—
282C2×C47C2×#39411yescharacteristic
283C2×C47#300411yescharacteristic
284C2×C47#300511yescharacteristic
285C2×C47#305612yes—
286C2×C47#309512yes—
287C23×Q8—471no—
288C24×C4C2×D4711yescharacteristic
289C22×M16C2×D4711yescharacteristic
290C22×M16C2×D4711yescharacteristic
291C2×C22⋊Q8—474no—
292C2×C22⋊Q8—944no—
293C2×C22⋊Q8—944no—
294C2×C22⋊Q8—942no—
295C2×#117—471no—
296C2×#117—471no—
297C22×C4⋊C4—471no—
298C22×C4⋊C4—941no—
299C22×C22⋊C4—941no—
300#428—944no—
301C2×#107—472no—
302C2×#106—472no—
303#419—944no—
304#419—944no—
305C2×#103—472no—
306C2×#103—474no—
307C2×#103—472no—
308#407—24no—
309C2×C22⋊C8C2×D4714yes—
310#381—941no—
311#380—472no—
312#380—472no—
313#379—941no—
314C4×C22⋊C4—942no—
315#340—474no—
316C47#2348111yescharacteristic, complemented, Sylow
317C25D18811yescharacteristic
318C22×Q8—474no—
319C22×Q8—472no—
320C22×Q8—941no—
321C22×Q8—942no—
322C22×Q8—942no—
323C22×Q8—942no—
324C22×Q8—942no—
325C22×Q8—942no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—941no—
333C23×C4—941no—
334C23×C4D18811yescharacteristic
335C23×C4D18812yes—
336C23×C4C22×D4712yes—
337C23×C4C22×D4711yescharacteristic
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—22no—
341C2×M16—22no—
342C2×M16—22no—
343C2×M16C22⋊D4714yes—
344C22×C8C22×D4712yes—
345C22×C8C22×D4712yes—
346C22⋊Q8—948no—
347C22⋊Q8—948no—
348C22⋊Q8—9416no—
349#117—942no—
350#117—942no—
351#117—942no—
352#117—942no—
353C2×C4⋊C4—474no—
354C2×C4⋊C4—472no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—471no—
357C2×C4⋊C4—471no—
358C2×C4⋊C4—948no—
359C2×C4⋊C4—941no—
360C2×C4⋊C4—944no—
361C2×C4⋊C4—942no—
362C2×C22⋊C4—948no—
363C2×C22⋊C4—944no—
364C2×C22⋊C4—941no—
365C2×C22⋊C4—941no—
366C2×C22⋊C4—942no—
367C2×C22⋊C4—942no—
368C2×C42—941no—
369C2×C42—941no—
370#107—944no—
371#106—944no—
372#103—944no—
373#103—944no—
374#103—948no—
375C22⋊C8—28no—
376#95—942no—
377#95—942no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24C2×D18812yes—
384C24C2×D18811yescharacteristic
385C2×Q8—474no—
386C2×Q8—944no—
387C2×Q8—942no—
388C2×Q8—948no—
389C2×Q8—948no—
390C2×Q8—942no—
391C2×Q8—942no—
392C2×Q8—942no—
393C2×Q8—942no—
394C2×Q8—1884no—
395C2×Q8—1882no—
396C2×Q8—1882no—
397C2×Q8—1884no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—944no—
415C22×C4—944no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—941no—
419C22×C4—941no—
420C22×C4—942no—
421C22×C4—1884no—
422C22×C4—1882no—
423C22×C4—1882no—
424C22×C4C2×D18812yes—
425C22×C4C2×D18811yescharacteristic
426C22×C4#1062535012yes—
427C22×C4D4×D4712yes—
428C22×C4D4×D4712yes—
429C22×C4#1062535212yes—
430C22×C4#1062535212yes—
431C22×C4C2×C22⋊D4712yes—
432C22×C4C23×D4711yescharacteristic
433M16—28no—
434M16—44no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8C2×C22⋊D4714yes—
441C4⋊C4—474no—
442C4⋊C4—9416no—
443C4⋊C4—944no—
444C4⋊C4—942no—
445C4⋊C4—941no—
446C4⋊C4—941no—
447C4⋊C4—941no—
448C4⋊C4—941no—
449C4⋊C4—942no—
450C4⋊C4—942no—
451C22⋊C4—9416no—
452C22⋊C4—1884no—
453C22⋊C4—1882no—
454C22⋊C4—1882no—
455C22⋊C4—1884no—
456C42—1882no—
457C42—1882no—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—48no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23#1309399412yes—
472C23#1309399812yes—
473C23#1309400812yes—
474C23#1309405714yes—
475C23C22×D18811yescharacteristic
476Q8—944no—
477Q8—944no—
478Q8—1888no—
479Q8—1882no—
480Q8—1882no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—944no—
491C2×C4—944no—
492C2×C4—1881no—
493C2×C4—1881no—
494C2×C4—1882no—
495C2×C4—1882no—
496C2×C4—1882no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4—1888no—
500C2×C4—1888no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1884no—
504C2×C4#1309399412yes—
505C2×C4#1309399514yes—
506C2×C4#1309399614yes—
507C2×C4#1309399812yes—
508C2×C4#1309400614yes—
509C2×C4#1309400812yes—
510C2×C4#1309401014yes—
511C2×C4C2×D4×D4711yescharacteristic
512C2×C4C2×D4×D4711yescharacteristic
513C2×C4C22×C22⋊D4711yescharacteristic
514C2×C4C2×#1062535211yescharacteristic
515C2×C4C2×#1062535211yescharacteristic
516C2×C4C2×#1062535011yescharacteristic
517C8—24no—
518C8—42no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#1568903914yes—
532C22C2×#1309399411yescharacteristic
533C22C2×#1309399811yescharacteristic
534C22C2×#1309400811yescharacteristic
535C22C2×#1309405712yes—
536C22#1568992312yes—
537C4—42no—
538C4—3762no—
539C4—3761no—
540C4—3761no—
541C4—1884no—
542C4—1884no—
543C4#1568903914yes—
544C4C2×#1309399511yescharacteristic
545C4C2×#1309399611yescharacteristic
546C4C2×#1309400611yescharacteristic
547C4C2×#1309401011yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×#1568903911yescharacteristic
551C2#2025262511yescharacteristic
552C2#2025263311yescharacteristic
553C2#2025301812yes—
554C2#2025333712yes—
subgroupG/H
centerC23C22×D188
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#407C2
socleC23×C47C22×D4
prime radicalC25×C47D4
hypercenterC2×#407D47
hypocenterC47#23481
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×#15689165C2semidirect
3. C2×#15689133C2semidirect
4. C2×#15689133C2semidirect
6. #20244592C2semidirect
7. #20244585C2semidirect
8. #20243108C2semidirect
21. C2×#13093958C22semidirect
22. C2×#13093956C22semidirect
25. C2×#13093949C22semidirect
26. C2×#13093949C22semidirect
27. C2×#13093949C22semidirect
33. #15688886C22semidirect
316. C47#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#130940032
C2×#156891652
C2×#156891332
#202457712
#202445922
#202445852
#202431082
C2×C47×#4072
#2348147