Finite Group Databasebeta

← #26243781

Subgroups — #26243781

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×#13094004C211yesmaximal, characteristic
2C2×#15689133C211yesmaximal, characteristic, complemented
3C2×#15689131C211yesmaximal, characteristic, complemented
4C2×#15689086C211yesmaximal, characteristic
5#20245767C211yesmaximal, characteristic
6#20244593C214yesmaximal, complemented
7#20244586C214yesmaximal, complemented
8#20243107C211yesmaximal, characteristic, complemented
9C2×C47×#428C211yesmaximal, characteristic, complemented
10C23×Dic94C2211yescharacteristic
11C23×C4×D47C2211yescharacteristic
12C22×#10625329C2211yescharacteristic
13C2×#13094004—24no—
14C2×#13094004—22no—
15C2×#13094004—24no—
16C2×#13094004C2214yes—
17C2×#13093983C2211yescharacteristic
18C22×C4⋊Dic47—21no—
19C22×(C2×D47)⋊C4—21no—
20#15689133—24no—
21#15689131—24no—
22C2×#13093949C2214yescomplemented
23C2×#13093947C2212yescomplemented
24C2×#13093947C2212yescomplemented
25#15689086—24no—
26C2×(C2×D47)⋊C8C2214yes—
27#15688984—21no—
28#15688980C2212yes—
29#15688980C2212yes—
30#15688978—21no—
31C4×(C2×D47)⋊C4—22no—
32#15688885C2214yescomplemented
33C22×M16×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C22×C4⋊C4×C47C2211yescharacteristic
36C47×#428—24no—
37C2×#107×C47C2212yescomplemented
38C2×#106×C47C2212yescomplemented
39C24×D47D411yescharacteristic
40C22×Dic94—22no—
41C22×Dic94—22no—
42C22×Dic94—22no—
43C22×Dic94—22no—
44C22×Dic94—21no—
45C22×Dic94—22no—
46C22×Dic94D414yes—
47C22×Dic94C2312yes—
48C23×Dic47—21no—
49C23×Dic47D411yescharacteristic
50C22×C4×D47—22no—
51C22×C4×D47—22no—
52C22×C4×D47—22no—
53C22×C4×D47—22no—
54C22×C4×D47—22no—
55C22×C4×D47—22no—
56C22×C4×D47D412yes—
57C22×C4×D47C2312yes—
58C2×#10625329—22no—
59C2×#10625329—22no—
60C2×#10625329—22no—
61C22×C47⋊C8C2312yes—
62#13094004—216no—
63#13094004—28no—
64#13094004—28no—
65#13093983—22no—
66#13093983—22no—
67C2×C4⋊Dic47—28no—
68C2×C4⋊Dic47—24no—
69C2×C4⋊Dic47C2312yes—
70C2×C22⋊Dic47—21no—
71C2×(C2×D47)⋊C4—28no—
72C2×(C2×D47)⋊C4—24no—
73C2×(C2×D47)⋊C4—22no—
74C2×(C2×D47)⋊C4—22no—
75C2×C4×Dic47—21no—
76#13093949—28no—
77#13093947—24no—
78#13093947—24no—
79(C2×D47)⋊C8—28no—
80C47⋊#95—22no—
81C47⋊#95—22no—
82C23×C4×C47—21no—
83C23×C4×C47C2311yescharacteristic
84C2×M16×C47—22no—
85C2×M16×C47—22no—
86C2×M16×C47D414yes—
87C22×C8×C47C2312yes—
88#117×C47—22no—
89#117×C47—22no—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47—22no—
92C2×C4⋊C4×C47D414yes—
93C2×C4⋊C4×C47C2311yescharacteristic
94C2×C4⋊C4×C47C2311yescharacteristic
95C2×C4⋊C4×C47C2312yes—
96C2×C22⋊C4×C47—21no—
97C2×C42×C47—21no—
98#107×C47—24no—
99#106×C47—24no—
100C24×C47C2×D411yescharacteristic
101C23×D47—22no—
102C23×D47—24no—
103C23×D47—22no—
104C23×D47—22no—
105C23×D47—42no—
106C23×D47C2×D412yes—
107C22×C4×C47—24no—
108C22×C4×C47—22no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—44no—
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2×D412yes—
116C22×C4×C47#4112yes—
117C22×C4×C47#4112yes—
118C22×C4×C47#4112yes—
119C22×C4×C47C2411yescharacteristic
120C22×Dic47—24no—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—42no—
127C22×Dic47—42no—
128C22×Dic47C2×D412yes—
129C2×C4×D47—24no—
130C2×C4×D47—24no—
131C2×C4×D47—24no—
132C2×C4×D47—22no—
133C2×C4×D47—22no—
134C2×C4×D47—22no—
135C2×C4×D47—22no—
136C2×C4×D47—22no—
137C2×C4×D47—22no—
138C2×C4×D47—44no—
139C2×C4×D47—44no—
140C2×C4×D47—44no—
141C2×C4×D47—44no—
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—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47C2×D414yes—
159C4⋊C4×C47—24no—
160C4⋊C4×C47—22no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47C2×D414yes—
166C22⋊C4×C47—42no—
167C42×C47—42no—
168C22⋊Dic47—42no—
169(C2×D47)⋊C4—216no—
170(C2×D47)⋊C4—44no—
171(C2×D47)⋊C4—44no—
172C4⋊Dic47—216no—
173C4⋊Dic47—22no—
174C4⋊Dic47—22no—
175C4×Dic47—42no—
176#10625329—44no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47#12012yes—
183C23×C47#12112yes—
184C23×C47#12312yes—
185C23×C47C22×D411yescharacteristic
186C22×D47—28no—
187C22×D47—22no—
188C22×D47—22no—
189C22×D47—22no—
190C22×D47—22no—
191C22×D47—22no—
192C22×D47—22no—
193C22×D47—48no—
194C22×D47—48no—
195C22×D47—44no—
196C22×D47—44no—
197C22×D47#13714yes—
198C2×C4×C47—24no—
199C2×C4×C47—48no—
200C2×C4×C47—44no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47#12012yes—
206C2×C4×C47#12114yes—
207C2×C4×C47#12114yes—
208C2×C4×C47#12112yes—
209C2×C4×C47#12314yes—
210C2×C4×C47#12312yes—
211C2×C4×C47#12414yes—
212C2×C4×C47C22×D411yescharacteristic
213C2×C4×C47C22×D411yescharacteristic
214C2×C4×C47C22×D411yescharacteristic
215C2×C4×C47C2×#4111yescharacteristic
216C2×C4×C47C2×#4111yescharacteristic
217C2×C4×C47C2×#4111yescharacteristic
218C2×Dic47—22no—
219C2×Dic47—24no—
220C2×Dic47—22no—
221C2×Dic47—22no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—48no—
225C2×Dic47—44no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C4×D47—42no—
230C4×D47—42no—
231C4×D47—42no—
232C4×D47—42no—
233C4×D47—48no—
234Dic94—24no—
235Dic94—24no—
236Dic94—42no—
237Dic94—42no—
238Dic94—48no—
239C8×C47—24no—
240C47⋊C8—42no—
241#23481—471nomaximal, Sylow
242C22×C47—24no—
243C22×C47—22no—
244C22×C47—22no—
245C22×C47—24no—
246C22×C47#39414yes—
247C22×C47C2×#12011yescharacteristic
248C22×C47C2×#12111yescharacteristic
249C22×C47C2×#12311yescharacteristic
250C22×C47C2×#13712yes—
251C22×C47#57512yes—
252C2×D47—24no—
253C2×D47—24no—
254C2×D47—24no—
255C2×D47—416no—
256C2×D47—42no—
257C2×D47—42no—
258C2×D47—42no—
259C2×D47—42no—
260C4×C47—44no—
261C4×C47—81no—
262C4×C47—81no—
263C4×C47#39414yes—
264C4×C47C2×#12111yescharacteristic
265C4×C47C2×#12111yescharacteristic
266C4×C47C2×#12311yescharacteristic
267C4×C47C2×#12411yescharacteristic
268Dic47—44no—
269Dic47—42no—
270Dic47—82no—
271C22×C22⋊Q8—471no—
272C2×#428D4711yescharacteristic, complemented
273C2×#419—471no—
274C2×#419—471no—
275C2×#407—471no—
276#2280—471no—
277#2036—474no—
278#2035—474no—
279#1736—471no—
280C2×C47—24no—
281C2×C47C2×#39411yescharacteristic
282C2×C47#300411yescharacteristic
283C2×C47#300511yescharacteristic
284C2×C47#305612yes—
285C2×C47#309512yes—
286D47—44no—
287C23×Q8—471no—
288C24×C4—471no—
289C22×M16—471no—
290C22×M16C2×D4711yescharacteristic
291C2×C22⋊Q8—474no—
292C2×C22⋊Q8—942no—
293C2×C22⋊Q8—944no—
294C2×C22⋊Q8—944no—
295C2×#117—471no—
296C2×#117C2×D4711yescharacteristic
297C22×C4⋊C4—941no—
298C22×C4⋊C4C2×D4711yescharacteristic
299C22×C22⋊C4—941no—
300#428—24no—
301C2×#107C2×D4712yescomplemented
302C2×#106C2×D4712yescomplemented
303#419—944no—
304#419—944no—
305C2×#103—474no—
306C2×#103—472no—
307C2×#103—472no—
308#407—944no—
309C2×C22⋊C8—474no—
310#381—941no—
311#380—472no—
312#380—472no—
313#379—941no—
314C4×C22⋊C4—942no—
315#340—474no—
316C47#2348111yescharacteristic, complemented, Sylow
317C25—471no—
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—21no—
327C23×C4—471no—
328C23×C4—472no—
329C23×C4—472no—
330C23×C4—941no—
331C23×C4—942no—
332C23×C4—942no—
333C23×C4—942no—
334C23×C4—942no—
335C23×C4—942no—
336C23×C4—942no—
337C23×C4C22×D4711yescharacteristic
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—942no—
341C2×M16—942no—
342C2×M16—942no—
343C2×M16D18814yes—
344C22×C8—472no—
345C22×C8C22×D4712yes—
346C22⋊Q8—948no—
347C22⋊Q8—9416no—
348C22⋊Q8—948no—
349#117—22no—
350#117—22no—
351#117—942no—
352#117—942no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—944no—
357C2×C4⋊C4—948no—
358C2×C4⋊C4C22⋊D4714yes—
359C2×C4⋊C4C22×D4711yescharacteristic
360C2×C4⋊C4C22×D4711yescharacteristic
361C2×C4⋊C4C22×D4712yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—944no—
364C2×C22⋊C4—948no—
365C2×C22⋊C4—942no—
366C2×C22⋊C4—942no—
367C2×C22⋊C4—941no—
368C2×C42—21no—
369C2×C42—941no—
370#107—24no—
371#106—24no—
372#103—944no—
373#103—944no—
374#103—948no—
375C22⋊C8—948no—
376#95—942no—
377#95—942no—
378C24—472no—
379C24—944no—
380C24—942no—
381C24—942no—
382C24—942no—
383C24—1882no—
384C24D4×D4711yescharacteristic
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—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—44no—
403C22×C4—472no—
404C22×C4—944no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—944no—
410C22×C4—944no—
411C22×C4—944no—
412C22×C4—942no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—1882no—
420C22×C4—1882no—
421C22×C4—1884no—
422C22×C4—1884no—
423C22×C4—1884no—
424C22×C4—1884no—
425C22×C4C2×D18812yes—
426C22×C4#1062535012yes—
427C22×C4D4×D4711yescharacteristic
428C22×C4D4×D4712yes—
429C22×C4#1062535412yes—
430C22×C4#1062535412yes—
431C22×C4C2×C22⋊D4712yes—
432C22×C4C23×D4711yescharacteristic
433M16—28no—
434M16—1884no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—942no—
438C2×C8—942no—
439C2×C8—942no—
440C2×C8C2×D18814yes—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—9416no—
448C4⋊C4—942no—
449C4⋊C4—942no—
450C4⋊C4C2×C22⋊D4714yes—
451C22⋊C4—42no—
452C22⋊C4—9416no—
453C22⋊C4—1884no—
454C22⋊C4—1884no—
455C22⋊C4—1882no—
456C42—42no—
457C42—1882no—
458C23—22no—
459C23—22no—
460C23—474no—
461C23—948no—
462C23—942no—
463C23—942no—
464C23—942no—
465C23—942no—
466C23—942no—
467C23—942no—
468C23—1888no—
469C23—1888no—
470C23—1884no—
471C23—1884no—
472C23#1309399212yes—
473C23#1309399912yes—
474C23#1309400912yes—
475C23C2×D4×D4711yescharacteristic
476Q8—944no—
477Q8—944no—
478Q8—1888no—
479Q8—1882no—
480Q8—1882no—
481C2×C4—24no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—944no—
489C2×C4—942no—
490C2×C4—942no—
491C2×C4—942no—
492C2×C4—1882no—
493C2×C4—1882no—
494C2×C4—1882no—
495C2×C4—1884no—
496C2×C4—1888no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4—1888no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
503C2×C4—1882no—
504C2×C4#1309399212yes—
505C2×C4#1309399814yes—
506C2×C4#1309399912yes—
507C2×C4#1309399914yes—
508C2×C4#1309400912yes—
509C2×C4#1309400914yes—
510C2×C4#1309401314yes—
511C2×C4C2×D4×D4711yescharacteristic
512C2×C4C22×C22⋊D4711yescharacteristic
513C2×C4C22×D18811yescharacteristic
514C2×C4C2×#1062535011yescharacteristic
515C2×C4C2×#1062535411yescharacteristic
516C2×C4C2×#1062535411yescharacteristic
517C8—24no—
518C8—1882no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—944no—
524C22—944no—
525C22—944no—
526C22—18816no—
527C22—1882no—
528C22—1882no—
529C22—1882no—
530C22—1882no—
531C22#1568904014yes—
532C22C2×#1309399211yescharacteristic
533C22C2×#1309399911yescharacteristic
534C22C2×#1309400911yescharacteristic
535C22#137×D4712yes—
536C22#1568991912yes—
537C4—44no—
538C4—81no—
539C4—81no—
540C4—3762no—
541C4—1882no—
542C4—1884no—
543C4#1568904014yes—
544C4C2×#1309399811yescharacteristic
545C4C2×#1309399911yescharacteristic
546C4C2×#1309400911yescharacteristic
547C4C2×#1309401311yescharacteristic
548C2—24no—
549C2—1884no—
550C2C2×#1568904011yescharacteristic
551C2#2025262011yescharacteristic
552C2#2025262711yescharacteristic
553C2#2025301912yes—
554C2#2025333812yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#428C2
socleC23×C47C22×D4
prime radicalC24×D47D4
hypercenterC2×#428D47
hypocenterC47#23481
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#15689133C2semidirect
3. C2×#15689131C2semidirect
6. #20244593C2semidirect
7. #20244586C2semidirect
8. #20243107C2semidirect
9. C2×C47×#428C2semidirect
22. C2×#13093949C22semidirect
23. C2×#13093947C22semidirect
24. C2×#13093947C22semidirect
32. #15688885C22semidirect
37. C2×#107×C47C22semidirect
38. C2×#106×C47C22semidirect
272. C2×#428D47semidirect
301. C2×#107C2×D47semidirect
302. C2×#106C2×D47semidirect
316. C47#23481semidirect

Maximal subgroups

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

subgroupindex
C22×#130940042
C2×#156891332
C2×#156891312
C2×#156890862
#202457672
#202445932
#202445862
#202431072
C2×C47×#4282
#2348147