Finite Group Databasebeta

← #26250357

Subgroups — #26250357

Order 12032.

1336 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689415C211yesmaximal, characteristic, complemented
2C2×#15689223C211yesmaximal, characteristic, complemented
3#20248297C211yesmaximal, characteristic, complemented
4#20244567C212yesmaximal, complemented
5#20244391C212yesmaximal, complemented
6#20244362C211yesmaximal, characteristic, complemented
7#20244187C211yesmaximal, characteristic, complemented
8#20243695C212yesmaximal, complemented
9#20243576C212yesmaximal, complemented
10#20243398C211yesmaximal, characteristic, complemented
11C2×C47×#547C211yesmaximal, characteristic
12C22×#10625335C2211yescharacteristic
13C2×#13094003C2212yes—
14C2×C4×Dic94C2211yescharacteristic
15#15689415—24no—
16#15689223—24no—
17C2×#13093957C2211yescharacteristic
18C2×#13093957C2212yes—
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093947C2211yescharacteristic
21C2×#13093947C2212yes—
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2212yes—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C22⋊(C47⋊C8)C2212yes—
26C2×#13093935C2211yescharacteristic
27#15689054—21no—
28#15689037—21no—
29#15689011—22no—
30#15688989C2211yescharacteristic
31#15688983C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C22×#41×C47—21no—
36C47×#547—24no—
37C47×#547—24no—
38C2×C4⋊Q8×C47C2211yescharacteristic
39C2×#123×C47—22no—
40C2×C22⋊Q8×C47—21no—
41C2×#121×C47C2212yes—
42C2×C4×D4×C47C2211yescharacteristic
43C2×C4×D4×C47C2211yescharacteristic
44C22×C4⋊C4×C47C2212yes—
45C22×Dic94C2311yescharacteristic
46C2×#10625335—22no—
47C2×#10625335—22no—
48C2×#10625335—22no—
49C22×C47⋊C8C2311yescharacteristic
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2311yescharacteristic
52#13094003—24no—
53#13094003—42no—
54C4×Dic94—22no—
55C4×Dic94—22no—
56C2×C4⋊Dic47C2311yescharacteristic
57C2×C4⋊Dic47C2312yes—
58C2×Dic47⋊C4—22no—
59C2×Dic47⋊C4—21no—
60C2×C22⋊Dic47—22no—
61C2×C22⋊Dic47—22no—
62C2×C4×Dic47—21no—
63#13093957—22no—
64#13093957—24no—
65C4⋊(C47⋊C8)—22no—
66#13093947—22no—
67#13093947—24no—
68#13093945—22no—
69#13093945—42no—
70C22⋊(C47⋊C8)—24no—
71C22⋊(C47⋊C8)—42no—
72#13093935—22no—
73C47⋊#95—22no—
74C47⋊#95—21no—
75C47⋊#95—21no—
76C2×#41×C47—28no—
77C2×#41×C47—48no—
78C22×Q8×C47—21no—
79C22×D4×C47—22no—
80C22×D4×C47D411yescharacteristic
81C22×D4×C47C2311yescharacteristic
82C23×C4×C47—22no—
83C23×C4×C47—21no—
84C23×C4×C47C2312yes—
85C4⋊Q8×C47—22no—
86C4⋊Q8×C47D414yes—
87#123×C47—216no—
88C22⋊Q8×C47—28no—
89#121×C47—24no—
90#121×C47—24no—
91C4×D4×C47—22no—
92C4×D4×C47—22no—
93C4×D4×C47—22no—
94C4×D4×C47D414yes—
95C2×C4⋊C4×C47—22no—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47—24no—
98C2×C4⋊C4×C47—24no—
99C2×C4⋊C4×C47D411yescharacteristic
100C2×C4⋊C4×C47C2311yescharacteristic
101C2×C4⋊C4×C47C2312yes—
102C2×C22⋊C4×C47—21no—
103C2×C22⋊C4×C47—22no—
104C2×C22⋊C4×C47D412yes—
105C2×C42×C47C2311yescharacteristic
106C24×C47—21no—
107C24×C47C2×D412yes—
108#41×C47—432no—
109C2×Q8×C47—24no—
110C2×Q8×C47—44no—
111C2×D4×C47—24no—
112C2×D4×C47—28no—
113C2×D4×C47—22no—
114C2×D4×C47—21no—
115C2×D4×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—44no—
120C2×D4×C47—44no—
121C2×D4×C47C2×D412yes—
122C22×C4×C47—28no—
123C22×C4×C47—21no—
124C22×C4×C47—22no—
125C22×C4×C47—24no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—44no—
133C22×C4×C47—44no—
134C22×C4×C47—44no—
135C22×C4×C47C2×D412yes—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2411yescharacteristic
139C22×Dic47—21no—
140C22×Dic47—22no—
141C22×Dic47—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94—21no—
146C2×Dic94—21no—
147C2×Dic94#4112yescomplemented
148C4⋊C4×C47—28no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—22no—
153C4⋊C4×C47C2×D412yes—
154C4⋊C4×C47C2×D414yes—
155C22⋊C4×C47—28no—
156C22⋊C4×C47—24no—
157C22⋊C4×C47—24no—
158C42×C47C2×D412yes—
159C42×C47C2×D412yes—
160C22⋊Dic47—44no—
161C22⋊Dic47—44no—
162#10625335—28no—
163C4⋊Dic47—21no—
164C4⋊Dic47—21no—
165C4⋊Dic47—22no—
166C4⋊Dic47—22no—
167Dic47⋊C4—44no—
168Dic47⋊C4—42no—
169C4×Dic47—42no—
170C2×C47⋊C8—21no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—22no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—21no—
176C2×C47⋊C8—22no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8#4112yescomplemented
179C23×C47—22no—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—41no—
187C23×C47—41no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47AGL(1,ℤ/8)14yes—
191C23×C47C22×D411yescharacteristic
192Q8×C47—48no—
193D4×C47—28no—
194D4×C47—22no—
195D4×C47—22no—
196D4×C47—416no—
197D4×C47—41no—
198D4×C47—41no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—22no—
202C2×C4×C47—22no—
203C2×C4×C47—24no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—48no—
207C2×C4×C47—416no—
208C2×C4×C47—42no—
209C2×C4×C47—42no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47#12012yes—
216C2×C4×C47#12012yes—
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47H(5,𝔽2)11yescharacteristic
222C2×C4×C472−1+411yescharacteristic
223C2×Dic47—22no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231Dic94—22no—
232Dic94—22no—
233Dic94—41no—
234Dic94—41no—
235C47⋊C8—22no—
236C47⋊C8—41no—
237C47⋊C8—42no—
238#24132—471nomaximal, Sylow
239C22×C47—24no—
240C22×C47—24no—
241C22×C47—24no—
242C22×C47—42no—
243C22×C47—42no—
244C22×C47—42no—
245C22×C47—42no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—42no—
249C22×C47—82no—
250C22×C47—82no—
251C22×C47C2×#12011yescharacteristic
252C22×C47#54611yescharacteristic
253C22×C47#54711yescharacteristic
254C22×C47C2×AGL(1,ℤ/8)12yes—
255C22×C47#57812yes—
256C4×C47—22no—
257C4×C47—22no—
258C4×C47—42no—
259C4×C47—42no—
260C4×C47—44no—
261C4×C47#52511yescharacteristic
262C4×C47#53711yescharacteristic
263C4×C47#54811yescharacteristic
264C4×C47#55011yescharacteristic
265Dic47—42no—
266Dic47—81no—
267Dic47—82no—
268C2×#547D4711yescharacteristic
269C2×#477—471no—
270C2×#440—471no—
271#2536—471no—
272#2033—472no—
273#2000—472no—
274#1993—471no—
275#1955—471no—
276#1872—472no—
277#1852—472no—
278#1804—471no—
279C2×C47—44no—
280C2×C47—82no—
281C2×C47#240711yescharacteristic
282C2×C47#300311yescharacteristic
283C2×C47#301111yescharacteristic
284C2×C47#330812yes—
285C2×C47#334312yes—
286C22×#41—21no—
287C22×SD16—471no—
288#547—24no—
289#547—24no—
290C2×C4⋊Q8C2×D4711yescharacteristic
291C2×#123—22no—
292C2×C22⋊Q8—21no—
293C2×C22⋊Q8—472no—
294C2×#121C2×D4712yes—
295C2×C4×Q8—471no—
296C2×C4×D4C2×D4711yescharacteristic
297C2×C4×D4C2×D4711yescharacteristic
298C22×C4⋊C4C2×D4712yes—
299#477—944no—
300#440—944no—
301C2×#107—472no—
302C2×#107—471no—
303C2×C4⋊C8—471no—
304C2×#103—472no—
305C2×#103—471no—
306C2×#102—471no—
307C2×#102—472no—
308C2×C22⋊C8—472no—
309C2×C22⋊C8—472no—
310C2×#97—471no—
311#398—941no—
312#394—941no—
313#388—942no—
314#382—471no—
315#380—472no—
316#340—472no—
317#340—472no—
318#340—472no—
319C47#2413211yescharacteristic, complemented, Sylow
320C2×#41—28no—
321C2×#41—48no—
322C22×Q8—21no—
323C22×Q8—471no—
324C22×D4—22no—
325C22×D4C22⋊D4711yescharacteristic
326C22×D4C22×D4711yescharacteristic
327C23×C4—22no—
328C23×C4—21no—
329C23×C4C22×D4712yes—
330C2×SD16—942no—
331C2×SD16—942no—
332C2×SD16—942no—
333C22×C8—471no—
334C22×C8—471no—
335C22×C8—472no—
336C4⋊Q8—22no—
337C4⋊Q8C22⋊D4714yes—
338#123—216no—
339C22⋊Q8—28no—
340C22⋊Q8—944no—
341C22⋊Q8—1882no—
342#121—24no—
343#121—24no—
344C4×Q8—942no—
345C4×Q8—942no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4C22⋊D4714yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—471no—
355C2×C4⋊C4—472no—
356C2×C4⋊C4—941no—
357C2×C4⋊C4—942no—
358C2×C4⋊C4C22⋊D4711yescharacteristic
359C2×C4⋊C4C22×D4711yescharacteristic
360C2×C4⋊C4C22×D4712yes—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—942no—
364C2×C22⋊C4—942no—
365C2×C22⋊C4C22⋊D4712yes—
366C2×C42—941no—
367C2×C42C22×D4711yescharacteristic
368#107—942no—
369#107—944no—
370C4⋊C8—942no—
371#103—942no—
372#103—944no—
373#102—942no—
374#102—1882no—
375C22⋊C8—944no—
376C22⋊C8—1882no—
377#97—942no—
378#95—942no—
379#95—941no—
380#95—941no—
381C24—21no—
382C24C2×C22⋊D4712yes—
383#41—432no—
384C2×Q8—24no—
385C2×Q8—44no—
386C2×Q8—472no—
387C2×Q8—941no—
388C2×Q8—941no—
389C2×Q8—941no—
390C2×Q8—941no—
391C2×Q8—941no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4C2×C22⋊D4712yes—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—941no—
417C22×C4—942no—
418C22×C4—941no—
419C22×C4C2×C22⋊D4712yes—
420C22×C4C2×C22⋊D4711yescharacteristic
421C22×C4C2×C22⋊D4711yescharacteristic
422C22×C4C23×D4711yescharacteristic
423SD16—948no—
424C2×C8—472no—
425C2×C8—941no—
426C2×C8—941no—
427C2×C8—941no—
428C2×C8—942no—
429C2×C8—942no—
430C2×C8—941no—
431C2×C8—942no—
432C2×C8—941no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—941no—
439C4⋊C4—941no—
440C4⋊C4—942no—
441C4⋊C4—942no—
442C4⋊C4—1884no—
443C4⋊C4—1882no—
444C4⋊C4C2×C22⋊D4712yes—
445C4⋊C4C2×C22⋊D4714yes—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—1884no—
450C22⋊C4—1884no—
451C42—1882no—
452C42C2×C22⋊D4712yes—
453C42C2×C22⋊D4712yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23#1309405014yes—
466C23C22×C22⋊D4711yescharacteristic
467Q8—48no—
468Q8—942no—
469Q8—942no—
470Q8—1881no—
471Q8—1881no—
472D4—28no—
473D4—22no—
474D4—22no—
475D4—416no—
476D4—41no—
477D4—41no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—48no—
486C2×C4—416no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—942no—
495C2×C4—1882no—
496C2×C4—1881no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4—1882no—
500C2×C4—1881no—
501C2×C4—1881no—
502C2×C4#1309399312yes—
503C2×C4#1309399312yes—
504C2×C4C22×C22⋊D4711yescharacteristic
505C2×C4C22×C22⋊D4711yescharacteristic
506C2×C4#41×D4711yescharacteristic
507C2×C4#41×D4711yescharacteristic
508C2×C4#1309407011yescharacteristic
509C2×C4#1309407111yescharacteristic
510C8—942no—
511C8—1882no—
512C8—1881no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22C2×#1309399311yescharacteristic
526C22#1568976111yescharacteristic
527C22#1568977211yescharacteristic
528C22C2×#1309405012yes—
529C22#1568993812yes—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—44no—
534C4—42no—
535C4—3762no—
536C4—3761no—
537C4—1882no—
538C4#1568963011yescharacteristic
539C4#1568970111yescharacteristic
540C4#1568978011yescharacteristic
541C4#1568979211yescharacteristic
542C2—44no—
543C2—82no—
544C2#2024672711yescharacteristic
545C2#2025261511yescharacteristic
546C2#2025268311yescharacteristic
547C2#2025537312yes—
548C2#2025582212yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#547C2
socleC23×C47C22×D4
hypercenterC2×#547D47
hypocenterC47#24132
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#15689415C2semidirect
2. C2×#15689223C2semidirect
3. #20248297C2semidirect
4. #20244567C2semidirect
5. #20244391C2semidirect
6. #20244362C2semidirect
7. #20244187C2semidirect
8. #20243695C2semidirect
9. #20243576C2semidirect
10. #20243398C2semidirect
147. C2×Dic94#41semidirect
178. C2×C47⋊C8#41semidirect
319. C47#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#156894152
C2×#156892232
#202482972
#202445672
#202443912
#202443622
#202441872
#202436952
#202435762
#202433982
C2×C47×#5472
#2413247