Finite Group Databasebeta

← #26252643

Subgroups — #26252643

Order 12032.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689400C211yesmaximal, characteristic, complemented
2C2×C4×#10625334C211yesmaximal, characteristic
3#20248306C211yesmaximal, characteristic, complemented
4#20244713C212yesmaximal, complemented
5#20244521C212yesmaximal, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244325C211yesmaximal, characteristic, complemented
8#20243736C212yesmaximal
9#20243635C212yesmaximal, complemented
10#20243388C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C47C211yesmaximal, characteristic, complemented
12C22×#10625334C2211yescharacteristic
13C2×#13094000—21no—
14C2×C4×D188C2211yescharacteristic
15#15689400—24no—
16C4×#10625334—24no—
17C2×#13093957C2212yes—
18C2×#13093957C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093945C2212yes—
21C2×#13093945C2211yescharacteristic
22C2×#13093944C2212yes—
23C2×#13093944C2211yescharacteristic
24C2×C4×C47⋊C8C2211yescharacteristic
25#15689045—22no—
26#15689025C2212yes—
27#15689021C2212yes—
28#15688993C2211yescharacteristic
29#15688980—21no—
30#15688885C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688878C2212yes—
34#15688878C2212yes—
35C23×Q8×C47—21no—
36D4×Q8×C47—24no—
37D4×Q8×C47—24no—
38C2×C4⋊Q8×C47C2212yes—
39C2×C4⋊Q8×C47C2211yescharacteristic
40C2×C22⋊Q8×C47—21no—
41C2×C22⋊Q8×C47—22no—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4×D4×C47C2212yes—
44C2×C4×D4×C47C2211yescharacteristic
45C22×D188C2311yescharacteristic
46C22×C4×D47—21no—
47C2×#10625334—22no—
48C2×#10625334—22no—
49C2×#10625334Q814yescomplemented
50C22×C47⋊C8C2311yescharacteristic
51C22×C47⋊C8C2312yes—
52C22×C47⋊C8C2311yescharacteristic
53#13094000—44no—
54C4×D188—22no—
55C4×D188—22no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47C2312yes—
59C2×C4⋊Dic47C2311yescharacteristic
60C2×Dic47⋊C4—22no—
61C2×(C2×D47)⋊C4—22no—
62C2×(C2×D47)⋊C4—21no—
63C2×(C2×D47)⋊C4—22no—
64#13093957—24no—
65#13093957—22no—
66C4⋊(C47⋊C8)—22no—
67#13093945—22no—
68#13093945—24no—
69#13093944—24no—
70#13093944—22no—
71C4×C47⋊C8—22no—
72C47⋊#95—22no—
73C22×Q8×C47—28no—
74C22×Q8×C47—22no—
75C22×Q8×C47—21no—
76C22×Q8×C47—44no—
77C22×Q8×C47—44no—
78C22×Q8×C47D411yescharacteristic
79C22×D4×C47C2311yescharacteristic
80C23×C4×C47—22no—
81C23×C4×C47—21no—
82C4⋊Q8×C47—24no—
83C4⋊Q8×C47—24no—
84C4⋊Q8×C47—22no—
85C4⋊Q8×C47D414yes—
86C22⋊Q8×C47—28no—
87C22⋊Q8×C47—216no—
88C4×Q8×C47—22no—
89C4×Q8×C47—22no—
90C4×D4×C47—24no—
91C4×D4×C47—24no—
92C4×D4×C47—22no—
93C4×D4×C47D414yes—
94C2×C4⋊C4×C47—22no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47D411yescharacteristic
97C2×C4⋊C4×C47D412yes—
98C2×C4⋊C4×C47C2312yes—
99C2×C4⋊C4×C47C2311yescharacteristic
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47—21no—
102C2×C42×C47C2312yes—
103C2×C42×C47C2311yescharacteristic
104C24×C47—21no—
105C23×D47—21no—
106C2×Q8×C47—28no—
107C2×Q8×C47—22no—
108C2×Q8×C47—24no—
109C2×Q8×C47—22no—
110C2×Q8×C47—44no—
111C2×Q8×C47—432no—
112C2×Q8×C47—44no—
113C2×Q8×C47—42no—
114C2×Q8×C47—42no—
115C2×Q8×C47D814yescomplemented
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47C2×D412yes—
120C2×D4×C47C2×Q812yes—
121C2×D4×C47#4112yes—
122C22×C4×C47—28no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
125C22×C4×C47—24no—
126C22×C4×C47—44no—
127C22×C4×C47—44no—
128C22×C4×C47—42no—
129C22×C4×C47—42no—
130C22×C4×C47C2×D412yes—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—21no—
136C22×Dic47—22no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188C2×Q812yescomplemented
143C2×C4×D47—42no—
144C2×C4×D47—42no—
145C2×C4×D47—42no—
146C4⋊C4×C47—28no—
147C4⋊C4×C47—24no—
148C4⋊C4×C47—24no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47—22no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47C2×D414yes—
153C4⋊C4×C47C2×D412yes—
154C22⋊C4×C47—28no—
155C22⋊C4×C47—24no—
156C42×C47—22no—
157C42×C47—22no—
158C42×C47C2×D412yes—
159C42×C47C2×D412yes—
160#10625334—28no—
161(C2×D47)⋊C4—42no—
162(C2×D47)⋊C4—44no—
163(C2×D47)⋊C4—82no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166C4⋊Dic47—22no—
167C4⋊Dic47—22no—
168C4⋊Dic47—42no—
169C4⋊Dic47—81no—
170Dic47⋊C4—44no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—21no—
178C2×C47⋊C8—22no—
179C2×C47⋊C8C2×Q812yescomplemented
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—41no—
184C23×C47—41no—
185C23×C47—41no—
186C23×C47—41no—
187C23×C47C22×D411yescharacteristic
188C22×D47—22no—
189C22×D47—41no—
190C22×D47—41no—
191C22×D47—41no—
192C22×D47—41no—
193C22×D47—41no—
194C22×D47—41no—
195Q8×C47—28no—
196Q8×C47—416no—
197Q8×C47—48no—
198D4×C47—22no—
199D4×C47—22no—
200D4×C47—22no—
201D4×C47C22⋊Q814yescomplemented
202C2×C4×C47—22no—
203C2×C4×C47—22no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—24no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—416no—
212C2×C4×C47—42no—
213C2×C4×C47—41no—
214C2×C4×C47—42no—
215C2×C4×C47—41no—
216C2×C4×C47—48no—
217C2×C4×C47#12012yes—
218C2×C4×C47#12012yes—
219C2×C4×C47C2×D814yes—
220C2×C4×C47C2×D812yes—
221C2×C4×C47AGL(1,ℤ/8)12yes—
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C22×D411yescharacteristic
224C2×C4×C47C22×Q811yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C472−1+411yescharacteristic
227C2×C4×C472−1+411yescharacteristic
228C2×Dic47—41no—
229C2×Dic47—42no—
230C2×Dic47—41no—
231C2×Dic47—42no—
232C2×Dic47—41no—
233C2×Dic47—42no—
234D188—22no—
235D188—22no—
236D188—41no—
237D188—41no—
238C4×D47—84no—
239C47⋊C8—22no—
240C47⋊C8—41no—
241C47⋊C8—42no—
242#24326—471nomaximal, Sylow
243C22×C47—24no—
244C22×C47—42no—
245C22×C47—42no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—42no—
249C22×C47—42no—
250C22×C47C2×#12011yescharacteristic
251C22×C47#54811yescharacteristic
252C22×C47D4×Q811yescharacteristic
253C22×C47C22×D811yescharacteristic
254C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C47#57711yescharacteristic
256C22×C47#57811yescharacteristic
257C2×D47—42no—
258C2×D47—42no—
259C2×D47—42no—
260C2×D47—81no—
261C2×D47—81no—
262C2×D47—81no—
263C2×D47—81no—
264C4×C47—24no—
265C4×C47—22no—
266C4×C47—44no—
267C4×C47—42no—
268C4×C47—42no—
269C4×C47#44714yes—
270C4×C47C2×C22⋊Q811yescharacteristic
271C4×C47#53611yescharacteristic
272C4×C47#54811yescharacteristic
273C4×C47D4×Q811yescharacteristic
274Dic47—82no—
275Dic47—81no—
276C2×D4×Q8D4711yescharacteristic, complemented
277C2×#474—471no—
278C2×C4×D8—471no—
279#2538—471no—
280#2057—472no—
281#2023—472no—
282#1990—471no—
283#1984—471no—
284#1879—472no—
285#1862—472no—
286#1802—471no—
287C2×C47—44no—
288C2×C47#240911yescharacteristic
289C2×C47C2×#44711yescharacteristic
290C2×C47#301311yescharacteristic
291C2×C47#335811yescharacteristic
292C2×C47#336711yescharacteristic
293C2×C47Q8×D811yescharacteristic
294C2×C47#338411yescharacteristic
295D47—82no—
296C23×Q8—21no—
297C22×D8—471no—
298D4×Q8—24no—
299D4×Q8—24no—
300C2×C4⋊Q8C2×D4712yes—
301C2×C4⋊Q8C2×D4711yescharacteristic
302C2×C22⋊Q8—21no—
303C2×C22⋊Q8—22no—
304C2×C22⋊Q8—941no—
305C2×C4×Q8C2×D4711yescharacteristic
306C2×C4×D4—471no—
307C2×C4×D4C2×D4712yes—
308C2×C4×D4C2×D4711yescharacteristic
309#474—944no—
310C4×D8—944no—
311C2×#107—471no—
312C2×#107—472no—
313C2×C4⋊C8—471no—
314C2×#102—472no—
315C2×#102—471no—
316C2×#102—472no—
317C2×#102—471no—
318C2×C4×C8—471no—
319#396—942no—
320#390—472no—
321#389—472no—
322#384—471no—
323#380—941no—
324#340—472no—
325#340—472no—
326#340—472no—
327#336—472no—
328#336—472no—
329C47#2432611yescharacteristic, complemented, Sylow
330C22×Q8—28no—
331C22×Q8—22no—
332C22×Q8—21no—
333C22×Q8—44no—
334C22×Q8—44no—
335C22×Q8C22⋊D4711yescharacteristic
336C22×D4—471no—
337C22×D4C22×D4711yescharacteristic
338C23×C4—22no—
339C23×C4—21no—
340C23×C4—941no—
341C2×D8—474no—
342C2×D8—942no—
343C2×D8—942no—
344C22×C8—471no—
345C22×C8—471no—
346C22×C8—472no—
347C4⋊Q8—24no—
348C4⋊Q8—24no—
349C4⋊Q8—22no—
350C4⋊Q8C22⋊D4714yes—
351C22⋊Q8—28no—
352C22⋊Q8—216no—
353C22⋊Q8—1884no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×D4—24no—
357C4×D4—24no—
358C4×D4—22no—
359C4×D4—942no—
360C4×D4—942no—
361C4×D4C22⋊D4714yes—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—471no—
365C2×C4⋊C4—472no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4C22⋊D4711yescharacteristic
370C2×C4⋊C4C22⋊D4712yes—
371C2×C4⋊C4C22×D4712yes—
372C2×C4⋊C4C22×D4711yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—942no—
376C2×C22⋊C4—941no—
377C2×C22⋊C4—942no—
378C2×C42C22×D4712yes—
379C2×C42C22×D4711yescharacteristic
380#107—942no—
381#107—944no—
382C4⋊C8—942no—
383#102—944no—
384#102—942no—
385#102—942no—
386#102—944no—
387C4×C8—942no—
388#95—942no—
389C24—21no—
390C24—941no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—432no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8#1062533414yescomplemented
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—472no—
405C2×D4—941no—
406C2×D4—941no—
407C2×D4—941no—
408C2×D4—941no—
409C2×D4—941no—
410C2×D4Q8×D4712yes—
411C2×D4#1062535412yes—
412C2×D4C2×C22⋊D4712yes—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—941no—
422C22×C4—942no—
423C22×C4—1882no—
424C22×C4—1882no—
425C22×C4—1882no—
426C22×C4C2×C22⋊D4712yes—
427C22×C4C2×C22⋊D4711yescharacteristic
428C22×C4C2×C22⋊D4712yes—
429C22×C4C2×C22⋊D4711yescharacteristic
430C22×C4C23×D4711yescharacteristic
431D8—948no—
432C2×C8—472no—
433C2×C8—941no—
434C2×C8—941no—
435C2×C8—941no—
436C2×C8—942no—
437C2×C8—942no—
438C2×C8—942no—
439C2×C8—941no—
440C2×C8—941no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—941no—
448C4⋊C4—941no—
449C4⋊C4—942no—
450C4⋊C4—942no—
451C4⋊C4—3761no—
452C4⋊C4—1884no—
453C4⋊C4—1882no—
454C4⋊C4C2×C22⋊D4714yes—
455C4⋊C4C2×C22⋊D4712yes—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—3762no—
459C22⋊C4—1882no—
460C22⋊C4—1884no—
461C42—22no—
462C42—22no—
463C42C2×C22⋊D4712yes—
464C42C2×C22⋊D4712yes—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—942no—
473C23—1881no—
474C23—1881no—
475C23—1881no—
476C23—1881no—
477C23—1881no—
478C23—1881no—
479C23C22×C22⋊D4711yescharacteristic
480Q8—28no—
481Q8—416no—
482Q8—48no—
483D4—22no—
484D4—22no—
485D4—22no—
486D4—942no—
487D4—942no—
488D4—1881no—
489D4—1881no—
490D4#1309400014yescomplemented
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—48no—
506C2×C4—3764no—
507C2×C4—1882no—
508C2×C4—1881no—
509C2×C4—1882no—
510C2×C4—1881no—
511C2×C4—1881no—
512C2×C4—1882no—
513C2×C4#1309399312yes—
514C2×C4#1309399312yes—
515C2×C4C2×#1062533414yes—
516C2×C4C2×#1062533412yes—
517C2×C4#1309405012yes—
518C2×C4C22×C22⋊D4711yescharacteristic
519C2×C4C22×C22⋊D4711yescharacteristic
520C2×C4C2×Q8×D4711yescharacteristic
521C2×C4C2×#1062535411yescharacteristic
522C2×C4#1309407211yescharacteristic
523C2×C4#1309407211yescharacteristic
524C8—942no—
525C8—1882no—
526C8—1881no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—3761no—
535C22—3761no—
536C22—3761no—
537C22—3761no—
538C22—1882no—
539C22—1882no—
540C22—1882no—
541C22C2×#1309399311yescharacteristic
542C22#1568978411yescharacteristic
543C22Q8×C22⋊D4711yescharacteristic
544C22C22×#1062533411yescharacteristic
545C22C2×#1309405011yescharacteristic
546C22#1568993111yescharacteristic
547C22#1568993811yescharacteristic
548C4—22no—
549C4—24no—
550C4—42no—
551C4—44no—
552C4—42no—
553C4—3762no—
554C4—3761no—
555C4#1568925714yes—
556C4C2×#1309400011yescharacteristic
557C4#1568969711yescharacteristic
558C4#1568978411yescharacteristic
559C4Q8×C22⋊D4711yescharacteristic
560C2—44no—
561C2—3762no—
562C2#2024674711yescharacteristic
563C2C2×#1568925711yescharacteristic
564C2#2025270311yescharacteristic
565C2#2025602311yescharacteristic
566C2#2025610611yescharacteristic
567C2Q8×#1062533411yescharacteristic
568C2#2025628311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×D4×Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×#10625334C22
hypercenterC2×D4×Q8D47
hypocenterC47#24326
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15689400C2semidirect
3. #20248306C2semidirect
4. #20244713C2semidirect
5. #20244521C2semidirect
6. #20244350C2semidirect
7. #20244325C2semidirect
9. #20243635C2semidirect
10. #20243388C2semidirect
11. C2×D4×Q8×C47C2semidirect
49. C2×#10625334Q8semidirect
115. C2×Q8×C47D8semidirect
142. C2×D188C2×Q8semidirect
179. C2×C47⋊C8C2×Q8semidirect
201. D4×C47C22⋊Q8semidirect
276. C2×D4×Q8D47semidirect
329. C47#24326semidirect
400. C2×Q8#10625334semidirect
490. D4#13094000semidirect

Maximal subgroups

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

subgroupindex
C2×#156894002
C2×C4×#106253342
#202483062
#202447132
#202445212
#202443502
#202443252
#202437362
#202436352
#202433882
C2×D4×Q8×C472
#2432647