Finite Group Databasebeta

← #26254626

Subgroups — #26254626

Order 12032.

1130 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689451C211yesmaximal, characteristic, complemented
2C22×#13093945C211yesmaximal, characteristic
3#20246255C211yesmaximal, characteristic, complemented
4#20244762C212yesmaximal, complemented
5#20244624C212yesmaximal, complemented
6#20244321C212yesmaximal, complemented
7#20244012C212yesmaximal, complemented
8#20243812C211yesmaximal, characteristic, complemented
9#20243568C211yesmaximal, characteristic, complemented
10#20243319C211yesmaximal, characteristic
11C47×#2435C211yesmaximal, characteristic
12C23×C47⋊C8C2211yescharacteristic
13C2×#13093980—21no—
14C22×C4⋊Dic47C2211yescharacteristic
15#15689451—24no—
16C2×#13093955C2211yescharacteristic, complemented
17C2×#13093955C2211yescharacteristic, complemented
18C2×C4⋊(C47⋊C8)C2212yescomplemented
19C2×#13093948C2211yescharacteristic, complemented
20C2×#13093948C2211yescharacteristic, complemented
21C2×#13093948C2212yescomplemented
22C2×#13093945—24no—
23C2×#13093945C2212yes—
24C2×#13093945C2212yes—
25C2×#13093945C2212yes—
26C2×#13093945C2212yes—
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28#15689021C2212yescomplemented
29#15689008—22no—
30#15688984—21no—
31#15688983C2211yescharacteristic
32C4×C4⋊Dic47C2212yescomplemented
33#15688885C2212yescomplemented
34#15688878C2212yescomplemented
35C23×D4×C47C2211yescharacteristic
36C2×#124×C47C2212yes—
37C2×C22⋊Q8×C47C2211yescharacteristic
38C47×#394—22no—
39C47×#390C2212yes—
40C47×#379—21no—
41C47×#379—21no—
42C4×C22⋊C4×C47C2211yescharacteristic
43C23×Dic47—21no—
44C22×C47⋊C8—22no—
45C22×C47⋊C8—42no—
46C22×C47⋊C8C2311yescharacteristic
47C22×C47⋊C8C2311yescharacteristic
48C22×C47⋊C8C2312yes—
49#13093980—44no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47—22no—
52C2×C4⋊Dic47D414yes—
53C2×C4⋊Dic47C2312yes—
54C2×C4⋊Dic47C2311yescharacteristic
55C2×C4⋊Dic47C2311yescharacteristic
56C2×Dic47⋊C4—22no—
57C2×C22⋊Dic47—21no—
58C2×C4×Dic47—22no—
59#13093955—22no—
60#13093955—22no—
61C4⋊(C47⋊C8)—42no—
62#13093948—22no—
63#13093948—22no—
64#13093948—42no—
65#13093945—24no—
66#13093945—24no—
67#13093945—24no—
68#13093945—42no—
69C22⋊(C47⋊C8)—22no—
70C47⋊#95—22no—
71C47⋊#95—21no—
72C47⋊#95—22no—
73C25×C47—21no—
74C22×Q8×C47C2311yescharacteristic
75C22×D4×C47—24no—
76C22×D4×C47—22no—
77C22×D4×C47—21no—
78C22×D4×C47—22no—
79C22×D4×C47—42no—
80C22×D4×C47C2312yes—
81C23×C4×C47C2311yescharacteristic
82#124×C47—24no—
83#124×C47—42no—
84C22⋊Q8×C47—22no—
85C22⋊Q8×C47—22no—
86C2×C4⋊C4×C47—21no—
87C2×C4⋊C4×C47C2311yescharacteristic
88C2×C22⋊C4×C47—21no—
89C2×C22⋊C4×C47—22no—
90C2×C22⋊C4×C47—22no—
91C2×C22⋊C4×C47—22no—
92C2×C22⋊C4×C47—22no—
93C2×C22⋊C4×C47D411yescharacteristic
94C2×C22⋊C4×C47D411yescharacteristic
95C2×C42×C47C2312yes—
96#95×C47D412yes—
97C24×C47—22no—
98C24×C47—44no—
99C24×C47—42no—
100C24×C47—42no—
101C24×C47—42no—
102C24×C47—82no—
103C24×C47C2×D411yescharacteristic
104C2×Q8×C47—21no—
105C2×Q8×C47—21no—
106C2×Q8×C47—22no—
107C2×Q8×C47—22no—
108C2×D4×C47—28no—
109C2×D4×C47—22no—
110C2×D4×C47—22no—
111C2×D4×C47—22no—
112C2×D4×C47—22no—
113C2×D4×C47—22no—
114C2×D4×C47—48no—
115C2×D4×C47—44no—
116C2×D4×C47—44no—
117C2×D4×C47—42no—
118C2×D4×C47#4114yes—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47C2×D412yes—
124C22×C4×C47C2×D412yes—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D412yes—
127C22×C4×C47#4112yes—
128C22×C4×C47C2411yescharacteristic
129C22×Dic47—24no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47—22no—
133C22×Dic47—42no—
134C22×Dic47—42no—
135C4⋊C4×C47—21no—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—42no—
138C22⋊C4×C47—22no—
139C22⋊C4×C47—22no—
140C22⋊C4×C47—44no—
141C22⋊C4×C47—44no—
142C22⋊C4×C47—42no—
143C22⋊C4×C47—82no—
144C22⋊C4×C47—82no—
145C42×C47—22no—
146C42×C47—22no—
147C22⋊Dic47—42no—
148C4⋊Dic47—24no—
149C4⋊Dic47—21no—
150C4⋊Dic47—21no—
151C4⋊Dic47—22no—
152C4⋊Dic47—22no—
153C4⋊Dic47C2×D412yes—
154C4⋊Dic47C2×D412yes—
155Dic47⋊C4—44no—
156C4×Dic47—44no—
157C2×C47⋊C8—21no—
158C2×C47⋊C8—21no—
159C2×C47⋊C8—22no—
160C2×C47⋊C8—22no—
161C2×C47⋊C8—22no—
162C2×C47⋊C8—22no—
163C2×C47⋊C8—22no—
164C2×C47⋊C8—44no—
165C23×C47—22no—
166C23×C47—22no—
167C23×C47—24no—
168C23×C47—48no—
169C23×C47—42no—
170C23×C47—42no—
171C23×C47—42no—
172C23×C47—42no—
173C23×C47—42no—
174C23×C47—42no—
175C23×C47—88no—
176C23×C47—88no—
177C23×C47—84no—
178C23×C47—84no—
179C23×C47#12112yes—
180C23×C47#13512yes—
181C23×C47AGL(1,ℤ/8)12yes—
182C23×C47C22×D411yescharacteristic
183Q8×C47—41no—
184Q8×C47—41no—
185Q8×C47—42no—
186D4×C47—24no—
187D4×C47—24no—
188D4×C47—48no—
189D4×C47—42no—
190D4×C47—42no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—24no—
195C2×C4×C47—22no—
196C2×C4×C47—22no—
197C2×C4×C47—22no—
198C2×C4×C47—22no—
199C2×C4×C47—22no—
200C2×C4×C47—22no—
201C2×C4×C47—41no—
202C2×C4×C47—41no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47#12112yes—
206C2×C4×C47C22×D411yescharacteristic
207C2×C4×C47C2×#4112yes—
208C2×C4×C47C2×#4111yescharacteristic
209C2×C4×C47H(5,𝔽2)12yes—
210C2×Dic47—22no—
211C2×Dic47—22no—
212C2×Dic47—41no—
213C2×Dic47—41no—
214C2×Dic47—41no—
215C2×Dic47—41no—
216C2×Dic47—48no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C47⋊C8—41no—
221C47⋊C8—41no—
222C47⋊C8—42no—
223#24477—471nomaximal, Sylow
224C22×C47—22no—
225C22×C47—24no—
226C22×C47—22no—
227C22×C47—24no—
228C22×C47—44no—
229C22×C47—44no—
230C22×C47—44no—
231C22×C47—816no—
232C22×C47—82no—
233C22×C47—82no—
234C22×C47—82no—
235C22×C47—82no—
236C22×C47#46314yes—
237C22×C47C2×#12111yescharacteristic
238C22×C47#54612yes—
239C22×C47C2×#13511yescharacteristic
240C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C47#57712yes—
242C4×C47—22no—
243C4×C47—42no—
244C4×C47—42no—
245C4×C47—81no—
246C4×C47—81no—
247C4×C47#52511yescharacteristic
248C4×C47#53511yescharacteristic
249C4×C47#55112yes—
250Dic47—42no—
251Dic47—42no—
252Dic47—82no—
253C2×#483—471no—
254C22×#102—471no—
255#2435D4711yescharacteristic
256#2344—471no—
257#2065—472no—
258#2042—472no—
259#1983—472no—
260#1926—472no—
261#1892—471no—
262#1850—471no—
263#1786—471no—
264C2×C47—24no—
265C2×C47—84no—
266C2×C47#241111yescharacteristic
267C2×C47C2×#46311yescharacteristic
268C2×C47#304311yescharacteristic
269C2×C47#329612yes—
270C2×C47#330612yes—
271C23×D4C2×D4711yescharacteristic
272C23×C8—471no—
273C2×#124C2×D4712yes—
274C2×C22⋊Q8C2×D4711yescharacteristic
275C2×#117—941no—
276C22×C4⋊C4—471no—
277#483—944no—
278C2×#106—471no—
279C2×#106—471no—
280C2×C4⋊C8—472no—
281C2×#103—471no—
282C2×#103—471no—
283C2×#103—472no—
284C2×#102—472no—
285C2×#102—472no—
286C2×#102—472no—
287C2×#102—472no—
288C2×#102—944no—
289C2×C22⋊C8—471no—
290#394—22no—
291#390C2×D4712yes—
292#389—472no—
293#387—942no—
294#381—941no—
295#380—471no—
296#379—21no—
297#379—21no—
298C4×C4⋊C4—472no—
299C4×C22⋊C4C2×D4711yescharacteristic
300#340—472no—
301#336—472no—
302C47#2447711yescharacteristic, complemented, Sylow
303C25—21no—
304C22×Q8C22×D4711yescharacteristic
305C22×D4—24no—
306C22×D4—22no—
307C22×D4—22no—
308C22×D4—21no—
309C22×D4—42no—
310C22×D4C22×D4712yes—
311C23×C4—941no—
312C23×C4C22×D4711yescharacteristic
313C22×C8—472no—
314C22×C8—471no—
315C22×C8—471no—
316C22×C8—942no—
317C22×C8—1882no—
318#124—24no—
319#124—42no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322#117—1884no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—474no—
325C2×C4⋊C4—471no—
326C2×C4⋊C4—471no—
327C2×C4⋊C4—472no—
328C2×C4⋊C4—942no—
329C2×C4⋊C4—941no—
330C2×C4⋊C4—942no—
331C2×C4⋊C4C22×D4711yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—941no—
338C2×C22⋊C4C22⋊D4711yescharacteristic
339C2×C22⋊C4C22⋊D4711yescharacteristic
340C2×C42—942no—
341C2×C42C22×D4712yes—
342#106—942no—
343#106—942no—
344C4⋊C8—1882no—
345#103—942no—
346#103—942no—
347#103—1882no—
348#102—944no—
349#102—944no—
350#102—944no—
351#102—1882no—
352C22⋊C8—942no—
353#95—942no—
354#95—941no—
355#95—942no—
356#95C22⋊D4712yes—
357C24—22no—
358C24—44no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—82no—
363C24C2×C22⋊D4711yescharacteristic
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4#1062535214yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—944no—
384C22×C4—941no—
385C22×C4—941no—
386C22×C4—942no—
387C22×C4—1882no—
388C22×C4—1882no—
389C22×C4D4×D4712yes—
390C22×C4#1062535212yes—
391C22×C4C2×C22⋊D4712yes—
392C22×C4C2×C22⋊D4712yes—
393C22×C4C2×C22⋊D4711yescharacteristic
394C22×C4C23×D4711yescharacteristic
395C2×C8—941no—
396C2×C8—941no—
397C2×C8—942no—
398C2×C8—942no—
399C2×C8—942no—
400C2×C8—942no—
401C2×C8—942no—
402C2×C8—1884no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—472no—
407C4⋊C4—472no—
408C4⋊C4—944no—
409C4⋊C4—941no—
410C4⋊C4—941no—
411C4⋊C4—942no—
412C4⋊C4—942no—
413C4⋊C4—1884no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4—1882no—
422C42—22no—
423C42—22no—
424C42—1884no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—48no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—88no—
436C23—88no—
437C23—84no—
438C23—84no—
439C23#1309399512yes—
440C23#1309404412yes—
441C23#1309404912yes—
442C23C22×C22⋊D4711yescharacteristic
443Q8—42no—
444Q8—41no—
445Q8—41no—
446D4—24no—
447D4—24no—
448D4—48no—
449D4—42no—
450D4—42no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—942no—
466C2×C4—942no—
467C2×C4—1882no—
468C2×C4—1882no—
469C2×C4—1882no—
470C2×C4—1888no—
471C2×C4—1881no—
472C2×C4—1881no—
473C2×C4—1881no—
474C2×C4—1881no—
475C2×C4#1309399512yes—
476C2×C4C2×D4×D4711yescharacteristic
477C2×C4C2×#1062535211yescharacteristic
478C2×C4C2×#1062535212yes—
479C2×C4#1309406912yes—
480C8—1881no—
481C8—1881no—
482C8—1882no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—816no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22#1568933514yes—
496C22C2×#1309399511yescharacteristic
497C22#1568976712yes—
498C22C2×#1309404411yescharacteristic
499C22C2×#1309404911yescharacteristic
500C22#1568993412yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—81no—
505C4—81no—
506C4—3762no—
507C4—1882no—
508C4—1882no—
509C4#1568962911yescharacteristic
510C4#1568968811yescharacteristic
511C4#1568980412yes—
512C2—24no—
513C2—84no—
514C2#2024676911yescharacteristic
515C2C2×#1568933511yescharacteristic
516C2#2025290511yescharacteristic
517C2#2025522312yes—
518C2#2025534912yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2435C2
socleC23×C47C22×D4
prime radicalC23×D4×C47C22
hypercenter#2435D47
hypocenterC47#24477
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
1. C2×#15689451C2semidirect
3. #20246255C2semidirect
4. #20244762C2semidirect
5. #20244624C2semidirect
6. #20244321C2semidirect
7. #20244012C2semidirect
8. #20243812C2semidirect
9. #20243568C2semidirect
16. C2×#13093955C22semidirect
17. C2×#13093955C22semidirect
18. C2×C4⋊(C47⋊C8)C22semidirect
19. C2×#13093948C22semidirect
20. C2×#13093948C22semidirect
21. C2×#13093948C22semidirect
28. #15689021C22semidirect
32. C4×C4⋊Dic47C22semidirect
33. #15688885C22semidirect
34. #15688878C22semidirect
302. C47#24477semidirect

Maximal subgroups

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

subgroupindex
C2×#156894512
C22×#130939452
#202462552
#202447622
#202446242
#202443212
#202440122
#202438122
#202435682
#202433192
C47×#24352
#2447747