Finite Group Databasebeta

← #26143021

Subgroups — #26143021

Order 12032.

1262 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688990C211yesmaximal, characteristic
2C2×#15688987C211yesmaximal, characteristic
3C2×C4⋊C4×Dic47C211yesmaximal, characteristic
4#20241653C211yesmaximal, characteristic
5#20241652C212yesmaximal
6#20241652C211yesmaximal, characteristic
7#20241647C211yesmaximal, characteristic
8#20241643C212yesmaximal
9#20241622C212yesmaximal
10#20241619C212yesmaximal
11C47×#1442C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2212yes—
14C22×Dic47⋊C4C2211yescharacteristic
15C22×Dic47⋊C4C2211yescharacteristic
16C22×Dic47⋊C4C2212yes—
17C22×C4×Dic47C2212yes—
18C22×C4×Dic47C2211yescharacteristic
19#15688990C418yescomplemented
20#15688987—24no—
21C4⋊C4×Dic47—24no—
22C2×C47⋊#95C2211yescharacteristic
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2212yes—
31C2×C47⋊#95C2212yes—
32C2×C47⋊#95C2211yescharacteristic
33C22×C4⋊C4×C47C2211yescharacteristic
34C2×#95×C47C2211yescharacteristic
35C2×#95×C47C2212yes—
36C2×#95×C47C2212yes—
37C2×#95×C47C2211yescharacteristic
38C23×Dic47C2311yescharacteristic
39C23×Dic47C2311yescharacteristic
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C23×Dic47C2311yescharacteristic
43C23×Dic47C2311yescharacteristic
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—24no—
46C2×C4⋊Dic47—24no—
47C2×C4⋊Dic47—42no—
48C2×C4⋊Dic47—42no—
49C2×Dic47⋊C4—22no—
50C2×Dic47⋊C4—22no—
51C2×Dic47⋊C4—24no—
52C2×Dic47⋊C4—42no—
53C2×Dic47⋊C4—42no—
54C2×Dic47⋊C4—42no—
55C2×Dic47⋊C4—42no—
56C2×C4×Dic47—24no—
57C2×C4×Dic47—42no—
58C2×C4×Dic47—42no—
59C2×C4×Dic47C2×C414yes—
60C2×C4×Dic47D414yes—
61C2×C4×Dic47Q814yes—
62C47⋊#95—22no—
63C47⋊#95—24no—
64C47⋊#95—24no—
65C47⋊#95—22no—
66C47⋊#95—22no—
67C47⋊#95—42no—
68C47⋊#95—42no—
69C47⋊#95—42no—
70C47⋊#95C2×C414yes—
71C47⋊#95C2×C418yes—
72C47⋊#95C2×C418yes—
73C23×C4×C47C2311yescharacteristic
74C23×C4×C47C2311yescharacteristic
75C23×C4×C47C2312yes—
76C23×C4×C47C2312yes—
77C23×C4×C47C2311yescharacteristic
78C2×C4⋊C4×C47—22no—
79C2×C4⋊C4×C47—24no—
80#95×C47—22no—
81#95×C47—24no—
82#95×C47—42no—
83#95×C47C2×C414yes—
84C24×C47C2411yescharacteristic
85C22×C4×C47—22no—
86C22×C4×C47—22no—
87C22×C4×C47—22no—
88C22×C4×C47—22no—
89C22×C4×C47—22no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—22no—
97C22×C4×C47—22no—
98C22×C4×C47—22no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47—21no—
107C22×C4×C47—21no—
108C22×C4×C47—21no—
109C22×C4×C47C22×C414yes—
110C22×C4×C47C22×C412yes—
111C22×C4×C47C2×D412yes—
112C22×C4×C47C2×Q812yes—
113C22×C4×C47#4114yes—
114C22×Dic47—21no—
115C22×Dic47—21no—
116C22×Dic47—22no—
117C22×Dic47—22no—
118C22×Dic47—21no—
119C22×Dic47—21no—
120C22×Dic47—22no—
121C22×Dic47—22no—
122C22×Dic47—22no—
123C22×Dic47—22no—
124C22×Dic47—22no—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—21no—
133C22×Dic47—21no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—21no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47C22×C414yes—
143C22×Dic47C22×C412yes—
144C22×Dic47C22×C412yes—
145C22×Dic47C2×D412yes—
146C22×Dic47C2×D412yes—
147C22×Dic47C2×Q812yes—
148C22×Dic47C2×Q812yes—
149C22×Dic47#4114yes—
150C4⋊C4×C47—28no—
151C4⋊Dic47—28no—
152C4⋊Dic47—48no—
153Dic47⋊C4—44no—
154Dic47⋊C4—44no—
155Dic47⋊C4—48no—
156C4×Dic47—48no—
157C4×Dic47C4⋊C4116yescomplemented
158C23×C47C23×C411yescharacteristic
159C23×C47C22×D411yescharacteristic
160C23×C47C22×Q811yescharacteristic
161C23×C47C2×#4112yes—
162C23×C47C2×#4112yes—
163C23×C47H(5,𝔽2)11yescharacteristic
164C23×C47H(5,𝔽2)11yescharacteristic
165C23×C47H(5,𝔽2)12yes—
166C23×C472−1+412yes—
167C23×C472−1+411yescharacteristic
168C23×C472−1+411yescharacteristic
169C2×C4×C47—24no—
170C2×C4×C47—22no—
171C2×C4×C47—22no—
172C2×C4×C47—24no—
173C2×C4×C47—24no—
174C2×C4×C47—24no—
175C2×C4×C47—24no—
176C2×C4×C47—24no—
177C2×C4×C47—24no—
178C2×C4×C47—24no—
179C2×C4×C47—42no—
180C2×C4×C47—42no—
181C2×C4×C47—42no—
182C2×C4×C47—42no—
183C2×C4×C47—42no—
184C2×C4×C47—42no—
185C2×C4×C47—42no—
186C2×C4×C47—42no—
187C2×C4×C47—41no—
188C2×C4×C47—41no—
189C2×C4×C47—41no—
190C2×C4×C47—41no—
191C2×C4×C47—41no—
192C2×C4×C47—41no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47—42no—
196C2×C4×C47—42no—
197C2×C4×C47C2×C4⋊C414yes—
198C2×Dic47—22no—
199C2×Dic47—24no—
200C2×Dic47—22no—
201C2×Dic47—22no—
202C2×Dic47—22no—
203C2×Dic47—22no—
204C2×Dic47—22no—
205C2×Dic47—22no—
206C2×Dic47—22no—
207C2×Dic47—24no—
208C2×Dic47—24no—
209C2×Dic47—24no—
210C2×Dic47—24no—
211C2×Dic47—24no—
212C2×Dic47—24no—
213C2×Dic47—24no—
214C2×Dic47—42no—
215C2×Dic47—42no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—41no—
223C2×Dic47—41no—
224C2×Dic47—41no—
225C2×Dic47—41no—
226C2×Dic47—41no—
227C2×Dic47—41no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47C2×C4⋊C414yes—
233C2×Dic47C2×C4⋊C414yes—
234#14298—471nomaximal, Sylow
235C22×C47C22×C4⋊C411yescharacteristic
236C22×C47C4×#4112yes—
237C22×C47#52011yescharacteristic
238C22×C47#52012yes—
239C22×C47#52011yescharacteristic
240C22×C47#52511yescharacteristic
241C22×C47#52511yescharacteristic
242C22×C47#53312yes—
243C22×C47#53511yescharacteristic
244C22×C47#53511yescharacteristic
245C22×C47#53611yescharacteristic
246C22×C47#53611yescharacteristic
247C22×C47#53812yes—
248C22×C47#53812yes—
249C22×C47#53912yes—
250C22×C47#53912yes—
251C22×C47#54112yes—
252C22×C47#54112yes—
253C22×C47#54212yes—
254C22×C47#54212yes—
255C22×C47#54411yescharacteristic
256C22×C47#54411yescharacteristic
257C22×C47#54412yes—
258C4×C47—24no—
259C4×C47—44no—
260C4×C47—44no—
261C4×C47—81no—
262C4×C47—81no—
263Dic47—24no—
264Dic47—24no—
265Dic47—44no—
266Dic47—44no—
267Dic47—81no—
268Dic47—81no—
269C2×#383—471no—
270C2×#382—471no—
271C2×C4×C4⋊C4—471no—
272#1444—471no—
273#1444—472no—
274#1444—471no—
275#1442—472no—
276#1442—471no—
277#1442D4711yescharacteristic
278#1436—472no—
279#1436—472no—
280C2×C47#229611yescharacteristic
281C2×C47#231711yescharacteristic
282C2×C47#231711yescharacteristic
283C2×C47#233412yes—
284C2×C47#233412yes—
285C2×C47#263111yescharacteristic
286C2×C47#263211yescharacteristic
287C2×C47#263311yescharacteristic
288C2×C47#263411yescharacteristic
289C2×C47#266612yes—
290C2×C47#266612yes—
291C22×C4⋊C4—471no—
292C22×C4⋊C4—472no—
293C22×C4⋊C4—471no—
294C22×C4⋊C4—471no—
295C22×C4⋊C4—472no—
296C22×C4⋊C4C2×D4711yescharacteristic
297C22×C42—471no—
298C22×C42—472no—
299#383—478no—
300#382—944no—
301C4×C4⋊C4—944no—
302C2×#95—472no—
303C2×#95—472no—
304C2×#95—471no—
305C2×#95—471no—
306C2×#95—471no—
307C2×#95—472no—
308C2×#95—472no—
309C2×#95—471no—
310C2×#95—472no—
311C2×#95—472no—
312C2×#95—472no—
313C2×#95C2×D4712yes—
314C2×#95C2×D4711yescharacteristic
315C2×#95C2×D4712yes—
316C2×#95C2×D4711yescharacteristic
317C47#1429811yescharacteristic, complemented, Sylow
318C23×C4—471no—
319C23×C4—471no—
320C23×C4—472no—
321C23×C4—472no—
322C23×C4—471no—
323C23×C4—471no—
324C23×C4C22×D4711yescharacteristic
325C23×C4C22×D4711yescharacteristic
326C23×C4C22×D4712yes—
327C23×C4C22×D4712yes—
328C23×C4C22×D4711yescharacteristic
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—944no—
332C2×C4⋊C4—942no—
333C2×C4⋊C4—944no—
334C2×C4⋊C4—942no—
335C2×C4⋊C4—942no—
336C2×C4⋊C4—944no—
337C2×C4⋊C4—1882no—
338C2×C4⋊C4—1882no—
339C2×C4⋊C4—1882no—
340C2×C4⋊C4—1882no—
341C2×C4⋊C4—1882no—
342C2×C4⋊C4—1882no—
343C2×C42—474no—
344C2×C42—474no—
345C2×C42—474no—
346C2×C42—944no—
347C2×C42—1882no—
348C2×C42—1882no—
349#95—22no—
350#95—24no—
351#95—42no—
352#95—478no—
353#95—478no—
354#95—474no—
355#95—944no—
356#95—942no—
357#95—942no—
358#95—942no—
359#95—944no—
360#95—1882no—
361#95—1882no—
362#95—1882no—
363#95C4×D4714yes—
364C24C23×D4711yescharacteristic
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—472no—
390C22×C4—472no—
391C22×C4—472no—
392C22×C4—472no—
393C22×C4—472no—
394C22×C4—472no—
395C22×C4—474no—
396C22×C4—474no—
397C22×C4—941no—
398C22×C4—941no—
399C22×C4—942no—
400C22×C4—941no—
401C22×C4—942no—
402C22×C4—941no—
403C22×C4—942no—
404C22×C4—942no—
405C22×C4—942no—
406C22×C4—942no—
407C22×C4—942no—
408C22×C4—942no—
409C22×C4—942no—
410C22×C4—942no—
411C22×C4—942no—
412C22×C4—942no—
413C22×C4—942no—
414C22×C4—942no—
415C22×C4—941no—
416C22×C4—941no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—941no—
420C22×C4—941no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—941no—
424C22×C4—941no—
425C22×C4C2×C4×D4714yes—
426C22×C4C2×C4×D4712yes—
427C22×C4D4×D4712yes—
428C22×C4#1062535214yes—
429C22×C4Q8×D4712yes—
430C4⋊C4—28no—
431C4⋊C4—948no—
432C4⋊C4—1884no—
433C4⋊C4—1884no—
434C4⋊C4—1888no—
435C4⋊C4—1888no—
436C42—4716no—
437C42—1888no—
438C23C22×C4×D4711yescharacteristic
439C23C2×D4×D4711yescharacteristic
440C23C2×Q8×D4711yescharacteristic
441C23#41×D4712yes—
442C23C2×#1062535212yes—
443C23#1309406911yescharacteristic
444C23#1309407012yes—
445C23#1309407011yescharacteristic
446C23#1309407111yescharacteristic
447C23#1309407112yes—
448C23#1309407211yescharacteristic
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—474no—
478C2×C4—474no—
479C2×C4—942no—
480C2×C4—942no—
481C2×C4—944no—
482C2×C4—942no—
483C2×C4—944no—
484C2×C4—942no—
485C2×C4—942no—
486C2×C4—942no—
487C2×C4—942no—
488C2×C4—942no—
489C2×C4—944no—
490C2×C4—944no—
491C2×C4—944no—
492C2×C4—944no—
493C2×C4—944no—
494C2×C4—944no—
495C2×C4—1882no—
496C2×C4—1882no—
497C2×C4—1881no—
498C2×C4—1881no—
499C2×C4—1881no—
500C2×C4—1881no—
501C2×C4—1881no—
502C2×C4—1881no—
503C2×C4—1882no—
504C2×C4—1882no—
505C2×C4—1882no—
506C2×C4—1882no—
507C2×C4—1882no—
508C2×C4—1882no—
509C2×C4—1882no—
510C2×C4—1882no—
511C2×C4—1882no—
512C2×C4—1882no—
513C2×C4C4⋊C4×D4714yes—
514C22C2×C4⋊C4×D4711yescharacteristic
515C22C4×#1062535212yes—
516C22#1568960211yescharacteristic
517C22#1568960312yes—
518C22#1568960511yescharacteristic
519C22#1568962911yescharacteristic
520C22#1568963111yescharacteristic
521C22#1568967512yes—
522C22#1568968911yescharacteristic
523C22#1568969111yescharacteristic
524C22#1568969311yescharacteristic
525C22#1568969611yescharacteristic
526C22#1568970912yes—
527C22#1568971212yes—
528C22#1568971712yes—
529C22#1568971812yes—
530C22#1568973012yes—
531C22#1568973212yes—
532C22#1568974212yes—
533C22#1568974612yes—
534C22#1568975011yescharacteristic
535C22#1568975311yescharacteristic
536C22#1568975312yes—
537C4—24no—
538C4—44no—
539C4—44no—
540C4—81no—
541C4—81no—
542C4—944no—
543C4—944no—
544C4—3761no—
545C4—3761no—
546C4—1884no—
547C4—1884no—
548C2#2024586711yescharacteristic
549C2#2024601011yescharacteristic
550C2#2024601311yescharacteristic
551C2#2024616012yes—
552C2#2024616812yes—
553C2#2024924211yescharacteristic
554C2#2024924711yescharacteristic
555C2#2024925511yescharacteristic
556C2#2024926611yescharacteristic
557C2#2024957012yes—
558C2#2024957212yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1442C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1442D47
hypocenterC47#14298
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
19. #15688990C4semidirect
157. C4×Dic47C4⋊C4semidirect
317. C47#14298semidirect

Maximal subgroups

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

subgroupindex
C2×#156889902
C2×#156889872
C2×C4⋊C4×Dic472
#202416532
#202416522
#202416472
#202416432
#202416222
#202416192
C47×#14422
#1429847