Finite Group Databasebeta

← #26143249

Subgroups — #26143249

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689062C211yesmaximal, characteristic
2C2×#15689021C211yesmaximal, characteristic
3#20241657C211yesmaximal, characteristic
4#20241652C211yesmaximal, characteristic
5#20241651C212yesmaximal
6#20241646C212yesmaximal
7#20241641C211yesmaximal, characteristic
8#20241621C212yesmaximal
9#20241615C212yesmaximal
10#20241615C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C47C211yesmaximal, characteristic
12C22×C4⋊Dic47C2211yescharacteristic
13C22×C4⋊Dic47C2211yescharacteristic
14C22×Dic47⋊C4C2212yes—
15C22×Dic47⋊C4C2212yes—
16#15689062C418yescomplemented
17#15689021—24no—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2211yescharacteristic
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2212yes—
24C2×C47⋊#95C2212yes—
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2211yescharacteristic
29C2×C47⋊#95C2212yes—
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×C42×C47C2211yescharacteristic
33C22×C42×C47C2211yescharacteristic
34C22×C42×C47C2211yescharacteristic
35C4×C4⋊C4×C47—24no—
36C2×#95×C47C2211yescharacteristic
37C2×#95×C47C2211yescharacteristic
38C23×Dic47C2312yes—
39C23×Dic47C2312yes—
40C23×Dic47C2312yes—
41C23×Dic47C2312yes—
42C2×C4⋊Dic47—22no—
43C2×C4⋊Dic47—22no—
44C2×C4⋊Dic47—22no—
45C2×C4⋊Dic47—42no—
46C2×C4⋊Dic47C2×C414yes—
47C2×Dic47⋊C4—24no—
48C2×Dic47⋊C4—24no—
49C2×Dic47⋊C4—42no—
50C2×Dic47⋊C4—42no—
51C2×Dic47⋊C4—42no—
52C2×Dic47⋊C4—42no—
53C47⋊#95—24no—
54C47⋊#95—24no—
55C47⋊#95—22no—
56C47⋊#95—24no—
57C47⋊#95—42no—
58C47⋊#95—42no—
59C47⋊#95—42no—
60C47⋊#95—42no—
61C47⋊#95—42no—
62C47⋊#95C2×C414yes—
63C47⋊#95C2×C418yes—
64C47⋊#95C2×C418yes—
65C23×C4×C47C2311yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C23×C4×C47C2311yescharacteristic
68C23×C4×C47C2311yescharacteristic
69C23×C4×C47C2311yescharacteristic
70C23×C4×C47C2311yescharacteristic
71C23×C4×C47C2311yescharacteristic
72C2×C4⋊C4×C47—22no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47—22no—
77C2×C4⋊C4×C47C2×C414yes—
78C2×C42×C47—22no—
79C2×C42×C47—22no—
80C2×C42×C47—22no—
81C2×C42×C47—42no—
82C2×C42×C47—42no—
83C2×C42×C47D414yes—
84C2×C42×C47Q814yes—
85#95×C47—22no—
86#95×C47—22no—
87C24×C47C2411yescharacteristic
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—22no—
93C22×C4×C47—22no—
94C22×C4×C47—22no—
95C22×C4×C47—22no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—21no—
103C22×C4×C47—21no—
104C22×C4×C47—22no—
105C22×C4×C47—22no—
106C22×C4×C47—22no—
107C22×C4×C47—22no—
108C22×C4×C47—22no—
109C22×C4×C47—22no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47C22×C412yes—
113C22×C4×C47C22×C412yes—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C2×D412yes—
116C22×C4×C47C2×D412yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×Q812yes—
119C22×C4×C47C2×Q812yes—
120C22×C4×C47C2×Q812yes—
121C22×C4×C47#4112yes—
122C22×C4×C47#4112yes—
123C22×C4×C47#4112yes—
124C22×C4×C47#4112yes—
125C22×Dic47—22no—
126C22×Dic47—22no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—22no—
131C22×Dic47—22no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47C22×C414yes—
152C22×Dic47C22×C414yes—
153C4⋊C4×C47—28no—
154C4⋊C4×C47—28no—
155C42×C47—28no—
156C42×C47—44no—
157C42×C47—44no—
158C4⋊Dic47—28no—
159C4⋊Dic47—44no—
160Dic47⋊C4—48no—
161Dic47⋊C4—48no—
162C23×C47C23×C411yescharacteristic
163C23×C47C22×D411yescharacteristic
164C23×C47C22×Q811yescharacteristic
165C23×C47C2×#4111yescharacteristic
166C23×C47C2×#4111yescharacteristic
167C23×C47C2×#4111yescharacteristic
168C23×C47C2×#4111yescharacteristic
169C23×C47H(5,𝔽2)12yes—
170C23×C47H(5,𝔽2)12yes—
171C23×C472−1+412yes—
172C23×C472−1+412yes—
173C2×C4×C47—22no—
174C2×C4×C47—22no—
175C2×C4×C47—24no—
176C2×C4×C47—22no—
177C2×C4×C47—22no—
178C2×C4×C47—22no—
179C2×C4×C47—22no—
180C2×C4×C47—22no—
181C2×C4×C47—22no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—24no—
185C2×C4×C47—24no—
186C2×C4×C47—24no—
187C2×C4×C47—22no—
188C2×C4×C47—24no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—22no—
192C2×C4×C47—22no—
193C2×C4×C47—22no—
194C2×C4×C47—24no—
195C2×C4×C47—24no—
196C2×C4×C47—41no—
197C2×C4×C47—41no—
198C2×C4×C47—41no—
199C2×C4×C47—41no—
200C2×C4×C47—41no—
201C2×C4×C47—41no—
202C2×C4×C47—42no—
203C2×C4×C47—42no—
204C2×C4×C47—42no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47C4×D414yes—
208C2×C4×C47C4×Q814yes—
209C2×C4×C47C22⋊Q814yes—
210C2×Dic47—24no—
211C2×Dic47—24no—
212C2×Dic47—24no—
213C2×Dic47—24no—
214C2×Dic47—24no—
215C2×Dic47—24no—
216C2×Dic47—42no—
217C2×Dic47—42no—
218C2×Dic47—42no—
219C2×Dic47—42no—
220C2×Dic47—42no—
221C2×Dic47—42no—
222C2×Dic47—42no—
223C2×Dic47—42no—
224C2×Dic47—42no—
225C2×Dic47—42no—
226C2×Dic47—42no—
227C2×Dic47—42no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238#14318—471nomaximal, Sylow
239C22×C47C2×C4×D411yescharacteristic
240C22×C47C2×C4×Q811yescharacteristic
241C22×C47C4×#4111yescharacteristic
242C22×C47#52012yes—
243C22×C47#52012yes—
244C22×C47C2×C22⋊Q811yescharacteristic
245C22×C47#52511yescharacteristic
246C22×C47#52911yescharacteristic
247C22×C47#53311yescharacteristic
248C22×C47#53411yescharacteristic
249C22×C47#53511yescharacteristic
250C22×C47#53611yescharacteristic
251C22×C47#53611yescharacteristic
252C22×C47#53812yes—
253C22×C47#53912yes—
254C22×C47#53912yes—
255C22×C47#54012yes—
256C22×C47#54112yes—
257C22×C47#54112yes—
258C22×C47#54212yes—
259C22×C47#54212yes—
260C22×C47#54412yes—
261C22×C47#54412yes—
262C4×C47—24no—
263C4×C47—24no—
264C4×C47—24no—
265C4×C47—42no—
266C4×C47—42no—
267C4×C47—42no—
268C4×C47—42no—
269Dic47—44no—
270Dic47—44no—
271Dic47—82no—
272Dic47—82no—
273C2×#400—471no—
274C2×#389—471no—
275C2×C4×C4⋊C4D4711yescharacteristic
276#1445—471no—
277#1444—471no—
278#1444—472no—
279#1442—472no—
280#1441—471no—
281#1436—472no—
282#1434—471no—
283#1434—472no—
284C2×C47C4×C22⋊Q811yescharacteristic
285C2×C47#232211yescharacteristic
286C2×C47#232611yescharacteristic
287C2×C47#233012yes—
288C2×C47#233412yes—
289C2×C47#262711yescharacteristic
290C2×C47#262811yescharacteristic
291C2×C47#263411yescharacteristic
292C2×C47#263511yescharacteristic
293C2×C47#265412yes—
294C2×C47#266612yes—
295C22×C4⋊C4—471no—
296C22×C4⋊C4—472no—
297C22×C4⋊C4—472no—
298C22×C4⋊C4—471no—
299C22×C4⋊C4C2×D4711yescharacteristic
300C22×C4⋊C4C2×D4711yescharacteristic
301C22×C42C2×D4711yescharacteristic
302C22×C42C2×D4711yescharacteristic
303C22×C42C2×D4711yescharacteristic
304#400—478no—
305#389—944no—
306C4×C4⋊C4—24no—
307C2×#95—472no—
308C2×#95—471no—
309C2×#95—472no—
310C2×#95—472no—
311C2×#95—472no—
312C2×#95—472no—
313C2×#95—471no—
314C2×#95—472no—
315C2×#95—472no—
316C2×#95—472no—
317C2×#95—472no—
318C2×#95—472no—
319C2×#95C2×D4711yescharacteristic
320C2×#95C2×D4711yescharacteristic
321C47#1431811yescharacteristic, complemented, Sylow
322C23×C4—472no—
323C23×C4—472no—
324C23×C4—472no—
325C23×C4—472no—
326C23×C4C22×D4711yescharacteristic
327C23×C4C22×D4711yescharacteristic
328C23×C4C22×D4711yescharacteristic
329C23×C4C22×D4711yescharacteristic
330C23×C4C22×D4711yescharacteristic
331C23×C4C22×D4711yescharacteristic
332C23×C4C22×D4711yescharacteristic
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—474no—
339C2×C4⋊C4—942no—
340C2×C4⋊C4—942no—
341C2×C4⋊C4—944no—
342C2×C4⋊C4—944no—
343C2×C4⋊C4—942no—
344C2×C4⋊C4—1882no—
345C2×C4⋊C4—1882no—
346C2×C4⋊C4—1882no—
347C2×C4⋊C4—1882no—
348C2×C4⋊C4—1882no—
349C2×C4⋊C4C4×D4714yes—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42Dic9414yes—
356C2×C42C22⋊D4714yes—
357#95—22no—
358#95—22no—
359#95—474no—
360#95—478no—
361#95—478no—
362#95—944no—
363#95—942no—
364#95—944no—
365#95—944no—
366#95—1882no—
367#95—1882no—
368#95—1882no—
369#95—1882no—
370#95—1882no—
371C24C23×D4711yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—474no—
397C22×C4—474no—
398C22×C4—942no—
399C22×C4—942no—
400C22×C4—942no—
401C22×C4—942no—
402C22×C4—942no—
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—942no—
416C22×C4—942no—
417C22×C4—942no—
418C22×C4—942no—
419C22×C4—942no—
420C22×C4—942no—
421C22×C4—942no—
422C22×C4—942no—
423C22×C4—942no—
424C22×C4C2×Dic9412yes—
425C22×C4C2×Dic9412yes—
426C22×C4C2×Dic9412yes—
427C22×C4C2×C4×D4712yes—
428C22×C4C2×C4×D4712yes—
429C22×C4C2×C4×D4712yes—
430C22×C4#1062535012yes—
431C22×C4#1062535012yes—
432C22×C4#1062535012yes—
433C22×C4#1062535012yes—
434C22×C4C2×C22⋊D4712yes—
435C22×C4C2×C22⋊D4712yes—
436C22×C4C2×C22⋊D4712yes—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—948no—
440C4⋊C4—1888no—
441C4⋊C4—1888no—
442C4⋊C4—1884no—
443C42—28no—
444C42—44no—
445C42—44no—
446C23C22×C4×D4711yescharacteristic
447C23C22×C22⋊D4711yescharacteristic
448C23C22×Dic9411yescharacteristic
449C23C2×#1062535011yescharacteristic
450C23C2×#1062535011yescharacteristic
451C23C2×#1062535011yescharacteristic
452C23C2×#1062535011yescharacteristic
453C23#1309406912yes—
454C23#1309407012yes—
455C23#1309407112yes—
456C23#1309407212yes—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—944no—
492C2×C4—944no—
493C2×C4—944no—
494C2×C4—944no—
495C2×C4—944no—
496C2×C4—944no—
497C2×C4—1882no—
498C2×C4—1882no—
499C2×C4—1882no—
500C2×C4—1882no—
501C2×C4—1882no—
502C2×C4—1882no—
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×C4—1882no—
514C2×C4—1882no—
515C2×C4—1882no—
516C2×C4—1882no—
517C2×C4—1882no—
518C2×C4—1882no—
519C2×C4C4×C22⋊D4714yes—
520C2×C4C4×Dic9414yes—
521C2×C4#1309400314yes—
522C22C2×C4×C22⋊D4711yescharacteristic
523C22C2×C4×Dic9411yescharacteristic
524C22C4×#1062535011yescharacteristic
525C22#1568960212yes—
526C22#1568960512yes—
527C22C2×#1309400311yescharacteristic
528C22#1568963411yescharacteristic
529C22#1568965711yescharacteristic
530C22#1568968011yescharacteristic
531C22#1568968211yescharacteristic
532C22#1568968711yescharacteristic
533C22#1568969411yescharacteristic
534C22#1568969711yescharacteristic
535C22#1568970612yes—
536C22#1568971512yes—
537C22#1568972012yes—
538C22#1568972412yes—
539C22#1568973112yes—
540C22#1568973112yes—
541C22#1568974412yes—
542C22#1568974512yes—
543C22#1568975112yes—
544C22#1568975412yes—
545C4—24no—
546C4—24no—
547C4—24no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—3762no—
553C4—3762no—
554C4—1884no—
555C4—1884no—
556C2C4×#1309400311yescharacteristic
557C2#2024605311yescharacteristic
558C2#2024609511yescharacteristic
559C2#2024612512yes—
560C2#2024615712yes—
561C2#2024920111yescharacteristic
562C2#2024920911yescharacteristic
563C2#2024926411yescharacteristic
564C2#2024927211yescharacteristic
565C2#2024946012yes—
566C2#2024956712yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C4×C4⋊C4×C47C2
socle, prime radical, cosocleC24×C47C24
hypercenterC2×C4×C4⋊C4D47
hypocenterC47#14318
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
16. #15689062C4semidirect
321. C47#14318semidirect

Maximal subgroups

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

subgroupindex
C2×#156890622
C2×#156890212
#202416572
#202416522
#202416512
#202416462
#202416412
#202416212
#202416152
C2×C4×C4⋊C4×C472
#1431847