Finite Group Databasebeta

← #26145233

Subgroups — #26145233

Order 12032.

1246 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689021C211yesmaximal, characteristic
2C2×#15689008C212yesmaximal
3#20241646C212yesmaximal
4#20241640C212yesmaximal
5#20241639C212yesmaximal
6#20241619C212yesmaximal
7Dic47⋊#95C212yesmaximal
8#20241615C211yesmaximal, characteristic
9C47×#1434C211yesmaximal, characteristic
10C22×C4⋊Dic47C2211yescharacteristic
11C22×C4⋊Dic47C2211yescharacteristic
12C22×Dic47⋊C4C2212yes—
13C22×Dic47⋊C4C2212yes—
14C22×Dic47⋊C4C2212yes—
15C22×C4×Dic47C2212yes—
16#15689021C418yescomplemented
17#15689008—28no—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C22×C42×C47C2211yescharacteristic
29C2×#95×C47C2212yes—
30C2×#95×C47C2212yes—
31C2×#95×C47C2211yescharacteristic
32C2×#95×C47C2211yescharacteristic
33C23×Dic47C2312yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C2×C4⋊Dic47—41no—
38C2×C4⋊Dic47—41no—
39C2×C4⋊Dic47C2×C414yes—
40C2×C4⋊Dic47C2×C414yes—
41C2×C4⋊Dic47D414yes—
42C2×C4⋊Dic47Q814yes—
43C2×Dic47⋊C4—24no—
44C2×Dic47⋊C4—24no—
45C2×Dic47⋊C4—24no—
46C2×Dic47⋊C4—42no—
47C2×Dic47⋊C4—42no—
48C2×Dic47⋊C4—42no—
49C2×Dic47⋊C4—42no—
50C2×Dic47⋊C4—42no—
51C2×Dic47⋊C4C2×C418yes—
52C2×C4×Dic47—24no—
53C2×C4×Dic47—24no—
54C2×C4×Dic47—42no—
55C47⋊#95—24no—
56C47⋊#95—24no—
57C47⋊#95—24no—
58C47⋊#95—24no—
59C47⋊#95—24no—
60C47⋊#95—41no—
61C47⋊#95—42no—
62C47⋊#95—42no—
63C47⋊#95—41no—
64C47⋊#95C2×C418yes—
65C23×C4×C47C2312yes—
66C23×C4×C47C2311yescharacteristic
67C23×C4×C47C2311yescharacteristic
68C23×C4×C47C2312yes—
69C23×C4×C47C2311yescharacteristic
70C2×C42×C47—41no—
71C2×C42×C47—41no—
72C2×C42×C47C2×C414yes—
73#95×C47—24no—
74#95×C47—24no—
75#95×C47—41no—
76#95×C47—41no—
77C24×C47C2411yescharacteristic
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
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—22no—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47C22×C412yes—
104C22×C4×C47C22×C412yes—
105C22×C4×C47C22×C412yes—
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2×Q812yes—
108C22×C4×C47#4114yes—
109C22×Dic47—22no—
110C22×Dic47—22no—
111C22×Dic47—22no—
112C22×Dic47—22no—
113C22×Dic47—22no—
114C22×Dic47—22no—
115C22×Dic47—22no—
116C22×Dic47—22no—
117C22×Dic47—22no—
118C22×Dic47—22no—
119C22×Dic47—22no—
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×Dic47C22×C414yes—
133C22×Dic47C22×C414yes—
134C22×Dic47C2×D414yes—
135C22×Dic47C2×Q814yes—
136C22×Dic47#4114yes—
137C42×C47—44no—
138C4⋊Dic47—44no—
139C4⋊Dic47C4⋊C4116yescomplemented
140Dic47⋊C4—48no—
141Dic47⋊C4—48no—
142Dic47⋊C4—48no—
143C4×Dic47—48no—
144C23×C47C23×C411yescharacteristic
145C23×C47C22×D411yescharacteristic
146C23×C47C22×Q811yescharacteristic
147C23×C47C2×#4112yes—
148C23×C47C2×#4112yes—
149C23×C47H(5,𝔽2)11yescharacteristic
150C23×C47H(5,𝔽2)11yescharacteristic
151C23×C47H(5,𝔽2)12yes—
152C23×C472−1+411yescharacteristic
153C23×C472−1+411yescharacteristic
154C23×C472−1+412yes—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—22no—
161C2×C4×C47—22no—
162C2×C4×C47—22no—
163C2×C4×C47—22no—
164C2×C4×C47—22no—
165C2×C4×C47—22no—
166C2×C4×C47—22no—
167C2×C4×C47—24no—
168C2×C4×C47—24no—
169C2×C4×C47—24no—
170C2×C4×C47—42no—
171C2×C4×C47—42no—
172C2×C4×C47—42no—
173C2×C4×C47—42no—
174C2×C4×C47—42no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—42no—
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×C47C2×C4⋊C414yes—
186C2×Dic47—24no—
187C2×Dic47—24no—
188C2×Dic47—24no—
189C2×Dic47—24no—
190C2×Dic47—24no—
191C2×Dic47—24no—
192C2×Dic47—24no—
193C2×Dic47—24no—
194C2×Dic47—24no—
195C2×Dic47—24no—
196C2×Dic47—24no—
197C2×Dic47—24no—
198C2×Dic47—42no—
199C2×Dic47—42no—
200C2×Dic47—42no—
201C2×Dic47—42no—
202C2×Dic47—42no—
203C2×Dic47—42no—
204C2×Dic47—42no—
205C2×Dic47—42no—
206C2×Dic47—42no—
207C2×Dic47—42no—
208C2×Dic47—42no—
209C2×Dic47—42no—
210C2×Dic47—42no—
211C2×Dic47—42no—
212C2×Dic47—42no—
213C2×Dic47C2×C4⋊C418yes—
214#14487—471nomaximal, Sylow
215C22×C47C22×C4⋊C411yescharacteristic
216C22×C47C4×#4111yescharacteristic
217C22×C47C4×#4111yescharacteristic
218C22×C47#51811yescharacteristic
219C22×C47#51811yescharacteristic
220C22×C47#51911yescharacteristic
221C22×C47#51911yescharacteristic
222C22×C47#53711yescharacteristic
223C22×C47#53711yescharacteristic
224C22×C47#53812yes—
225C22×C47#53912yes—
226C22×C47#54012yes—
227C22×C47#54212yes—
228C22×C47#54411yescharacteristic
229C22×C47#54411yescharacteristic
230C22×C47#54612yes—
231C22×C47#54812yes—
232C22×C47#55212yes—
233C22×C47#55212yes—
234C22×C47#55412yes—
235C22×C47#55712yes—
236C22×C47#56312yes—
237C22×C47#56312yes—
238C4×C47—24no—
239C4×C47—42no—
240C4×C47—42no—
241C4×C47—44no—
242C4×C47—82no—
243Dic47—28no—
244Dic47—44no—
245Dic47—44no—
246Dic47—82no—
247C2×#389—471no—
248C2×#387—472no—
249#1442—472no—
250#1441—472no—
251#1441—472no—
252#1436—472no—
253C4⋊#95—472no—
254#1434—471no—
255#1434D4711yescharacteristic
256C2×C47#231611yescharacteristic
257C2×C47#233211yescharacteristic
258C2×C47#233311yescharacteristic
259C2×C47#234911yescharacteristic
260C2×C47#235111yescharacteristic
261C2×C47#237111yescharacteristic
262C2×C47#238111yescharacteristic
263C2×C47#270212yes—
264C2×C47#270712yes—
265C2×C47#277412yes—
266C2×C47#279112yes—
267C22×C4⋊C4—471no—
268C22×C4⋊C4—472no—
269C22×C4⋊C4—471no—
270C22×C4⋊C4—472no—
271C22×C4⋊C4—472no—
272C22×C42—472no—
273C22×C42C2×D4711yescharacteristic
274#389—478no—
275#387—948no—
276C2×#95—472no—
277C2×#95—472no—
278C2×#95—472no—
279C2×#95—472no—
280C2×#95—472no—
281C2×#95—472no—
282C2×#95—471no—
283C2×#95—471no—
284C2×#95—472no—
285C2×#95—472no—
286C2×#95C2×D4712yes—
287C2×#95C2×D4712yes—
288C2×#95C2×D4711yescharacteristic
289C2×#95C2×D4711yescharacteristic
290C47#1448711yescharacteristic, complemented, Sylow
291C23×C4—472no—
292C23×C4—472no—
293C23×C4—472no—
294C23×C4—472no—
295C23×C4C22×D4712yes—
296C23×C4C22×D4711yescharacteristic
297C23×C4C22×D4711yescharacteristic
298C23×C4C22×D4712yes—
299C23×C4C22×D4711yescharacteristic
300C2×C4⋊C4—474no—
301C2×C4⋊C4—474no—
302C2×C4⋊C4—474no—
303C2×C4⋊C4—478no—
304C2×C4⋊C4—474no—
305C2×C4⋊C4—944no—
306C2×C4⋊C4—944no—
307C2×C4⋊C4—944no—
308C2×C4⋊C4—1882no—
309C2×C4⋊C4—1882no—
310C2×C4⋊C4—1882no—
311C2×C4⋊C4—1882no—
312C2×C4⋊C4—1882no—
313C2×C4⋊C4—1881no—
314C2×C4⋊C4—1881no—
315C2×C42—41no—
316C2×C42—41no—
317C2×C42—944no—
318C2×C42—944no—
319C2×C42—1882no—
320C2×C42C4×D4714yes—
321#95—24no—
322#95—24no—
323#95—41no—
324#95—41no—
325#95—478no—
326#95—944no—
327#95—944no—
328#95—944no—
329#95—944no—
330#95—944no—
331#95—1882no—
332#95—1881no—
333#95—1881no—
334#95—1882no—
335C24C23×D4711yescharacteristic
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—474no—
362C22×C4—474no—
363C22×C4—474no—
364C22×C4—474no—
365C22×C4—474no—
366C22×C4—942no—
367C22×C4—942no—
368C22×C4—942no—
369C22×C4—942no—
370C22×C4—942no—
371C22×C4—942no—
372C22×C4—942no—
373C22×C4—942no—
374C22×C4—942no—
375C22×C4—942no—
376C22×C4—942no—
377C22×C4—942no—
378C22×C4—942no—
379C22×C4—942no—
380C22×C4—942no—
381C22×C4—942no—
382C22×C4—942no—
383C22×C4—942no—
384C22×C4—942no—
385C22×C4—942no—
386C22×C4—942no—
387C22×C4—942no—
388C22×C4—942no—
389C22×C4C2×C4×D4712yes—
390C22×C4C2×C4×D4712yes—
391C22×C4C2×C4×D4712yes—
392C22×C4#1062535014yes—
393C22×C4D4×D4712yes—
394C22×C4Q8×D4712yes—
395C4⋊C4—4716no—
396C4⋊C4—1888no—
397C4⋊C4—1884no—
398C4⋊C4—1888no—
399C4⋊C4—1888no—
400C42—44no—
401C42—1888no—
402C23C22×C4×D4711yescharacteristic
403C23C2×D4×D4711yescharacteristic
404C23C2×Q8×D4711yescharacteristic
405C23#41×D4712yes—
406C23C2×#1062535012yes—
407C23#1309406911yescharacteristic
408C23#1309406911yescharacteristic
409C23#1309407012yes—
410C23#1309407112yes—
411C23#1309407111yescharacteristic
412C23#1309407111yescharacteristic
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—478no—
444C2×C4—944no—
445C2×C4—944no—
446C2×C4—944no—
447C2×C4—944no—
448C2×C4—944no—
449C2×C4—944no—
450C2×C4—944no—
451C2×C4—944no—
452C2×C4—944no—
453C2×C4—944no—
454C2×C4—944no—
455C2×C4—944no—
456C2×C4—1882no—
457C2×C4—1882no—
458C2×C4—1882no—
459C2×C4—1882no—
460C2×C4—1882no—
461C2×C4—1882no—
462C2×C4—1882no—
463C2×C4—1882no—
464C2×C4—1882no—
465C2×C4—1882no—
466C2×C4—1882no—
467C2×C4—1882no—
468C2×C4—1882no—
469C2×C4—1882no—
470C2×C4—1882no—
471C2×C4C4⋊C4×D4714yes—
472C22C2×C4⋊C4×D4711yescharacteristic
473C22#1568959111yescharacteristic
474C22C4×#1062535011yescharacteristic
475C22#1568959611yescharacteristic
476C22#1568959711yescharacteristic
477C22#1568960011yescharacteristic
478C22#1568960011yescharacteristic
479C22#1568970011yescharacteristic
480C22#1568970011yescharacteristic
481C22#1568970912yes—
482C22#1568971512yes—
483C22#1568972512yes—
484C22#1568973812yes—
485C22#1568975211yescharacteristic
486C22#1568975211yescharacteristic
487C22#1568976912yes—
488C22#1568978112yes—
489C22#1568980612yes—
490C22#1568981212yes—
491C22#1568983312yes—
492C22#1568985012yes—
493C22#1568987512yes—
494C22#1568987512yes—
495C4—24no—
496C4—42no—
497C4—42no—
498C4—44no—
499C4—82no—
500C4—948no—
501C4—3762no—
502C4—1884no—
503C4—1884no—
504C2#2024600511yescharacteristic
505C2#2024614411yescharacteristic
506C2#2024615011yescharacteristic
507C2#2024629911yescharacteristic
508C2#2024630511yescharacteristic
509C2#2024649311yescharacteristic
510C2#2024656811yescharacteristic
511C2#2025001112yes—
512C2#2025008612yes—
513C2#2025094012yes—
514C2#2025117112yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×#1434C2
socle, prime radical, cosocleC24×C47C24
hypercenter#1434D47
hypocenterC47#14487
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
16. #15689021C4semidirect
139. C4⋊Dic47C4⋊C4semidirect
290. C47#14487semidirect

Maximal subgroups

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

subgroupindex
C2×#156890212
C2×#156890082
#202416462
#202416402
#202416392
#202416192
Dic47⋊#952
#202416152
C47×#14342
#1448747