Finite Group Databasebeta

← #26107172

Subgroups — #26107172

Order 12032.

804 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243150C211yesmaximal, characteristic, complemented
2C2×#15688857C212yesmaximal
3C4×C47⋊#95C211yesmaximal, characteristic, complemented
4#20241266C212yesmaximal, complemented
5C4×#121×C47C211yesmaximal, characteristic
6C22×C4×Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×#13093935—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12#15688857C418yescomplemented
13C2×#121×C47C2211yescharacteristic
14C2×C4×D4×C47—21no—
15C2×C4×D4×C47C411yescharacteristic, complemented
16C22×C42×C47C2211yescharacteristic
17C47×#390C411yescharacteristic, complemented
18C47×#389C411yescharacteristic, complemented
19C47×#388—21no—
20C47×#386—21no—
21C47×#380C2211yescharacteristic
22C4×C4⋊C4×C47C411yescharacteristic, complemented
23C4×C22⋊C4×C47—21no—
24C23×Dic47—22no—
25C22×C47⋊C8—22no—
26C2×C4×Dic47—24no—
27C2×C4×Dic47—42no—
28C2×C4×Dic47—42no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)C2×C414yescomplemented
31#13093935—42no—
32C47⋊#95—24no—
33C47⋊#95—22no—
34C47⋊#95C2×C414yescomplemented
35C22×D4×C47—21no—
36C22×D4×C47C2×C411yescharacteristic
37C23×C4×C47—21no—
38C23×C4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C23×C4×C47C2311yescharacteristic
41#121×C47—22no—
42#121×C47C2×C414yes—
43C4×D4×C47—42no—
44C4×D4×C47—44no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47C2×C411yescharacteristic
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C42×C47—21no—
55C2×C42×C47—42no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47C2×C411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60#95×C47—21no—
61#95×C47—21no—
62#95×C47—21no—
63C24×C47—21no—
64C24×C47C2×D411yescharacteristic
65C2×D4×C47—24no—
66C2×D4×C47—21no—
67C2×D4×C47—41no—
68C2×D4×C47—41no—
69C2×D4×C47—42no—
70C2×D4×C47—41no—
71C2×D4×C47—41no—
72C2×D4×C47C4212yescomplemented
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—21no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47—42no—
86C22×C4×C47—42no—
87C22×C4×C47—42no—
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C22⋊C411yescharacteristic
91C22×C4×C47C22⋊C411yescharacteristic
92C22×C4×C47C22×C411yescharacteristic
93C22×C4×C47C22×C412yes—
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47#4112yes—
96C22×Dic47—24no—
97C22×Dic47—24no—
98C22×Dic47—24no—
99C22×Dic47—42no—
100C22×Dic47—42no—
101C22×Dic47—42no—
102C22×Dic47—42no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—42no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47C4212yescomplemented
108C22⋊C4×C47—24no—
109C22⋊C4×C47—42no—
110C22⋊C4×C47—42no—
111C22⋊C4×C47—42no—
112C22⋊C4×C47—42no—
113C42×C47—22no—
114C42×C47—22no—
115C42×C47—41no—
116C42×C47—41no—
117C42×C47—42no—
118C4×Dic47—48no—
119C2×C47⋊C8—24no—
120C2×C47⋊C8—42no—
121C2×C47⋊C8—42no—
122C23×C47—21no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—22no—
127C23×C47—41no—
128C23×C47—41no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47C2×C22⋊C411yescharacteristic
134C23×C47C4×D412yes—
135C23×C47AGL(1,ℤ/8)11yescharacteristic
136C23×C47AGL(1,ℤ/8)12yes—
137C23×C47#13711yescharacteristic
138D4×C47—42no—
139D4×C47—42no—
140D4×C47—44no—
141D4×C47—44no—
142C2×C4×C47—24no—
143C2×C4×C47—24no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—22no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—42no—
159C2×C4×C47—42no—
160C2×C4×C47—44no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—84no—
170C2×C4×C47#9912yes—
171C2×C4×C47#10412yes—
172C2×C4×C47#10412yes—
173C2×C4×C47C2×C4211yescharacteristic
174C2×C4×C47#11711yescharacteristic
175C2×C4×C47C4×D412yes—
176C2×Dic47—28no—
177C2×Dic47—44no—
178C2×Dic47—44no—
179C2×Dic47—44no—
180C2×Dic47—44no—
181C2×Dic47—44no—
182C2×Dic47—44no—
183C47⋊C8—44no—
184#8218—471nomaximal, Sylow
185C22×C47—22no—
186C22×C47—22no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—22no—
190C22×C47—22no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—42no—
194C22×C47—42no—
195C22×C47—42no—
196C22×C47—81no—
197C22×C47—81no—
198C22×C47—81no—
199C22×C47—81no—
200C22×C47C4×C22⋊C411yescharacteristic
201C22×C47C2×#9911yescharacteristic
202C22×C47#41011yescharacteristic
203C22×C47#41711yescharacteristic
204C22×C47#41811yescharacteristic
205C22×C47C2×#10411yescharacteristic
206C22×C47C2×#10411yescharacteristic
207C22×C47#44012yes—
208C22×C47#44212yes—
209C4×C47—22no—
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—22no—
213C4×C47—22no—
214C4×C47—22no—
215C4×C47—42no—
216C4×C47—41no—
217C4×C47—81no—
218C4×C47—81no—
219Dic47—48no—
220C4×#121D4711yescharacteristic
221#1749—471no—
222C2×#327—472no—
223C4×#95—471no—
224#1294—472no—
225C2×C47—22no—
226C2×C47—22no—
227C2×C47—82no—
228C2×C47#150411yescharacteristic
229C2×C47#150511yescharacteristic
230C2×C47#150711yescharacteristic
231C2×C47#150911yescharacteristic
232C2×C47C4×#9911yescharacteristic
233C2×C47C4×#10411yescharacteristic
234C2×C47#176211yescharacteristic
235C2×#121C2×D4711yescharacteristic
236C2×C4×D4—21no—
237C2×C4×D4Dic4711yescharacteristic, complemented
238C22×C42—941no—
239C22×C42C2×D4711yescharacteristic
240C2×C22⋊C8—472no—
241C2×#97—941no—
242#390Dic4711yescharacteristic, complemented
243#389Dic4711yescharacteristic, complemented
244#388—21no—
245#386—21no—
246#380C2×D4711yescharacteristic
247C4×C4⋊C4Dic4711yescharacteristic, complemented
248C4×C22⋊C4—21no—
249C2×#95—472no—
250C2×#95—941no—
251#336—941no—
252#327—478no—
253C47#821811yescharacteristic, complemented, Sylow
254C22×D4—21no—
255C22×D4C2×Dic4711yescharacteristic
256C23×C4—21no—
257C23×C4—942no—
258C23×C4C22⋊D4711yescharacteristic
259C23×C4C22⋊D4711yescharacteristic
260C23×C4C22×D4711yescharacteristic
261C22×C8—942no—
262#121—22no—
263#121C4×D4714yes—
264C4×D4—42no—
265C4×D4—44no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C2×Dic4711yescharacteristic
268C2×C4⋊C4C2×Dic4711yescharacteristic
269C2×C4⋊C4C2×Dic4711yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—21no—
276C2×C42—42no—
277C2×C42—944no—
278C2×C42—1882no—
279C2×C42—1882no—
280C2×C42C2×Dic4711yescharacteristic
281C2×C42C2×Dic4711yescharacteristic
282C2×C42C22⋊D4711yescharacteristic
283C2×C42C22⋊D4711yescharacteristic
284C22⋊C8—474no—
285C22⋊C8—942no—
286#97—1882no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—474no—
291#95—944no—
292#95—942no—
293C24—21no—
294C24C2×C22⋊D4711yescharacteristic
295C2×D4—24no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4C4×Dic4712yescomplemented
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—944no—
319C22×C4—944no—
320C22×C4—944no—
321C22×C4—1882no—
322C22×C4—1882no—
323C22×C4—1882no—
324C22×C4—1882no—
325C22×C4C22⋊Dic4711yescharacteristic
326C22×C4C22⋊Dic4711yescharacteristic
327C22×C4C22⋊Dic4711yescharacteristic
328C22×C4C22⋊Dic4711yescharacteristic
329C22×C4C2×C4×D4712yes—
330C22×C4#1062535012yes—
331C22×C4C22×Dic4711yescharacteristic
332C22×C4C2×C22⋊D4711yescharacteristic
333C2×C8—944no—
334C2×C8—1882no—
335C2×C8—1882no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4C4×Dic4712yescomplemented
341C22⋊C4—24no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42—42no—
351C42—1888no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C2×C22⋊Dic4711yescharacteristic
364C23C4×C22⋊D4712yes—
365C23#1309404911yescharacteristic
366C23#1309405012yes—
367C23#1309405411yescharacteristic
368D4—42no—
369D4—42no—
370D4—44no—
371D4—44no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—84no—
400C2×C4—948no—
401C2×C4—1884no—
402C2×C4—1884no—
403C2×C4—1884no—
404C2×C4—1884no—
405C2×C4—1884no—
406C2×C4—1884no—
407C2×C4#1309393912yes—
408C2×C4#1309395112yes—
409C2×C4#1309395112yes—
410C2×C4C2×C4×Dic4711yescharacteristic
411C2×C4#1309398311yescharacteristic
412C2×C4C4×C22⋊D4712yes—
413C8—1884no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22C4×C22⋊Dic4711yescharacteristic
430C22C2×#1309393911yescharacteristic
431C22#1568909711yescharacteristic
432C22#1568912411yescharacteristic
433C22#1568912811yescharacteristic
434C22C2×#1309395111yescharacteristic
435C22C2×#1309395111yescharacteristic
436C22#1568922312yes—
437C22#1568923312yes—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—22no—
444C4—42no—
445C4—41no—
446C4—81no—
447C4—81no—
448C4—1888no—
449C2—22no—
450C2—22no—
451C2—82no—
452C2#2024189211yescharacteristic
453C2#2024189611yescharacteristic
454C2#2024190411yescharacteristic
455C2#2024191311yescharacteristic
456C2C4×#1309393911yescharacteristic
457C2C4×#1309395111yescharacteristic
458C2#2024321311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC4×#121×C47C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenterC4×#121D47
hypocenterC47#8218
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #20243150C2semidirect
3. C4×C47⋊#95C2semidirect
4. #20241266C2semidirect
12. #15688857C4semidirect
15. C2×C4×D4×C47C4semidirect
17. C47×#390C4semidirect
18. C47×#389C4semidirect
22. C4×C4⋊C4×C47C4semidirect
30. C22⋊(C47⋊C8)C2×C4semidirect
34. C47⋊#95C2×C4semidirect
72. C2×D4×C47C42semidirect
107. C4⋊C4×C47C42semidirect
237. C2×C4×D4Dic47semidirect
242. #390Dic47semidirect
243. #389Dic47semidirect
247. C4×C4⋊C4Dic47semidirect
253. C47#8218semidirect
302. C2×D4C4×Dic47semidirect
340. C4⋊C4C4×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202431502
C2×#156888572
C4×C47⋊#952
#202412662
C4×#121×C472
#821847