Finite Group Databasebeta

← #26199850

Subgroups — #26199850

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254700C211yesmaximal, characteristic, complemented
2#20254326C211yesmaximal, characteristic, complemented
3#20253840C211yesmaximal, characteristic
4#20242872C212yesmaximal, complemented
5#20242860C212yesmaximal, complemented
6#20242748C212yesmaximal, complemented
7#20242736C212yesmaximal, complemented
8#20242140C211yesmaximal, characteristic, complemented
9#20242038C211yesmaximal, characteristic, complemented
10C2×#15688866C211yesmaximal, characteristic, complemented
11C47×#2962C211yesmaximal, characteristic, complemented
12C2×#13094052—21no—
13C2×#13094052—21no—
14#15689673C2211yescharacteristic
15C2×#13094021C2211yescharacteristic
16#15689512—22no—
17#15689500C2212yes—
18#15689497—21no—
19#15689496C2212yes—
20#15689471—21no—
21#15689410—21no—
22#15689408C2212yescomplemented
23#15689354—22no—
24#15689325—21no—
25#15689323C2212yes—
26#15689323C2212yes—
27#15689323C2212yescomplemented
28#15689273—22no—
29#15689176C2211yescharacteristic
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31#15688876C2212yescomplemented
32#15688874C2212yescomplemented
33#15688870C2211yescharacteristic, complemented
34#15688870C2211yescharacteristic, complemented
35#15688866C2212yes—
36#15688866C2212yes—
37#15688866C2211yescharacteristic, complemented
38#15688866C2211yescharacteristic, complemented
39C47×#533C2211yescharacteristic
40C47×#446—22no—
41C47×#446C2212yescomplemented
42C8×Q8×C47C2212yescomplemented
43C47×#424—21no—
44C2×C4⋊C8×C47C2211yescharacteristic
45C47×#405—21no—
46C4×M16×C47C2211yescharacteristic
47C2×#10625350—21no—
48C22×D188—21no—
49C22×D188—21no—
50C22×D188—21no—
51#13094052—44no—
52#13094052—44no—
53C2×#10625325—22no—
54C2×#10625325—22no—
55C2×#10625336—22no—
56C2×#10625336—22no—
57C2×#10625336—22no—
58C2×D376—22no—
59C2×D376—22no—
60C22×C47⋊C8—21no—
61#13094024C2311yescharacteristic
62#13094021—21no—
63#13094021—21no—
64#13094021—21no—
65#13094021C2312yes—
66#13094021C2311yescharacteristic
67#13094012—21no—
68#13093998—21no—
69#13093998—21no—
70C4×D188—21no—
71C4×D188—21no—
72#13093955—22no—
73C4⋊(C47⋊C8)—21no—
74C4⋊(C47⋊C8)C2311yescharacteristic
75C4⋊(C47⋊C8)C2311yescharacteristic
76C4⋊(C47⋊C8)C2312yes—
77#13093948—22no—
78#13093944—22no—
79#13093944—22no—
80#13093944—22no—
81C22⋊(C47⋊C8)—21no—
82C4×C47⋊C8—21no—
83C2×M16×C47D411yescharacteristic
84C2×M16×C47D411yescharacteristic
85C22×C8×C47—21no—
86C4⋊Q8×C47D411yescharacteristic
87C4⋊Q8×C47D411yescharacteristic
88#125×C47—21no—
89C22⋊Q8×C47—21no—
90C22⋊Q8×C47D412yes—
91C4×Q8×C47—21no—
92C4×Q8×C47C2312yes—
93#117×C47—21no—
94C2×C42×C47C2311yescharacteristic
95C4⋊C8×C47—22no—
96C4⋊C8×C47—22no—
97C4⋊C8×C47—21no—
98C4⋊C8×C47C2311yescharacteristic
99C4⋊C8×C47C2311yescharacteristic
100#97×C47—22no—
101#97×C47D412yes—
102C4×C8×C47—22no—
103C4×C8×C47C2312yes—
104C23×D47—41no—
105C2×Q8×C47—21no—
106C2×Q8×C47C2×D412yes—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C2×C22⋊D47—41no—
111#10625350—42no—
112#10625350—42no—
113C2×D188—21no—
114C2×D188—22no—
115C2×D188—21no—
116C2×D188—22no—
117C2×D188—21no—
118C2×D188—22no—
119C2×D188—42no—
120C2×D188—42no—
121C2×D188—41no—
122C2×D188—41no—
123C2×D188—41no—
124C2×D188—81no—
125C2×D188—81no—
126C2×D188—82no—
127C2×C4×D47—41no—
128C2×Dic94—21no—
129M16×C47—22no—
130M16×C47SD1614yescomplemented
131C2×C8×C47—21no—
132C2×C8×C47—21no—
133C2×C8×C47—42no—
134C2×C8×C47C2×D412yes—
135C2×C8×C47C2×D412yes—
136C4⋊C4×C47—21no—
137C4⋊C4×C47—21no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47C2×D412yes—
142C22⋊C4×C47—22no—
143C42×C47—22no—
144C42×C47C2×D411yescharacteristic
145C42×C47C2×D411yescharacteristic
146C42×C47#4111yescharacteristic
147C42×C47C2411yescharacteristic
148#10625336—82no—
149#10625336—82no—
150#10625336—82no—
151(C2×D47)⋊C4—41no—
152(C2×D47)⋊C4—41no—
153C4⋊Dic47—21no—
154C4⋊Dic47—21no—
155C2×C47⋊C8—21no—
156C2×C47⋊C8—21no—
157C2×C47⋊C8—22no—
158C2×C47⋊C8—81no—
159D376—44no—
160D376—44no—
161#10625325—44no—
162#10625325—44no—
163C23×C47#12111yescharacteristic
164C22×D47—41no—
165C22×D47—42no—
166C22×D47—82no—
167C22×D47—82no—
168C22×D47—82no—
169Q8×C47—22no—
170Q8×C47—22no—
171Q8×C47—81no—
172C2×C4×C47—21no—
173C2×C4×C47—21no—
174C2×C4×C47—22no—
175C2×C4×C47—22no—
176C2×C4×C47—41no—
177C2×C4×C47#12111yescharacteristic
178C2×C4×C47#12111yescharacteristic
179C2×C4×C47#12111yescharacteristic
180C2×C4×C47C2×SD1612yes—
181C2×C4×C47AGL(1,ℤ/8)12yes—
182C2×C4×C47C22×D411yescharacteristic
183C2×C4×C47C22×D411yescharacteristic
184C2×C4×C47C2×#4111yescharacteristic
185C22⋊D47—82no—
186C2×Dic47—41no—
187D188—42no—
188D188—41no—
189D188—42no—
190D188—41no—
191D188—41no—
192D188—42no—
193D188—42no—
194D188—41no—
195D188—81no—
196D188—82no—
197D188—82no—
198D188—82no—
199D188—82no—
200C4×D47—82no—
201Dic94—41no—
202Dic94—41no—
203C8×C47—22no—
204C8×C47—41no—
205C8×C47—41no—
206C8×C47C2×SD1614yes—
207C47⋊C8—42no—
208C47⋊C8—41no—
209C47⋊C8—41no—
210#19238—471nomaximal, Sylow
211C22×C47—21no—
212C22×C47—21no—
213C22×C47#46012yes—
214C22×C47C2×#12111yescharacteristic
215C2×D47—81no—
216C2×D47—82no—
217C2×D47—82no—
218C2×D47—81no—
219C2×D47—82no—
220C2×D47—81no—
221C2×D47—164no—
222C4×C47—21no—
223C4×C47—21no—
224C4×C47—21no—
225C4×C47—42no—
226C4×C47—42no—
227C4×C47#46012yes—
228C4×C47C22×SD1611yescharacteristic
229C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C47#57611yescharacteristic
231C4×C47#57711yescharacteristic
232Dic47—81no—
233#3246—471no—
234#3208—471no—
235#3151—471no—
236#2962D4711yescharacteristic, complemented
237#1688—472no—
238#1686—472no—
239#1668—472no—
240#1666—472no—
241#1565—471no—
242#1540—471no—
243C2×#331—471no—
244C2×C47—22no—
245C2×C47C2×#46011yescharacteristic
246C2×C47#306211yescharacteristic
247C2×C47#306811yescharacteristic
248D47—161no—
249D47—162no—
250C2×AGL(1,ℤ/8)—941no—
251C2×AGL(1,ℤ/8)—941no—
252#533C2×D4711yescharacteristic
253#532—471no—
254C2×#127—471no—
255#496—942no—
256#493—472no—
257#492—472no—
258#492—941no—
259#486—941no—
260#476—941no—
261#475—472no—
262#465—942no—
263#461—941no—
264#460—472no—
265#460—472no—
266#460—472no—
267#450—942no—
268#446—22no—
269#446C2×D4712yescomplemented
270C8×Q8C2×D4712yescomplemented
271#431—471no—
272#424—21no—
273C2×C4⋊C8—471no—
274C2×C4⋊C8C2×D4711yescharacteristic
275#405—21no—
276C4×M16C2×D4711yescharacteristic
277#335—472no—
278#334—472no—
279#332—471no—
280#332—471no—
281#331—472no—
282#331—472no—
283#331—471no—
284#331—471no—
285C47#1923811yescharacteristic, complemented, Sylow
286C2×#41—941no—
287C22×D4—941no—
288C22×D4—941no—
289C22×D4—941no—
290AGL(1,ℤ/8)—1884no—
291AGL(1,ℤ/8)—1884no—
292C2×SD16—942no—
293C2×SD16—942no—
294C2×SD16—942no—
295C2×SD16—942no—
296C2×SD16—942no—
297C2×D8—942no—
298C2×D8—942no—
299C2×M16C22⋊D4711yescharacteristic
300C2×M16C22⋊D4711yescharacteristic
301C22×C8—21no—
302C22×C8—941no—
303C4⋊Q8—471no—
304C4⋊Q8D18811yescharacteristic
305C4⋊Q8D18811yescharacteristic
306#127—472no—
307#127—471no—
308#127—941no—
309#127—941no—
310#127—941no—
311#125—21no—
312#124—941no—
313C22⋊Q8—21no—
314C22⋊Q8D18812yes—
315#121—941no—
316#121—941no—
317C4×Q8—21no—
318C4×Q8C22×D4712yes—
319C4×D4—941no—
320C4×D4—941no—
321#117—21no—
322C2×C42C22×D4711yescharacteristic
323#106—942no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—21no—
327C4⋊C8—472no—
328C4⋊C8—471no—
329C4⋊C8—471no—
330C4⋊C8—941no—
331C4⋊C8C22×D4711yescharacteristic
332C4⋊C8C22×D4711yescharacteristic
333#103—942no—
334#102—942no—
335#102—942no—
336#102—942no—
337C22⋊C8—941no—
338#97—22no—
339#97C22⋊D4712yes—
340C4×C8—22no—
341C4×C8—941no—
342C4×C8C22×D4712yes—
343C24—1881no—
344#41—1882no—
345#41—1882no—
346C2×Q8—21no—
347C2×Q8—941no—
348C2×Q8C2×D18812yes—
349C2×D4—941no—
350C2×D4—942no—
351C2×D4—941no—
352C2×D4—942no—
353C2×D4—941no—
354C2×D4—942no—
355C2×D4—3761no—
356C2×D4—3761no—
357C2×D4—3762no—
358C2×D4—1881no—
359C2×D4—1881no—
360C2×D4—1881no—
361C2×D4—1881no—
362C2×D4—1882no—
363C2×D4—1882no—
364C22×C4—1881no—
365C22×C4C2×D18811yescharacteristic
366C22×C4#1062535011yescharacteristic
367C22×C4C2×C22⋊D4711yescharacteristic
368SD16—3762no—
369SD16—3762no—
370SD16—3762no—
371SD16—1884no—
372SD16—1884no—
373D8—1884no—
374D8—1884no—
375M16—22no—
376M16#1062533614yescomplemented
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—42no—
380C2×C8—941no—
381C2×C8—941no—
382C2×C8—942no—
383C2×C8—3761no—
384C2×C8C2×C22⋊D4712yes—
385C2×C8C2×C22⋊D4712yes—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—941no—
392C4⋊C4—941no—
393C4⋊C4C2×D18812yes—
394C22⋊C4—22no—
395C22⋊C4—1881no—
396C22⋊C4—1881no—
397C42—22no—
398C42C2×D18811yescharacteristic
399C42#1062535011yescharacteristic
400C42C2×C22⋊D4711yescharacteristic
401C42C23×D4711yescharacteristic
402C23—3762no—
403C23—3762no—
404C23—3762no—
405C23—1881no—
406C23—1882no—
407C23#1309399811yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—81no—
411Q8—1881no—
412Q8—1881no—
413D4—3761no—
414D4—3762no—
415D4—3762no—
416D4—3762no—
417D4—3762no—
418D4—3762no—
419D4—1881no—
420D4—1882no—
421D4—1881no—
422D4—1882no—
423D4—1881no—
424D4—1882no—
425D4—1882no—
426D4—1881no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—3762no—
433C2×C4—1881no—
434C2×C4#1309399811yescharacteristic
435C2×C4#1309399811yescharacteristic
436C2×C4#1309399811yescharacteristic
437C2×C4C2×#1062533612yes—
438C2×C4#1309404912yes—
439C2×C4C22×C22⋊D4711yescharacteristic
440C2×C4C22×D18811yescharacteristic
441C2×C4C2×#1062535011yescharacteristic
442C8—22no—
443C8—41no—
444C8—41no—
445C8—1882no—
446C8—1881no—
447C8—1881no—
448C8C2×#1062533614yes—
449C22—21no—
450C22—21no—
451C22—3762no—
452C22—3761no—
453C22—3761no—
454C22—3762no—
455C22—3762no—
456C22—3761no—
457C22—7524no—
458C22#1568932312yes—
459C22C2×#1309399811yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—3761no—
466C4#1568932312yes—
467C4C22×#1062533611yescharacteristic
468C4C2×#1309404911yescharacteristic
469C4#1568992811yescharacteristic
470C4#1568993511yescharacteristic
471C2—22no—
472C2—7522no—
473C2—7521no—
474C2C2×#1568932311yescharacteristic
475C2#2025306811yescharacteristic
476C2#2025311611yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2962C2
socleC22×C47C2×#121
prime radical#20253840C2
hypercenter#2962D47
hypocenterC47#19238
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #20254700C2semidirect
2. #20254326C2semidirect
4. #20242872C2semidirect
5. #20242860C2semidirect
6. #20242748C2semidirect
7. #20242736C2semidirect
8. #20242140C2semidirect
9. #20242038C2semidirect
10. C2×#15688866C2semidirect
11. C47×#2962C2semidirect
22. #15689408C22semidirect
27. #15689323C22semidirect
31. #15688876C22semidirect
32. #15688874C22semidirect
33. #15688870C22semidirect
34. #15688870C22semidirect
37. #15688866C22semidirect
38. #15688866C22semidirect
41. C47×#446C22semidirect
42. C8×Q8×C47C22semidirect
130. M16×C47SD16semidirect
236. #2962D47semidirect
269. #446C2×D47semidirect
270. C8×Q8C2×D47semidirect
285. C47#19238semidirect
376. M16#10625336semidirect

Maximal subgroups

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

subgroupindex
#202547002
#202543262
#202538402
#202428722
#202428602
#202427482
#202427362
#202421402
#202420382
C2×#156888662
C47×#29622
#1923847