Finite Group Databasebeta

← #26224731

Subgroups — #26224731

Order 12032.

1398 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689499C211yesmaximal, characteristic
2C2×#15689233C214yesmaximal, complemented
3C4×#13094021C211yesmaximal, characteristic, complemented
4#20244263C214yesmaximal, complemented
5#20244195C211yesmaximal, characteristic, complemented
6#20244193C211yesmaximal, characteristic, complemented
7#20243544C211yesmaximal, characteristic, complemented
8C4×#13093935C211yesmaximal, characteristic, complemented
9C4×#127×C47C211yesmaximal, characteristic, complemented
10C22×#10625334C2214yes—
11C2×#13094021C2211yescharacteristic
12C2×C4×D188—22no—
13C2×C4×D188C2212yes—
14#15689499C418yescomplemented
15#15689233—216no—
16C2×#13093955C2214yes—
17C2×#13093945C2214yes—
18C2×#13093944C2214yes—
19C2×#13093935C2212yes—
20C2×#13093935C2212yes—
21C2×C4×C47⋊C8C2211yescharacteristic
22C2×C4×C47⋊C8C2211yescharacteristic
23#15689025—22no—
24#15689025C2212yes—
25#15688993C2211yescharacteristic
26C2×#127×C47C2211yescharacteristic
27C2×C4×D4×C47—22no—
28C2×C4×D4×C47C2212yes—
29C47×#390—22no—
30C47×#390C2212yes—
31C47×#384C2211yescharacteristic
32C43×C47C2211yescharacteristic
33C22×D188—22no—
34C22×D188C2312yes—
35C22×C4×D47—22no—
36C2×#10625334—28no—
37C2×#10625334—28no—
38C2×#10625334C2×C4116yescomplemented
39C22×C47⋊C8C2312yes—
40C22×C47⋊C8C2312yes—
41#13094021—22no—
42#13094021C2×C414yescomplemented
43C4×D188—28no—
44C4×D188—48no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47C2312yes—
47C2×(C2×D47)⋊C4—22no—
48C2×(C2×D47)⋊C4—24no—
49#13093955—28no—
50#13093945—28no—
51#13093944—28no—
52#13093935—24no—
53#13093935D418yescomplemented
54C4×C47⋊C8—22no—
55C4×C47⋊C8C2×C414yescomplemented
56C22×D4×C47—22no—
57C22×D4×C47C2312yes—
58C23×C4×C47—22no—
59#127×C47—22no—
60#127×C47C2×C414yescomplemented
61C4×D4×C47—28no—
62C4×D4×C47—48no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47C2312yes—
65C2×C22⋊C4×C47—24no—
66C2×C22⋊C4×C47—22no—
67C2×C42×C47D412yes—
68C2×C42×C47D412yes—
69C2×C42×C47C2311yescharacteristic
70C2×C42×C47C2311yescharacteristic
71C2×C42×C47C2311yescharacteristic
72C24×C47—22no—
73C23×D47—22no—
74C2×D4×C47—28no—
75C2×D4×C47—24no—
76C2×D4×C47—24no—
77C2×D4×C47—22no—
78C2×D4×C47—44no—
79C2×D4×C47—44no—
80C2×D4×C47C22×C414yes—
81C22×C4×C47—22no—
82C22×C4×C47—48no—
83C22×C4×C47—44no—
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2411yescharacteristic
89C22×Dic47—22no—
90C2×D188—24no—
91C2×D188—24no—
92C2×D188—28no—
93C2×D188—22no—
94C2×D188—44no—
95C2×D188—44no—
96C2×D188C22×C414yes—
97C2×C4×D47—48no—
98C2×C4×D47—44no—
99C4⋊C4×C47—24no—
100C4⋊C4×C47—44no—
101C22⋊C4×C47—48no—
102C22⋊C4×C47—44no—
103C42×C47—24no—
104C42×C47—24no—
105C42×C47—22no—
106C42×C47C22×C412yes—
107C42×C47C2×D414yes—
108C42×C47#4112yes—
109#10625334—232no—
110(C2×D47)⋊C4—48no—
111(C2×D47)⋊C4—44no—
112C4⋊Dic47—24no—
113C4⋊Dic47—44no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—22no—
116C2×C47⋊C8C22×C414yes—
117C2×C47⋊C8C2×D414yes—
118C2×C47⋊C8C2×D414yes—
119C2×C47⋊C8#4114yes—
120C23×C47—24no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47—44no—
124C23×C47—44no—
125C23×C47C22×D411yescharacteristic
126C22×D47—24no—
127C22×D47—42no—
128C22×D47—42no—
129C22×D47—44no—
130C22×D47—44no—
131D4×C47—28no—
132D4×C47—44no—
133D4×C47—416no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—42no—
140C2×C4×C47—44no—
141C2×C4×C47—88no—
142C2×C4×C47C4×D414yes—
143C2×C4×C47#12712yes—
144C2×C4×C47#12712yes—
145C2×C4×C47AGL(1,ℤ/8)14yes—
146C2×C4×C47AGL(1,ℤ/8)14yes—
147C2×C4×C47C23×C411yescharacteristic
148C2×C4×C47C22×D412yes—
149C2×C4×C47C2×#4112yes—
150C2×C4×C47C2×#4111yescharacteristic
151C2×Dic47—42no—
152C2×Dic47—44no—
153D188—28no—
154D188—44no—
155D188—416no—
156C4×D47—88no—
157C47⋊C8—24no—
158C47⋊C8C4×D418yescomplemented
159#21690—471nomaximal, Sylow
160C22×C47—44no—
161C22×C47—48no—
162C22×C47—82no—
163C22×C47—82no—
164C22×C47—84no—
165C22×C47C2×C4×D411yescharacteristic
166C22×C47C2×#12711yescharacteristic
167C22×C47#53211yescharacteristic
168C22×C47C2×AGL(1,ℤ/8)12yes—
169C22×C47C2×AGL(1,ℤ/8)12yes—
170C2×D47—44no—
171C2×D47—48no—
172C2×D47—82no—
173C2×D47—82no—
174C2×D47—84no—
175C4×C47—22no—
176C4×C47—22no—
177C4×C47—42no—
178C4×C47—82no—
179C4×C47#44218yescomplemented
180C4×C47C2×C4×D412yes—
181C4×C47#53212yes—
182Dic47—82no—
183C2×#493—471no—
184C2×#442—474no—
185C4×#127—471no—
186C4×#127D4711yescharacteristic, complemented
187#1972—474no—
188#1957—471no—
189#1957—471no—
190#1844—471no—
191C4×#97—471no—
192C2×C47—84no—
193C2×C47C4×#12711yescharacteristic
194C2×C47C2×#44212yes—
195C2×C47#315111yescharacteristic
196C2×C47#315111yescharacteristic
197C2×C47#318412yes—
198D47—84no—
199C22×D8—474no—
200C2×#127—471no—
201C2×#127C2×D4711yescharacteristic
202C2×C4×D4—22no—
203C2×C4×D4—472no—
204C2×C4×D4—942no—
205C2×C4×D4C2×D4712yes—
206#493—478no—
207#442—9416no—
208C2×#106—474no—
209C2×#102—474no—
210C2×#102—474no—
211C2×#97—472no—
212C2×#97—472no—
213C2×C4×C8—471no—
214C2×C4×C8—471no—
215#390—22no—
216#390—472no—
217#390—942no—
218#390C2×D4712yes—
219#384—471no—
220#384C2×D4711yescharacteristic
221C43C2×D4711yescharacteristic
222C47#2169011yescharacteristic, complemented, Sylow
223C22×D4—22no—
224C22×D4—472no—
225C22×D4—942no—
226C22×D4C22×D4712yes—
227C23×C4—22no—
228C23×C4—942no—
229C2×D8—4716no—
230C2×D8—948no—
231C2×D8—948no—
232C22×C8—472no—
233C22×C8—472no—
234#127—22no—
235#127—474no—
236#127—942no—
237#127C4×D4714yescomplemented
238C4×D4—28no—
239C4×D4—48no—
240C4×D4—948no—
241C4×D4—1888no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4—942no—
245C2×C4⋊C4C22×D4712yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—942no—
249C2×C22⋊C4—944no—
250C2×C42C22⋊D4712yes—
251C2×C42C22⋊D4712yes—
252C2×C42C22×D4711yescharacteristic
253C2×C42C22×D4711yescharacteristic
254C2×C42C22×D4711yescharacteristic
255#106—948no—
256#102—948no—
257#102—948no—
258#97—478no—
259#97—944no—
260C4×C8—474no—
261C4×C8—942no—
262C24—22no—
263C24—942no—
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—474no—
271C2×D4—948no—
272C2×D4—944no—
273C2×D4—944no—
274C2×D4—942no—
275C2×D4—1884no—
276C2×D4—1884no—
277C2×D4C2×C4×D4714yes—
278C22×C4—22no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—942no—
282C22×C4—1888no—
283C22×C4—1884no—
284C22×C4C2×C22⋊D4712yes—
285C22×C4C2×C22⋊D4711yescharacteristic
286C22×C4C2×C22⋊D4711yescharacteristic
287C22×C4C2×C22⋊D4712yes—
288C22×C4C23×D4711yescharacteristic
289D8—9432no—
290C2×C8—474no—
291C2×C8—474no—
292C2×C8—474no—
293C2×C8—474no—
294C2×C8—942no—
295C2×C8—942no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—944no—
299C4⋊C4—1884no—
300C22⋊C4—48no—
301C22⋊C4—44no—
302C22⋊C4—1888no—
303C22⋊C4—1884no—
304C42—24no—
305C42—24no—
306C42—22no—
307C42C2×C4×D4712yes—
308C42#1062535012yes—
309C42D4×D4714yes—
310C23—24no—
311C23—44no—
312C23—44no—
313C23—42no—
314C23—42no—
315C23—944no—
316C23—1884no—
317C23—1884no—
318C23—1882no—
319C23—1882no—
320C23C22×C22⋊D4711yescharacteristic
321D4—28no—
322D4—416no—
323D4—44no—
324D4—948no—
325D4—18816no—
326D4—1884no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—88no—
335C2×C4—3768no—
336C2×C4—1884no—
337C2×C4—1882no—
338C2×C4C4×C22⋊D4714yes—
339C2×C4#1309402012yes—
340C2×C4#1309402012yes—
341C2×C4#1309404914yes—
342C2×C4#1309405014yes—
343C2×C4C22×C4×D4711yescharacteristic
344C2×C4C2×D4×D4712yes—
345C2×C4#41×D4712yes—
346C2×C4C2×#1062535011yescharacteristic
347C8—478no—
348C8—944no—
349C22—48no—
350C22—44no—
351C22—82no—
352C22—82no—
353C22—84no—
354C22—3762no—
355C22—3762no—
356C22—3764no—
357C22—1888no—
358C22—1884no—
359C22C2×C4×C22⋊D4711yescharacteristic
360C22C2×#1309402011yescharacteristic
361C22#1568966911yescharacteristic
362C22C2×#1309404912yes—
363C22C2×#1309405012yes—
364C4—22no—
365C4—22no—
366C4—42no—
367C4—82no—
368C4—3762no—
369C4#1568923318yescomplemented
370C4C4×D4×D4712yes—
371C4#1568967012yes—
372C2—84no—
373C2—3764no—
374C2C4×#1309402011yescharacteristic
375C2C2×#1568923312yes—
376C2#2025383811yescharacteristic
377C2#2025384111yescharacteristic
378C2#2025412212yes—
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC4×#127×C47C2
socleC23×C47C22×D4
prime radicalC2×#15689499C2
cosocleC22×C4×C47C24
hypercenterC4×#127D47
hypocenterC47#21690
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C2×#15689233C2semidirect
3. C4×#13094021C2semidirect
4. #20244263C2semidirect
5. #20244195C2semidirect
6. #20244193C2semidirect
7. #20243544C2semidirect
8. C4×#13093935C2semidirect
9. C4×#127×C47C2semidirect
14. #15689499C4semidirect
38. C2×#10625334C2×C4semidirect
42. #13094021C2×C4semidirect
53. #13093935D4semidirect
55. C4×C47⋊C8C2×C4semidirect
60. #127×C47C2×C4semidirect
158. C47⋊C8C4×D4semidirect
179. C4×C47#442semidirect
186. C4×#127D47semidirect
222. C47#21690semidirect
237. #127C4×D47semidirect
369. C4#15689233semidirect

Maximal subgroups

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

subgroupindex
C2×#156894992
C2×#156892332
C4×#130940212
#202442632
#202441952
#202441932
#202435442
C4×#130939352
C4×#127×C472
#2169047