Finite Group Databasebeta

← #26121891

Subgroups — #26121891

Order 12032.

918 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688878C211yesmaximal, characteristic
2C2×#15688877C212yesmaximal
3C2×#15688877C212yesmaximal
4C4×C47⋊#95C211yesmaximal, characteristic
5C22×C4×C8×C47C211yesmaximal, characteristic
6C23×C47⋊C8C2212yes—
7C22×C4×Dic47C2212yes—
8C2×C47⋊#95C412yes—
9C2×C47⋊#95C412yes—
10#15688878C414yescomplemented
11#15688877—24no—
12#15688877—24no—
13C23×C8×C47C2211yescharacteristic
14C23×C8×C47C2211yescharacteristic
15C22×C42×C47C2211yescharacteristic
16C2×C4×C8×C47—24no—
17C2×C4×C8×C47C414yescomplemented
18C23×Dic47C2×C414yes—
19C22×C47⋊C8—22no—
20C22×C47⋊C8—22no—
21C22×C47⋊C8—22no—
22C22×C47⋊C8—22no—
23C22×C47⋊C8—22no—
24C22×C47⋊C8—22no—
25C22×C47⋊C8C2×C414yes—
26C2×C4×Dic47—22no—
27C2×C4×Dic47—22no—
28C2×C4×Dic47—22no—
29C2×C4×Dic47—22no—
30C2×C4×Dic47—22no—
31C2×C4×Dic47—22no—
32C47⋊#95—24no—
33C47⋊#95C818yescomplemented
34C23×C4×C47C2×C411yescharacteristic
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C22×C8×C47—22no—
38C22×C8×C47—22no—
39C22×C8×C47—22no—
40C22×C8×C47—21no—
41C22×C8×C47—21no—
42C22×C8×C47C2×C412yes—
43C22×C8×C47C2×C412yes—
44C22×C8×C47D414yes—
45C22×C8×C47D412yes—
46C22×C8×C47Q812yes—
47C2×C42×C47—22no—
48C2×C42×C47—22no—
49C2×C42×C47C2×C412yes—
50C2×C42×C47C2×C412yes—
51C4×C8×C47—28no—
52C24×C47C22×C411yescharacteristic
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47C4212yes—
58C22×C4×C47C4212yes—
59C22×C4×C47C2×C814yes—
60C22×C4×C47M1614yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×Q811yescharacteristic
66C22×C4×C47#4112yes—
67C22×C4×C47#4112yes—
68C22×C4×C47#4111yescharacteristic
69C22×C4×C47#4112yes—
70C22×C4×C47#4111yescharacteristic
71C22×Dic47—24no—
72C22×Dic47—24no—
73C22×Dic47—24no—
74C22×Dic47—24no—
75C22×Dic47—24no—
76C22×Dic47—24no—
77C22×Dic47C2×C818yes—
78C2×C8×C47—22no—
79C2×C8×C47—22no—
80C2×C8×C47—22no—
81C2×C8×C47—22no—
82C2×C8×C47—24no—
83C2×C8×C47—24no—
84C2×C8×C47—24no—
85C2×C8×C47—24no—
86C2×C8×C47C22⋊C414yes—
87C2×C8×C47C4⋊C414yes—
88C42×C47—22no—
89C42×C47—22no—
90C42×C47—24no—
91C4×Dic47—42no—
92C4×Dic47—42no—
93C4×Dic47—42no—
94C4×Dic47—42no—
95C2×C47⋊C8—24no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C23×C47C2×C4211yescharacteristic
103C23×C47#11712yes—
104C23×C47C4×D412yes—
105C23×C47C4×D411yescharacteristic
106C23×C47C4×Q811yescharacteristic
107C23×C47C22×C812yes—
108C23×C47C2×M1612yes—
109C23×C47#13112yes—
110C23×C47#13112yes—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—28no—
116C2×C4×C47—28no—
117C2×C4×C47C4×C814yes—
118C2×C4×C47#9714yes—
119C2×C4×C47C2×C22⋊C411yescharacteristic
120C2×C4×C47C2×C4⋊C411yescharacteristic
121C2×C4×C47#11712yes—
122C2×C4×C47#11711yescharacteristic
123C2×C4×C47#11711yescharacteristic
124C2×C4×C47#11711yescharacteristic
125C2×C4×C47#11711yescharacteristic
126C2×C4×C47C4×D412yes—
127C2×C4×C47C4×D411yescharacteristic
128C2×C4×C47C4×Q811yescharacteristic
129C2×C4×C47#12112yes—
130C2×C4×C47C22⋊Q812yes—
131C2×C4×C47#12312yes—
132C2×C4×C47#12312yes—
133C2×C4×C47#12412yes—
134C2×C4×C47#12512yes—
135C2×C4×C47#12612yes—
136C2×C4×C47#12612yes—
137C2×Dic47—28no—
138C2×Dic47—28no—
139C2×Dic47—28no—
140C2×Dic47—44no—
141C2×Dic47—44no—
142C2×Dic47—44no—
143C2×Dic47—44no—
144C8×C47—24no—
145C8×C47—24no—
146C47⋊C8—44no—
147#10797—471nomaximal, Sylow
148C22×C47#37611yescharacteristic
149C22×C47C4×C22⋊C411yescharacteristic
150C22×C47C4×C4⋊C411yescharacteristic
151C22×C47#38712yes—
152C22×C47#38812yes—
153C22×C47C2×C4×C811yescharacteristic
154C22×C47C2×#9711yescharacteristic
155C22×C47#40511yescharacteristic
156C22×C47#40511yescharacteristic
157C22×C47#43114yes—
158C22×C47#43314yes—
159C22×C47C8×D414yes—
160C22×C47C8×D412yes—
161C22×C47#43514yes—
162C22×C47#43612yes—
163C22×C47C8×Q812yes—
164C22×C47#44612yes—
165C4×C47—24no—
166C4×C47—24no—
167C4×C47#38011yescharacteristic
168C4×C47#38111yescharacteristic
169C4×C47#38211yescharacteristic
170C4×C47#38311yescharacteristic
171C4×C47#38712yes—
172C4×C47#38812yes—
173Dic47—48no—
174C22×C4×C8D4711yescharacteristic
175C2×#336—471no—
176C2×#336—472no—
177C2×#336—472no—
178C4×#95—471no—
179C2×C47#143411yescharacteristic
180C2×C47#174411yescharacteristic
181C2×C47#174611yescharacteristic
182C2×C47C8×C22⋊C411yescharacteristic
183C2×C47#175211yescharacteristic
184C2×C47C8×C4⋊C411yescharacteristic
185C2×C47#177011yescharacteristic
186C2×C47#191614yes—
187C2×C47#192314yes—
188C23×C8—472no—
189C23×C8C2×D4711yescharacteristic
190C23×C8C2×D4711yescharacteristic
191C22×C42—472no—
192C22×C42C2×D4711yescharacteristic
193C2×C4×C8—24no—
194C2×C4×C8Dic4714yescomplemented
195C2×#95—472no—
196C2×#95—472no—
197#336—474no—
198#336—944no—
199#336—944no—
200C47#1079711yescharacteristic, complemented, Sylow
201C23×C4—474no—
202C23×C4C4×D4711yescharacteristic
203C23×C4C4×D4711yescharacteristic
204C23×C4C22×D4711yescharacteristic
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—474no—
211C22×C8—942no—
212C22×C8—942no—
213C22×C8—942no—
214C22×C8—942no—
215C22×C8—942no—
216C22×C8—942no—
217C22×C8Dic9412yes—
218C22×C8D18812yes—
219C22×C8C2×Dic4712yes—
220C22×C8C2×Dic4712yes—
221C22×C8C22⋊D4714yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—942no—
225C2×C42—942no—
226C2×C42—942no—
227C2×C42—942no—
228C2×C42—942no—
229C2×C42—942no—
230C2×C42C4×D4712yes—
231C2×C42C2×Dic4712yes—
232C4×C8—28no—
233#95—478no—
234#95—944no—
235C24C2×C4×D4711yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—478no—
241C22×C4—944no—
242C22×C4—944no—
243C22×C4—944no—
244C22×C4—944no—
245C22×C4—944no—
246C22×C4—944no—
247C22×C4C8×D4714yes—
248C22×C4#1062532414yes—
249C22×C4C4×Dic4712yes—
250C22×C4C4×Dic4712yes—
251C22×C4C2×Dic9411yescharacteristic
252C22×C4C2×C4×D4711yescharacteristic
253C22×C4C2×D18811yescharacteristic
254C22×C4#1062535012yes—
255C22×C4#1062535012yes—
256C22×C4#1062535012yes—
257C22×C4#1062535011yescharacteristic
258C22×C4#1062535011yescharacteristic
259C22×C4C22×Dic4711yescharacteristic
260C22×C4C2×C22⋊D4712yes—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—944no—
270C2×C8—944no—
271C2×C8—944no—
272C2×C8—1882no—
273C2×C8—1882no—
274C2×C8—1882no—
275C2×C8—1882no—
276C2×C8C4⋊Dic4714yes—
277C2×C8C22⋊Dic4714yes—
278C42—22no—
279C42—22no—
280C42—24no—
281C42—1882no—
282C42—1882no—
283C42—1882no—
284C42—1882no—
285C23C2×C4×Dic4711yescharacteristic
286C23#1309398112yes—
287C23C4×C22⋊D4712yes—
288C23C4×D18811yescharacteristic
289C23C4×Dic9411yescharacteristic
290C23C2×C8×D4712yes—
291C23C2×#1062532412yes—
292C23#1309403012yes—
293C23#1309403012yes—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—948no—
301C2×C4—948no—
302C2×C4—948no—
303C2×C4—1884no—
304C2×C4—1884no—
305C2×C4—1884no—
306C2×C4—1884no—
307C2×C4C8×Dic4714yes—
308C2×C4#1309393414yes—
309C2×C4C2×C22⋊Dic4711yescharacteristic
310C2×C4C2×C4⋊Dic4711yescharacteristic
311C2×C4#1309398112yes—
312C2×C4#1309398311yescharacteristic
313C2×C4#1309398311yescharacteristic
314C2×C4#1309398311yescharacteristic
315C2×C4#1309398311yescharacteristic
316C2×C4C4×C22⋊D4712yes—
317C2×C4C4×D18811yescharacteristic
318C2×C4C4×Dic9411yescharacteristic
319C2×C4#1309399812yes—
320C2×C4#1309400312yes—
321C2×C4#1309400712yes—
322C2×C4#1309400712yes—
323C2×C4#1309401212yes—
324C2×C4#1309401612yes—
325C2×C4#1309401812yes—
326C2×C4#1309401812yes—
327C8—24no—
328C8—24no—
329C8—1884no—
330C22#1568896911yescharacteristic
331C22C4×C22⋊Dic4711yescharacteristic
332C22C4×C4⋊Dic4711yescharacteristic
333C22#1568900712yes—
334C22#1568901412yes—
335C22C2×C8×Dic4711yescharacteristic
336C22C2×#1309393411yescharacteristic
337C22#1568907811yescharacteristic
338C22#1568907811yescharacteristic
339C22#1568917414yes—
340C22#1568918214yes—
341C22C8×C22⋊D4714yes—
342C22C8×D18812yes—
343C22#1568919414yes—
344C22#1568920212yes—
345C22C8×Dic9412yes—
346C22#1568925312yes—
347C4—24no—
348C4—24no—
349C4—1888no—
350C4#1568898311yescharacteristic
351C4#1568898611yescharacteristic
352C4#1568898911yescharacteristic
353C4#1568899111yescharacteristic
354C4#1568900712yes—
355C4#1568901412yes—
356C2#2024161511yescharacteristic
357C2#2024313311yescharacteristic
358C2#2024313911yescharacteristic
359C2C8×C22⋊Dic4711yescharacteristic
360C2#2024316411yescharacteristic
361C2C8×C4⋊Dic4711yescharacteristic
362C2#2024325511yescharacteristic
363C2#2024394514yes—
364C2#2024399414yes—
subgroupG/H
center, FrattiniC23×C4C22×D47
commutatorC22×C47C2×C4×C8
FittingC22×C4×C8×C47C2
socle, prime radicalC24×C47C22×C4
cosocleC23×C4×C47C23
hypercenterC22×C4×C8D47
hypocenterC47#10797
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C22×C4×C8

Complements

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

NH ≅ G/Ntype
10. #15688878C4semidirect
17. C2×C4×C8×C47C4semidirect
33. C47⋊#95C8semidirect
194. C2×C4×C8Dic47semidirect
200. C47#10797semidirect

Maximal subgroups

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

subgroupindex
C2×#156888782
C2×#156888772
C4×C47⋊#952
C22×C4×C8×C472
#1079747