Finite Group Databasebeta

← #26132277

Subgroups — #26132277

Order 12032.

500 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688920C212yesmaximal
2#20243384C212yesmaximal, complemented
3#20241515C212yesmaximal, complemented
4C4×#124×C47C211yesmaximal, characteristic
5#15689147—22no—
6C2×#13093943C2214yes—
7#15688920—24no—
8#15688892—22no—
9C2×#124×C47C2211yescharacteristic
10C2×C4×Q8×C47C2211yescharacteristic
11C2×C4×D4×C47C411yescharacteristic
12C47×#391C2211yescharacteristic
13C47×#390C411yescharacteristic
14C47×#388—22no—
15C47×#382C411yescharacteristic
16C4×C22⋊C4×C47—22no—
17C43×C47C411yescharacteristic
18C2×#10625329—24no—
19C4⋊(C47⋊C8)—42no—
20C4⋊(C47⋊C8)—42no—
21#13093943—24no—
22#13093943—24no—
23C22×Q8×C47C2311yescharacteristic
24C22×D4×C47C2×C411yescharacteristic
25C23×C4×C47—21no—
26#124×C47—22no—
27#124×C47D412yes—
28#124×C47Q812yes—
29C4×Q8×C47—41no—
30C4×Q8×C47—41no—
31C4×D4×C47—41no—
32C4×D4×C47—41no—
33C2×C4⋊C4×C47C2×C411yescharacteristic
34C2×C4⋊C4×C47C2×C411yescharacteristic
35C2×C4⋊C4×C47D412yes—
36C2×C22⋊C4×C47—22no—
37C2×C22⋊C4×C47—21no—
38C2×C22⋊C4×C47—22no—
39C2×C42×C47—21no—
40C2×C42×C47—21no—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D412yes—
45#95×C47—22no—
46#95×C47—22no—
47C24×C47—21no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47C2×D411yescharacteristic
51C2×Q8×C47C2×Q811yescharacteristic
52C2×Q8×C47#4112yes—
53C2×D4×C47—21no—
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C2×D4×C47C4⋊C412yes—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—44no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D412yes—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—41no—
71C22⋊C4×C47—24no—
72C22⋊C4×C47—24no—
73C22⋊C4×C47—44no—
74C22⋊C4×C47—42no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—22no—
78C42×C47—22no—
79C42×C47—21no—
80C42×C47—42no—
81C42×C47—42no—
82C42×C47—41no—
83C42×C47C4⋊C412yes—
84#10625329—44no—
85#10625329—44no—
86C2×C47⋊C8—44no—
87C23×C47—22no—
88C23×C47—42no—
89C23×C47—42no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47C2×C22⋊C411yescharacteristic
93Q8×C47—41no—
94Q8×C47—41no—
95Q8×C47—42no—
96D4×C47—42no—
97D4×C47—42no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—82no—
116C2×C4×C47—82no—
117C2×C4×C47#9911yescharacteristic
118C2×C4×C47#9911yescharacteristic
119C2×C4×C47C2×C4⋊C411yescharacteristic
120C2×C4×C47#11711yescharacteristic
121C2×C4×C47#12112yes—
122C2×C4×C47C22⋊Q812yes—
123C47⋊C8—84no—
124#12801—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—44no—
128C22×C47—42no—
129C22×C47—82no—
130C22×C47—81no—
131C22×C47—81no—
132C22×C47#38011yescharacteristic
133C22×C47C2×#9911yescharacteristic
134C22×C47#41011yescharacteristic
135C4×C47—22no—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—22no—
139C4×C47—21no—
140C4×C47—42no—
141C4×C47—42no—
142C4×C47—41no—
143C4×C47—42no—
144C4×C47—81no—
145C4×C47#35512yescomplemented
146C4×#124D4711yescharacteristic
147C2×#355—472no—
148#1801—472no—
149#1394—472no—
150C2×C47—21no—
151C2×C47—21no—
152C2×C47—82no—
153C2×C47#179911yescharacteristic
154C2×C47C2×#35511yescharacteristic
155C2×C47#213211yescharacteristic
156C2×#124C2×D4711yescharacteristic
157C2×C4×Q8C2×D4711yescharacteristic
158C2×C4×D4Dic4711yescharacteristic
159#423—942no—
160C2×#101—474no—
161#391C2×D4711yescharacteristic
162#390Dic4711yescharacteristic
163#388—22no—
164#382Dic4711yescharacteristic
165C4×C22⋊C4—22no—
166C43Dic4711yescharacteristic
167#355—944no—
168#343—942no—
169C47#1280111yescharacteristic, complemented, Sylow
170C22×Q8C22×D4711yescharacteristic
171C22×D4C2×Dic4711yescharacteristic
172C23×C4—21no—
173C2×M16—944no—
174#124—22no—
175#124Dic9412yes—
176#124D18812yes—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4C2×Dic4711yescharacteristic
182C2×C4⋊C4C2×Dic4711yescharacteristic
183C2×C4⋊C4C22⋊D4712yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42C2×Dic4711yescharacteristic
190C2×C42C2×Dic4711yescharacteristic
191C2×C42C2×Dic4711yescharacteristic
192C2×C42C22⋊D4712yes—
193C4⋊C8—1882no—
194C4⋊C8—1882no—
195#101—944no—
196#101—944no—
197#95—22no—
198#95—22no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×Dic9411yescharacteristic
203C2×Q8C2×D18811yescharacteristic
204C2×Q8#1062535012yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊Dic4712yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—44no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4C22⋊Dic4712yes—
216C22×C4C22⋊Dic4711yescharacteristic
217C22×C4C22⋊Dic4711yescharacteristic
218C22×C4C22×Dic4711yescharacteristic
219C22×C4C2×C22⋊D4712yes—
220M16—1884no—
221M16—1884no—
222C2×C8—1884no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—44no—
229C22⋊C4—42no—
230C42—22no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—41no—
236C42—42no—
237C42—42no—
238C42C4⋊Dic4712yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23C2×C22⋊Dic4711yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4#1309393911yescharacteristic
270C2×C4#1309393911yescharacteristic
271C2×C4C2×C4⋊Dic4711yescharacteristic
272C2×C4#1309398311yescharacteristic
273C2×C4#1309399812yes—
274C2×C4#1309400312yes—
275C8—3764no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22#1568898311yescharacteristic
284C22C2×#1309393911yescharacteristic
285C22#1568909711yescharacteristic
286C4—21no—
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—81no—
296C4#1568892012yescomplemented
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2024337611yescharacteristic
301C2C2×#1568892011yescharacteristic
302C2#2024507311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC4×#124×C47C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC22×Q8×C47C23
hypercenterC4×#124D47
hypocenterC47#12801
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. #20243384C2semidirect
3. #20241515C2semidirect
145. C4×C47#355semidirect
169. C47#12801semidirect
296. C4#15688920semidirect

Maximal subgroups

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

subgroupindex
C2×#156889202
#202433842
#202415152
C4×#124×C472
#1280147