Finite Group Databasebeta

← #26107213

Subgroups — #26107213

Order 12032.

488 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243137C211yesmaximal, characteristic
2C4×#13093935C211yesmaximal, characteristic
3C2×#15688865C212yesmaximal
4#20241269C212yesmaximal
5C4×#125×C47C211yesmaximal, characteristic
6C2×#13093935—21no—
7C2×#13093935C2212yes—
8C2×#13093935C2212yes—
9C2×C4×C47⋊C8—21no—
10#15688878—21no—
11#15688878—21no—
12#15688865C418yescomplemented
13C2×#125×C47C2211yescharacteristic
14C47×#389C411yescharacteristic
15C47×#389C411yescharacteristic
16C47×#387—21no—
17C47×#387—21no—
18C47×#382C2211yescharacteristic
19C4×C4⋊C4×C47—21no—
20C4×C4⋊C4×C47—21no—
21C4×C4⋊C4×C47C411yescharacteristic
22C4×C4⋊C4×C47C411yescharacteristic
23C43×C47C2211yescharacteristic
24C22×C47⋊C8—22no—
25C22×C47⋊C8—22no—
26#13093935—24no—
27#13093935—22no—
28#13093935—22no—
29#13093935C2×C414yes—
30#13093935C2×C414yes—
31C4×C47⋊C8—24no—
32#125×C47—22no—
33#125×C47C2×C414yes—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C42×C47—21no—
43C2×C42×C47—21no—
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47C2×C411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47C2311yescharacteristic
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C4⋊C4×C47—21no—
69C4⋊C4×C47—21no—
70C4⋊C4×C47—24no—
71C4⋊C4×C47—24no—
72C4⋊C4×C47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47C4212yes—
77C4⋊C4×C47C4212yes—
78C42×C47—22no—
79C42×C47—22no—
80C42×C47—22no—
81C42×C47—41no—
82C42×C47—41no—
83C42×C47—41no—
84C42×C47—42no—
85C42×C47—42no—
86C42×C47C22×C412yes—
87C42×C47#4112yes—
88C2×C47⋊C8—24no—
89C2×C47⋊C8—24no—
90C2×C47⋊C8—24no—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—42no—
94C23×C47C2×C22⋊C411yescharacteristic
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47#10112yes—
123C2×C4×C47#10412yes—
124C2×C4×C47#10412yes—
125C2×C4×C47C2×C4211yescharacteristic
126C2×C4×C47#11711yescharacteristic
127C2×C4×C47C4×D412yes—
128C2×C4×C47C4×D412yes—
129C47⋊C8—28no—
130C47⋊C8—44no—
131#8227—471nomaximal, Sylow
132C22×C47C4×C22⋊C411yescharacteristic
133C22×C47C2×#10111yescharacteristic
134C22×C47#41311yescharacteristic
135C22×C47C2×#10411yescharacteristic
136C22×C47C2×#10411yescharacteristic
137C22×C47#42111yescharacteristic
138C22×C47#42111yescharacteristic
139C4×C47—21no—
140C4×C47—21no—
141C4×C47—22no—
142C4×C47—22no—
143C4×C47—22no—
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—42no—
147C4×C47—42no—
148C4×C47—41no—
149C4×C47—81no—
150C4×C47—81no—
151C4×#125D4711yescharacteristic
152#1745—471no—
153C4×#97—471no—
154C2×#330—472no—
155#1295—472no—
156C2×C47#152811yescharacteristic
157C2×C47#152911yescharacteristic
158C2×C47#153111yescharacteristic
159C2×C47#153211yescharacteristic
160C2×C47C4×#10111yescharacteristic
161C2×C47C4×#10411yescharacteristic
162C2×C47#175911yescharacteristic
163C2×#125C2×D4711yescharacteristic
164C2×#97—472no—
165C2×#97—472no—
166C2×#97—941no—
167C2×C4×C8—941no—
168#389Dic4711yescharacteristic
169#389Dic4711yescharacteristic
170#387—21no—
171#387—21no—
172#382C2×D4711yescharacteristic
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4Dic4711yescharacteristic
176C4×C4⋊C4Dic4711yescharacteristic
177C43C2×D4711yescharacteristic
178#336—941no—
179#336—941no—
180#330—478no—
181C47#822711yescharacteristic, complemented, Sylow
182C22×C8—942no—
183C22×C8—942no—
184#125—22no—
185#125C4×D4714yes—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×Dic4711yescharacteristic
191C2×C4⋊C4C2×Dic4711yescharacteristic
192C2×C4⋊C4C2×Dic4711yescharacteristic
193C2×C4⋊C4C2×Dic4711yescharacteristic
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C2×Dic4711yescharacteristic
197C2×C42C2×Dic4711yescharacteristic
198C2×C42C22⋊D4711yescharacteristic
199C2×C42C22⋊D4711yescharacteristic
200C2×C42C22⋊D4711yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C2×C42C22×D4711yescharacteristic
203#97—474no—
204#97—474no—
205#97—944no—
206#97—942no—
207#97—942no—
208C4×C8—944no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4C22⋊Dic4711yescharacteristic
220C22×C4C22⋊Dic4711yescharacteristic
221C22×C4C22⋊Dic4711yescharacteristic
222C22×C4C22⋊Dic4711yescharacteristic
223C22×C4C22×Dic4711yescharacteristic
224C22×C4C2×C22⋊D4711yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226C2×C8—944no—
227C2×C8—944no—
228C2×C8—944no—
229C2×C8—944no—
230C2×C8—1882no—
231C2×C8—1882no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4C4×Dic4712yes—
241C4⋊C4C4×Dic4712yes—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—42no—
249C42—42no—
250C42C2×C4×D4712yes—
251C42#1062535012yes—
252C23C2×C22⋊Dic4711yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4#1309394312yes—
281C2×C4#1309395112yes—
282C2×C4#1309395112yes—
283C2×C4C2×C4×Dic4711yescharacteristic
284C2×C4#1309398311yescharacteristic
285C2×C4C4×C22⋊D4712yes—
286C2×C4C4×C22⋊D4712yes—
287C8—948no—
288C8—1884no—
289C22C4×C22⋊Dic4711yescharacteristic
290C22C2×#1309394311yescharacteristic
291C22#1568910711yescharacteristic
292C22C2×#1309395111yescharacteristic
293C22C2×#1309395111yescharacteristic
294C22#1568914011yescharacteristic
295C22#1568914011yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—21no—
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—42no—
305C4—42no—
306C4—81no—
307C4—81no—
308C2#2024199011yescharacteristic
309C2#2024199511yescharacteristic
310C2#2024200311yescharacteristic
311C2#2024200811yescharacteristic
312C2C4×#1309394311yescharacteristic
313C2C4×#1309395111yescharacteristic
314C2#2024319811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC4×#125×C47C2
socle, prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenterC4×#125D47
hypocenterC47#8227
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
12. #15688865C4semidirect
181. C47#8227semidirect

Maximal subgroups

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

subgroupindex
#202431372
C4×#130939352
C2×#156888652
#202412692
C4×#125×C472
#822747