Finite Group Databasebeta

← #26107214

Subgroups — #26107214

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
1C4×#13094016C211yesmaximal, characteristic
2#20243137C211yesmaximal, characteristic
3C2×#15688864C212yesmaximal
4#20241270C212yesmaximal
5C4×#97×C47C211yesmaximal, characteristic
6C2×#13094016C2211yescharacteristic
7C2×#13093935C2212yes—
8#15689021C411yescharacteristic
9#15689021C411yescharacteristic
10#15689007—21no—
11#15689007—21no—
12#15688989C2211yescharacteristic
13C4×Dic47⋊C4—21no—
14C4×Dic47⋊C4—21no—
15C4×C4⋊Dic47C411yescharacteristic
16C4×C4⋊Dic47C411yescharacteristic
17#15688878—21no—
18#15688878—21no—
19#15688864C418yescomplemented
20C2×#97×C47—21no—
21C2×#97×C47C2212yes—
22C2×C4×C8×C47—21no—
23C43×C47C2211yescharacteristic
24C22×C47⋊C8—22no—
25#13094016—22no—
26#13094016C2×C414yes—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×C4×Dic47—21no—
36C2×C4×Dic47—21no—
37#13093935—22no—
38#13093935C2×C414yes—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C22×C8×C47—22no—
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53#97×C47—24no—
54#97×C47—22no—
55#97×C47C2×C414yes—
56C4×C8×C47—24no—
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—21no—
65C22×Dic47—21no—
66C22×Dic47—21no—
67C22×Dic47—21no—
68C2×C8×C47—24no—
69C2×C8×C47—24no—
70C2×C8×C47—24no—
71C42×C47—22no—
72C42×C47—22no—
73C42×C47—22no—
74C42×C47—41no—
75C42×C47—41no—
76C42×C47—41no—
77C42×C47C22×C412yes—
78C42×C47#4112yes—
79C4⋊Dic47—21no—
80C4⋊Dic47—21no—
81C4⋊Dic47—41no—
82C4⋊Dic47—41no—
83C4⋊Dic47C4212yes—
84C4⋊Dic47C4212yes—
85Dic47⋊C4—24no—
86Dic47⋊C4—24no—
87Dic47⋊C4—42no—
88Dic47⋊C4—42no—
89C4×Dic47—42no—
90C4×Dic47—42no—
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—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47#10112yes—
111C2×C4×C47#10412yes—
112C2×C4×C47#10412yes—
113C2×C4×C47C2×C4211yescharacteristic
114C2×C4×C47#11711yescharacteristic
115C2×C4×C47C4×D412yes—
116C2×C4×C47C4×D412yes—
117C2×Dic47—22no—
118C2×Dic47—22no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C8×C47—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—41no—
147Dic47—42no—
148Dic47—42no—
149Dic47—81no—
150Dic47—81no—
151C4×#125—471no—
152#1745—471no—
153C4×#97D4711yescharacteristic
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×#125—471no—
164C2×#97—21no—
165C2×#97—472no—
166C2×#97C2×D4712yes—
167C2×C4×C8—21no—
168#389—471no—
169#389—471no—
170#387—941no—
171#387—941no—
172#382—471no—
173C4×C4⋊C4—471no—
174C4×C4⋊C4—471no—
175C4×C4⋊C4—941no—
176C4×C4⋊C4—941no—
177C43C2×D4711yescharacteristic
178#336—941no—
179#336—941no—
180#330—478no—
181C47#822711yescharacteristic, complemented, Sylow
182C22×C8—22no—
183C22×C8—942no—
184#125—474no—
185#125—942no—
186C2×C4⋊C4—471no—
187C2×C4⋊C4—471no—
188C2×C4⋊C4—471no—
189C2×C4⋊C4—471no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C42—941no—
195C2×C42—941no—
196C2×C42C4×D4711yescharacteristic
197C2×C42C4×D4711yescharacteristic
198C2×C42D18811yescharacteristic
199C2×C42D18811yescharacteristic
200C2×C42C22⋊D4711yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C2×C42C22×D4711yescharacteristic
203#97—24no—
204#97—22no—
205#97—474no—
206#97—942no—
207#97C4×D4714yes—
208C4×C8—24no—
209#95—941no—
210#95—941no—
211#95—941no—
212#95—941no—
213#95—941no—
214#95—941no—
215C22×C4—941no—
216C22×C4—941no—
217C22×C4—941no—
218C22×C4—941no—
219C22×C4(C2×D47)⋊C411yescharacteristic
220C22×C4(C2×D47)⋊C411yescharacteristic
221C22×C4(C2×D47)⋊C411yescharacteristic
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4C2×C4×D4711yescharacteristic
224C22×C4C2×D18811yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—944no—
230C2×C8—1882no—
231C2×C8—1882no—
232C4⋊C4—472no—
233C4⋊C4—472no—
234C4⋊C4—941no—
235C4⋊C4—941no—
236C4⋊C4—944no—
237C4⋊C4—944no—
238C4⋊C4—1881no—
239C4⋊C4—1881no—
240C4⋊C4—1882no—
241C4⋊C4—1882no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—1882no—
249C42—1882no—
250C42C2×C4×D4712yes—
251C42#1062535012yes—
252C23C2×(C2×D47)⋊C411yescharacteristic
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—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—942no—
269C2×C4—942no—
270C2×C4—1881no—
271C2×C4—1881no—
272C2×C4—1881no—
273C2×C4—1881no—
274C2×C4—1881no—
275C2×C4—1881no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4—1881no—
279C2×C4—1881no—
280C2×C4#1309394212yes—
281C2×C4#1309395212yes—
282C2×C4#1309395212yes—
283C2×C4C42×D4711yescharacteristic
284C2×C4#1309398111yescharacteristic
285C2×C4C4×C22⋊D4712yes—
286C2×C4C4×D18812yes—
287C8—28no—
288C8—1884no—
289C22C4×(C2×D47)⋊C411yescharacteristic
290C22C2×#1309394211yescharacteristic
291C22#1568910511yescharacteristic
292C22C2×#1309395211yescharacteristic
293C22C2×#1309395211yescharacteristic
294C22#1568913911yescharacteristic
295C22#1568913911yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—21no—
301C4—21no—
302C4—22no—
303C4—41no—
304C4—3761no—
305C4—3761no—
306C4—1882no—
307C4—1882no—
308C2#2024198911yescharacteristic
309C2#2024199311yescharacteristic
310C2#2024200211yescharacteristic
311C2#2024200611yescharacteristic
312C2C4×#1309394211yescharacteristic
313C2C4×#1309395211yescharacteristic
314C2#2024320111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC4×#97×C47C2
socle, prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenterC4×#97D47
hypocenterC47#8227
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
19. #15688864C4semidirect
181. C47#8227semidirect

Maximal subgroups

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

subgroupindex
C4×#130940162
#202431372
C2×#156888642
#202412702
C4×#97×C472
#822747