Finite Group Databasebeta

← #26109587

Subgroups — #26109587

Order 12032.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093958C211yesmaximal, characteristic, complemented
2#20242151C211yesmaximal, characteristic, complemented
3#20241913C212yesmaximal
4#20241245C212yesmaximal, complemented
5C47×#2966C211yesmaximal, characteristic
6#15689180C2212yes—
7C2×#13093958—21no—
8C4×C4⋊Dic47C2212yes—
9#15688886—21no—
10#15688876—21no—
11#15688874—21no—
12#15688860—22no—
13#15688857—22no—
14C47×#529C2211yescharacteristic
15C47×#446C411yescharacteristic, complemented
16C8×Q8×C47C411yescharacteristic, complemented
17C47×#436C411yescharacteristic, complemented
18C47×#435—21no—
19C8×D4×C47—21no—
20C8×D4×C47C411yescharacteristic, complemented
21C47×#433—21no—
22C47×#432C2211yescharacteristic
23C47×#405—21no—
24C2×C4×C8×C47C2211yescharacteristic
25C2×C4⋊Dic47—22no—
26C2×C4×Dic47—22no—
27#13093958—24no—
28C4⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30#13093935—22no—
31C47⋊#95—22no—
32C2×M16×C47—21no—
33C22×C8×C47—21no—
34C22×C8×C47D411yescharacteristic
35C22×C8×C47D411yescharacteristic
36#126×C47—21no—
37#125×C47Q811yescharacteristic
38#124×C47Q811yescharacteristic
39#123×C47—21no—
40C22⋊Q8×C47D411yescharacteristic
41#121×C47D411yescharacteristic
42C4×Q8×C47C2×C411yescharacteristic
43C4×D4×C47—21no—
44C4×D4×C47C2×C411yescharacteristic
45#117×C47—21no—
46C2×C42×C47C2311yescharacteristic
47C4⋊C8×C47—21no—
48C4⋊C8×C47—21no—
49C4⋊C8×C47C2×C411yescharacteristic
50C4⋊C8×C47C2×C411yescharacteristic
51C22⋊C8×C47—21no—
52C22⋊C8×C47—21no—
53C22⋊C8×C47—21no—
54#97×C47—21no—
55#97×C47—21no—
56C4×C8×C47—21no—
57C4×C8×C47C2×C411yescharacteristic
58C4×C8×C47C2×C411yescharacteristic
59C4×C8×C47D411yescharacteristic
60C4×C8×C47D411yescharacteristic
61C2×Q8×C47C4⋊C411yescharacteristic
62C2×D4×C47—21no—
63C2×D4×C47C4⋊C411yescharacteristic
64C22×C4×C47—21no—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47#4111yescharacteristic
68C22×Dic47—42no—
69M16×C47—42no—
70C2×C8×C47—22no—
71C2×C8×C47—21no—
72C2×C8×C47—21no—
73C2×C8×C47—41no—
74C2×C8×C47—42no—
75C2×C8×C47C22⋊C411yescharacteristic
76C2×C8×C47C22⋊C411yescharacteristic
77C2×C8×C47C22⋊C411yescharacteristic
78C2×C8×C47C22⋊C411yescharacteristic
79C4⋊C4×C47—21no—
80C4⋊C4×C47—21no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47C4⋊C411yescharacteristic
84C4⋊C4×C47C4⋊C411yescharacteristic
85C22⋊C4×C47—21no—
86C22⋊C4×C47—21no—
87C22⋊C4×C47—21no—
88C22⋊C4×C47—21no—
89C22⋊C4×C47—21no—
90C42×C47—21no—
91C42×C47C22×C411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47C2×Q811yescharacteristic
94C42×C47#4111yescharacteristic
95C4⋊Dic47—24no—
96C4⋊Dic47—42no—
97C4×Dic47—82no—
98C2×C47⋊C8—42no—
99C23×C47—21no—
100C23×C47#12111yescharacteristic
101Q8×C47—41no—
102D4×C47—41no—
103D4×C47—41no—
104D4×C47—41no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
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×C47C2×C22⋊C411yescharacteristic
116C2×C4×C47C2×C4⋊C411yescharacteristic
117C2×C4×C47#11711yescharacteristic
118C2×C4×C47#12111yescharacteristic
119C2×C4×C47C22⋊Q811yescharacteristic
120C2×C4×C47C22⋊Q811yescharacteristic
121C2×Dic47—44no—
122C2×Dic47—82no—
123C2×Dic47—82no—
124C8×C47—21no—
125C8×C47—21no—
126C8×C47—21no—
127C8×C47—41no—
128C8×C47—41no—
129C8×C47#10412yes—
130C47⋊C8—82no—
131#8695—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47#38011yescharacteristic
138C22×C47#47211yescharacteristic
139C22×C47#47311yescharacteristic
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47C2×#10411yescharacteristic
149C4×C47#42111yescharacteristic
150Dic47—84no—
151#2966D4711yescharacteristic
152C4×#107—471no—
153#1568—471no—
154#1509—472no—
155#1285—472no—
156C2×C47—21no—
157C2×C47—21no—
158C2×C47—81no—
159C2×C47#180611yescharacteristic
160C2×C47#181211yescharacteristic
161C2×C47#181611yescharacteristic
162#529C2×D4711yescharacteristic
163#446Dic4711yescharacteristic, complemented
164C8×Q8Dic4711yescharacteristic, complemented
165#436Dic4711yescharacteristic, complemented
166#435—21no—
167C8×D4—21no—
168C8×D4Dic4711yescharacteristic, complemented
169#433—21no—
170#432—472no—
171#432C2×D4711yescharacteristic
172C2×#107—941no—
173#405—21no—
174C2×C4×C8C2×D4711yescharacteristic
175C4×C4⋊C4—472no—
176#340—941no—
177#335—941no—
178#334—941no—
179#328—942no—
180#327—942no—
181C47#869511yescharacteristic, complemented, Sylow
182C2×M16—21no—
183C22×C8—21no—
184C22×C8C22⋊D4711yescharacteristic
185C22×C8C22⋊D4711yescharacteristic
186#126—21no—
187#125Dic9411yescharacteristic
188#124Dic9411yescharacteristic
189#123—21no—
190C22⋊Q8D18811yescharacteristic
191#121D18811yescharacteristic
192C4×Q8C2×Dic4711yescharacteristic
193C4×D4—21no—
194C4×D4C2×Dic4711yescharacteristic
195#117—21no—
196C2×C4⋊C4—942no—
197C2×C42—942no—
198C2×C42C22×D4711yescharacteristic
199#107—944no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—942no—
203C4⋊C8C2×Dic4711yescharacteristic
204C4⋊C8C2×Dic4711yescharacteristic
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—942no—
209#97—21no—
210#97—21no—
211#97—942no—
212C4×C8—21no—
213C4×C8C2×Dic4711yescharacteristic
214C4×C8C2×Dic4711yescharacteristic
215C4×C8C22⋊D4711yescharacteristic
216C4×C8C22⋊D4711yescharacteristic
217#95—942no—
218C2×Q8C4⋊Dic4711yescharacteristic
219C2×D4—21no—
220C2×D4C4⋊Dic4711yescharacteristic
221C22×C4—21no—
222C22×C4—1882no—
223C22×C4C2×D18811yescharacteristic
224C22×C4#1062535011yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226M16—42no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—1882no—
233C2×C8C22⋊Dic4711yescharacteristic
234C2×C8C22⋊Dic4711yescharacteristic
235C2×C8C22⋊Dic4711yescharacteristic
236C2×C8C22⋊Dic4711yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—944no—
242C4⋊C4—1882no—
243C4⋊C4C4⋊Dic4711yescharacteristic
244C4⋊C4C4⋊Dic4711yescharacteristic
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—21no—
248C22⋊C4—21no—
249C22⋊C4—21no—
250C42—21no—
251C42—3762no—
252C42C2×Dic9411yescharacteristic
253C42#1062535011yescharacteristic
254C42C22×Dic4711yescharacteristic
255C42C2×C22⋊D4711yescharacteristic
256C23—21no—
257C23#1309399811yescharacteristic
258Q8—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—3762no—
273C2×C4—3762no—
274C2×C4—1884no—
275C2×C4C2×C22⋊Dic4711yescharacteristic
276C2×C4C2×C4⋊Dic4711yescharacteristic
277C2×C4#1309398311yescharacteristic
278C2×C4#1309399811yescharacteristic
279C2×C4#1309400311yescharacteristic
280C2×C4#1309400311yescharacteristic
281C8—21no—
282C8—21no—
283C8—21no—
284C8—41no—
285C8—41no—
286C8—3762no—
287C8#1309395112yes—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22#1568898311yescharacteristic
294C22#1568939311yescharacteristic
295C22#1568939811yescharacteristic
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—3764no—
305C4C2×#1309395111yescharacteristic
306C4#1568914011yescharacteristic
307C2—21no—
308C2—21no—
309C2—81no—
310C2#2024340811yescharacteristic
311C2#2024343411yescharacteristic
312C2#2024345211yescharacteristic
subgroupG/H
centerC22#15688983
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2966C2
socleC22×C47#380
prime radical#121×C47D4
cosocleC2×C42×C47C23
hypercenter#2966D47
hypocenterC47#8695
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. C4×#13093958C2semidirect
2. #20242151C2semidirect
4. #20241245C2semidirect
15. C47×#446C4semidirect
16. C8×Q8×C47C4semidirect
17. C47×#436C4semidirect
20. C8×D4×C47C4semidirect
163. #446Dic47semidirect
164. C8×Q8Dic47semidirect
165. #436Dic47semidirect
168. C8×D4Dic47semidirect
181. C47#8695semidirect

Maximal subgroups

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

subgroupindex
C4×#130939582
#202421512
#202419132
#202412452
C47×#29662
#869547