Finite Group Databasebeta

← #26125188

Subgroups — #26125188

Order 12032.

1426 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 259 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×#15688890C214yesmaximal
2#20241611C212yesmaximal
3C22×C22⋊Q8×C47C211yesmaximal, characteristic
4C22×C22⋊Dic47C2214yes—
5C2×C47⋊#95—24no—
6#15688890—28no—
7C23×Q8×C47C411yescharacteristic, complemented
8C24×C4×C47C2211yescharacteristic
9C2×C22⋊Q8×C47—24no—
10C2×C22⋊Q8×C47—28no—
11C22×C4⋊C4×C47C411yescharacteristic, complemented
12C22×C4⋊C4×C47C412yescomplemented
13C22×C22⋊C4×C47C2212yes—
14C23×Dic47—24no—
15C2×C22⋊Dic47—28no—
16C2×C22⋊Dic47—28no—
17C2×C22⋊Dic47—24no—
18C2×C22⋊Dic47Q818yescomplemented
19C47⋊#95—48no—
20C25×C47C2311yescharacteristic
21C22×Q8×C47—24no—
22C22×Q8×C47—22no—
23C22×Q8×C47—44no—
24C23×C4×C47—24no—
25C23×C4×C47—44no—
26C23×C4×C47C2×C411yescharacteristic
27C23×C4×C47C2×C412yes—
28C23×C4×C47C2×C412yes—
29C23×C4×C47C2×C411yescharacteristic
30C23×C4×C47D414yes—
31C22⋊Q8×C47—232no—
32C2×C4⋊C4×C47—24no—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—24no—
35C2×C4⋊C4×C47—24no—
36C2×C4⋊C4×C47—24no—
37C2×C22⋊C4×C47—24no—
38C2×C22⋊C4×C47—24no—
39C2×C22⋊C4×C47D418yes—
40C24×C47—42no—
41C24×C47—42no—
42C24×C47C22×C411yescharacteristic
43C24×C47C2×D412yes—
44C24×C47C2×D414yes—
45C24×C47C2×Q814yes—
46C24×C47#4114yes—
47C2×Q8×C47—28no—
48C2×Q8×C47—416no—
49C2×Q8×C47—44no—
50C2×Q8×C47—44no—
51C22×C4×C47—22no—
52C22×C4×C47—24no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—28no—
59C22×C4×C47—28no—
60C22×C4×C47—42no—
61C22×C4×C47—44no—
62C22×C4×C47—416no—
63C22×C4×C47—44no—
64C22×C4×C47—44no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22⋊C414yes—
68C22×C4×C47C22⋊C414yes—
69C22×Dic47—28no—
70C22×Dic47—48no—
71C22×Dic47—48no—
72C22×Dic47—44no—
73C22×Dic47—44no—
74C4⋊C4×C47—28no—
75C4⋊C4×C47—216no—
76C22⋊C4×C47—216no—
77C22⋊Dic47—216no—
78C22⋊Dic47—48no—
79C23×C47—24no—
80C23×C47—22no—
81C23×C47—24no—
82C23×C47—22no—
83C23×C47—44no—
84C23×C47—44no—
85C23×C47—48no—
86C23×C47—48no—
87C23×C47C2×C22⋊C412yes—
88C23×C47C2×C22⋊C411yescharacteristic
89C23×C47C4×Q814yes—
90C23×C47#12014yes—
91C23×C47C22⋊Q814yes—
92C23×C47C22⋊Q818yes—
93C23×C47#12412yes—
94C23×C47#12414yes—
95C23×C47C4⋊Q812yes—
96C23×C47C4⋊Q814yes—
97Q8×C47—416no—
98C2×C4×C47—28no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—24no—
102C2×C4×C47—24no—
103C2×C4×C47—22no—
104C2×C4×C47—24no—
105C2×C4×C47—416no—
106C2×C4×C47—44no—
107C2×C4×C47—44no—
108C2×C4×C47—44no—
109C2×C4×C47—44no—
110C2×C4×C47—44no—
111C2×C4×C47—44no—
112C2×C4×C47—44no—
113C2×C4×C47#9914yes—
114C2×Dic47—416no—
115C2×Dic47—48no—
116C2×Dic47—88no—
117C2×Dic47—84no—
118C2×Dic47—84no—
119#11381—471nomaximal, Sylow
120C22×C47—28no—
121C22×C47—24no—
122C22×C47—28no—
123C22×C47—24no—
124C22×C47—24no—
125C22×C47—22no—
126C22×C47—24no—
127C22×C47—22no—
128C22×C47—48no—
129C22×C47—44no—
130C22×C47—44no—
131C22×C47#37911yescharacteristic
132C22×C47#39114yes—
133C22×C47#39112yes—
134C22×C47#39314yes—
135C22×C47#39514yes—
136C22×C47C2×#9912yes—
137C22×C47#41012yes—
138C4×C47—24no—
139C4×C47—42no—
140C4×C47—44no—
141C4×C47—44no—
142Dic47—88no—
143C22×C22⋊Q8D4711yescharacteristic
144C2×#342—474no—
145C22⋊#95—472no—
146C2×C47—24no—
147C2×C47—24no—
148C2×C47—24no—
149C2×C47#144611yescharacteristic
150C2×C47#178111yescharacteristic
151C2×C47#178211yescharacteristic
152C2×C47#199614yes—
153C23×Q8Dic4711yescharacteristic, complemented
154C24×C4C2×D4711yescharacteristic
155C2×C22⋊Q8—24no—
156C2×C22⋊Q8—28no—
157C22×C4⋊C4Dic4711yescharacteristic, complemented
158C22×C4⋊C4Dic4712yescomplemented
159C22×C22⋊C4—474no—
160C22×C22⋊C4C2×D4712yes—
161C2×#95—944no—
162#342—948no—
163C47#1138111yescharacteristic, complemented, Sylow
164C25C22×D4711yescharacteristic
165C22×Q8—24no—
166C22×Q8—22no—
167C22×Q8—44no—
168C23×C4—24no—
169C23×C4—44no—
170C23×C4—944no—
171C23×C4C2×Dic4711yescharacteristic
172C23×C4C2×Dic4712yes—
173C23×C4C2×Dic4712yes—
174C23×C4C2×Dic4711yescharacteristic
175C23×C4C22⋊D4714yes—
176C22⋊Q8—232no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—478no—
185C2×C22⋊C4—948no—
186C2×C22⋊C4—948no—
187C2×C22⋊C4—944no—
188C2×C22⋊C4C22⋊D4718yes—
189#95—1888no—
190C24—42no—
191C24—42no—
192C24Q8×D4714yes—
193C24#1062535414yes—
194C24C22×Dic4711yescharacteristic
195C24C2×C22⋊D4712yes—
196C24C2×C22⋊D4714yes—
197C2×Q8—28no—
198C2×Q8—416no—
199C2×Q8—44no—
200C2×Q8—44no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—28no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—416no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—948no—
216C22×C4—1888no—
217C22×C4—1888no—
218C22×C4—1884no—
219C22×C4—1884no—
220C22×C4C22⋊Dic4712yes—
221C22×C4C22⋊Dic4712yes—
222C22×C4C22⋊Dic4714yes—
223C22×C4C22⋊Dic4714yes—
224C4⋊C4—28no—
225C4⋊C4—216no—
226C22⋊C4—216no—
227C22⋊C4—9416no—
228C22⋊C4—1888no—
229C23—22no—
230C23—24no—
231C23—22no—
232C23—24no—
233C23—44no—
234C23—48no—
235C23—48no—
236C23—44no—
237C23C2×C22⋊Dic4712yes—
238C23C2×C22⋊Dic4711yescharacteristic
239C23Q8×Dic4714yes—
240C23#1309399314yes—
241C23#1309400014yes—
242C23#1309400018yes—
243C23#1309401312yes—
244C23#1309401314yes—
245C23#1309402212yes—
246C23#1309402214yes—
247Q8—416no—
248C2×C4—28no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—416no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—3764no—
264C2×C4—3764no—
265C2×C4—3768no—
266C2×C4—18816no—
267C2×C4—1888no—
268C2×C4#1309393914yes—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—24no—
273C22—24no—
274C22—28no—
275C22—28no—
276C22—24no—
277C22—48no—
278C22—44no—
279C22—44no—
280C22#1568897911yescharacteristic
281C22#1568903014yes—
282C22#1568903012yes—
283C22#1568903414yes—
284C22#1568904214yes—
285C22C2×#1309393912yes—
286C22#1568909712yes—
287C4—24no—
288C4—42no—
289C4—44no—
290C4—44no—
291C4—3768no—
292C2—24no—
293C2—24no—
294C2—24no—
295C2#2024165911yescharacteristic
296C2#2024330311yescharacteristic
297C2#2024330611yescharacteristic
298C2#2024437414yes—
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC24×C47C22×C4
FrattiniC25C22×D47
FittingC22×C22⋊Q8×C47C2
socleC23×C47C2×C22⋊C4
prime radical, cosocleC25×C47C23
hypercenterC22×C22⋊Q8D47
hypocenterC47#11381
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
7. C23×Q8×C47C4semidirect
11. C22×C4⋊C4×C47C4semidirect
12. C22×C4⋊C4×C47C4semidirect
18. C2×C22⋊Dic47Q8semidirect
153. C23×Q8Dic47semidirect
157. C22×C4⋊C4Dic47semidirect
158. C22×C4⋊C4Dic47semidirect
163. C47#11381semidirect

Maximal subgroups

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

subgroupindex
C2×#156888902
#202416112
C22×C22⋊Q8×C472
#1138147